diff --git "a/title_31K_G/test_title_long_2405.02426v1.json" "b/title_31K_G/test_title_long_2405.02426v1.json"
new file mode 100644--- /dev/null
+++ "b/title_31K_G/test_title_long_2405.02426v1.json"
@@ -0,0 +1,101 @@
+{
+    "url": "http://arxiv.org/abs/2405.02426v1",
+    "title": "Generalized Solution for Double-Porosity Flow through a Graded Excavation Damaged Zone",
+    "abstract": "Prediction of flow to boreholes or excavations in fractured low-permeability\nrocks is important for resource extraction and disposal or sequestration\nactivities. Analytical solutions for fluid pressure and flowrate, when\navailable, are powerful, insightful, and efficient tools enabling parameter\nestimation and uncertainty quantification. A flexible porous media flow\nsolution for arbitrary physical dimension is derived and extended to double\nporosity for converging radial flow when permeability and porosity decrease\nradially as a power law away from a borehole or opening. This distribution can\narise from damage accumulation due to stress relief associated with drilling or\nmining. The single-porosity graded conductivity solution was initially found\nfor heat conduction, the arbitrary dimension flow solution comes from\nhydrology, and the solution with both arbitrary dimension and graded\npermeability distribution appeared in reservoir engineering. These existing\nsolutions are here combined and extended to two implementations of the\ndouble-porosity conceptual model, for both a simpler thin-film mass transfer\nand more physically realistic diffusion between fracture and matrix. This work\npresents a new specified-flowrate solution with wellbore storage for the\nsimpler double-porosity model, and a new more physically realistic solution for\nany wellbore boundary condition. A new closed-form expression is derived for\nthe matrix diffusion solution (applicable to both homogeneous and graded\nproblems), improving on previous infinite series expressions.",
+    "authors": "Kristopher L. Kuhlman",
+    "published": "2024-05-03",
+    "updated": "2024-05-03",
+    "primary_cat": "physics.flu-dyn",
+    "cats": [
+        "physics.flu-dyn",
+        "physics.geo-ph",
+        "86A05"
+    ],
+    "label": "Original Paper",
+    "paper_cat": "Diffusion AND Model",
+    "gt": "Generalized Solution for Double-Porosity Flow through a Graded Excavation Damaged Zone",
+    "main_content": "Introduction Fluid flow through damage-induced fracture networks in otherwise low-permeability crystalline rocks (e.g., granite, argillite or halite) is of interest to geothermal energy production (Tao et al, 2021), radioactive waste disposal (Tsang et al, 2005), hydrogen storage (AbuAisha and Billiotte, 2021), and compressed air energy storage (Kim et al, 2012). Rock damage around an excavation (i.e., the Excavation Damaged Zone, EDZ; Davies and Bernier (2005)) increases the connected porosity, and leads to increased permeability. Fractured rock often has higher porosity and permeability than intact rock. Damage near a borehole or excavation will decrease the relative contribution from flow in the lower-permeability farfield, and will confound the estimation of hydrologic properties using approaches that assume uniform homogeneous distributions of permeability and porosity. There is a need for a flexible analytical solution for flow to a borehole or excavation in the presence of damage, that includes wellbore storage, doubleporosity flow, and variable flow dimension. This is most evident in a mechanically weak, low-permeability medium like salt, but should also apply to other low-permeability fractured rocks like granite or shale. 1 arXiv:2405.02426v1  [physics.flu-dyn]  3 May 2024 \fIn salt, the far-field (i.e., undamaged) permeability is unmeasurably low (Beauheim and Roberts, 2002) due to salt\u2019s tendency to creep shut any unsupported openings. The permeability around a borehole in salt is derived from accumulated damage due to stress redistribution around the excavation itself (Wallace et al, 1990; Stormont et al, 1991; Cosenza, 1996; Hou, 2003; Kuhlman, 2014). Stormont et al (1991) presented brine and gas permeability data measured in salt for packer-isolated intervals of small boreholes before and after a central 1-meter diameter borehole was drilled (i.e., a mineby experiment). Figure 1 shows these data support the conceptual model of permeability and porosity decaying away from an excavation. Cosenza (1996) proposed the power-law model for permeability and porosity plotted in the figure. These data show porosity and permeability decrease with distance from the central excavation. Two lines are shown with to the data; one is a monomial power-law, the other includes an additive background term. The two curves differ primarily away from the excavation (r/rw \u22653), where larger uncertainties in estimated porosity and permeability exist, for three reasons. First, the access drift EDZ (test conducted in the floor of a 5-m wide room) is superimposed on the 1-m borehole EDZ. Second, the small-diameter (2.5-cm) measurement boreholes themselves each have a small EDZ overprinted on the 1-m borehole EDZ. Lastly, the apparent background permeability may represent the measurement limit of the packer system used (i.e., compliance of the packer inflation elements and working fluid). Especially in salt, the undisturbed background permeability is near zero, and is difficult to measure consistently in the field (Beauheim and Roberts, 2002). The power-law distribution of permeability matches the more certain near-field permeability distribution, and is conceptually more elegant than a finite domain or a flow domain with piece-wise heterogeneous properties (i.e., a higher-permeability EDZ adjacent to lowerpermeability intact rock). Other investigations have also shown porosity and permeability decaying away with distance from an excavation in crystalline rocks (Shen et al, 2011; Cho et al, 2013; Ghazvinian, 2015) and sedimentary rocks (Perras et al, 2010; Perras and Diederichs, 2016). Fig. 1 Permeability and porosity observations around a 1-m borehole (radial distance scaled by excavation radius) in salt from small-scale mine-by experiment (data from Stormont et al (1991)) Salt permeability has been related to both the confining and shear stresses (Reynolds and Gloyna, 1960; Lai, 1971; Stormont and Fuenkajorn, 1994; Alkan, 2009). Confining stresses reduce fracture aperture and bulk permeability, while shear stresses are associated with increased bulk permeability. Aydan et al (1993) present solutions for radial and tangential plane stress and strain (i.e., dilatation or a change in porosity) around a circular excavation. Strain is proportional to r\u22122 D or r\u22123 D (where rD is radial distance 2 \finto the formation scaled by the excavation size), depending on whether the region is experiencing elastic (exponent 2) or plastic (exponent \u22483) deformation. These relationships illustrate a possible behavior of rock in the EDZ. The true extent of the EDZ depends on drilling or excavation method, borehole or tunnel geometry, state of stress, and rock mechanical properties (Hudson et al, 2009). Softer or weaker sedimentary rocks like argillite or halite typically have a larger EDZ than stiffer or stronger rocks like granite. There are several well-known empirical power-law relationships between porosity and permeability in fractured or granular media (e.g., Kozeny, 1927; Carman, 1937) and many studies have discussed their applicability (David et al, 1994; Kuhlman and Matteo, 2018). Permeability in fractured rocks is more sensitive to small changes in porosity than granular rocks (i.e., fractured rocks have higher pore compressibility resulting in larger exponents in porosity-permeability relationships). Based on evidence from these observations, graded dimensionless porosity is assumed to follow n(r) = n0 \u0012 r rw \u0013\u2212\u03b7 , (1) where rw is the borehole or excavation radius [m], n0 = n(rw) is maximum porosity at the borehole wall, and \u03b7 is a dimensionless exponent (see Table 1 for a list of physical variables and notation). Using the same form, the graded permeability can be represented with the form k(r) = k0 \u0012 r rw \u0013\u2212\u03ba , (2) where k0 = k(rw) is the maximum permeability [m2] at the borehole wall and \u03ba is another dimensionless exponent. Based on lab measurements on fractured granite, the empirical relationship \u03ba \u22483\u03b7 has been proposed (Kranz et al, 1979; David et al, 1994). The Stormont et al (1991) salt data (Figure 1) support \u03b7 = 4.5 and \u03ba = 17, which shows a somewhat faster-decaying permeability (\u03ba = 3.8\u03b7) than seen in granitic rocks. The power-law permeability and porosity distribution conceptual model presented here is an alternative to flow models using wellbore skin (Streltsova, 1988; Pasandi et al, 2008), finite domain (Gelbard, 1992; Lin et al, 2016), or low-permeability non-Darcy flow with a threshold gradient (Liu, 2014, 2017). These three conceptualizations all lead to reduced contributions of flow from the far field, but only borehole skin can account for observed distributions of higher porosity or permeability near the excavation, which are important when analyzing pressure or flowrate data at early time. The contribution from lower permeability in the far field are more important at late time. Finite domains and skin can have analytical flow solutions, but low-permeability non-Darcy flow does not typically lend itself to analytical solutions. Barker (1988) developed a generalized solution for converging flow to a borehole with variable noninteger dimension, D. This conceptualization has been used to characterize flow in fractured systems, where lower-dimension (i.e., D < 3) results associated with discrete fractures are more common than higher dimension results (Beauheim et al, 2004; Le Borgne et al, 2004; Bowman et al, 2013; Ferroud et al, 2018). Doe (1991) extended the solution of Barker (1988) to the conceptualization where permeability varies with radial distance, through analogy with the heat conduction literature (Carslaw and Jaeger, 1959). A single-porosity flow solution is derived here with power-law variable properties, like the approach of Doe (1991) (who did not present a derivation). The single-porosity solution is then readily extended to a double-porosity conceptualization, using first the approach of Warren and Root (1963) for thin-film mass transfer between fractures and matrix, then the more physically realistic matrix diffusion approach of Kazemi (1969). Double-porosity flow is a common and efficient conceptualization in fractured rocks (Aguilera, 1980; van Golf-Racht, 1982; Da Prat, 1990). The medium is conceptualized as two communicating physically overlapping continua including fractures with high permeability (but little to no storage) and matrix or intact rock with significant storage (but little to no flow) (Barenblatt and Zheltov, 1960; Barenblatt et al, 1960). Many extensions to the basic double-porosity conceptual model exist, including multiple matrix or fracture porosities, and different assumptions about the geometry or underlying physics governing flow in the fractures or matrix (Chen, 1989; Kuhlman and Heath, 2021). The Warren and Root (1963) 3 \fsolution simplifies the exchange between matrix and fractures to a mass-transfer thin-film approximation, leading to numerous analytical solutions (Aguilera, 1980; Chen, 1989). It is commonly used for this reason, even though it is well-known that spatial pressure gradients in matrix blocks are important, as the matrix is low-permeability and would therefore be expected to experience steep, slow-changing gradients. A series representation of the Kazemi (1969) solution is used here, an extension of the multirate mass transfer model to double-porosity flow (Kuhlman et al, 2015). The more physically correct (but more difficult to solve) solution can be represented by an infinite series of porosities, which can either represent an infinite number of Warren-Root type matrix porosities, or if the coefficients are chosen specifically, a single Kazemi-type matrix diffusion porosity. More recently, Wang et al (2021) has developed a semi-analytical solution for flow in a double-porosity formation, for the case when non-Darcian flow is significant. Moutsopoulos et al (2022) have provided analytical and semi-analytical solutions for two classical problems in flow of unconfined double-porosity aquifers, based on Moutsopoulos (2021). De-Smedt (2022) presented an analytical solution for flow in double-porosity media for fractional flow dimensions, which is a generalization of De-Smedt (2011). Hayek et al (2018) presented a semi-analytical solution for flow due to pumping a double-porosity aquifer via a constant-pressure boundary condition (without wellbore storage) where permeability varied as a power law. The fractal reservoir flow problem (Chang and Yortsos, 1990) is also analogous to the radially variable properties approach presented here, but the governing equations of the two problems are only equivalent when the spectral exponent (\u03b8 in Chang and Yortsos (1990)) in the fractal problem is zero. The fractal reservoir governing equation is typically solved approximately, since the additional terms due to non-zero spectral exponent in the governing equation do not readily allow closed-form analytical solution. In the next section, the governing equations and boundary conditions are developed for the variabledimension single-porosity flow problem (Doe, 1991). This solution is mapped onto the modified Bessel equation, allowing solution for flow to both specified pressure (type-I) and specified flowrate with wellbore storage (type-III). These more general single-porosity solutions are shown to degenerate down to several well-known cases. The single-porosity solutions are then extended to a simpler Warren-Root type doubleporosity model for type-I (Hayek et al, 2018) and type-III (new) and then a new Kazemi type doubleporosity model. The Kazemi series solution approach is then summed analytically to arrive at a new closed-form expression for the response in Laplace space, a solution that is new for both graded and homogeneous domains. Finally, a summary and discussion of limitations is given for the new solutions. The approach taken here, representing the porosity and permeability of fractured rocks as power-law distributions, was first developed by Delay et al (2007), and first pursued by the author for applications in deep (> 3 km) borehole disposal of radioactive waste in basement rock (Brady et al, 2017; Kuhlman et al, 2019). The approach is also applicable to flow in salt surrounding excavations, like those in mine-by experiments (Stormont et al, 1991). 2 Development of Flow Problem To introduce and contrast with the dual-porosity solution, the single-porosity solution is developed first. To make a single solution for Cartesian linear, cylindrical, and spherical geometries, a variable-dimension approach like Barker (1988) is used, including variable permeability and porosity, like Doe (1991). The governing equation for slightly compressible time-dependent change in pressure p [Pa] in a general 1D coordinate (Barker, 1988) is n(r)c\u2202p \u2202t = 1 rm \u2202 \u2202r \u0014k(r)rm \u00b5 \u2202p \u2202r \u0015 , (3) where c is bulk compressibility [1/Pa] and the dimensionless parameter m is 0 for a Cartesian strip, 1 for a cylinder, and 2 for a sphere (i.e., m = D \u22121, where D is the dimension). The derivative of the bracketed term in (3) is expanded via chain rule; starting from (2), dk dr = \u2212\u03bak(r)/r is substituted with the definitions of k(r) and n(r), to get n0c \u0012 r rw \u0013\u2212\u03b7 \u2202p \u2202t = k0 \u00b5 \u0012 r rw \u0013\u2212\u03ba \u0014m \u2212\u03ba r \u2202p \u2202r + \u22022p \u2202r2 \u0015 . (4) For converging radial flow in a semi-infinite domain, the relevant wellbore boundary conditions are constant-pressure (type-I), constant-flux (type-II), or constant-flux with wellbore storage (type-III in 4 \fLaplace space). The initial, far-field, and source borehole boundary conditions for a borehole in an infinite symmetric domain are initial p(r, t = 0) = 0 far \u2212field p(r \u2192\u221e, t) < \u221e wellbore type \u2212I pI(r = rw, t) = p1(t); or (5) wellbore type \u2212II Amk0 \u00b5 \u2202pII(t) \u2202r \f \f \f \f r=rw = Q(t); or wellbore type \u2212III Amk0 \u00b5 \u2202pIII(t) \u2202r \f \f \f \f r=rw = Q(t) + Ac \u03c1g \u2202pw(t) \u2202t , respectively. See Appendix A for definition of source borehole boundary condition terms. These boundary conditions represent a homogeneous uniform initial condition, a requirement that the solution remains finite at large distance, and a specified pressure or pressure gradient at the source (r = rw). The Type-II boundary condition (specified flowrate) is a special case (\u03c3 = 0) of the wellbore storage boundary condition (flowrate linearly proportional to change in pressure), so it is not developed further. 2.1 Dimensional Analysis A solution is derived for equation (4), using the approach of Doe (1991), which was based on analogy with the heat conduction literature (Carslaw and Jaeger, 1959). Reducing the governing equation (4) to dimensionless form using characteristic time, Tc = n0cL2 c\u00b5/k0, and characteristic length, Lc = rw, leads to r\u03ba\u2212\u03b7 D \u2202pD \u2202tD = m \u2212\u03ba rD \u2202pD \u2202rD + \u22022pD \u2202r2 D , (6) where the dimensionless quantities rD = r/Lc, tD = t/Tc, and p{I,III} D = p/p{I,III} c are used (see Table 2 for a summary of dimensionless quantities). The characteristic pressure change is given by pI c = \u02c6 p1, where p1(t) = \u02c6 p1ft separates the timedependent specified pressure into a constant characteristic pressure and a dimensionless variable time behavior (for a constant specified pressure, ft = 1). The dimensionless type-I initial and boundary conditions are pD(rD, tD = 0) = 0 pD(rD \u2192\u221e, tD) < \u221e (7) pI D(rD = 1, tD) = ft. Using pIII c = rw \u02c6 Q\u00b5 Amk0 , where Q(t) = \u02c6 Qft similarly separates the time-dependent volumetric flowrate into a constant characteristic flowrate and a dimensionless time behavior. The dimensionless type-III source borehole boundary condition is \u2202pIII D \u2202rD \f \f \f \f rD=1 = ft + \u03c3 \u2202pIII D \u2202t , (8) where \u03c3 is a dimensionless wellbore storage coefficient (see Appendix A) and the same initial and far-field conditions apply as the type-I case. 2.2 Laplace Transform Taking the dimensionless Laplace transform \u0000 \u00af f(s) = R \u221e 0 e\u2212stDf(tD) dtD \u0001 of the governing partial differential equation (6) (without loss of generality assuming zero initial condition) leads to the ordinary differential equation d2\u00af pD dr2 D + m \u2212\u03ba rD d\u00af pD drD \u2212s\u00af pDr\u03ba\u2212\u03b7 D = 0, (9) 5 \fassuming \u03ba, \u03b7, and m are not functions of time, and s is the dimensionless Laplace transform parameter. The transformed type-I and far-field boundary conditions (7) are \u00af pD(rD \u2192\u221e) < \u221e (10) \u00af pI D(rD = 1) = \u00af ft, where \u00af ft represents the Laplace transform of the boundary condition\u2019s time behavior. For a unit step change at t = 0 (where ft = 1, a typical assumption), \u00af ft = 1 s. Other temporal behaviors are simply handled, including a step change at a non-zero time, an exponentially decaying source term, an arbitrary piecewise-constant or piecewise-linear behavior, or a sinusoidal source term (Kruseman and de Ridder, 1994; Mishra et al, 2013). The transformed wellbore-storage boundary condition is d\u00af pIII D drD \f \f \f \f rD=1 = \u00af ft + \u03c3s\u00af pIII D , (11) which now more clearly resembles a Type-III boundary condition. 2.3 Numerical Inverse Laplace Transform The governing equations and associated boundary conditions are solved exactly in Laplace space, then numerically inverted back to the time domain using one of several viable approaches (Kuhlman, 2013). The equations were rapidly prototyped and inverted using the Python library mpmath (Johansson et al, 2017), which provides arbitrary precision special functions and numerical inverse Laplace transform algorithms. A Fortran program was also developed to facilitate plotting and parameter estimation, implementing the inversion algorithm of de Hoog et al (1982). Python and Fortran implementations of the solution are available at https://github.com/klkuhlm/graded. 3 Solution of Flow Problem 3.1 Mapping onto Modified Bessel Equation The governing ordinary differential equation (9) can be made equivalent to a form of the modified Bessel equation after a change of variables first used by Lommel (1868) for the standard Bessel equation. Appendix B illustrates an analogous change of variables to the modified Bessel equation. Comparing (9) to this scaled version of the modified Bessel equation (41), they are equivalent given the following correspondences \u03b1 =1 2 (\u03ba \u2212m + 1) \u03b3 =1 2 (\u03ba \u2212\u03b7 + 2) (12) \u03bd = s \u03b12 \u03b32 = \u03ba \u2212m + 1 \u03ba \u2212\u03b7 + 2 \u03b2 = r s \u03b32 = s 4s (\u03ba \u2212\u03b7 + 2)2 . The transformed modified Bessel equation has the general solution (37) y = z\u03b1 [AI\u03bd (\u03b2z\u03b3) + BK\u03bd (\u03b2z\u03b3)] , (\u03b3 \u0338= 0) , (13) where A and B are constants determined by the boundary conditions and I\u03bd(z) and K\u03bd(z) are the firstand second-kind modified Bessel functions of non-integer order and real argument (McLachlan, 1955; Bowman, 1958; Spanier and Oldham, 1987; DLMF, 2023). The finiteness boundary condition (10) requires A = 0 to keep the solution finite as rD \u2192\u221e, since the first-kind modified Bessel function grows exponentially with increasing real argument, leaving \u00af pD (rD) = r\u03b1 DBK\u03bd (\u03b2r\u03b3 D) , (14) 6 \fwhich is not defined for \u03b3 = 0 (i.e., \u03ba\u2212\u03b7 = \u22122, which is unrealistic because \u03ba is larger than \u03b7 for physical reasons), and B is determined by the Laplace-space source borehole boundary conditions. 3.2 Constant-Pressure (Type-I) at Borehole The borehole boundary condition (rD = 1) for specified change in pressure leads to the solution (the Warren and Root (1963) double porosity solution for this wellbore boundary condition is equivalent to Hayek et al (2018)) \u00af pI D(rD) = \u00af ftr\u03b1 D K\u03bd (\u03b2r\u03b3 D) K\u03bd (\u03b2) (15) and its radial gradient (i.e., proportional to flow of fluid into the borehole) d\u00af pI D drD = \u00af ftr\u03b1\u22121 D \u0014 (\u03b1 \u2212\u03b3\u03bd) K\u03bd (\u03b2r\u03b3 D) K\u03bd (\u03b2) + \u03b2\u03b3r\u03b3 D K\u03bd\u22121 (\u03b2r\u03b3 D) K\u03bd (\u03b2) \u0015 , (16) using a recurrence relationship for the derivative of the Bessel function in terms of Bessel functions of adjacent orders (DLMF, 2023, \u00a710.29.2). Restricting \u03ba \u2265\u03b7 (i.e., permeability decreases as fast or faster than porosity), then \u03b3 > 0 and \u03b1 = \u03b3\u03bd (for \u03b3 < 0, \u03b1 \u2212\u03b3\u03bd = 2\u03b1). This physically motivated restriction on parameters simplifies (16) to d\u00af pI D drD = \u221as \u00af ftr\u03b1+\u03b3\u22121 D K\u03bd\u22121 (\u03b2r\u03b3 D) K\u03bd (\u03b2) , (17) since \u03b2\u03b3 = \u221as for \u03b3 > 0. When evaluated in the source borehole (rD = 1), the solution simplifies further. Figure 2 shows plots of the predicted pressure gradient at rD = 1 due to a constant-pressure condition there (top row) and the predicted decrease in pressure radially away from the boundary (values of \u03b7, \u03ba, and m for each simulation are listed in the caption and title of each figure). Both rows of plots show the variability with the porosity exponent (\u03b7, given by the line color) and the permeability exponent (\u03ba = \u03b7\u03c4, given by the line type). The same results are shown for Cartesian linear (m = 0), cylindrical (m = 1), and spherical (m = 2) geometries in three columns. For a given set of parameters, a higher-dimensional domain (larger m) leads to a slower drop in produced fluids at any time. The highest sustained flowrate for all dimensions is achieved with constant properties in space (i.e., the red curve \u03b7 = \u03ba = 0). More negative exponents in the porosity and permeability power-laws lead to more rapid decrease in flowrate, as the contribution to flow from large radius vanishes when the exponent increases in magnitude. These types of responses might be mis-interpreted as being associated with lower permeability (which would also lead to a faster decrease in flowrate) using a model with constant properties and a fixed dimension. In the source well (top row of subplots), the effect of \u03ba is different and are predicted to reverse between dimensions. For \u03b7 = 3 (black lines), the \u03ba = {3, 6, 9} cases are swapped between m = 1 and m = 2. For \u03b7 = 2 (blue lines), the \u03ba cases are swapped between m = 0 and m = 1. The bottom row of figures shows the predicted pressure with distance at tD = 10. At locations away from the source well (rD > 1), changes in the porosity exponent, \u03b7, have relatively less impact than changes in the permeability exponent, \u03ba (different colored solid lines are close together, while colored lines of different line type are widely separated). The dimensionality (m) has a smaller effect at locations away from the source borehole than it had on the gradient predicted at the source borehole. 3.3 Constant-Flowrate with Wellbore Storage (Type-III) The wellbore-storage boundary condition for the specified flowrate solution at rD = 1 results in the general solution (that is new for any double-porosity solution with power-law variation in material properties) \u00af pIII D (rD) = \u00af ftr\u03b1 D K\u03bd (\u03b2r\u03b3 D) (\u03b1 \u2212\u03b3\u03bd + \u03c3s) K\u03bd (\u03b2) + \u03b2\u03b3K\u03bd\u22121 (\u03b2), (18) which can be simplified using \u03b1 = \u03b3\u03bd and \u03b2\u03b3 = \u221as to \u00af pIII D (rD) = \u00af ftr\u03b1 D K\u03bd (\u03b2r\u03b3 D) \u221asK\u03bd\u22121 (\u03b2) + \u03c3sK\u03bd (\u03b2). (19) 7 \fFig. 2 Type-I flowrate (top row at rD = 1) and pressure (bottom row at rD > 1 and tD = 10) solution at borehole for m = 0, 1, 2 (Cartesian, cylindrical, and spherical) and at different radial distances. Line color indicates \u03b7; line type indicates \u03ba/\u03b7. Line segments in top row illustrate slopes of 1/2, 1, and 3/2. Analogous to the results for the Type-I solution but only showing the m = 1 and m = 2 cases, Figure 3 shows the predicted pressure through time at the boundary for a specified flowrate at the boundary. Figure 3 results are for no wellbore storage (\u03c3 = 0), while Figure 4 shows the same results with nonzero wellbore storage (all model parameters listed in caption or title of each figure). Wellbore storage is important at early time, leading to a smaller predicted change in pressure, with the predicted response giving a characteristic 1 : 1 slope on log-log plots before formation storage contributes significantly to the flow (i.e., pumping in a bathtub). Wellbore storage makes more of a difference (i.e., shows a larger deviation from \u03c3 = 0 case) for larger \u03b7 (and \u03ba, since \u03ba = 2\u03b7). 3.4 Parameter Combinations Yielding Simpler Solutions When \u03b7 = \u03ba = 0, permeability and porosity are constant in space; in this case (9) simplifies to d2\u00af pD dr2 D + m rD d\u00af pD drD \u2212s\u00af pD = 0, (20) 8 \fFig. 3 Type-II solution (Type-III with \u03c3 = 0) at borehole for m = 1, 2 (cylindrical and spherical). Line color indicates \u03b7; line type indicates \u03ba/\u03b7. which is the dimensionless form of the equation solved by Barker (1988). In this case \u03b3 = 1, \u03b1 = (1\u2212m)/2, \u03bd = \u03b1, and \u03b2 = \u221as. The solution in Laplace-space under these conditions becomes \u00af pD (rD) = r\u03bd DBK\u03bd \u0000\u221asrD \u0001 , (21) which was found by Barker (1988, Eqn. 15). When \u03b7 = \u03ba = m = 0 the time-domain solution simplifies to pD(t) = 1/ \u221a \u03c0t, because \u03bd = 1/2 and \u03bd \u22121 = \u22121/2, the numerator and denominator of (17) are equal since K\u03bd(z) \u2261K\u2212\u03bd(z). Another simplification occurs when m = \u03ba = \u03b7, not necessarily zero. In this case, the permeability and porosity decrease at the same rate radially that the surface area of the domain grows in size (A0 \u221d1, A1 \u221drD, A2 \u221dr2 D), resulting in an equivalent Cartesian coordinate system, d2\u00af pD dr2 D \u2212s\u00af pD = 0, (22) which has a solution in terms of sin(\u221asrD) and cos(\u221asrD) or exp(\u00b1\u221asrD) and typically has an explicit inverse Laplace transform. In this case \u03b1 = \u03bd = 1/2, \u03b3 = 0, and \u03b2 = \u221as. When \u03bd = n \u00b1 1 2 (for n integer), the modified Bessel functions become modified spherical Bessel functions (DLMF, 2023, \u00a710.47), and when \u03bd = \u00b1 1 3, they become Airy functions (DLMF, 2023, \u00a79.6). These additional special cases are not handled differently here (i.e., the more general solution in terms of modified Bessel functions is still valid), since in the case given here \u03bd varies with \u03ba, \u03b7, and m (12). 4 Extension of Solution to Double Porosity 4.1 Mass-Transfer Coefficient Approximation Beginning with the Warren and Root (1963) formulation for double-porosity (i.e., high-conductance fractures and high-capacity matrix), the power-law permeability and porosity distributions are incorporated. 9 \fFig. 4 Type-III solution at borehole (rD = 1), for m = 1, 2 (cylindrical and spherical). Line color indicates \u03b7; line type indicates \u03c3. All curves for \u03ba/\u03b7 = 2. The equations for double-porosity flow in the fractures and matrix are 1 rm \u2202 \u2202r \u0014kf \u00b5 \u2202pf \u2202r \u0015 = nrcr \u2202pr \u2202t + nfcf \u2202pf \u2202t \u02c6 \u03b1kr \u00b5 (pf \u2212pr) = nrcr \u2202pr \u2202t (23) where \u02c6 \u03b1 is the shape factor [1/m2] of Warren and Root (1963), subscript f indicates fracture, and subscript r indicates matrix (rock). The matrix equation does not involve a spatial gradient of pressure, nor a matching of pressure and flux at the boundary, but simply a difference between the fracture and matrix pressure (i.e., the mass transfer coefficient approximation often used for heat transfer across thin films). This behavior is sometimes referred to in the petroleum engineering literature as \u201csteady-state\u201d flow between the fracture and matrix (Da Prat, 1990), but it also represents one-dimensional diffusion in the matrix with a thin-film mass-transfer approximation between the fracture and matrix reservoirs, analogous to Newton\u2019s law of cooling. Substituting the permeability ki = ki0 \u0010 r rw \u0011\u2212\u03bai and porosity ni = ni0 \u0010 r rw \u0011\u2212\u03b7i (i \u2208{f, r}), then converting to dimensionless form using an analogous approach to Warren and Root (1963), where \u03c9 = nf0cf/ (nr0cr + nf0cf) is the dimensionless fracture storage coefficient and \u03bb = \u02c6 \u03b1krr2 w/kf is the dimensionless interporosity exchange coefficient. Finally, taking the Laplace transform of both equations results in the pair of ordinary differential equations \u0014d2\u00af pfD dr2 D + m \u2212\u03baf rD d\u00af pfD drD \u0015 r\u2212\u03baf = (1 \u2212\u03c9)r\u2212\u03b7r D \u00af pmDs + \u03c9r\u2212\u03b7f D \u00af pfDs \u03bb (\u00af pfD \u2212\u00af prD) r\u2212\u03bar D = (1 \u2212\u03c9)r\u2212\u03b7r D \u00af prDs. (24) Solving for matrix pressure in the matrix equation, \u00af prD = \u00af pfD\u03bbr\u2212\u03bar D / \u0002 (1 \u2212\u03c9)sr\u2212\u03b7r D + \u03bbr\u2212\u03bar D \u0003 , and substituting this into the fracture equation leads to a single equation solely in terms of dimensionless 10 \fFig. 5 Type-I flowrate solution at borehole (left) and Type-II solution for pressure (\u03c3 = 0, right), for m = 1 (cylindrical). Line color indicates \u03bb; line type indicates \u03c9. Laplace-domain fracture pressure \u0014d2\u00af pfD dr2 D + m \u2212\u03baf rD d\u00af pfD drD \u0015 r\u2212\u03baf = r\u2212\u03b7r D \u00af pfD ( (1 \u2212\u03c9)sr\u2212\u03bar D \u03bb (1 \u2212\u03c9)sr\u2212\u03b7r D + \u03bbr\u2212\u03bar D ) + \u03c9r\u2212\u03b7f D \u00af pfDs. (25) To force the term in curly brackets in (25) to be independent of rD, \u03bar = \u03b7r is assumed. Setting \u03bar and \u03b7r equal to \u03b7f allows rD and \u00af pfD to be similar form to previous solutions. Simplifying the subsequent notation \u03baf \u2192\u03ba, \u03b7r \u2192\u03b7, and \u00af pfD \u2192\u00af pD results in d2\u00af pD dr2 D + m \u2212\u03ba rD d\u00af pD drD = r\u03ba\u2212\u03b7 D \u00af pD \u0014 (1 \u2212\u03c9)s\u03bb (1 \u2212\u03c9)s + \u03bb + \u03c9s \u0015 , (26) which is the same form as (9). This solution corresponds to the same scaled Bessel equation, with only the definition of \u03b2 changing to \u03b2W R = s\u0014 \u03bb \u03bb/(1 \u2212\u03c9) + s + \u03c9 \u0015 s \u03b32 . (27) Any more general spatial behavior of matrix properties (e.g., \u03b7r \u0338= \u03bar) would not be solvable with the same approach. This limitation still makes physical sense, as the the most important terms to vary with space are the fracture permeability and the matrix storage. Setting \u03ba = \u03b7 = 0 and m = 1 results in the Warren and Root (1963) solution. Figure 5 shows typical solution behaviors for the cylindrical (m = 1) case for Type-I and Type-II wellbore boundary conditions, for \u03b7 = 3 and \u03ba = 6. Figure 6 shows behavior from the \u201cmiddle\u201d curve in Figure 5 (\u03bb = 10\u22125 and \u03c9 = 10\u22124), for a range of porosity and permeability exponents similar to those shown in Warren and Root (1963), listed in the figure caption. 11 \fFig. 6 Type-I flowrate solution at borehole (left) and Type-II solution for pressure (\u03c3 = 0, right), for m = 1 (cylindrical). All curves are for \u03bb = 10\u22125 and \u03c9 = 10\u22124 (middle curves shown in Figure 5). Line color indicates \u03b7; line type indicates \u03ba/\u03b7. 4.2 Matrix Diffusion The matrix diffusion problem of Kazemi (1969) is more physically realistic (Aguilera, 1980; Da Prat, 1990), but it is typically solved numerically or via late-time approximations (De Swaan, 1976), rather than analytically like Warren and Root (1963). The series approach of Kuhlman et al (2015) is used here to represent matrix diffusion in a single matrix continuum through the sum of an infinite series of Warren-Root matrix continua, and the infinite sum is then analytically summed. The generalization of (23) to multiple matrix continua starts with 1 rm \u2202 \u2202r \u0014kf \u00b5 \u2202pf \u2202r \u0015 = N X j=1 njcj \u2202pj \u2202t + nfcf \u2202pf \u2202t \u02c6 \u03b1jkj \u00b5 (pf \u2212pj) = njcj \u2202pj \u2202t j = 1, 2, . . . N, (28) where N is the number of matrix continua (one additional equation for each continuum). Similarly taking the Laplace transform of this set of equations, solving for \u00af pf, substituting the matrix equations into the fracture equation, and simplifying the notation leads to d2\u00af pD dr2 D + m \u2212\u03ba rD d\u00af pD drD = r\u03ba\u2212\u03b7 D \u00af pD\u03c9s(1 + \u00af g), (29) where \u00af g = N X j=1 \u02c6 \u03bejuj s + uj (30) is a matrix memory kernel (Haggerty and Gorelick, 1995), \u02c6 \u03bej is related to the storage properties of each matrix continuum (analogous to \u03c9 of Warren and Root (1963)), and uj is related to the interporosity flow coefficient of each matrix continuum (analogous to \u03bb of Warren and Root (1963)). The Laplacespace memory kernel approach is flexible, and is used elsewhere in hydrology and reservoir engineering (Herrera and Yates, 1977; Haggerty et al, 2000; Schumer et al, 2003). Equation (29) can be simplified to 12 \fWarren and Root (1963) with a particular choice of \u00af g and N = 1, and to the solution for a triple-porosity reservoir (Clossman, 1975) with a different choice of \u00af g and N = 2 (Kuhlman et al, 2015). When N \u2192\u221ein (30), the it is more convenient to specify the mean and variance of the parameter distributions than the individual parameters associated with each porosity. Several different distributions are possible (Haggerty and Gorelick, 1995). In the form presented by Kuhlman et al (2015), the parameters are specified as the infinite series uj = (2j \u22121)2\u03c02\u03bb 4(1 \u2212\u03c9) \u02c6 \u03bej = 8(1 \u2212\u03c9) (2j \u22121)2\u03c9\u03c02 j = 1, 2, . . . N \u2192\u221e (31) which leads to the Kazemi (1969) solution for matrix diffusion. The parameters \u03bb and \u03c9 have the same definitions as in Warren and Root (1963). Setting \u03ba = \u03b7 = 0 results in the solution of Kuhlman et al (2015). The new governing equation is the same form and the modified Bessel function solution, only requiring re-definition of \u03b2 as \u03b2KZ = v u u t \" N X j=1 \u03c9\u02c6 \u03bejuj uj + s + \u03c9 # s \u03b32 , N \u2192\u221e. (32) Substituting the definitions of u and \u02c6 \u03be from (31) and simplifying leads to \u03b2KZ = v u u t \" N X j=1 2\u03bb W 2 j \u03bb/(1 \u2212\u03c9) + s + \u03c9 # s \u03b32 , N \u2192\u221e, (33) where Wj = \u03c0(2j \u22121)/2. This is similar in form to (27) but the term in the denominator grows as the index increases, illustrating how the series solution approximates the Kazemi (1969) solution through an infinite series of modified Warren and Root (1963) matrix porosities. Further simplifying the approach of Kuhlman et al (2015), the infinite series in (33) can be evaluated in closed form using residue methods (Wolfram Research, Inc., 2021), resulting in \u03b2KZ = v u u t \"r \u03bb(1 \u2212\u03c9) s tanh  r s(1 \u2212\u03c9) \u03bb ! + \u03c9 # s \u03b32 , (34) where tanh(\u00b7) is the hyperbolic tangent. This closed-form expression derived here is more accurate and numerically more efficient than truncating or accelerating the infinite series in (32), which is an improvement over the series presented in Kuhlman et al (2015) for graded or homogeneous domains. Figure 7 illustrates the transition from the Warren and Root (1963) (N = 1) to the Kazemi (1969) series approximation for increasing terms (N = {2, 10, 100, 1000}, heavy colored solid lines) and the expression for the infinite sum (34) (heavy black dashed line) for flow to a specified flux (type-II, \u03c3 = 0) cylindrical (m = 1) borehole of constant material properties (\u03ba = \u03b7 = 0). The bounding Theis (1935) behavior is shown for the fracture and matrix compressibilities (thin red dashed lines). 5 Applications and Limitations A general converging radial flow solution for specified flowrate or specified wellhead pressure was derived for domains with power-law variability in porosity and permeability due to damage. The single-porosity version has already been presented by Doe (1991), and a solution for constant-pressure condition without wellbore storage was derived by Hayek et al (2018), but the specified-flowrate double-porosity solution with wellbore storage presented here is new. The infinite series approximation to Kazemi was summed analytically, resulting in a new closed-form expression of the series presented in Kuhlman et al (2015), which is an improvement for both graded and homogeneous properties. The newly developed analytical solutions are more general (i.e., several existing solutions are special cases of the new solution) and include more behaviors typical in well-test solutions (i.e., wellbore storage, positive skin, double porosity), 13 \fFig. 7 Type-II solution for pressure at source borehole (\u03c3 = 0), for m = 1 (cylindrical) for different number of terms. All curves are for \u03bb = 10\u22125, \u03c9 = 10\u22124, \u03ba = \u03b7 = 0. while still being straightforward and parsimonious (i.e., as few free parameters as possible) in their implementation. The basic flow solution assumes linear single-phase flow of a fluid in a slightly compressible formation. The double-porosity solution assumes the fractures are high permeability, with low storage capacity, while the matrix (i.e., intact rock between fractures) is high storage capacity with low permeability. These assumptions are representative for analytical solutions to subsurface porous media flow problems in the hydrology and petroleum engineering literature, and are shared by the solutions of Barker (1988), Doe (1991), Warren and Root (1963), Kazemi (1969), and Kuhlman et al (2015). To apply this analytical solution to observed data, either observed data would be transformed into dimensionless space, or the analytical solution could be transformed to dimensional space, then a parameter estimation routine would be used to minimize the model-data misfit, and possibly explore the uncertainty or uniqueness of the solution. The solution method developed to solve these solutions uses numerical inverse Laplace transforms and runs quickly enough to be used in parameter estimation (e.g., Monte Carlo methods that require hundreds of thousands of evaluations). The analytical solution might be of most use with parameter estimation to fit observations, but the non-uniqueness of the curves may make estimation of unique physical parameters difficult, without further physical or site-specific constraints. Realistically, the parameters in the Bessel equation may be estimable (i.e., \u03b1, \u03b2, \u03b3, and \u03bd defined in (12)), but without defining the flow dimension (m) or the relationship between the porosity and permeability exponents (\u03c4 = \u03ba/\u03b7), it may be difficult to identify all the parameters from data alone, since many the curves have similar shapes, unlike classical Type curves (Bourdet et al, 1989). 14 \fAc borehole cross-sectional area m2 Am borehole cylindrical surface area m2 c bulk compressibility 1/Pa ft time variability \u2212 g gravitational acceleration m/s2 h hydraulic head m k permeability m2 Lc characteristic length (rw) m m dimension (D \u22121) \u2212 n porosity \u2212 p change in pressure Pa s Laplace transform parameter \u2212 Q volumetric flowrate m3/s r distance coordinate m rw borehole or excavation radius m \u02c6 \u03b1 Warren and Root (1963) shape factor 1/m2 \u03b7 porosity power-law exponent \u2212 \u03ba permeability power-law exponent \u2212 \u03c1 fluid density kg/m3 \u00b5 fluid viscosity Pa \u00b7 s Table 1 Physical Properties and Parameters pD scaled pressure p/pc tD scaled time tk0/n0cL2 c\u00b5 rD scaled distance r/Lc \u03bb interporosity exchange coefficient \u02c6 \u03b1krr2 w/kf \u03c3 wellbore storage coefficient Ac/(rwn0c\u03c1gAm) \u03c9 fracture storage coefficient nf0cf/(nr0cr + nf0cf) Table 2 Dimensionless Quantities Statements and Declarations Funding The author thanks the U.S. Department of Energy Office of Nuclear Energy\u2019s Spent Fuel and Waste Science and Technology program for funding. Conflicts of Interest The author has no competing interests to declare. Availability of Data and Material No data or materials were used by the author in the preparation of the manuscript. Code Availability The source code of Fortran and Python implementations of the program are available from the author upon request. Acknowledgments This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish 15 \for reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.energy.gov/downloads/doe-public-access-plan. The author thanks Tara LaForce from Sandia for technically reviewing the manuscript. 6 Appendix A: Wellbore Storage Boundary Condition The wellbore-storage boundary condition accounts for the storage in the finite borehole arising from the mass balance Qin \u2212Qout = Ac \u2202hw \u2202t . Qin [m3/s] is volumetric flow into the borehole from the formation, Qout is possibly time-variable flow out of the well through the pump (Q(t) [m3/s]), and \u2202hw \u2202t is the change in hydraulic head [m] (hw = pw \u03c1g + z) of water standing in the borehole through time, pw is change in pressure [Pa] of water in the borehole, \u03c1 is fluid density [kg/m3], z is an elevation datum [m], and g is gravitational acceleration [m/s2]. Ac is the cross-sectional surface area of the pipe, sphere or box providing storage (it may be a constant or a function of elevation); for a typical pipe, it becomes Ac = \u03c0r2 c, where rc is the radius of the casing where the water level is changing. The mass balance is then Amk0 \u00b5 \u2202p \u2202r \f \f \f \f r=rw \u2212Q(t) = Ac \u03c1g \u2202pw \u2202t , (35) where Am is the area of the borehole communicating with the formation. For the integer m considered here these are A0 = b2, A1 = 2\u03c0rwb, A2 = 4\u03c0r2 w (b is a length independent of the borehole radius). Assuming the change in water level in the borehole (hw = pw/ (\u03c1g)) is equal to the change in formation water level (h = p/ (\u03c1g)), this can be converted into dimensionless form as \u2202pD \u2202rD \f \f \f \f rD=1 \u2212ft = \u03c3 \u2202pD \u2202t , (36) where \u03c3 = Ac/ (rwn0c\u03c1gAm) is a dimensionless ratio of formation to wellbore storage; \u03c3 \u21920 is an infinitesimally small well with only formation response, while \u03c3 \u2192\u221eis a well with no formation response (i.e., a bathtub). 7 Appendix B: Transformation of Modified Bessel Equation Following the approach of Bowman (1958), alternative forms of the Bessel equation are found, this approach is a simplification of the original approach of Lommel (1868). An analogous approach is applied here to \u201cback into\u201d the desired modified Bessel equation. The equation satisfied by the pair of functions y1 = x\u03b1I\u03bd (\u03b2x\u03b3) , y2 = x\u03b1K\u03bd (\u03b2x\u03b3) (37) is sought, where \u03b1, \u03b2, \u03b3, and \u03bd are constants. Using the substitutions \u03b6 = yx\u2212\u03b1 and \u03be = \u03b2x\u03b3 gives \u03b61 = I\u03bd (\u03be) and \u03b62 = K\u03bd (\u03be), which are the two solutions to the modified Bessel equation (DLMF, 2023, \u00a710.25.1), \u03be d d\u03be \u0012 \u03be d\u03b6 d\u03be \u0013 \u2212(\u03be2 + \u03bd)\u03b6 = 0. (38) Given \u03be d d\u03be \u0012 \u03be d\u03b6 d\u03be \u0013 = x \u03b32 d dx \u0012 x d\u03b6 dx \u0013 , (39) and x d dx \u0012 x d\u03b6 dx \u0013 = y\u2032\u2032 x\u03b1\u22122 \u2212(2\u03b1 \u22121) y\u2032 x\u03b1\u22121 + \u03b12y x\u03b1 , (40) the standard-form equation satisfied by y is y\u2032\u2032 + (1 \u22122\u03b1) y\u2032 + \u03b12y x\u03b1 \u2212 \u0012 \u03b22\u03b32x2\u03b3\u22122 \u2212\u03b12 \u2212\u03bd2\u03b32 x2 \u0013 y = 0. (41) 16 \fThis equation can be compared to the Laplace-space ordinary differential equation (9), allowing direct use of the product of powers and modified Bessel function (37) as solutions (13).",
+    "additional_graph_info": {
+        "graph": [
+            [
+                "Kristopher L. Kuhlman",
+                "Bwalya Malama"
+            ]
+        ],
+        "node_feat": {
+            "Kristopher L. Kuhlman": [
+                {
+                    "url": "http://arxiv.org/abs/2405.02426v1",
+                    "title": "Generalized Solution for Double-Porosity Flow through a Graded Excavation Damaged Zone",
+                    "abstract": "Prediction of flow to boreholes or excavations in fractured low-permeability\nrocks is important for resource extraction and disposal or sequestration\nactivities. Analytical solutions for fluid pressure and flowrate, when\navailable, are powerful, insightful, and efficient tools enabling parameter\nestimation and uncertainty quantification. A flexible porous media flow\nsolution for arbitrary physical dimension is derived and extended to double\nporosity for converging radial flow when permeability and porosity decrease\nradially as a power law away from a borehole or opening. This distribution can\narise from damage accumulation due to stress relief associated with drilling or\nmining. The single-porosity graded conductivity solution was initially found\nfor heat conduction, the arbitrary dimension flow solution comes from\nhydrology, and the solution with both arbitrary dimension and graded\npermeability distribution appeared in reservoir engineering. These existing\nsolutions are here combined and extended to two implementations of the\ndouble-porosity conceptual model, for both a simpler thin-film mass transfer\nand more physically realistic diffusion between fracture and matrix. This work\npresents a new specified-flowrate solution with wellbore storage for the\nsimpler double-porosity model, and a new more physically realistic solution for\nany wellbore boundary condition. A new closed-form expression is derived for\nthe matrix diffusion solution (applicable to both homogeneous and graded\nproblems), improving on previous infinite series expressions.",
+                    "authors": "Kristopher L. Kuhlman",
+                    "published": "2024-05-03",
+                    "updated": "2024-05-03",
+                    "primary_cat": "physics.flu-dyn",
+                    "cats": [
+                        "physics.flu-dyn",
+                        "physics.geo-ph",
+                        "86A05"
+                    ],
+                    "main_content": "Introduction Fluid flow through damage-induced fracture networks in otherwise low-permeability crystalline rocks (e.g., granite, argillite or halite) is of interest to geothermal energy production (Tao et al, 2021), radioactive waste disposal (Tsang et al, 2005), hydrogen storage (AbuAisha and Billiotte, 2021), and compressed air energy storage (Kim et al, 2012). Rock damage around an excavation (i.e., the Excavation Damaged Zone, EDZ; Davies and Bernier (2005)) increases the connected porosity, and leads to increased permeability. Fractured rock often has higher porosity and permeability than intact rock. Damage near a borehole or excavation will decrease the relative contribution from flow in the lower-permeability farfield, and will confound the estimation of hydrologic properties using approaches that assume uniform homogeneous distributions of permeability and porosity. There is a need for a flexible analytical solution for flow to a borehole or excavation in the presence of damage, that includes wellbore storage, doubleporosity flow, and variable flow dimension. This is most evident in a mechanically weak, low-permeability medium like salt, but should also apply to other low-permeability fractured rocks like granite or shale. 1 arXiv:2405.02426v1  [physics.flu-dyn]  3 May 2024 \fIn salt, the far-field (i.e., undamaged) permeability is unmeasurably low (Beauheim and Roberts, 2002) due to salt\u2019s tendency to creep shut any unsupported openings. The permeability around a borehole in salt is derived from accumulated damage due to stress redistribution around the excavation itself (Wallace et al, 1990; Stormont et al, 1991; Cosenza, 1996; Hou, 2003; Kuhlman, 2014). Stormont et al (1991) presented brine and gas permeability data measured in salt for packer-isolated intervals of small boreholes before and after a central 1-meter diameter borehole was drilled (i.e., a mineby experiment). Figure 1 shows these data support the conceptual model of permeability and porosity decaying away from an excavation. Cosenza (1996) proposed the power-law model for permeability and porosity plotted in the figure. These data show porosity and permeability decrease with distance from the central excavation. Two lines are shown with to the data; one is a monomial power-law, the other includes an additive background term. The two curves differ primarily away from the excavation (r/rw \u22653), where larger uncertainties in estimated porosity and permeability exist, for three reasons. First, the access drift EDZ (test conducted in the floor of a 5-m wide room) is superimposed on the 1-m borehole EDZ. Second, the small-diameter (2.5-cm) measurement boreholes themselves each have a small EDZ overprinted on the 1-m borehole EDZ. Lastly, the apparent background permeability may represent the measurement limit of the packer system used (i.e., compliance of the packer inflation elements and working fluid). Especially in salt, the undisturbed background permeability is near zero, and is difficult to measure consistently in the field (Beauheim and Roberts, 2002). The power-law distribution of permeability matches the more certain near-field permeability distribution, and is conceptually more elegant than a finite domain or a flow domain with piece-wise heterogeneous properties (i.e., a higher-permeability EDZ adjacent to lowerpermeability intact rock). Other investigations have also shown porosity and permeability decaying away with distance from an excavation in crystalline rocks (Shen et al, 2011; Cho et al, 2013; Ghazvinian, 2015) and sedimentary rocks (Perras et al, 2010; Perras and Diederichs, 2016). Fig. 1 Permeability and porosity observations around a 1-m borehole (radial distance scaled by excavation radius) in salt from small-scale mine-by experiment (data from Stormont et al (1991)) Salt permeability has been related to both the confining and shear stresses (Reynolds and Gloyna, 1960; Lai, 1971; Stormont and Fuenkajorn, 1994; Alkan, 2009). Confining stresses reduce fracture aperture and bulk permeability, while shear stresses are associated with increased bulk permeability. Aydan et al (1993) present solutions for radial and tangential plane stress and strain (i.e., dilatation or a change in porosity) around a circular excavation. Strain is proportional to r\u22122 D or r\u22123 D (where rD is radial distance 2 \finto the formation scaled by the excavation size), depending on whether the region is experiencing elastic (exponent 2) or plastic (exponent \u22483) deformation. These relationships illustrate a possible behavior of rock in the EDZ. The true extent of the EDZ depends on drilling or excavation method, borehole or tunnel geometry, state of stress, and rock mechanical properties (Hudson et al, 2009). Softer or weaker sedimentary rocks like argillite or halite typically have a larger EDZ than stiffer or stronger rocks like granite. There are several well-known empirical power-law relationships between porosity and permeability in fractured or granular media (e.g., Kozeny, 1927; Carman, 1937) and many studies have discussed their applicability (David et al, 1994; Kuhlman and Matteo, 2018). Permeability in fractured rocks is more sensitive to small changes in porosity than granular rocks (i.e., fractured rocks have higher pore compressibility resulting in larger exponents in porosity-permeability relationships). Based on evidence from these observations, graded dimensionless porosity is assumed to follow n(r) = n0 \u0012 r rw \u0013\u2212\u03b7 , (1) where rw is the borehole or excavation radius [m], n0 = n(rw) is maximum porosity at the borehole wall, and \u03b7 is a dimensionless exponent (see Table 1 for a list of physical variables and notation). Using the same form, the graded permeability can be represented with the form k(r) = k0 \u0012 r rw \u0013\u2212\u03ba , (2) where k0 = k(rw) is the maximum permeability [m2] at the borehole wall and \u03ba is another dimensionless exponent. Based on lab measurements on fractured granite, the empirical relationship \u03ba \u22483\u03b7 has been proposed (Kranz et al, 1979; David et al, 1994). The Stormont et al (1991) salt data (Figure 1) support \u03b7 = 4.5 and \u03ba = 17, which shows a somewhat faster-decaying permeability (\u03ba = 3.8\u03b7) than seen in granitic rocks. The power-law permeability and porosity distribution conceptual model presented here is an alternative to flow models using wellbore skin (Streltsova, 1988; Pasandi et al, 2008), finite domain (Gelbard, 1992; Lin et al, 2016), or low-permeability non-Darcy flow with a threshold gradient (Liu, 2014, 2017). These three conceptualizations all lead to reduced contributions of flow from the far field, but only borehole skin can account for observed distributions of higher porosity or permeability near the excavation, which are important when analyzing pressure or flowrate data at early time. The contribution from lower permeability in the far field are more important at late time. Finite domains and skin can have analytical flow solutions, but low-permeability non-Darcy flow does not typically lend itself to analytical solutions. Barker (1988) developed a generalized solution for converging flow to a borehole with variable noninteger dimension, D. This conceptualization has been used to characterize flow in fractured systems, where lower-dimension (i.e., D < 3) results associated with discrete fractures are more common than higher dimension results (Beauheim et al, 2004; Le Borgne et al, 2004; Bowman et al, 2013; Ferroud et al, 2018). Doe (1991) extended the solution of Barker (1988) to the conceptualization where permeability varies with radial distance, through analogy with the heat conduction literature (Carslaw and Jaeger, 1959). A single-porosity flow solution is derived here with power-law variable properties, like the approach of Doe (1991) (who did not present a derivation). The single-porosity solution is then readily extended to a double-porosity conceptualization, using first the approach of Warren and Root (1963) for thin-film mass transfer between fractures and matrix, then the more physically realistic matrix diffusion approach of Kazemi (1969). Double-porosity flow is a common and efficient conceptualization in fractured rocks (Aguilera, 1980; van Golf-Racht, 1982; Da Prat, 1990). The medium is conceptualized as two communicating physically overlapping continua including fractures with high permeability (but little to no storage) and matrix or intact rock with significant storage (but little to no flow) (Barenblatt and Zheltov, 1960; Barenblatt et al, 1960). Many extensions to the basic double-porosity conceptual model exist, including multiple matrix or fracture porosities, and different assumptions about the geometry or underlying physics governing flow in the fractures or matrix (Chen, 1989; Kuhlman and Heath, 2021). The Warren and Root (1963) 3 \fsolution simplifies the exchange between matrix and fractures to a mass-transfer thin-film approximation, leading to numerous analytical solutions (Aguilera, 1980; Chen, 1989). It is commonly used for this reason, even though it is well-known that spatial pressure gradients in matrix blocks are important, as the matrix is low-permeability and would therefore be expected to experience steep, slow-changing gradients. A series representation of the Kazemi (1969) solution is used here, an extension of the multirate mass transfer model to double-porosity flow (Kuhlman et al, 2015). The more physically correct (but more difficult to solve) solution can be represented by an infinite series of porosities, which can either represent an infinite number of Warren-Root type matrix porosities, or if the coefficients are chosen specifically, a single Kazemi-type matrix diffusion porosity. More recently, Wang et al (2021) has developed a semi-analytical solution for flow in a double-porosity formation, for the case when non-Darcian flow is significant. Moutsopoulos et al (2022) have provided analytical and semi-analytical solutions for two classical problems in flow of unconfined double-porosity aquifers, based on Moutsopoulos (2021). De-Smedt (2022) presented an analytical solution for flow in double-porosity media for fractional flow dimensions, which is a generalization of De-Smedt (2011). Hayek et al (2018) presented a semi-analytical solution for flow due to pumping a double-porosity aquifer via a constant-pressure boundary condition (without wellbore storage) where permeability varied as a power law. The fractal reservoir flow problem (Chang and Yortsos, 1990) is also analogous to the radially variable properties approach presented here, but the governing equations of the two problems are only equivalent when the spectral exponent (\u03b8 in Chang and Yortsos (1990)) in the fractal problem is zero. The fractal reservoir governing equation is typically solved approximately, since the additional terms due to non-zero spectral exponent in the governing equation do not readily allow closed-form analytical solution. In the next section, the governing equations and boundary conditions are developed for the variabledimension single-porosity flow problem (Doe, 1991). This solution is mapped onto the modified Bessel equation, allowing solution for flow to both specified pressure (type-I) and specified flowrate with wellbore storage (type-III). These more general single-porosity solutions are shown to degenerate down to several well-known cases. The single-porosity solutions are then extended to a simpler Warren-Root type doubleporosity model for type-I (Hayek et al, 2018) and type-III (new) and then a new Kazemi type doubleporosity model. The Kazemi series solution approach is then summed analytically to arrive at a new closed-form expression for the response in Laplace space, a solution that is new for both graded and homogeneous domains. Finally, a summary and discussion of limitations is given for the new solutions. The approach taken here, representing the porosity and permeability of fractured rocks as power-law distributions, was first developed by Delay et al (2007), and first pursued by the author for applications in deep (> 3 km) borehole disposal of radioactive waste in basement rock (Brady et al, 2017; Kuhlman et al, 2019). The approach is also applicable to flow in salt surrounding excavations, like those in mine-by experiments (Stormont et al, 1991). 2 Development of Flow Problem To introduce and contrast with the dual-porosity solution, the single-porosity solution is developed first. To make a single solution for Cartesian linear, cylindrical, and spherical geometries, a variable-dimension approach like Barker (1988) is used, including variable permeability and porosity, like Doe (1991). The governing equation for slightly compressible time-dependent change in pressure p [Pa] in a general 1D coordinate (Barker, 1988) is n(r)c\u2202p \u2202t = 1 rm \u2202 \u2202r \u0014k(r)rm \u00b5 \u2202p \u2202r \u0015 , (3) where c is bulk compressibility [1/Pa] and the dimensionless parameter m is 0 for a Cartesian strip, 1 for a cylinder, and 2 for a sphere (i.e., m = D \u22121, where D is the dimension). The derivative of the bracketed term in (3) is expanded via chain rule; starting from (2), dk dr = \u2212\u03bak(r)/r is substituted with the definitions of k(r) and n(r), to get n0c \u0012 r rw \u0013\u2212\u03b7 \u2202p \u2202t = k0 \u00b5 \u0012 r rw \u0013\u2212\u03ba \u0014m \u2212\u03ba r \u2202p \u2202r + \u22022p \u2202r2 \u0015 . (4) For converging radial flow in a semi-infinite domain, the relevant wellbore boundary conditions are constant-pressure (type-I), constant-flux (type-II), or constant-flux with wellbore storage (type-III in 4 \fLaplace space). The initial, far-field, and source borehole boundary conditions for a borehole in an infinite symmetric domain are initial p(r, t = 0) = 0 far \u2212field p(r \u2192\u221e, t) < \u221e wellbore type \u2212I pI(r = rw, t) = p1(t); or (5) wellbore type \u2212II Amk0 \u00b5 \u2202pII(t) \u2202r \f \f \f \f r=rw = Q(t); or wellbore type \u2212III Amk0 \u00b5 \u2202pIII(t) \u2202r \f \f \f \f r=rw = Q(t) + Ac \u03c1g \u2202pw(t) \u2202t , respectively. See Appendix A for definition of source borehole boundary condition terms. These boundary conditions represent a homogeneous uniform initial condition, a requirement that the solution remains finite at large distance, and a specified pressure or pressure gradient at the source (r = rw). The Type-II boundary condition (specified flowrate) is a special case (\u03c3 = 0) of the wellbore storage boundary condition (flowrate linearly proportional to change in pressure), so it is not developed further. 2.1 Dimensional Analysis A solution is derived for equation (4), using the approach of Doe (1991), which was based on analogy with the heat conduction literature (Carslaw and Jaeger, 1959). Reducing the governing equation (4) to dimensionless form using characteristic time, Tc = n0cL2 c\u00b5/k0, and characteristic length, Lc = rw, leads to r\u03ba\u2212\u03b7 D \u2202pD \u2202tD = m \u2212\u03ba rD \u2202pD \u2202rD + \u22022pD \u2202r2 D , (6) where the dimensionless quantities rD = r/Lc, tD = t/Tc, and p{I,III} D = p/p{I,III} c are used (see Table 2 for a summary of dimensionless quantities). The characteristic pressure change is given by pI c = \u02c6 p1, where p1(t) = \u02c6 p1ft separates the timedependent specified pressure into a constant characteristic pressure and a dimensionless variable time behavior (for a constant specified pressure, ft = 1). The dimensionless type-I initial and boundary conditions are pD(rD, tD = 0) = 0 pD(rD \u2192\u221e, tD) < \u221e (7) pI D(rD = 1, tD) = ft. Using pIII c = rw \u02c6 Q\u00b5 Amk0 , where Q(t) = \u02c6 Qft similarly separates the time-dependent volumetric flowrate into a constant characteristic flowrate and a dimensionless time behavior. The dimensionless type-III source borehole boundary condition is \u2202pIII D \u2202rD \f \f \f \f rD=1 = ft + \u03c3 \u2202pIII D \u2202t , (8) where \u03c3 is a dimensionless wellbore storage coefficient (see Appendix A) and the same initial and far-field conditions apply as the type-I case. 2.2 Laplace Transform Taking the dimensionless Laplace transform \u0000 \u00af f(s) = R \u221e 0 e\u2212stDf(tD) dtD \u0001 of the governing partial differential equation (6) (without loss of generality assuming zero initial condition) leads to the ordinary differential equation d2\u00af pD dr2 D + m \u2212\u03ba rD d\u00af pD drD \u2212s\u00af pDr\u03ba\u2212\u03b7 D = 0, (9) 5 \fassuming \u03ba, \u03b7, and m are not functions of time, and s is the dimensionless Laplace transform parameter. The transformed type-I and far-field boundary conditions (7) are \u00af pD(rD \u2192\u221e) < \u221e (10) \u00af pI D(rD = 1) = \u00af ft, where \u00af ft represents the Laplace transform of the boundary condition\u2019s time behavior. For a unit step change at t = 0 (where ft = 1, a typical assumption), \u00af ft = 1 s. Other temporal behaviors are simply handled, including a step change at a non-zero time, an exponentially decaying source term, an arbitrary piecewise-constant or piecewise-linear behavior, or a sinusoidal source term (Kruseman and de Ridder, 1994; Mishra et al, 2013). The transformed wellbore-storage boundary condition is d\u00af pIII D drD \f \f \f \f rD=1 = \u00af ft + \u03c3s\u00af pIII D , (11) which now more clearly resembles a Type-III boundary condition. 2.3 Numerical Inverse Laplace Transform The governing equations and associated boundary conditions are solved exactly in Laplace space, then numerically inverted back to the time domain using one of several viable approaches (Kuhlman, 2013). The equations were rapidly prototyped and inverted using the Python library mpmath (Johansson et al, 2017), which provides arbitrary precision special functions and numerical inverse Laplace transform algorithms. A Fortran program was also developed to facilitate plotting and parameter estimation, implementing the inversion algorithm of de Hoog et al (1982). Python and Fortran implementations of the solution are available at https://github.com/klkuhlm/graded. 3 Solution of Flow Problem 3.1 Mapping onto Modified Bessel Equation The governing ordinary differential equation (9) can be made equivalent to a form of the modified Bessel equation after a change of variables first used by Lommel (1868) for the standard Bessel equation. Appendix B illustrates an analogous change of variables to the modified Bessel equation. Comparing (9) to this scaled version of the modified Bessel equation (41), they are equivalent given the following correspondences \u03b1 =1 2 (\u03ba \u2212m + 1) \u03b3 =1 2 (\u03ba \u2212\u03b7 + 2) (12) \u03bd = s \u03b12 \u03b32 = \u03ba \u2212m + 1 \u03ba \u2212\u03b7 + 2 \u03b2 = r s \u03b32 = s 4s (\u03ba \u2212\u03b7 + 2)2 . The transformed modified Bessel equation has the general solution (37) y = z\u03b1 [AI\u03bd (\u03b2z\u03b3) + BK\u03bd (\u03b2z\u03b3)] , (\u03b3 \u0338= 0) , (13) where A and B are constants determined by the boundary conditions and I\u03bd(z) and K\u03bd(z) are the firstand second-kind modified Bessel functions of non-integer order and real argument (McLachlan, 1955; Bowman, 1958; Spanier and Oldham, 1987; DLMF, 2023). The finiteness boundary condition (10) requires A = 0 to keep the solution finite as rD \u2192\u221e, since the first-kind modified Bessel function grows exponentially with increasing real argument, leaving \u00af pD (rD) = r\u03b1 DBK\u03bd (\u03b2r\u03b3 D) , (14) 6 \fwhich is not defined for \u03b3 = 0 (i.e., \u03ba\u2212\u03b7 = \u22122, which is unrealistic because \u03ba is larger than \u03b7 for physical reasons), and B is determined by the Laplace-space source borehole boundary conditions. 3.2 Constant-Pressure (Type-I) at Borehole The borehole boundary condition (rD = 1) for specified change in pressure leads to the solution (the Warren and Root (1963) double porosity solution for this wellbore boundary condition is equivalent to Hayek et al (2018)) \u00af pI D(rD) = \u00af ftr\u03b1 D K\u03bd (\u03b2r\u03b3 D) K\u03bd (\u03b2) (15) and its radial gradient (i.e., proportional to flow of fluid into the borehole) d\u00af pI D drD = \u00af ftr\u03b1\u22121 D \u0014 (\u03b1 \u2212\u03b3\u03bd) K\u03bd (\u03b2r\u03b3 D) K\u03bd (\u03b2) + \u03b2\u03b3r\u03b3 D K\u03bd\u22121 (\u03b2r\u03b3 D) K\u03bd (\u03b2) \u0015 , (16) using a recurrence relationship for the derivative of the Bessel function in terms of Bessel functions of adjacent orders (DLMF, 2023, \u00a710.29.2). Restricting \u03ba \u2265\u03b7 (i.e., permeability decreases as fast or faster than porosity), then \u03b3 > 0 and \u03b1 = \u03b3\u03bd (for \u03b3 < 0, \u03b1 \u2212\u03b3\u03bd = 2\u03b1). This physically motivated restriction on parameters simplifies (16) to d\u00af pI D drD = \u221as \u00af ftr\u03b1+\u03b3\u22121 D K\u03bd\u22121 (\u03b2r\u03b3 D) K\u03bd (\u03b2) , (17) since \u03b2\u03b3 = \u221as for \u03b3 > 0. When evaluated in the source borehole (rD = 1), the solution simplifies further. Figure 2 shows plots of the predicted pressure gradient at rD = 1 due to a constant-pressure condition there (top row) and the predicted decrease in pressure radially away from the boundary (values of \u03b7, \u03ba, and m for each simulation are listed in the caption and title of each figure). Both rows of plots show the variability with the porosity exponent (\u03b7, given by the line color) and the permeability exponent (\u03ba = \u03b7\u03c4, given by the line type). The same results are shown for Cartesian linear (m = 0), cylindrical (m = 1), and spherical (m = 2) geometries in three columns. For a given set of parameters, a higher-dimensional domain (larger m) leads to a slower drop in produced fluids at any time. The highest sustained flowrate for all dimensions is achieved with constant properties in space (i.e., the red curve \u03b7 = \u03ba = 0). More negative exponents in the porosity and permeability power-laws lead to more rapid decrease in flowrate, as the contribution to flow from large radius vanishes when the exponent increases in magnitude. These types of responses might be mis-interpreted as being associated with lower permeability (which would also lead to a faster decrease in flowrate) using a model with constant properties and a fixed dimension. In the source well (top row of subplots), the effect of \u03ba is different and are predicted to reverse between dimensions. For \u03b7 = 3 (black lines), the \u03ba = {3, 6, 9} cases are swapped between m = 1 and m = 2. For \u03b7 = 2 (blue lines), the \u03ba cases are swapped between m = 0 and m = 1. The bottom row of figures shows the predicted pressure with distance at tD = 10. At locations away from the source well (rD > 1), changes in the porosity exponent, \u03b7, have relatively less impact than changes in the permeability exponent, \u03ba (different colored solid lines are close together, while colored lines of different line type are widely separated). The dimensionality (m) has a smaller effect at locations away from the source borehole than it had on the gradient predicted at the source borehole. 3.3 Constant-Flowrate with Wellbore Storage (Type-III) The wellbore-storage boundary condition for the specified flowrate solution at rD = 1 results in the general solution (that is new for any double-porosity solution with power-law variation in material properties) \u00af pIII D (rD) = \u00af ftr\u03b1 D K\u03bd (\u03b2r\u03b3 D) (\u03b1 \u2212\u03b3\u03bd + \u03c3s) K\u03bd (\u03b2) + \u03b2\u03b3K\u03bd\u22121 (\u03b2), (18) which can be simplified using \u03b1 = \u03b3\u03bd and \u03b2\u03b3 = \u221as to \u00af pIII D (rD) = \u00af ftr\u03b1 D K\u03bd (\u03b2r\u03b3 D) \u221asK\u03bd\u22121 (\u03b2) + \u03c3sK\u03bd (\u03b2). (19) 7 \fFig. 2 Type-I flowrate (top row at rD = 1) and pressure (bottom row at rD > 1 and tD = 10) solution at borehole for m = 0, 1, 2 (Cartesian, cylindrical, and spherical) and at different radial distances. Line color indicates \u03b7; line type indicates \u03ba/\u03b7. Line segments in top row illustrate slopes of 1/2, 1, and 3/2. Analogous to the results for the Type-I solution but only showing the m = 1 and m = 2 cases, Figure 3 shows the predicted pressure through time at the boundary for a specified flowrate at the boundary. Figure 3 results are for no wellbore storage (\u03c3 = 0), while Figure 4 shows the same results with nonzero wellbore storage (all model parameters listed in caption or title of each figure). Wellbore storage is important at early time, leading to a smaller predicted change in pressure, with the predicted response giving a characteristic 1 : 1 slope on log-log plots before formation storage contributes significantly to the flow (i.e., pumping in a bathtub). Wellbore storage makes more of a difference (i.e., shows a larger deviation from \u03c3 = 0 case) for larger \u03b7 (and \u03ba, since \u03ba = 2\u03b7). 3.4 Parameter Combinations Yielding Simpler Solutions When \u03b7 = \u03ba = 0, permeability and porosity are constant in space; in this case (9) simplifies to d2\u00af pD dr2 D + m rD d\u00af pD drD \u2212s\u00af pD = 0, (20) 8 \fFig. 3 Type-II solution (Type-III with \u03c3 = 0) at borehole for m = 1, 2 (cylindrical and spherical). Line color indicates \u03b7; line type indicates \u03ba/\u03b7. which is the dimensionless form of the equation solved by Barker (1988). In this case \u03b3 = 1, \u03b1 = (1\u2212m)/2, \u03bd = \u03b1, and \u03b2 = \u221as. The solution in Laplace-space under these conditions becomes \u00af pD (rD) = r\u03bd DBK\u03bd \u0000\u221asrD \u0001 , (21) which was found by Barker (1988, Eqn. 15). When \u03b7 = \u03ba = m = 0 the time-domain solution simplifies to pD(t) = 1/ \u221a \u03c0t, because \u03bd = 1/2 and \u03bd \u22121 = \u22121/2, the numerator and denominator of (17) are equal since K\u03bd(z) \u2261K\u2212\u03bd(z). Another simplification occurs when m = \u03ba = \u03b7, not necessarily zero. In this case, the permeability and porosity decrease at the same rate radially that the surface area of the domain grows in size (A0 \u221d1, A1 \u221drD, A2 \u221dr2 D), resulting in an equivalent Cartesian coordinate system, d2\u00af pD dr2 D \u2212s\u00af pD = 0, (22) which has a solution in terms of sin(\u221asrD) and cos(\u221asrD) or exp(\u00b1\u221asrD) and typically has an explicit inverse Laplace transform. In this case \u03b1 = \u03bd = 1/2, \u03b3 = 0, and \u03b2 = \u221as. When \u03bd = n \u00b1 1 2 (for n integer), the modified Bessel functions become modified spherical Bessel functions (DLMF, 2023, \u00a710.47), and when \u03bd = \u00b1 1 3, they become Airy functions (DLMF, 2023, \u00a79.6). These additional special cases are not handled differently here (i.e., the more general solution in terms of modified Bessel functions is still valid), since in the case given here \u03bd varies with \u03ba, \u03b7, and m (12). 4 Extension of Solution to Double Porosity 4.1 Mass-Transfer Coefficient Approximation Beginning with the Warren and Root (1963) formulation for double-porosity (i.e., high-conductance fractures and high-capacity matrix), the power-law permeability and porosity distributions are incorporated. 9 \fFig. 4 Type-III solution at borehole (rD = 1), for m = 1, 2 (cylindrical and spherical). Line color indicates \u03b7; line type indicates \u03c3. All curves for \u03ba/\u03b7 = 2. The equations for double-porosity flow in the fractures and matrix are 1 rm \u2202 \u2202r \u0014kf \u00b5 \u2202pf \u2202r \u0015 = nrcr \u2202pr \u2202t + nfcf \u2202pf \u2202t \u02c6 \u03b1kr \u00b5 (pf \u2212pr) = nrcr \u2202pr \u2202t (23) where \u02c6 \u03b1 is the shape factor [1/m2] of Warren and Root (1963), subscript f indicates fracture, and subscript r indicates matrix (rock). The matrix equation does not involve a spatial gradient of pressure, nor a matching of pressure and flux at the boundary, but simply a difference between the fracture and matrix pressure (i.e., the mass transfer coefficient approximation often used for heat transfer across thin films). This behavior is sometimes referred to in the petroleum engineering literature as \u201csteady-state\u201d flow between the fracture and matrix (Da Prat, 1990), but it also represents one-dimensional diffusion in the matrix with a thin-film mass-transfer approximation between the fracture and matrix reservoirs, analogous to Newton\u2019s law of cooling. Substituting the permeability ki = ki0 \u0010 r rw \u0011\u2212\u03bai and porosity ni = ni0 \u0010 r rw \u0011\u2212\u03b7i (i \u2208{f, r}), then converting to dimensionless form using an analogous approach to Warren and Root (1963), where \u03c9 = nf0cf/ (nr0cr + nf0cf) is the dimensionless fracture storage coefficient and \u03bb = \u02c6 \u03b1krr2 w/kf is the dimensionless interporosity exchange coefficient. Finally, taking the Laplace transform of both equations results in the pair of ordinary differential equations \u0014d2\u00af pfD dr2 D + m \u2212\u03baf rD d\u00af pfD drD \u0015 r\u2212\u03baf = (1 \u2212\u03c9)r\u2212\u03b7r D \u00af pmDs + \u03c9r\u2212\u03b7f D \u00af pfDs \u03bb (\u00af pfD \u2212\u00af prD) r\u2212\u03bar D = (1 \u2212\u03c9)r\u2212\u03b7r D \u00af prDs. (24) Solving for matrix pressure in the matrix equation, \u00af prD = \u00af pfD\u03bbr\u2212\u03bar D / \u0002 (1 \u2212\u03c9)sr\u2212\u03b7r D + \u03bbr\u2212\u03bar D \u0003 , and substituting this into the fracture equation leads to a single equation solely in terms of dimensionless 10 \fFig. 5 Type-I flowrate solution at borehole (left) and Type-II solution for pressure (\u03c3 = 0, right), for m = 1 (cylindrical). Line color indicates \u03bb; line type indicates \u03c9. Laplace-domain fracture pressure \u0014d2\u00af pfD dr2 D + m \u2212\u03baf rD d\u00af pfD drD \u0015 r\u2212\u03baf = r\u2212\u03b7r D \u00af pfD ( (1 \u2212\u03c9)sr\u2212\u03bar D \u03bb (1 \u2212\u03c9)sr\u2212\u03b7r D + \u03bbr\u2212\u03bar D ) + \u03c9r\u2212\u03b7f D \u00af pfDs. (25) To force the term in curly brackets in (25) to be independent of rD, \u03bar = \u03b7r is assumed. Setting \u03bar and \u03b7r equal to \u03b7f allows rD and \u00af pfD to be similar form to previous solutions. Simplifying the subsequent notation \u03baf \u2192\u03ba, \u03b7r \u2192\u03b7, and \u00af pfD \u2192\u00af pD results in d2\u00af pD dr2 D + m \u2212\u03ba rD d\u00af pD drD = r\u03ba\u2212\u03b7 D \u00af pD \u0014 (1 \u2212\u03c9)s\u03bb (1 \u2212\u03c9)s + \u03bb + \u03c9s \u0015 , (26) which is the same form as (9). This solution corresponds to the same scaled Bessel equation, with only the definition of \u03b2 changing to \u03b2W R = s\u0014 \u03bb \u03bb/(1 \u2212\u03c9) + s + \u03c9 \u0015 s \u03b32 . (27) Any more general spatial behavior of matrix properties (e.g., \u03b7r \u0338= \u03bar) would not be solvable with the same approach. This limitation still makes physical sense, as the the most important terms to vary with space are the fracture permeability and the matrix storage. Setting \u03ba = \u03b7 = 0 and m = 1 results in the Warren and Root (1963) solution. Figure 5 shows typical solution behaviors for the cylindrical (m = 1) case for Type-I and Type-II wellbore boundary conditions, for \u03b7 = 3 and \u03ba = 6. Figure 6 shows behavior from the \u201cmiddle\u201d curve in Figure 5 (\u03bb = 10\u22125 and \u03c9 = 10\u22124), for a range of porosity and permeability exponents similar to those shown in Warren and Root (1963), listed in the figure caption. 11 \fFig. 6 Type-I flowrate solution at borehole (left) and Type-II solution for pressure (\u03c3 = 0, right), for m = 1 (cylindrical). All curves are for \u03bb = 10\u22125 and \u03c9 = 10\u22124 (middle curves shown in Figure 5). Line color indicates \u03b7; line type indicates \u03ba/\u03b7. 4.2 Matrix Diffusion The matrix diffusion problem of Kazemi (1969) is more physically realistic (Aguilera, 1980; Da Prat, 1990), but it is typically solved numerically or via late-time approximations (De Swaan, 1976), rather than analytically like Warren and Root (1963). The series approach of Kuhlman et al (2015) is used here to represent matrix diffusion in a single matrix continuum through the sum of an infinite series of Warren-Root matrix continua, and the infinite sum is then analytically summed. The generalization of (23) to multiple matrix continua starts with 1 rm \u2202 \u2202r \u0014kf \u00b5 \u2202pf \u2202r \u0015 = N X j=1 njcj \u2202pj \u2202t + nfcf \u2202pf \u2202t \u02c6 \u03b1jkj \u00b5 (pf \u2212pj) = njcj \u2202pj \u2202t j = 1, 2, . . . N, (28) where N is the number of matrix continua (one additional equation for each continuum). Similarly taking the Laplace transform of this set of equations, solving for \u00af pf, substituting the matrix equations into the fracture equation, and simplifying the notation leads to d2\u00af pD dr2 D + m \u2212\u03ba rD d\u00af pD drD = r\u03ba\u2212\u03b7 D \u00af pD\u03c9s(1 + \u00af g), (29) where \u00af g = N X j=1 \u02c6 \u03bejuj s + uj (30) is a matrix memory kernel (Haggerty and Gorelick, 1995), \u02c6 \u03bej is related to the storage properties of each matrix continuum (analogous to \u03c9 of Warren and Root (1963)), and uj is related to the interporosity flow coefficient of each matrix continuum (analogous to \u03bb of Warren and Root (1963)). The Laplacespace memory kernel approach is flexible, and is used elsewhere in hydrology and reservoir engineering (Herrera and Yates, 1977; Haggerty et al, 2000; Schumer et al, 2003). Equation (29) can be simplified to 12 \fWarren and Root (1963) with a particular choice of \u00af g and N = 1, and to the solution for a triple-porosity reservoir (Clossman, 1975) with a different choice of \u00af g and N = 2 (Kuhlman et al, 2015). When N \u2192\u221ein (30), the it is more convenient to specify the mean and variance of the parameter distributions than the individual parameters associated with each porosity. Several different distributions are possible (Haggerty and Gorelick, 1995). In the form presented by Kuhlman et al (2015), the parameters are specified as the infinite series uj = (2j \u22121)2\u03c02\u03bb 4(1 \u2212\u03c9) \u02c6 \u03bej = 8(1 \u2212\u03c9) (2j \u22121)2\u03c9\u03c02 j = 1, 2, . . . N \u2192\u221e (31) which leads to the Kazemi (1969) solution for matrix diffusion. The parameters \u03bb and \u03c9 have the same definitions as in Warren and Root (1963). Setting \u03ba = \u03b7 = 0 results in the solution of Kuhlman et al (2015). The new governing equation is the same form and the modified Bessel function solution, only requiring re-definition of \u03b2 as \u03b2KZ = v u u t \" N X j=1 \u03c9\u02c6 \u03bejuj uj + s + \u03c9 # s \u03b32 , N \u2192\u221e. (32) Substituting the definitions of u and \u02c6 \u03be from (31) and simplifying leads to \u03b2KZ = v u u t \" N X j=1 2\u03bb W 2 j \u03bb/(1 \u2212\u03c9) + s + \u03c9 # s \u03b32 , N \u2192\u221e, (33) where Wj = \u03c0(2j \u22121)/2. This is similar in form to (27) but the term in the denominator grows as the index increases, illustrating how the series solution approximates the Kazemi (1969) solution through an infinite series of modified Warren and Root (1963) matrix porosities. Further simplifying the approach of Kuhlman et al (2015), the infinite series in (33) can be evaluated in closed form using residue methods (Wolfram Research, Inc., 2021), resulting in \u03b2KZ = v u u t \"r \u03bb(1 \u2212\u03c9) s tanh  r s(1 \u2212\u03c9) \u03bb ! + \u03c9 # s \u03b32 , (34) where tanh(\u00b7) is the hyperbolic tangent. This closed-form expression derived here is more accurate and numerically more efficient than truncating or accelerating the infinite series in (32), which is an improvement over the series presented in Kuhlman et al (2015) for graded or homogeneous domains. Figure 7 illustrates the transition from the Warren and Root (1963) (N = 1) to the Kazemi (1969) series approximation for increasing terms (N = {2, 10, 100, 1000}, heavy colored solid lines) and the expression for the infinite sum (34) (heavy black dashed line) for flow to a specified flux (type-II, \u03c3 = 0) cylindrical (m = 1) borehole of constant material properties (\u03ba = \u03b7 = 0). The bounding Theis (1935) behavior is shown for the fracture and matrix compressibilities (thin red dashed lines). 5 Applications and Limitations A general converging radial flow solution for specified flowrate or specified wellhead pressure was derived for domains with power-law variability in porosity and permeability due to damage. The single-porosity version has already been presented by Doe (1991), and a solution for constant-pressure condition without wellbore storage was derived by Hayek et al (2018), but the specified-flowrate double-porosity solution with wellbore storage presented here is new. The infinite series approximation to Kazemi was summed analytically, resulting in a new closed-form expression of the series presented in Kuhlman et al (2015), which is an improvement for both graded and homogeneous properties. The newly developed analytical solutions are more general (i.e., several existing solutions are special cases of the new solution) and include more behaviors typical in well-test solutions (i.e., wellbore storage, positive skin, double porosity), 13 \fFig. 7 Type-II solution for pressure at source borehole (\u03c3 = 0), for m = 1 (cylindrical) for different number of terms. All curves are for \u03bb = 10\u22125, \u03c9 = 10\u22124, \u03ba = \u03b7 = 0. while still being straightforward and parsimonious (i.e., as few free parameters as possible) in their implementation. The basic flow solution assumes linear single-phase flow of a fluid in a slightly compressible formation. The double-porosity solution assumes the fractures are high permeability, with low storage capacity, while the matrix (i.e., intact rock between fractures) is high storage capacity with low permeability. These assumptions are representative for analytical solutions to subsurface porous media flow problems in the hydrology and petroleum engineering literature, and are shared by the solutions of Barker (1988), Doe (1991), Warren and Root (1963), Kazemi (1969), and Kuhlman et al (2015). To apply this analytical solution to observed data, either observed data would be transformed into dimensionless space, or the analytical solution could be transformed to dimensional space, then a parameter estimation routine would be used to minimize the model-data misfit, and possibly explore the uncertainty or uniqueness of the solution. The solution method developed to solve these solutions uses numerical inverse Laplace transforms and runs quickly enough to be used in parameter estimation (e.g., Monte Carlo methods that require hundreds of thousands of evaluations). The analytical solution might be of most use with parameter estimation to fit observations, but the non-uniqueness of the curves may make estimation of unique physical parameters difficult, without further physical or site-specific constraints. Realistically, the parameters in the Bessel equation may be estimable (i.e., \u03b1, \u03b2, \u03b3, and \u03bd defined in (12)), but without defining the flow dimension (m) or the relationship between the porosity and permeability exponents (\u03c4 = \u03ba/\u03b7), it may be difficult to identify all the parameters from data alone, since many the curves have similar shapes, unlike classical Type curves (Bourdet et al, 1989). 14 \fAc borehole cross-sectional area m2 Am borehole cylindrical surface area m2 c bulk compressibility 1/Pa ft time variability \u2212 g gravitational acceleration m/s2 h hydraulic head m k permeability m2 Lc characteristic length (rw) m m dimension (D \u22121) \u2212 n porosity \u2212 p change in pressure Pa s Laplace transform parameter \u2212 Q volumetric flowrate m3/s r distance coordinate m rw borehole or excavation radius m \u02c6 \u03b1 Warren and Root (1963) shape factor 1/m2 \u03b7 porosity power-law exponent \u2212 \u03ba permeability power-law exponent \u2212 \u03c1 fluid density kg/m3 \u00b5 fluid viscosity Pa \u00b7 s Table 1 Physical Properties and Parameters pD scaled pressure p/pc tD scaled time tk0/n0cL2 c\u00b5 rD scaled distance r/Lc \u03bb interporosity exchange coefficient \u02c6 \u03b1krr2 w/kf \u03c3 wellbore storage coefficient Ac/(rwn0c\u03c1gAm) \u03c9 fracture storage coefficient nf0cf/(nr0cr + nf0cf) Table 2 Dimensionless Quantities Statements and Declarations Funding The author thanks the U.S. Department of Energy Office of Nuclear Energy\u2019s Spent Fuel and Waste Science and Technology program for funding. Conflicts of Interest The author has no competing interests to declare. Availability of Data and Material No data or materials were used by the author in the preparation of the manuscript. Code Availability The source code of Fortran and Python implementations of the program are available from the author upon request. Acknowledgments This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish 15 \for reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.energy.gov/downloads/doe-public-access-plan. The author thanks Tara LaForce from Sandia for technically reviewing the manuscript. 6 Appendix A: Wellbore Storage Boundary Condition The wellbore-storage boundary condition accounts for the storage in the finite borehole arising from the mass balance Qin \u2212Qout = Ac \u2202hw \u2202t . Qin [m3/s] is volumetric flow into the borehole from the formation, Qout is possibly time-variable flow out of the well through the pump (Q(t) [m3/s]), and \u2202hw \u2202t is the change in hydraulic head [m] (hw = pw \u03c1g + z) of water standing in the borehole through time, pw is change in pressure [Pa] of water in the borehole, \u03c1 is fluid density [kg/m3], z is an elevation datum [m], and g is gravitational acceleration [m/s2]. Ac is the cross-sectional surface area of the pipe, sphere or box providing storage (it may be a constant or a function of elevation); for a typical pipe, it becomes Ac = \u03c0r2 c, where rc is the radius of the casing where the water level is changing. The mass balance is then Amk0 \u00b5 \u2202p \u2202r \f \f \f \f r=rw \u2212Q(t) = Ac \u03c1g \u2202pw \u2202t , (35) where Am is the area of the borehole communicating with the formation. For the integer m considered here these are A0 = b2, A1 = 2\u03c0rwb, A2 = 4\u03c0r2 w (b is a length independent of the borehole radius). Assuming the change in water level in the borehole (hw = pw/ (\u03c1g)) is equal to the change in formation water level (h = p/ (\u03c1g)), this can be converted into dimensionless form as \u2202pD \u2202rD \f \f \f \f rD=1 \u2212ft = \u03c3 \u2202pD \u2202t , (36) where \u03c3 = Ac/ (rwn0c\u03c1gAm) is a dimensionless ratio of formation to wellbore storage; \u03c3 \u21920 is an infinitesimally small well with only formation response, while \u03c3 \u2192\u221eis a well with no formation response (i.e., a bathtub). 7 Appendix B: Transformation of Modified Bessel Equation Following the approach of Bowman (1958), alternative forms of the Bessel equation are found, this approach is a simplification of the original approach of Lommel (1868). An analogous approach is applied here to \u201cback into\u201d the desired modified Bessel equation. The equation satisfied by the pair of functions y1 = x\u03b1I\u03bd (\u03b2x\u03b3) , y2 = x\u03b1K\u03bd (\u03b2x\u03b3) (37) is sought, where \u03b1, \u03b2, \u03b3, and \u03bd are constants. Using the substitutions \u03b6 = yx\u2212\u03b1 and \u03be = \u03b2x\u03b3 gives \u03b61 = I\u03bd (\u03be) and \u03b62 = K\u03bd (\u03be), which are the two solutions to the modified Bessel equation (DLMF, 2023, \u00a710.25.1), \u03be d d\u03be \u0012 \u03be d\u03b6 d\u03be \u0013 \u2212(\u03be2 + \u03bd)\u03b6 = 0. (38) Given \u03be d d\u03be \u0012 \u03be d\u03b6 d\u03be \u0013 = x \u03b32 d dx \u0012 x d\u03b6 dx \u0013 , (39) and x d dx \u0012 x d\u03b6 dx \u0013 = y\u2032\u2032 x\u03b1\u22122 \u2212(2\u03b1 \u22121) y\u2032 x\u03b1\u22121 + \u03b12y x\u03b1 , (40) the standard-form equation satisfied by y is y\u2032\u2032 + (1 \u22122\u03b1) y\u2032 + \u03b12y x\u03b1 \u2212 \u0012 \u03b22\u03b32x2\u03b3\u22122 \u2212\u03b12 \u2212\u03bd2\u03b32 x2 \u0013 y = 0. (41) 16 \fThis equation can be compared to the Laplace-space ordinary differential equation (9), allowing direct use of the product of powers and modified Bessel function (37) as solutions (13)."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1502.03067v1",
+                    "title": "Multiporosity Flow in Fractured Low-Permeability Rocks",
+                    "abstract": "A multiporosity extension of classical double and triple porosity fractured\nrock flow models for slightly compressible fluids is presented. The\nmultiporosity model is an adaptation of the multirate solute transport model of\nHaggerty and Gorelick (1995) to viscous flow in fractured rock reservoirs. It\nis a generalization of both pseudo-steady-state and transient interporosity\nflow double porosity models. The model includes a fracture continuum and an\noverlapping distribution of multiple rock matrix continua, whose\nfracture-matrix exchange coefficients are specified through a discrete\nprobability mass function. Semi-analytical cylindrically symmetric solutions to\nthe multiporosity mathematical model are developed using the Laplace transform\nto illustrate its behavior. The multiporosity model presented here is\nconceptually simple, yet flexible enough to simulate common conceptualizations\nof double and triple porosity flow. This combination of generality and\nsimplicity makes the multiporosity model a good choice for flow in\nlow-permeability fractured rocks.",
+                    "authors": "Kristopher L. Kuhlman, Bwalya Malama, Jason E. Heath",
+                    "published": "2015-01-19",
+                    "updated": "2015-01-19",
+                    "primary_cat": "physics.geo-ph",
+                    "cats": [
+                        "physics.geo-ph"
+                    ],
+                    "main_content": "Introduction The \ufb02ow of slightly compressible \ufb02uids through fractured rocks is of fundamental importance to groundwater and hydrocarbon production and e\ufb00ective isolation of radioactive waste and CO2. In low-permeability rocks, fractures are often the primary source of bulk permeability. Reservoir and storage rocks often include multiple overlapping and intersecting fracture 1 arXiv:1502.03067v1  [physics.geo-ph]  19 Jan 2015 \fsets. Discrete representations of fractures in numerical models is possible if information is available about fracture spacing, orientation, and aperture. Typically, spatial distributions of these data are not available, and fractured rocks must be approximated as porous media (Gringarten, 1982; Sahimi, 2011). Multiple porosity types can simultaneously participate in \ufb02ow through fractured and unfractured rocks. Shales have macroscopic porosity associated with clastic particles and microscopic porosity associated with their organic fraction (Akkutlu and Fathi, 2012). The Culebra Dolomite has illustrated e\ufb00ects of multiple porosity types (fractures, vugs, and intergranular porosity) during \ufb02ow and solute transport at both the \ufb01eld and laboratory scale (McKenna et al., 2001; Malama et al., 2013). The e\ufb00ect of multiple types and scales of porosities can be masked by or mistaken for the e\ufb00ects of multiple types and scales of heterogeneity (Altman et al., 2002). Dual or multiple porosity models are common simpli\ufb01ed conceptualiztaions of this complex reality of interacting porosity types and the spatially heterogeneous nature of rocks. The treatment of fractured rock as a system of interacting and overlapping continua has been used successfully in its di\ufb00erent forms (e.g., Gringarten, 1982; Aguilera, 1980; Streltsova, 1988; Chen, 1989; Da Prat, 1990; or Bourdet, 2002) since its \ufb01rst introduction by Barenblatt and Zheltov (1960). The conceptualization is based on a low-permeability but high storage-capacity rock matrix drained via natural and man-made high-permeability but low storage-capacity fractures. Slightly compressible \ufb02uids include water, brine, oil, and other liquids for which density can be assumed constant, allowing a formulation in terms of head (rather than pressure). Gases can be treated as a slightly compressible \ufb02uids after a transformation to create a pseudo-pressure and pseudo-time, which accounts for pressure-dependence of \ufb02uid properties (Friedmann, 1958; Al Hussainy et al., 1966). Slightly compressible \ufb02ow models can also approximate more complex non-linear \ufb02ow (i.e., multiphase \ufb02ow, time-variable permeability, or gas desorption) by transforming observations into one of several possible integrated pseudo variables (see review by Clarkson, 2013). Flow prediction for slightly compressible \ufb02uids in fractured low-permeability rocks at a macroscopic (wellbore or reservoir) scale is of great importance to applications in groundwater supply, hydrocarbon production, and underground sequestration of nuclear waste or CO2. Fractured reservoirs exhibit pressure drawdown due to production at a speci\ufb01ed pressure or production at a speci\ufb01ed \ufb02owrate (e.g., during recovery or shut-in), in a manner characteristic of multiple interacting porosities (e.g., Crawford et al., 1976; Gringarten, 1982; Moench, 1984). Fractures provide high-permeability pathways, often orders of magnitude more permeable than the unfractured rock itself, but comprise only a small portion of the rock volume. Natural fracture porosity is often less than 0.1%, but the fracture network is a much more e\ufb03cient \ufb02uid conductor per unit porosity than the matrix (Streltsova, 1988). Several well-known double porosity conceptualizations for \u201cuniformly\u201d or \u201cnaturally\u201d fractured reservoirs have been developed (e.g., Barenblatt et al., 1960; Warren and Root, 1963; and Kazemi, 1969), which approximate the matrix as being storage only (no advection outside of fractures). These solutions represent the \ufb02ow domain with overlapping fracture and matrix continua. These models assume the domain of interest is large enough and the fracture 2 \fdensity is uniform enough to treat both fractures and matrix as continua. Fractures are the main source of permeability and spatial connectivity, while the matrix is the main source of storage. Triple porosity solutions with a single fracture continuum and two matrix continua (only connected through the fracture porosity) are a logical extension of dual porosity (e.g., Clossman, 1975; Liu, 1981; Al Ahmadi and Wattenbarger, 2011; and Tivayanonda et al., 2012). Beginning from the system proposed by Barenblatt and Zheltov (1960), but equally allowing advection through fractures and matrix results in the dual permeability model, which requires a coupled set of governing equations and boundary conditions. These were \ufb01rst solved analytically by Chen and Jiang (1980) in terms of quasi-Bessel functions, but this solution has not seen wide use. Although the dual-permability approach is more physically realistic than the simpli\ufb01ed dual-porosity approach, when matrix permeability is much smaller than the fractures, dual porosity is an adequate approximation (Chen, 1989). The multiple interacting continua (MINC) approach taken by TOUGH2 is a numerical approach capable of simulating both dual-porosity and dual-permeability systems (Pruess and Narasimhan, 1985; Pruess et al., 1999). Many analytical solutions for variations on double and triple porosity conceptual models have been developed (Moench, 1984; Chen, 1989; Da Prat, 1990), with di\ufb00erent con\ufb01gurations and relationships between fracture and matrix continua, but a generalized solution is needed. We present multiporosity (the name coming from multirate and dual-porosity) as a generalization and extension of the Warren and Root (1963) solution to any number of matrix continua interacting with a fracture continuum. Multiporosity is an adaptation of the multirate solute transport theory to viscous \ufb02ow, for computing pressure-driven \ufb02ow through low-permeability rocks with heterogeneity and multiple porosities. The multirate (i.e., multiple reaction rates) conceptual model for solute transport (Haggerty and Gorelick, 1995, 1998) has been successfully used to simulate di\ufb00usion of solutes from fast-\ufb02owing fractures into a distribution of di\ufb00usion-dominated matrix block sizes (Haggerty et al., 2000, 2001; McKenna et al., 2001; Malama et al., 2013). 2 Multiporosity Model It is useful to conceptualize slightly compressible \ufb02ow in uniformly fractured domains as diffusion in a porous medium. Some deviations from this ideal behavior can be accommodated through use of pseudo-time and -pressure (Clarkson, 2013). 2.1 Conceptualization Two primary \ufb02ow conceptualizations used in slightly compressible dual-porosity systems are: 1. The pseudo-steady-state Warren and Root (1963) (WR) interporosity \ufb02ow conceptualization (similar to Barenblatt et al., 1960) assumes \ufb02ow from matrix blocks is proportional to the pressure di\ufb00erence between the fracture and the average matrix pressures. 3 \f2. The Kazemi (1969) (KZ) transient interporosity \ufb02ow conceptualization allows transient di\ufb00usion from the fracture to the matrix, and couples \ufb02ow in the fracture and matrix domains through a source term proportional to \ufb02ux in the fracture \ufb02ow governing equation. The WR matrix \ufb02ow conceptualization is much simpler than the KZ conceptualization, but leads to analytical \ufb02ow solutions more readily than the KZ approach (e.g., see review by Chen, 1989). The KZ approach has been solved using \ufb01nite di\ufb00erences (Kazemi, 1969), \ufb01nite volumes (Pruess et al., 1999), and various analytical (e.g., Serra et al., 1983, Chen et al., 1985, or Ozkan et al., 1987) and approximate (Walton and McLennan, 2013) approaches. The WR solution produces physically unrealistic pressure transient solutions during the transition from early fracture-dominated \ufb02ow to later matrix-dominated \ufb02ow (Gringarten, 1982; Moench, 1984). It produces a nearly \ufb02at transition between early fracture \ufb02ow and late-time matrix \ufb02ow on a semi-log plot, equivalent to a vanishing log time derivative \u2013 \u2202/\u2202(ln t). Moench (1984) showed the WR model could be improved by adding a fracture skin, which delayed communication between the fracture and matrix. The WR approach assumes pseudo-steady-state \ufb02ow between fracture and matrix, which may not occur physically until late time in low-permeability rocks. The KZ model produces a half-slope transition between the early and late-time \ufb02ow, which is more in agreement with typical \ufb01eld observations (Moench, 1984). The multiporosity approach is a generalization of the WR double-porosity model and the pseudo-steady-state triple-porosity model of Clossman (1975). Triple-porosity models can represent two systems of natural preexisting microand macro-fractures (Al Ahmadi and Wattenbarger, 2011). The multiporosity model is presented here as a spatial distribution of natural fractures and matrix materials (i.e., matrix heterogeneity). When moving between models with di\ufb00erent numbers of porosities (e.g., doubleor tripleporosity compared to single-porosity models), a model with more parameters is typically more \ufb02exible and able to \ufb01t a wider range of observed behaviors, but more free parameters must be estimated. Data collected from typical low-permeability wells are inadequate to uniquely constrain models with many estimable parameters. To constrain the number of free parameters in the multiporosity model, the interporosity \ufb02ow parameters can be speci\ufb01ed with a distribution function (e.g., Ranjbar et al. (2012)). Haggerty and Gorelick (1995) derived a distribution comprised of an in\ufb01nite series of porosities. Their distribution is equivalent to a distribution of pseudo-steady-state WR matrix porosities that behave in total as a transient KZ-type system. The multiporosity model can be made equivalent to di\ufb00usion into a slab (i.e., the KZ conceptualization), cylinder, or sphere. During parameter estimation, the multiporosity model can be matched to data either with \ufb02exible but parsimonious property distributions (e.g., lognormal or beta), or with an arbitrary number of individual porosities and their associated parameters. The multiporosity distribution can be \ufb02exible, but with certain distributions it can also be shown to be a generalization of two well-known physically based end members. Porosity distributions have been utilized in some dual-porosity solutions based upon different block-size distribution function (e.g., McGuinness, 1986; Chen, 1989; and Ranjbar 4 \fet al., 2012), which is similar to the approach taken here. Through parameter estimation, matrix block distributions can provide a fracture distribution, which is much more \ufb02exible than typically double or triple porosity models that assume more uniformly spaced fracture distributions. Existing block-size distribution solutions have focused primarily on the derivation of shape factors. We show how certain distributions of porosities lead to the well-known special cases of WR and KZ double-porosity \ufb02ow. A multiporosity conceptual model for \ufb02ow has two possible meanings. Multiporosity can represent a single matrix continuum with the inter-porosity exchange coe\ufb03cient between the fracture and matrix continua treated as heterogeneous (i.e., a random variable). This is a simpli\ufb01cation of small-scale spatial heterogeneity (e.g., Haggerty and Gorelick, 1995; Haggerty and Gorelick, 1998; or Akkutlu and Fathi, 2012). Alternatively, it can represent a set of multiple physically-distinct matrix continua with the porosity and permeability of each as random variables. Our discussion takes the former approach, we believe it to be the most physically realistic. 2.2 Mathematical Model We present the multiporosity \ufb02ow model, which is a logical extension of the WR conceptualization (Warren and Root, 1963), to an arbitrary number of pseudo-steady-state matrix domains. The model is an adaptation of the multirate advection-dispersion solute transport solution (Haggerty and Gorelick, 1995, 1998) to viscous \ufb02ow. It conceptualizes random porosity spatial variability in rock matrix as a random distribution of uniform matrix porosities communicating with the primary fracture porosity. The governing equation for pressure head drawdown \u2206p(x, t) = p0(x) \u2212p(x, t) (p0 is initial pressure head) in the fracture continuum is \u03c6fcf \u2202(\u2206pf) \u2202t + N X j=0 \u03c6jcj\u03c7j \u2202(\u2206pj) \u2202t = kf \u00b5 \u22072 (\u2206pf) , (1) where t is time, index j and subscript f denote quantities related to the jth matrix and fracture continua, the sum is across N matrix porosities (N may be in\ufb01nite), \u03c6 is dimensionless porosity, c is a compressibility or storage coe\ufb03cient, k is permeability, \u00b5 is \ufb02uid viscosity, and \u03c7 is a dimensionless probability mass function (PMF \u2013 the discrete form of a probability density function) of interporosity exchange coe\ufb03cients. See Table 2 for a summary of physical quantities and their units. In this multiporosity conceptualization, matrix properties and dependent variables are implicitly discrete functions of the distribution of matrix continua index. The properties controlling matrix-fracture \ufb02uid exchange across a potentially in\ufb01nite number of matrixfracture interfaces can be considered a random variable, but the resulting governing equations are deterministic because only their sum or bulk behavior appears in (1). Flow in each of the matrix domains is generally governed by the di\ufb00usion equation, viz. \u03c6jcj \u2202(\u2206pj) \u2202t = kj \u00b5 \u22072 (\u2206pj) j = 1, . . . , N; (2) 5 \fthe governing equations will be non-dimensionalized and solved using the Laplace transform. The governing multiporosity fracture \ufb02ow equation (1) can be non-dimensionalized by dividing through by the total formation compressible storage \u03c6c = \u03c6fcf + PN j=1 \u03c6jcj, and a characteristic pressure Pc, to produce the dimensionless fracture \ufb02ow equation \u03c9f \u2202\u03c8f \u2202tD + N X j=1 \u03c9j\u03c7j \u2202\u03c8j \u2202tD = \u22072 D\u03c8f, (3) where \u03c8\u2113= \u2206p\u2113/Pc (\u2113\u2208{f, j}) is dimensionless pressure change, \u03c9\u2113= \u03c6\u2113c\u2113/ (\u03c6c) is the fractional storage of an individual continuum (0 \u2264\u03c9\u2113\u22641 and P \u03c9 = 1), \u22072 D/L2 c = \u22072 is the dimensionless Laplacian, Lc is a characteristic length (often de\ufb01ned as the pumping well radius, rw), tD = t/Tc, and Tc = L2 c\u00b5\u03c6c/kf is a characteristic time. Alternatively, \u03c9\u2113 can be related to the volume-weighted storage coe\ufb03cient commonly used in hydrogeology, \u03c9\u2113= Ss\u2113V\u2113/ \u0010 SsfVf + P j SsjVj \u0011 , where V is a fraction of the total volume (Gringarten, 1982; Moench, 1984). Table 3 de\ufb01nes dimensionless quantities for the current problem. The matrix \ufb02ow equation (2) can analogously be non-dimensionalized into \u03c9j \u2202\u03c8j \u2202tD = \u03baj\u22072 D\u03c8j j = 1, . . . , N, (4) where \u03baj = kj/kf is the jth matrix-to-fracture permeability ratio. To simplify from the transient to the pseudo-steady-state interporosity \ufb02ow conceptualization we use average matrix pressure by integrating the matrix governing equation (4) across the matrix domain. We assume matrix blocks are comprised of one-dimensional slabs (i.e., rectangular radially symmetric blocks with \ufb02ow perpendicular to fractures, see Figure 1). The layered idealization in Figure 1 results in an equivalent solution to the dual porosity conceptualization when continua exist at each physical location (Streltsova, 1988). We choose slabs for simplicity; other possible matrix block geometries (e.g., spheres or cylinders) do not result in markedly di\ufb00erent predicted results for the double porosity conceptualization (e.g., Gringarten, 1982; Moghadam et al., 2010). Integrating (4) across a one-dimensional slab results in \u2202\u27e8\u03c8j\u27e9 \u2202tD = \u03baj LD\u03c9j \" \u2202\u03c8j \u2202yD \f \f \f \f yD=LD \u2212\u2202\u03c8j \u2202yD \f \f \f \f yD=0 # , (5) where LD = L/Lc is the dimensionless half-distance between evenly spaced fractures, yD = y/Lc is the dimensionless matrix space coordinate perpendicular to the fracture (0 \u2264yD \u2264 LD), and \u27e8\u03c8j\u27e9(t) = 1 LD Z LD 0 \u03c8j(x, t) dx is the spatially averaged dimensionless change in matrix pressure. The \ufb01rst term on the right-hand-side of (5) is \ufb02ux at the matrix-fracture interface, while the second term is \ufb02ux at the center of the block. The latter vanishes identically by symmetry. 6 \fThe two no-\ufb02ow boundaries perpendicular to the borehole in Figure 1 are symmetry boundary conditions, which de\ufb01ne the unit cell represented by the model given here. In the case of a systematically fractured horizontal well one plane is in the midplane of the fracture and the other halfway between equally spaced fractures. The cylindrical boundary parallel to the borehole represents the edge of the reservoir \u2013 potentially at in\ufb01nite radial distance. At this far edge of the domain we implement a general linear Type-III boundary condition. Continuing with the WR pseudo-steady-state interporosity \ufb02ow assumption, \ufb02ux from each matrix continuum to the fracture continuum is proportional to the di\ufb00erence between the dimensionless fracture and average matrix pressure changes, \u2202\u03c8j \u2202yD \f \f \f \f yD=LD = \u03f5j LD [\u03c8f \u2212\u27e8\u03c8j\u27e9] j = 1, . . . , N, (6) where \u03f5j is a dimensionless constant of proportionality. Substituting this expression for \ufb02ux (6) into the integrated matrix \ufb02ow equation (5) leads to \u2202\u27e8\u03c8j\u27e9 \u2202tD = \u03f5j\u03baj L2 D\u03c9j [\u03c8f \u2212\u27e8\u03c8j\u27e9] j = 1, . . . , N. (7) Comparing the j = 1 matrix porosity from (7) to the standard WR solution for matrix \ufb02ow in a double-porosity system (Warren and Root, 1963, Eqn. 11), results in the equivalence \u03b1r2 wkm (1 \u2212\u03c9)kf = \u03f51k1L2 c L2\u03c91kf , (8) which suggests \u03f51 = L2 D\u03b1, where \u03b1 is WR\u2019s shape parameter [L\u22122], km = k1 is the WR matrix permeability, \u03c9 = \u03c9f, \u03c91 = 1 \u2212\u03c9, and WR used Lc = rw as a characteristic length. We choose the group uj = \u03f5j\u03baj \u03c9j to characterize the \ufb02ow problem across the distribution of matrix continua (0 \u2264uj < \u221e). Taking the Laplace transform of (7) results in   \u00af \u03c8j \u000b s = uj \u0002 \u00af \u03c8f \u2212   \u00af \u03c8j \u000b\u0003 j = 1, . . . , N (9) where s is the dimensionless Laplace transform parameter and an overbar indicates a transformed dependent variable, i.e., \u00af f = R \u221e 0 e\u2212stDf(tD) dtD. The averaged change in dimensionless matrix pressure due to changes in the fracture pressure is   \u00af \u03c8j \u000b = uj \u00af \u03c8f s + uj . This can be substituted into the Laplace-transformed form of (3) after similarly integrating (3) across the matrix blocks (equivalent to replacing \u03c8j with   \u00af \u03c8j \u000b ), resulting in s\u03c9f \u00af \u03c8f + s\u03c9f N X j=1 \u0014 \u03b2j\u03c7j uj s + uj \u0015 \u00af \u03c8f = \u22072 D \u00af \u03c8f, (10) 7 \fwhere \u03b2j = \u03c9j/\u03c9f is a dimensionless matrix/fracture storage capacity ratio. Equation (10) can be further simpli\ufb01ed into \u22072 D \u00af \u03c8f \u2212\u00af \u03c8f\u03c9fs (1 + \u00af g) = 0, (11) where \u00af g = N X j=1 \u02c6 \u03c7juj s + uj , (12) is the matrix memory kernel (Haggerty and Gorelick, 1995) and \u02c6 \u03c7j = \u03c7j\u03b2j is a scaled PMF. 2.3 Distributions To compute a solution to (11), the matrix memory kernel must be speci\ufb01ed. Any valid PMF can be speci\ufb01ed for \u03c7j, but we present three special cases. The multiporsity solution simpli\ufb01es to the dual-porosity WR solution through \u00af gWR = \u0000 1 \u03c9 \u22121 \u0001 \u03bb 1\u2212\u03c9 s + \u03bb 1\u2212\u03c9 , (13) which is u1 = \u03bb/(1 \u2212\u03c9) = \u03bb/\u03c91 = \u03b1r2 w\u03ba/\u03c91, \u03c71 = 1, and N = 1. Here \u03bb = \u03b1\u03bar2 w is WR\u2019s dimensionless interporosity \ufb02ow parameter. Similarly, the multiporosity solution is equivalent to the pseudo-steady-state triple-porosity solution of Clossman (1975) or Liu (1981) using (12), uj = \u03baj/\u03c9j \u03c7j = \u03b3j(1 \u2212\u03b3) \u03b3 and N = 2, where \u03b3 is the bulk volume fraction of \ufb01ssures, and \u03b3{1,2} are the volume fraction of good and poor rock in the matrix (Odeh, 1965; Clossman, 1975). Haggerty and Gorelick (1995, Table 1) presented in\ufb01nite discrete distributions which make the overall multiporosity system (comprised of a sum of WR pseudo-steady-state continua) behave like transient interporosity \ufb02ow (i.e., the KZ model). They presented series of coe\ufb03cients for a matrix memory kernel, which are mathematically equivalent to di\ufb00usion into matrix blocks of di\ufb00erent geometry through analogies between the series solution and analytical solutions (e.g., similar to those for dual porosity by de Swaan O. (1976) and Najurieta (1980)). Di\ufb00usion into a slab is equivalent to an in\ufb01nite distribution of pseudo-steady-state matrix domains given by ui = (2i \u22121)2\u03c02\u03f5i\u03bai 4\u03c9i i = 0, 1, . . . (14) and \u02c6 \u03c7i = 8\u03c9i (2i \u22121)2\u03c9f\u03c02 i = 0, 1, . . . . (15) 8 \fHaggerty and Gorelick (1995) presented similar series of coe\ufb03cients representing di\ufb00usion into cylinders or spheres (as did de Swaan O., 1976). For their problem, they found truncating the series at N = 100 resulted in pseudo-steady-state solutions within 1% of matrix di\ufb00usion, with errors con\ufb01ned to early time. The series are quick to compute, and using 1000 or more terms is not computationally expensive. Alternative distributions of matrix continua and properties can be speci\ufb01ed by other widely used probability distributions, such as exponential, normal, lognormal, or linear (e.g., Ranjbar et al., 2012; Malama et al., 2013). 3 Solutions The solution to (11) in cylindrical coordinates can be found directly in Laplace space, analogous to solutions in the literature for traditional dual-porosity problems (Warren and Root, 1963; Kazemi, 1969; Mavor and Cinco-Ley, 1979) for either Type I (speci\ufb01ed down-hole pressure) or Type II (speci\ufb01ed \ufb02owrate) wellbore boundary conditions. 3.1 Flow problem of interest The following assumptions and boundary conditions are used to develop Laplace-space analytical solutions to the governing equations derived in the previous section: 1. single-phase slightly compressible \ufb02ow (i.e., water, brine, oil, or gas treated using appropriate pseudo-variable methods), 2. a speci\ufb01ed \ufb02owrate Q(t) (Type II) or pressure change \u2206p(t) (Type I) at the well completion, 3. the completion only intersects or interacts with fractures, with the matrix connected to the completion through fractures, 4. a symmetry no-\ufb02ow boundary conditions parallel to the fracture midway between two fractures, and 5. a Type-III boundary condition at far edge of the domain, r = R (Figure 1). The TypeIII boundary condition can either represent no-\ufb02ow (bounded reservoirs), speci\ufb01ed head (a circular island or laboratory sand tank with a speci\ufb01ed head condition), or a linear combination of the two. The pressure solution is developed for drawdown, \u03c8 = (p \u2212p0) /Pc (change from an initial state), therefore only the homogeneous initial condition is considered. The Laplace-transformed governing \ufb02ow equation (11) in cylindrical coordinates with radial symmetry is the modi\ufb01ed Helmholtz equation for radial coordinates, \u22022\u03c8f \u2202r2 D + 1 rD \u2202\u03c8f \u2202rD \u2212\u03c8f\u03b72 = 0, (16) 9 \fwhere rD = r/Lc and \u03b72 = s\u03c9f (1 + \u00af g) is a purely imaginary wave number, also called the fracture function (Al Ahmadi and Wattenbarger, 2011). We pick Lc = rw as a characteristic length for radial \ufb02ow to a well. 3.2 Approach Analogous to Mavor and Cinco-Ley (1979), we compute time-domain numerical values from solutions to (16) using a numerical inverse Laplace transform algorithm (de Hoog et al., 1982). Several feasible alternative numerical inversion approaches exist (Kuhlman, 2013), some which only require real values of s; the de Hoog et al. (1982) approach requires complex s. We present solutions for both \ufb01nite and in\ufb01nite domains as well as for speci\ufb01ed bottomhole pressure or \ufb02owrate using modi\ufb01ed Bessel functions \u2013 the radial eigenfunctions of the modi\ufb01ed Helmholtz equation in cylindrical coordinates. The multiporosity conceptualization may be appropriate for \ufb02ow in fractured rock in which hydraulic fracturing is being performed, but the simple homogeneous cylindrical geometry of the radial solution presented here cannot adequately represent pumping discrete fractures under typical \ufb01eld conditions (Gringarten, 1982; Clarkson, 2013). The governing equations presented here could be solved for discrete linear fractures using analytical solutions for single fractures (e.g., Gringarten et al., 1974) or using analytic element superposition-based combinations of line element solutions (Bakker and Kuhlman, 2011; Biryukov and Kuchuk, 2012). These approaches can better represent the geometry encountered during hydraulic fracturing. Line sinks and sources, or narrow ellipses and polygons of high permeability can be used to represent discrete fractures added to a uniformly fractured multiporosity rock. 3.3 In\ufb01nite Domain Solutions For an in\ufb01nite radial domain we assume a homogeneous far-\ufb01eld Type-I boundary condition, limrD\u2192\u221e\u03c8f(rD) = 0, while the wellbore boundary condition is speci\ufb01ed in two di\ufb00erent ways as follows. For speci\ufb01ed down-hole \ufb02owrate, we choose Pc = \u00b5Bq/(2\u03c0hfkf), where Q(t) = q \u00af ft is a volume \ufb02owrate at the surface, q is a constant characteristic \ufb02owrate, hf is the portion of the well completion open to fractures, and B is a dimensionless formation volume factor correcting for the di\ufb00erence between down-hole and surface \ufb02owrates due to compressibility (in low-pressure or incompressible \ufb02ow systems B = 1). This choice of Pc results in the wellbore boundary condition \u2202\u03c8f \u2202rD \f \f \f \f \f rD=1 = \u2212\u00af ft, where \u00af ft is the Laplace-transformed temporal behavior of the boundary condition (i.e., temporal \ufb02uctuations from q). Using \u00af ft, we consider general time behavior using the Laplace-space form of Duhamel\u2019s theorem (\u00a8 Ozi\u00b8 sik, 1993, Chap. 5). For example, a step function on at tD = tD0 is \u00af ft = e\u2212tD0s/s 10 \f(with the typical case of \u00af ft = 1/s for tD0 = 0), and a pulse function during tD0 \u2264tD \u2264tD1 is \u00af ff = (e\u2212tD0s \u2212e\u2212tD1s) /s. The temporal behavior of an arbitrary \ufb02owrate, Q(t), or down-hole pressure, \u2206p(t), is readily approximated using piecewise constant or linear behavior (e.g., Streltsova, 1988 or Mishra et al., 2013). For nearly constant or monotonically declining speci\ufb01ed pressures and \ufb02owrates, the numerical inverse Laplace transform algorithm converges quickly (e.g., 21 Laplace-space function evaluations for each time is typical). Many more Laplace-space function evaluations are needed to ensure accuracy when using periodic or rapidly \ufb02uctuating rates (Kuhlman, 2013). For this reason, we recommend \ufb01tting \u00af ft to follow general observed trends, rather than every observed \ufb02uctuation. The Laplace-space solution to (16) for \u00af \u03c8f in the completion (rD = 1), given this boundary condition at the wellbore, is \u00af \u03c8(q,\u221e) f = \u2212\u00af ft K0(\u03b7) \u03b7K1(\u03b7), (17) where Kn(x) is the second-kind modi\ufb01ed Bessel function of order n (DLMF, \u00a710). We do not consider wellbore storage explicitly. Mavor and Cinco-Ley (1979) presented similar solutions including both wellbore storage and skin e\ufb00ects; these solutions could be used with the multiporosity conceptualization to approximately include wellbore storage and skin e\ufb00ects. To rigorously include wellbore storage e\ufb00ects would additionally require including terms related to the time derivative of \u00af ft. For speci\ufb01ed downhole pressure, we chose Pc(tD) = p0 \u00af ft. The Laplace-space solution to (16) for \ufb02owrate in the completion, given the Type-I boundary condition at the wellbore, is \u00af q(P,\u221e) = \u03c0hfD\u03b7 \u00af ft K1(\u03b7) K0(\u03b7). (18) 3.4 Finite Domain Solutions For a \ufb01nite cylindrical domain centered on the well completion, with a homogeneous TypeIII boundary condition at r = R (rD = RD), the Laplace-space solution to (16) is given generally by Carslaw and Jaeger (1959, \u00a713.4). The speci\ufb01c Laplace-space solution to (16) for \u00af \u03c8f in the completion, given the speci\ufb01ed \ufb02owrate boundary condition at the wellbore and the Type-III boundary condition kf \u00b5 \u2202\u2206pf \u2202r + H\u2206pf = 0 at r = R (nondimensionalized in Laplace space to \u2202\u00af \u03c8f \u2202rD + HD \u00af \u03c8f = 0, where HD = H\u00b5Lc/kf), is \u00af \u03c8(q,R) f = \u00af ft \u03b7 I0(\u03b7)\u03be + K0(\u03b7)\u03b6 I1(\u03b7)\u03be \u2212K1(\u03b7)\u03b6 (19) where \u03be = \u03b7K1(\u03b7RD) \u2212HDK0(\u03b7RD), \u03b6 = \u03b7I1(\u03b7RD) + HDI0(\u03b7RD), H is the Type-III boundary surface conductivity, and In(x) is the \ufb01rst-kind modi\ufb01ed Bessel function of order n (DLMF, \u00a710). 11 \fThe Laplace-space solution to (16) for \ufb02owrate, given the speci\ufb01ed pressure boundary condition at the wellbore and a homogeneous Type-III boundary condition at the outer boundary, is \u00af q(P,R) = \u03c0hfD\u03b7 \u00af ft I1(\u03b7)\u03be \u2212K1(\u03b7)\u03b6 I0(\u03b7)\u03be + K0(\u03b7)\u03b6 . (20) In the limiting case where HD = 0, the boundary condition at rD = RD becomes a no-\ufb02ow condition. When HD \u2192\u221e, the outer boundary condition becomes a speci\ufb01ed-pressure condition. In the more general case, HD represents \u201cleakiness\u201d of the outer boundary; a second reservoir beyond rD = RD is providing a non-zero \ufb02ux into the domain, proportional to the change in pressure at the boundary. 4 Model Behavior We present examples of some typical behavior of the multiporosity model. First we illustrate how the series of matrix porosities (Equations 14 and 15) can approximate both WR and KZ \ufb02ow as end members for the in\ufb01nite domain solution. Second, we illustrate model predictions for a range of di\ufb00erent matrix and fracture properties. Finally, we illustrate the nature of the di\ufb00erent boundary conditions (H \u2208{0, H, \u221e}) at the radial extent of the domain rD = RD. For constant matrix properties, the multiporosity model can reproduce both pseudosteady-state (WR) and transient dual-porosity (KZ) \ufb02ow, as well as a range of intermediate behaviors. Figure 2 shows the behavior of the in\ufb01nite domain solution (17) for uj = u, \u03b2j = \u03b2, and di\ufb00erent values of N. Using constant formation properties for an in\ufb01nite number of pseudo-steady-state matrix porosities results in an equivalent transient KZ solution with similar properties. The \ufb01gure shows the increase in dimensionless pressure drawdown due to production at a constant rate. The line types indicate the model, while the line colors indicate the matrix/fracture permeability ratio \u03ba. The right plot in Figure 2 shows the slope with respect to ln(t), illustrating how the KZ model (solid line) has an intermediate slope which is 1/2 of its slope at early or late time. The WR model has essentially zero slope at intermediate time. As more terms are added to the multiporosity model, the slope increases from zero to 1/2 the late time slope, beginning from the later-time portion of the intermediate time portion and moving back towards earlier time. Figure 3 shows the in\ufb01nite domain solution for \ufb02owrate, given a speci\ufb01ed bottomhole pressure (18). This \ufb01gure shows the decay in \ufb02owrate, due to production at a speci\ufb01ed bottomhole pressure, illustrating the variation between the WR and KZ endmembers. Figure 4 shows multiporosity solutions with di\ufb00erent matrix properties for each matrix continuum, rather than uniform properties, as shown in Figures 2 and 3. These solutions are analogous to classical doubleand triple-porosity models, where the two matrix permeabilities represent \u201cgood\u201d or \u201cbad\u201d quality rock (Clossman, 1975), or possibly heterogeneity in the matrix system (Al Ahmadi and Wattenbarger, 2011). Table 1 shows the values of uj and \u02c6 \u03c7j used in (12). These values are derived from \u03c9f = 0.001 and \u03baj = {0.5, 0.05, 5\u00d710\u22125, 5\u00d710\u22128}. Similar to Figure 2 for constant parameters across the di\ufb00erent porosities, the \ufb02at transition regions beyond the \ufb01rst transition disappear with increasing number of matrix continua for 12 \fthis selection of matrix properties. The double-porosity model has one transition zone, the triple-porosity model has two, and the quad-porosity has three transition periods (a single porosity model has no transitions). Figure 5 shows the e\ufb00ects of boundary conditions at the radial extent of the domain. The solid curves represent the solution for H = 0 (a no-\ufb02ow boundary condition). For the smaller domain (red lines), the e\ufb00ects of the boundary are seen before the e\ufb00ects of the matrix porosity. In larger domains (green and blue), the e\ufb00ects of the boundary aren\u2019t observed until later time. The black curve shows the solution for the in\ufb01nite domain. The dotted lines represent the solution for H \u2192\u221e(implemented as 108), which is e\ufb00ectively a no-drawdown condition at the edge of the domain. Type I and III boundary conditions are less applicable to bounded geologic reservoirs, but these solutions may be useful in other applications, where a high-permeability reservoir surrounds a doubleor multiple-porosity medium. The linear relationship between \ufb02ux and change in pressure between the domain and boundary condition at RD is analogous to the pseudo-steady-state assumption for \ufb02ux between the fracture and matrix. 5 Summary The multiporosity model is shown to be a generalization of pseudo-steady-state double and triple porosity solutions of Warren and Root (1963), Clossman (1975), and Liu (1983), as well as the transient double porosity solution of Kazemi (1969), given di\ufb00erent distributions of matrix properties. Previous solutions for interporosity \ufb02ow with variable block sizes (Ranjbar et al., 2012) have not included the connection to the solution of Kazemi (1969), and have focused on the block-size distribution properties. The cylindrically symmetric solutions presented here for continuously fractured and discretely fractured systems show how the multiporosity model can be presented as a unifying approach to solve multiple \ufb02ow regimes using a single di\ufb00usion-based model, when the same properties are used across the di\ufb00erent matrix continua. When di\ufb00erent properties are used in each matrix porosity, the solution can produce results based upon the interaction of multiple physical reservoirs (i.e., macroand micro-fractures). The radially symmetry Laplace-domain solutions presented are given for speci\ufb01ed downhole pressure and \ufb02owrate, with \ufb01nite or in\ufb01nite domains, and a general Type-III boundary condition at the far extent of the domain. Although there have already been many approximate and analytical solutions proposed for \ufb02ow in fractured rock (Gringarten, 1982; Chen, 1989), we propose the multiporosity solution as a simple yet unifying approach to include the e\ufb00ects of multiple interacting reservoirs. The conceptual solute transport approach of Haggerty and Gorelick (1994, 1995, 1998) has been adapted to the di\ufb00usion of pressure between fracture and matrix reservoirs. 13 \facknowledgments This research was funded by Sandia Laboratory-Directed Research and Development. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy\u2019s National Nuclear Security Administration under contract DE-AC0494AL85000. A Fortran program implementing the multiporosity model is available from the corresponding author."
+                }
+            ],
+            "Bwalya Malama": [
+                {
+                    "url": "http://arxiv.org/abs/1607.05128v1",
+                    "title": "Modeling cross-hole slug tests in an unconfined aquifer",
+                    "abstract": "A modified version of a published slug test model for unconfined aquifers is\napplied to cross-hole slug test data collected in field tests conducted at the\nWiden site in Switzerland. The model accounts for water-table effects using the\nlinearised kinematic condition. The model also accounts for inertial effects in\nsource and observation wells. The primary objective of this work is to\ndemonstrate applicability of this semi-analytical model to multi-well and\nmulti-level pneumatic slug tests. The pneumatic perturbation was applied at\ndiscrete intervals in a source well and monitored at discrete vertical\nintervals in observation wells. The source and observation well pairs were\nseparated by distances of up to 4 m. The analysis yielded vertical profiles of\nhydraulic conductivity, specific storage, and specific yield at observation\nwell locations. The hydraulic parameter estimates are compared to results from\nprior pumping and single-well slug tests conducted at the site, as well as to\nestimates from particle size analyses of sediment collected from boreholes\nduring well installation. The results are in general agreement with results\nfrom prior tests and are indicative of a sand and gravel aquifer. Sensitivity\nanalysis show that model identification of specific yield is strongest at\nlate-time. However, the usefulness of late-time data is limited due to the low\nsignal-to-noise ratios.",
+                    "authors": "Bwalya Malama, Kristopher L. Kuhlman, Ralf Brauchler, Peter Bayer",
+                    "published": "2016-07-07",
+                    "updated": "2016-07-07",
+                    "primary_cat": "physics.geo-ph",
+                    "cats": [
+                        "physics.geo-ph"
+                    ],
+                    "main_content": "Introduction Slug tests are a common tool in hydrogeology for hydraulic characterization of aquifers because they are quick, obviate the need for waste water disposal, require less equipment, and are not as labor intensive as pumping tests. Fundamentally, they involve instantaneous (step) perturbation of \ufb02uid pressure in an interval followed by continuous monitoring of the pressure change as it dissipates by \ufb02uid \ufb02ow through the aquifer. This is typically achieved by either dropping a slug mass into a well (Cooper et al., 1967) or pneumatically pressurizing the water column in a well (Butler Jr., 1998; Malama et al., 2011), a con\ufb01guration referred to as a single well test. Several mathematical models are available in the hydrogeology literature for analyzing con\ufb01ned (Cooper et al., 1967; Bredehoeft and Papadopulos, 1980; Zurbuchen et al., 2002; Butler Jr. and Zhan, 2004) and uncon\ufb01ned (Bouwer and Rice, 1976; Springer and Gelhar, 1991; Hyder et al., 1994; Spane, 1996; Zlotnik and McGuire, 1998; Malama et al., 2011) aquifer slug test data under the Darcian \ufb02ow regime. Consideration of slug tests under non-Darcian \ufb02ow regimes may be found in Quinn et al. (2013) and Wang et al. (2015). Slug tests have the advantage of only involving limited contact with and minimal disposal of e\ufb04uent formation water. As such, they have found wide application for characterizing heterogeneous formations at contaminated sites (Shapiro and Hsieh, 1998) and for investigating \ufb02ow in fractured rock (Quinn et al., 2013; \u2217Corresponding author. Tel.: + 1 805 756 2971; fax: + 1 805 756 1402 Email address: bmalama@scalpoly.edu (Bwalya Malama) Preprint submitted to Elsevier July 19, 2016 \fJi and Koh, 2015; Ostendorf et al., 2015). However, the small volumes of water involved impose a physical limit on the volume of the formation interrogated during tests (Shapiro and Hsieh, 1998; Beckie and Harvey, 2002) because the resulting pressure perturbations often do not propagate far enough to be measurable in observation wells. As a result, hydraulic parameters estimated from single well slug-test data can only be associated with the formation volume within the immediate vicinity of the source well (Beckie and Harvey, 2002; Butler Jr, 2005). Cross-hole (or multi-well) slug tests are less common but have been applied to interrogate relatively large formation volumes in what has come to be known as hydraulic tomography (Yeh and Liu, 2000; Illman et al., 2009). For example, Vesselinov et al. (2001) and Illman and Neuman (2001) used pneumatic cross-hole injection tests to hydraulically characterized a fractured unsaturated rock formation with dimensions of 30\u00d7 30\u00d730 m3. Barker and Black (1983) presented evidence of measurable pressure responses in observation wells several meters from the source well. Audouin and Bodin (2008) reported cross-hole slug tests conducted in fractured rock, where they collected data in observations wells at radial distances 30 to about 120 m from the source well, and observed maximum peak amplitudes ranging from 5 to 20 cm. This demonstrated empirically that slug test pressure perturbations can propagate over relatively large distances beyond the immediate vicinity of the source well, albeit for fractured rocks, which have high hydraulic di\ufb00usivities. Brauchler et al. (2010) attempted to intensively apply cross-hole slug tests to obtained a detailed image of con\ufb01ned aquifer heterogeneity. They used the model of Butler Jr. and Zhan (2004) to estimate aquifer hydraulic conductivity, speci\ufb01c storage and anisotropy. Cross-hole slug tests in uncon\ufb01ned aquifers, neglecting wellbore inertial e\ufb00ects, have been reported by Spane (1996), Spane et al. (1996), and Belitz and Dripps (1999) for sourceto-observation well distances not exceeding 15 m. Recently Paradis and Lefebvre (2013) and Paradis et al. (2014, 2015) analysed synthetic cross-hole slug test data using a model for over-damped observation well responses. The need, therefore, still exists to analyse \ufb01eld data and characterize high permeability heterogeneous uncon\ufb01ned aquifers using cross-hole slug tests where source and observation well inertial e\ufb00ects may not be neglected. Malama et al. (2011) developed a slug test model for uncon\ufb01ned aquifers using the linearised kinematic condition of Neuman (1972) at the water-table, and accounting for inertial e\ufb00ects of the source well. They analysed data from single-well tests performed in a shallow uncon\ufb01ned aquifer. This work extends the application of the model of Malama et al. (2011) to multi-well tests and to response data collected in observation wells. The data analysed were collected at multiple vertical intervals in an observation well about 4 m from the source well, which itself was perturbed at multiple intervals. The model and data are used to estimate hydraulic conductivity, speci\ufb01c storage, and speci\ufb01c yield. The sensitivity of predicted model behaviour to these parameters is also analysed. In the following, the mathematical model is presented, the multi-level multi-well tests are described, and data analysed. The work concludes with an analysis of the sensitivity coe\ufb03cients for the hydraulic and well characteristic parameters. 2. Slug Test Model Malama et al. (2011) developed a model for formation and source well response to slug tests performed in uncon\ufb01ned aquifers using the linearised kinematic condition at the water-table. The model allows for estimation of speci\ufb01c yield in addition to hydraulic conductivity and speci\ufb01c storage. The model also accounts for source-well wellbore storage and inertial e\ufb00ects. Wellbore storage in the source well is treated in the manner of Cooper et al. (1967). A schematic of the conceptual model used to derive the semi-analytical solution is shown in Figure 1. Whereas the solution of (Malama et al., 2011) was obtained for and applied to source wells, here a more complete solution is presented that applies to observation wells. The complete aquifer response for both source and observation wells is given by (see Appendix A and Malama et al. (2011) for details) \u02c6 sD = \u02c6 uD \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 \u0002 1 \u2212\u02c6 vD (dD) \u0003 \u02c6 f1(zD) \u2200zD \u2208[0, dD] 1 \u2212\u02c6 vD \u2200zD \u2208[dD, lD] \u0002 1 \u2212\u02c6 vD (lD) \u0003 \u02c6 f2(zD) \u2200zD \u2208[lD, 1], (1) 2 \fK2 K1 K3 2rc obs r 1 2 3 d l z = 0 Ground surface Packer Observation well r Water table CMT system Observation zones z = B z Source well \u03b4 \u03b4 \u03b4 Figure 1: Schematic of a typical cross-hole slug test set-up for an uncon\ufb01ned aquifer. For the tests reported herein, the source and observation intervals were isolated with a multi chamber well not a multi-packer system. where \u02c6 sD is the double Laplace-Hankel transform of the dimensionless formation head response sD = s/H0, dD = d/B and lD = l/B are dimensionless depths to the top and bottom of the test interval, zD = z/B (z \u2208[0, B]) is dimensionless depth below the water-table, B is initial saturated thickness, \u02c6 uD = CD(1 \u2212pHD) \u03ba\u03b72\u03bewK1(\u03bew) , (2) \u02c6 vD = \u22060(dD) \u22060(1) cosh(\u03b7z\u2217 D) + sinh(\u03b7l\u2217 D)\u2206\u2032 0(zD) \u03b7\u22060(1) , (3) \u02c6 f1(zD) = \u2206\u2032 0(zD) \u2206\u2032 0(dD), (4) \u02c6 f2(zD) = cosh(\u03b7z\u2217 D) cosh(\u03b7l\u2217 D) , (5) \u22060(zD) = sinh(\u03b7zD) + \u03b5 cosh(\u03b7zD), (6) and \u2206\u2032 0(zD) = \u03b7 [cosh (\u03b7zD) + \u03b5 sinh (\u03b7zD)] . (7) Additionally, z\u2217 D = 1 \u2212zD, l\u2217 D = 1 \u2212lD, \u03b7 = p (p + a2 i )/\u03ba, p and ai are the dimensionless Laplace and \ufb01nite Hankel transform parameters, CD = r2 D,c/(bsSs) is the dimensionless wellbore storage coe\ufb03cient of the source well, Ss is formation speci\ufb01c (elastic) storage, bs = l \u2212d the length of the source well completion interval, \u03ba = Kz/Kr is the formation anisotropy ratio, Kz and Kr are vertical and radial hydraulic conductivities, \u03bew = rD,w\u221ap, \u03b5 = p/(\u03b7\u03b1D), and K1() is the \ufb01rst-order second-kind modi\ufb01ed Bessel function (Olver et al., 2010, \u00a710.25). The relevant dimensionless parameters are listed in Table 1. The function HD(p) in (2) is the Laplace transform of HD(tD) = H(t)/H0, the normalized response in 3 \fTable 1: Dimensionless variables and parameters sD,i = si/H0 HD = H(t)/H0 rD = r/B rD,w = rw/B rD,c = rc/B rD,s = rs/B RD = R/B zD = z/B dD = d/B tD = \u03b1r,1t/B2 CD = r2 D,c/(bSs) \u03b1D = \u03ba\u03c3 \u03b21 = 8\u03bdL/(r2 cgTc) \u03b22 = Le/(gT 2 c ) \u03b2D = \u03b21/\u221a\u03b22 \u03bai = Kz,i/Kr,i \u03c3 = BSs/Sy \u03b3 = Kr,2/Kr,1 \u03d1 = 2bSs,2(rw/rc)2 \u03besk = rsk/rw \u03bew = rD,w\u221ap \u03b72 = (p + a2 i )/\u03ba the source well, and is given by HD(p) = \u03c81(p) \u03c92 D,s + p\u03c81(p), (8) where \u03c9D,s = \u03c9sTc, \u03c9s = p g/Le is the source well frequency, Le is a characteristic length associated with the source well oscillatory term, Tc = B2/\u03b1r is a characteristic time, g is the acceleration due to gravity, and \u03c81(p) = p + \u03b3D,s + \u03c92 D,s 2 \u2126(rD,w, p) . (9) The function \u2126is de\ufb01ned by \u2126(rD,w, p) = H\u22121 0 n\u02c6 \u2126(ai, p) o\f \f \f rD,w , (10) where H\u22121 0 {} denotes the inverse zeroth-order \ufb01nite Hankel transform operator, rD,w = rw/B is the dimensionless wellbore radius, \u03b3D,s = \u03b3sTc, \u03b3s is the source well damping coe\ufb03cient, and \u02c6 \u2126(ai, p) = CD \u0002 1 \u2212  \u02c6 wD (ai, p) \u000b\u0003 \u03ba\u03b72\u03bewK1(\u03bew) . (11) Malama et al. (2011) showed that \u27e8\u02c6 wD\u27e9= 1 \u03b7bD,s \u22060(dD) \u22060(1) \u001a sinh(\u03b7d\u2217 D) \u2212 \u0014 2 \u2212\u22060(lD) \u22060(dD) \u0015 sinh(\u03b7l\u2217 D) \u001b , (12) where d\u2217 D = 1 \u2212dD, and bD,s = bs/B. According to Butler Jr. and Zhan (2004), the source well damping coe\ufb03cient is \u03b3s = 8\u03bdL/(Ler2 c), where \u03bd is the kinematic viscosity of water and L is a characteristic length 4 \fassociated with the perturbed column of water in the source well. Whereas Malama et al. (2011) used the in\ufb01nite Hankel transform, here a \ufb01nite Hankel transform (Sneddon, 1951; Miles, 1971) is used for inversion, with the transform pair de\ufb01ned as \u02c6 f(ai) = H0 {f(rD)} = Z RD 0 rDf(rD)J0(rDai) drD, f(rD) = H\u22121 0 n \u02c6 f(ai) o = 2 R2 D \u221e X i=0 \u02c6 f(ai) J0(rDai) [J1(RDai)]2 , (13) where ai are the roots of J0(RDai) = 0, RD = R/B, R is the radius of in\ufb02uence of the source well, and Jn() is the nth-order \ufb01rst-kind Bessel function (Olver et al., 2010, \u00a710.2). For the speci\ufb01ed roots and Hankel transform pair given in (13), a homogeneous Dirichlet boundary condition is enforced at rD = RD. Due to the short duration of the signal, a radius of in\ufb02uence such that R \u22652robs is su\ufb03cient. The \ufb01nite Hankel transform is chosen for computational expedience; it is simpler to invert numerically than the in\ufb01nite Hankel transform (Malama, 2013). Laplace transform inversion is performed using the algorithm of de Hoog et al. (1982). The software used to implement the analytical solution described here is released under an open-source MIT license and is available from a public Bitbucket repository (https://bitbucket.org/klkuhlm/slug-osc). 2.1. Approximation of observation well skin It is assumed here that the slug test response at the observation well is due to \ufb02uid \ufb02ow through the sub-domains associated with the source and observation wells and the formation shown in Figure 1. The well skin and formation hydraulic conductivities, Ki, i = 1, 2, 3, are arranged in series for radial \ufb02ow, and in parallel for vertical \ufb02ow. Hence, the e\ufb00ective radial and vertical hydraulic conductivity, \u27e8Kr\u27e9and \u27e8Kz\u27e9, of the formation between the source and observation wells are approximated as \u27e8Kr\u27e9= \u03b4T / 3 X n=1 \u03b4\u2217 i Ki , and \u27e8Kz\u27e9= 1 \u03b4T 3 X n=1 \u03b4iKi, where \u03b4\u2217 1 = (\u02c6 r/r1)\u03b41 and \u03b4\u2217 2 = (\u02c6 r/r2)\u03b42, \u03b4T = P3 n=1 \u03b4i, \u03b4i is the radial thickness of zone i, r1 = (rw +rskin)/2, r2 = (rskin+robs)/2, and \u02c6 r = (rw+robs)/2. This approximate approach follows the work of Shapiro and Hsieh (1998) for using the equivalent hydraulic conductivity approach to account for simple heterogeneity. It is based on the simplifying assumption of a piecewise linear head distribution in the skin and formation. It follows directly from an application of mass conservation and Darcy\u2019s law in a radial (cylindrical) \ufb02ow system. The result may also be obtained using a centered \ufb01nite di\ufb00erence approximation of the hydraulic gradient at r1 and r2 for a head distribution given by Theim equation. 2.2. Observation well storage & inertial e\ufb00ects The column of water in the observation well oscillates in response to a source well perturbation. It is reasonable to assume that the e\ufb00ective weight of the water column in the observation well controls its head response and damping of the oscillations. Mass balance in the manner of Black and Kipp Jr. (1977) and momentum balance (Kipp Jr., 1985; Butler Jr. and Zhan, 2004) in the observation well account for wellbore storage and inertial e\ufb00ects. In non-dimensional form, the momentum balance equation is given by d2sD,obs dt2 D + \u03b3D,o dsD,obs dtD + \u03c92 D,o sD,obs = \u03c92 D,o\u27e8sD\u27e9 (14) 5 \fwhere \u03b3D,o is dimensionless observation well damping coe\ufb03cient and \u03c9D,o is dimensionless observation well characteristic frequency. where sD,obs is the dimensionless observation well response, \u27e8sD\u27e9is the depthaveraged dimensionless formation response across the observation interval. It follows from Butler Jr. and Zhan (2004) that \u03b3D,o = Tc8\u03bdLobs/(Le,obsr2 c,obs), \u03c9D,o = Tc p g/Le,obs, where Le,obs and Lobs are the characteristic length scales for observation well inertial e\ufb00ects. Here we estimate \u03b3o and \u03c9o through Lobs and Le,obs from observation well data. Applying the Laplace transform and solving for sobs gives sD,obs = \u03c82(p) \u27e8sD (rD, p)\u27e9, (15) where \u03c82(p) = \u03c92 D,o/(p2 + p\u03b3D,o + \u03c92 D,o), \u27e8\u02c6 sD\u27e9= 1 bD,o Z lD,o dD,o \u02c6 sD(ai, p, zD) dzD, (16) and lD,o = lo/B and dD,o = do/B are the dimensionless depths to the top and bottom of the observation well interval from the water-table. Upon inverting the Laplace transform, one obtains sD,obs = Z tD 0 \u03c82(tD \u2212\u03c4) \u27e8sD (rD, \u03c4)\u27e9d\u03c4 (17) with \u03c82(t) = L\u22121 \b \u03c82(p) \t . Equation 17 is the solution accounting for observation well inertial e\ufb00ects. It is used in the subsequent analysis to estimate hydraulic parameters. 3. Model Application to Cross-hole Slug Test Data The model described above is applied to observations collected in a series of multi-level cross-hole pneumatic slug tests performed in June 2013 at the Widen site in north-east Switzerland. The site is on the \ufb02oodplain of the Thur River, a tributary of the Rhine river (Diem et al., 2010). The multi-well layout of the test site is depicted schematically in Figure 2(a). The wells used in the experiments are completed in an uncon\ufb01ned sand and gravel aquifer with a saturated thickness of 5.8 m. The aquifer is quaternary post-glacial sediment underlain by an aquitard of low permeability lacustrine sediment comprising \ufb01ne silt and clay (Diem et al., 2010; Coscia et al., 2011). It is overlain with alluvial loam that constitutes the top soil. The aquifer itself can be further subdivided into a silty sand top layer underlain with silty gravel and a sand layer to a thickness of about 7 m (Diem et al., 2010). The source well is screened across the entire saturated thickness (see Figure 2(a)). Straddle packers were used to sequentially isolate discrete intervals in the source well. The pressure responses were recorded in three observation wells, which were equipped with a Continuous Multichannel Tubing (CMT) system (Einarson and Cherry, 2002) in which pressure transducers were installed. This system was originally designed for multi-level sampling. It consists of a PVC pipe with seven continuous separate channels or chambers (inner diameter 0.014 m), which are arranged in a honeycomb structure. Each individual chamber has a 0.08 m long slot covered with a sand \ufb01lter and allows for hydraulic contact with the formation. 3.1. Experimental procedure The cross-hole pneumatic slug tests were initiated by applying gas pressure to the water column in a chosen interval, then releasing the gas pressure through an out\ufb02ow valve to provide the instantaneous initial slug perturbation. A double-packer system straddling the test interval (bs = 0.35 m) was used with the pneumatic slug applied through a smaller tubing (rc = 1.55 \u00d7 10\u22122 m). The source well used in these tests was well P13, with a wellbore radius of rw = 3.15 \u00d7 10\u22122 m. The dissipation of the slug was monitored with a pressure transducer in the source well positioned at the top of the water column above the test interval. The data considered here was obtained in three observations wells labelled MC1, MC2, and MC4 (in Figure 2(a)) and located at radial distances of 3.9, 2.9, and 2.8 m, respectively, from the source well. The responses at multiple vertical positions in each observation well were monitored with pressure transducers in 6 \f(a) (b) Figure 2: (a) Multi-well layout and (b) example experimental system setup for cross-hole slug tests at the Widen site, Switzerland. For the example shown, data from interval i is denoted P13-MC1-i a seven-channel CMT system with screen intervals of bo = 8\u00d710\u22122 m. Each channel in the CMT system has an equivalent radius of rc,o = 6.5 \u00d7 10\u22123 m; installation of a pressure transducer in these channels reduces their e\ufb00ective radii (and e\ufb00ective wellbore storage) signi\ufb01cantly. The CMT system allows for simultaneous monitoring of the response at seven vertical positions for each slug test. Pressure responses were recorded at a frequency of 50 Hz (every 0.02 s) for a period of about 20 seconds from slug initiation using miniature submersible level transmitters MTM/N 10 manufactured by STS Sensor Technik in Switzerland. The housing diameter of 0.39 inches allowed for pressure measurements in small diameter (1/2 inch) monitoring wells, stand pipes and bore holes. The stainless steel construction and integral polyurethane cable is ideal for long term installation. The transducer cable is reinforced with Kevlar to avoid elongation in deep boreholes. The experiments reported herein were performed in shallow wells and over a relatively short duration to make cable elongation is negligible. Only data from the observation intervals at approximately the same vertical position as the source-well test interval are analysed here because of their favourable signal-to-noise ratio (SNR). Data from ports not directly in line with the tested interval showed signi\ufb01cant decay for the magnitudes of the perturbation used in the \ufb01eld tests. Transducers with greater precision and accuracy or larger source well perturbation are needed to obtain analysable responses in such ports. A schematic of the experimental setup for tests between wells P13 and MC1 is shown in Figure 2(b). 7 \f3.2. Observation well data The typical slug test responses observed during tests at the Widen site are shown in Figure 3. The plots in Figure 3(a) are the source well responses, and those in (b) are the corresponding responses in an observation well about 3 m radially from the source well. The results clearly show damped oscillations generated in the source well are measurable in an observation well a few meters away. Comparing the results in (a) and (b) also shows the maximum amplitude of the signal decays about two orders of magnitude from the source to the observation well, which decreases the SNR. -0.5  0  0.5  1  1.5  0  2  4  6  8  10  12  14  16  18  20 Source well response, HD Time since slug initiation, t (s) (a) Source wells test 1 test 2 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  0  2  4  6  8  10  12  14  16  18  20 Observation well response, sD (\u00d7 102) Time since slug initiation, t (s) (b) P13-MC1 z = 1.11 m z = 5.11 m -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  0  2  4  6  8  10  12  14  16  18  20 Observation well response, sD (\u00d7 102) Time since slug initiation, t (s) (c) P13-MC2 z = 0.63 m z = 4.63 m -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  0  2  4  6  8  10  12  14  16  18  20 Observation well response, sD (\u00d7 102) Time since slug initiation, t (s) (d) P13-MC4 z = 1.46 m z = 4.46 m Figure 3: Typical (a) source and (b-d) observation well responses measured during cross-hole slug tests. Observation well data show increasing damping when approaching the watertable for all three pro\ufb01les. The observation well response pairs generally are increasingly damped moving towards the water-table, even when the initial displacements from the equilibrium position are comparable. This is evident in the data from all three pro\ufb01les shown in Figure 3, where observation well data collected closer to the water-table appear to be more damped than those at greater depths. Measurable observation well displacements are still obtainable near the water-table (i.e., interval 9 in Figure 2(b)). The con\ufb01guration of the equipment made it physically impossible to record the response at the water-table. Placing a pressure transducer at the watertable would be useful to con\ufb01rm the appropriate type of boundary condition to represent the water-table. While Malama et al. (2011) and the modi\ufb01ed model presented here use the linearised kinematic water-table representation, Hyder et al. (1994) use a constant-head boundary condition to represent the water-table. 8 \f3.3. Parameter estimation The modi\ufb01ed model was used to estimate model parameters from data collected in observation wells during the tests at the Widen site. For the present study, to reduce the number of estimated parameters, it is su\ufb03cient to assume the aquifer is isotropic (Kr = Kz = K), and the skin conductivities of the source and observation wells are equal (K1 = K3 = Kskin). Using the non-linear optimization software PEST (Doherty, 2010, 2015), we estimated skin hydraulic conductivity (Kskin), formation hydraulic conductivity (K), speci\ufb01c storage (Ss), and the length parameters L and Le that characterize the source and observation well damping coe\ufb03cients and frequencies. It is typical to compute L and Le using the formulas (Butler Jr., 2002; Kipp Jr., 1985; Zurbuchen et al., 2002) L = d + b 2 \u0012 rc rw \u00134 , (18) and Le = L + b 2 \u0012 rc rw \u00132 . (19) The values of L and Le computed with these formulas were used as initial guesses during the parameter estimation procedure. The parameters Le,obs and Lobs, which determine the frequency and damping coef\ufb01cient of the observation well were also estimated with initial guesses determined similarly. The non-linear optimization software PEST (Doherty, 2010, 2015) was used to estimate the optimal parameters and the model parameter sensitivity at the optimal solution. The \ufb01t of the model to observed cross-hole responses was very sensitive to the time of the initial observation (i.e., the syncing of the clocks at the source and observation wells). Initially it was di\ufb03cult to get model/data agreement to both early and late-time data without assigning non-physical parameter values. Estimating a modest time shift (o\ufb00-set) for each test greatly improved model \ufb01ts to the data. Estimated observation data time delays were between 4 and 6 tenths of a second, which is a permissible time o\ufb00-set between two synced transducer clocks. PEST-estimated parameters are summarized in Table 2. A subset of the complete dataset (25% of the 50 Hz data stream) was used in the PEST optimization; this subset is shown in Figure 4. The corresponding model \ufb01ts to observation well data are shown in Figure 4. The relatively large average value of skin conductivity (averaging Kskin = 8.5 \u00d7 10\u22122 m/s) estimated from tests is consistent with a disturbed zone resulting from well installation by direct-push. The technology uses a hydraulic hammer supplemented with weight of the direct-push unit to push down drive rods to the desired depth of the projected well. The well casing is then lowered into the drive rods (inner diameter: 0.067 m, outer diameter 0.083 m). By retracting the drive rods, the formation is allowed to collapse back against the casing. The negative skin estimates (Kskin greater than formation K) are indicative of formation collapse due to material bridging resulting in a disturbed zone around the well casing. Skin conductivity estimation variances range from 10\u22122 m2/s2 for low noise data to 103 m2/s2 noisy data and are indicative of dependence of estimation uncertainties on measurement errors. 9 \fTable 2: PEST-estimated model parameters. K Kskin Ss Sy L Le Lobs Le,obs Test [m \u00b7 s\u22121] [m \u00b7 s\u22121] [m\u22121] [-] [m] [m] [m] [m] P13-MC1-1 7.81 \u00d7 10\u22124 2.27 \u00d7 10\u22121 3.39 \u00d7 10\u22125 0.037 1.90 5.71 4.07 1.87 \u00d7 10\u22122 P13-MC1-3 8.85 \u00d7 10\u22124 1.07 \u00d7 10\u22121 1.25 \u00d7 10\u22125 0.40 1.18 4.31 2.39 1.80 \u00d7 10\u22122 P13-MC1-5 7.70 \u00d7 10\u22124 2.21 \u00d7 10\u22121 1.70 \u00d7 10\u22125 0.36 2.53 3.21 3.10 1.76 \u00d7 10\u22122 P13-MC1-7 1.28 \u00d7 10\u22123 1.02 \u00d7 10\u22122 3.85 \u00d7 10\u22125 0.018 0.23 2.26 11.4 9.74 \u00d7 10\u22122 P13-MC1-9 1.48 \u00d7 10\u22123 6.42 \u00d7 102 2.79 \u00d7 10\u22128 0.001 2.95 0.83 8.83 6.04 \u00d7 10\u22122 P13-MC2-2 7.67 \u00d7 10\u22124 1.73 \u00d7 10\u22121 2.76 \u00d7 10\u22125 0.40 1.07 5.07 3.92 5.73 \u00d7 10\u22121 P13-MC2-4 1.36 \u00d7 10\u22123 2.15 \u00d7 10\u22121 5.11 \u00d7 10\u22125 0.04 8.01 3.66 4.63 4.84 \u00d7 10\u22122 P13-MC2-6 1.08 \u00d7 10\u22123 6.42 \u00d7 10\u22122 1.95 \u00d7 10\u22125 0.40 1.38 2.91 2.55 1.79 \u00d7 10\u22122 P13-MC2-8 2.22 \u00d7 10\u22123 4.44 \u00d7 10\u22121 3.06 \u00d7 10\u22125 0.001 2.26 1.80 5.34 1.64 \u00d7 10\u22123 P13-MC4-2 1.60 \u00d7 10\u22123 3.17 \u00d7 10\u22121 7.41 \u00d7 10\u22125 0.005 2.39 4.81 6.42 1.35 \u00d7 10\u22122 P13-MC4-4 5.27 \u00d7 10\u22124 3.37 \u00d7 10\u22121 9.16 \u00d7 10\u22125 0.40 4.56 3.49 9.69 1.38 \u00d7 10\u22122 P13-MC4-6 3.79 \u00d7 10\u22123 2.04 \u00d7 10\u22122 7.22 \u00d7 10\u22125 0.001 3.24 2.82 3.09 4.85 \u00d7 10\u22121 P13-MC4-8 1.46 \u00d7 10\u22123 6.15 \u00d7 10\u22122 1.80 \u00d7 10\u22124 0.40 0.78 1.76 9.24 3.48 \u00d7 10\u22122 10 \f0 2 4 6 8 10 12 14 time (s) 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 Observation well response, sD  ( \u00d7102 ) P13-MC1-7 0 2 4 6 8 10 time (s) 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Observation well response, sD  ( \u00d7102 ) P13-MC2-08 0 1 2 3 4 5 6 7 8 time (s) 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 Observation well response, sD  ( \u00d7102 ) P13-MC4-8 0 2 4 6 8 10 12 14 16 time (s) 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 Observation well response, sD  ( \u00d7102 ) P13-MC1-5 0 2 4 6 8 10 12 14 16 time (s) 0.4 0.2 0.0 0.2 0.4 0.6 Observation well response, sD  ( \u00d7102 ) P13-MC2-06 0 2 4 6 8 10 12 14 16 time (s) 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 Observation well response, sD  ( \u00d7102 ) P13-MC4-6 0 5 10 15 time (s) 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Observation well response, sD  ( \u00d7102 ) P13-MC1-3 0 5 10 15 time (s) 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 Observation well response, sD  ( \u00d7102 ) P13-MC2-04 0 2 4 6 8 10 12 14 16 time (s) 0.2 0.1 0.0 0.1 0.2 0.3 0.4 Observation well response, sD  ( \u00d7102 ) P13-MC4-4 0 2 4 6 8 10 12 14 16 time (s) 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 Observation well response, sD  ( \u00d7102 ) P13-MC1-1 0 2 4 6 8 10 12 14 16 time (s) 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Observation well response, sD  ( \u00d7102 ) P13-MC2-02 0 2 4 6 8 10 12 14 16 time (s) 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 Observation well response, sD  ( \u00d7102 ) P13-MC4-2 Figure 4: Model \ufb01ts to cross-hole slug test data collected along vertical pro\ufb01les in three observation wells at the Widen Site, Switzerland. The columns correspond to pro\ufb01les in observation wells 1, 2, and 4. 11 \fDrilling logs and previous hydrogeophysical investigations at the site (Lochb\u00a8 uhler et al., 2013; Coscia et al., 2011) indicate a sand and gravel aquifer. The formation hydraulic conductivities estimated here are on the order of 10\u22124 to 10\u22123 m/s, and in general agreement with the \ufb01ndings from earlier studies at the site. Coscia et al. (2011) report estimates of the order of 10\u22123 to 10\u22122 m/s from multiple pumping and single-well slug tests conducted at the site by Diem et al. (2010). The average values estimated here range from a low of 7.1 \u00d7 10\u22124 m/s to a high value of 3.8 \u00d7 10\u22123 m/s. These and estimates from earlier studies at the site are reasonable for unconsolidated well-sorted sand and gravel aquifers (Bear, 1972; Fetter, 2001). The vertical variability in the estimates is re\ufb02ective of site heterogeneity. The objective of multi-level slug tests is to characterize such heterogeneity using a physically based \ufb02ow model. It should be understood that the model used in this analysis was developed for a homogeneous but anisotropic aquifer. Its application to characterizing heterogeneity is thus limited and only approximate, with data collected at discrete depth intervals assumed to yield hydraulic parameter values associated with that interval. Estimation variances for formation hydraulic conductivity range in magnitude from 6 \u00d7 10\u22122 to 1.2 \u00d7 101 m2/s2. Estimates of speci\ufb01c storage, Ss, also show only modest variability and are generally of the order of 10\u22125 m\u22121, with the largest value being about 10\u22125 m\u22121 and the smallest 10\u22127 m\u22121. The estimated values are indicative of poorly consolidated shallow alluvium, and variability may re\ufb02ect uncertainty or non-uniqueness in the solution for this con\ufb01guration and dataset. Estimates of Sy were quite variable, with estimation variances of the order of 10\u22127 to 102. Estimated values of 0.4 correspond to the upper bound during optimization. P13-MC1-1, 7, and 9 resulted in estimated Sy values of a few percent, which are physically realistic for these types of sediments and for the linearized kinematic condition at the watertable. In this parameter estimation analysis no signi\ufb01cant physical constraints where introduced on the objective function; the observations were allowed to freely constrain the estimates of model parameters. Estimates of the parameter Le from data are comparable to those predicted by equation (19). However, estimates of L from data are consistently larger than the values predicted by (18). 3.4. Sensitivity analysis The model sensitivity or Jacobian matrix, J, of dimensions N \u00d7M, where N is the number of observations and M is the number of estimated parameters, is of central importance to parameter estimation. The sensitivity coe\ufb03cients are simply the elements of the Jacobian matrix; they are the partial derivatives of the model-predicted aquifer head response, s, with respect to the estimated parameter \u03b8m. Sensitivity coe\ufb03cients are represented here as functions of time using the nomenclature J\u03b8m(t) = \u2202sD,obs \u2202\u03b8m , (20) where m = 1, 2, ..., M. They describe the sensitivity of predicted model behavior (head response) to the model parameters. They provide a measure of the ease of estimation (identi\ufb01ability) of the parameters from system state observations (Jacquez and Greif, 1985). The Jacobian matrix J has to satisfy the identi\ufb01ability condition, |JT J| \u0338= 0, for parameters to be estimable. This condition is typically satis\ufb01ed for linearly independent sensitivity coe\ufb03cients with appreciably large magnitudes. For this work, the number of parameters estimated is M = 8, and the vector of estimated parameters is (\u03b81, ..., \u03b88) = (K, Kskin, Ss, Sy, L, Le, Lobs, Le,obs) . (21) Sensitivity coe\ufb03cients for tests P13-MC1-1 (deepest) and P13-MC1-5 (intermediate depth) are shown as functions of time in Figures 5 and 6. Semi-log plots of the same information are included to more clearly show the non-zero sensitivity values at late-time. Generally, the sensitivities are oscillatory functions of time with decaying amplitudes that vary over several orders of magnitude among the parameters. Figure 5(a) shows the sensitivity to the parameters K, Kskin, Ss, and Sy. It is clear that well skin conductivity, Kskin, has the highest peak sensitivity at earlytime, and is therefore the most easily identi\ufb01able parameter from early-time data. Speci\ufb01c yield, Sy, has the 12 \f-6 -4 -2  0  2  4  0  2  4  6  8  10  12  14  16  18 Sensitivity\t\tJ\u03b8m\t\t (\u00d7 10-3) Time, t (s) (a) K Kskin Ss Sy obs 10-5 10-4 10-3 10-2  0  2  4  6  8  10  12  14  16  18 Sensitivity\t\t|J\u03b8m| Time, t (s) (b) K Kskin Ss Sy -20 -15 -10 -5  0  5  10  15  0  2  4  6  8  10  12  14  16  18 Sensitivity\t\tJ\u03b8m Time, t (s) (c) L (\u00d7 10-4) Le (\u00d7 10-2) Lobs (\u00d7 10-4) Le,obs (\u00d7 10-5) 10-7 10-6 10-5 10-4 10-3 10-2  0  2  4  6  8  10  12  14  16  18 Sensitivity\t\t|J\u03b8m| Time, t (s) (d) L Le Lobs Le,obs Figure 5: Temporal variation of the sensitivity coe\ufb03cients (linear scale (a & c) and log scale (b & d)) for the indicated parameters at the source-observation pair P13-MC1-1 (5.1 m below watertable). Subplot (a) shows observed response. smallest sensitivities (about an order of magnitude smaller than Kskin) and was the least identi\ufb01able (most di\ufb03cult to estimate) of all the parameters. Figures 5(a) and (b) also show that the sensitivity functions are generally out of phase with each other as well as with the observed response. For example, the sensitivity function JK(t) is almost completely out of phase (phase-shift of \u223c\u03c0) with JKskin. The same is true for JSs(t) and JSy(t). This indicates linearindependence of the sensitivity coe\ufb03cient among all four parameters. This is desirable as it implies that the identi\ufb01ability condition is satis\ufb01ed, permitting concomitant estimation of all these four parameters. Figure 5(a) shows the JSy is oscillatory with the small amplitudes and does not change sign, but decay more slowly than the other sensitivity responses. The predicted model response showed only modest sensitivity to speci\ufb01c yield, Sy, but the sensitivity becomes appreciably dominant at late-time (Figure 5(b)). Malama et al. (2011) showed that slug tests are more sensitive to Sy at late-time, and for relatively large initial perturbation. At late-time slug test head data are typically of low SNR (diminished data quality) making it di\ufb03cult to discern e\ufb00ects of speci\ufb01c yield. However, with measurements such as those reported in Malama et al. (2011) for a site in Montana, it is possible to obtain single-well slug tests data with clear e\ufb00ects due to Sy. The cross-hole slug test data analysed herein showed only modest watertable e\ufb00ects and the late-time data were not of su\ufb03cient quality. This suggests the importance of late-time data to maximize Sy identi\ufb01ability and estimability as also noted in Malama et al. (2011). Figures 5(c) and (d) show scaled slug response sensitivities to parameters L, Le, Lobs, and Le,obs. They show orders of magnitude of variability, with sensitivity Le being three orders of magnitude larger than 13 \f-3.5 -3 -2.5 -2 -1.5 -1 -0.5  0  0.5  1  1.5  0  2  4  6  8  10  12  14  16  18 Sensitivity\t\tJ\u03b8m\t\t (\u00d7 10-3) Time, t (s) (a) K Kskin Ss Sy obs 10-7 10-6 10-5 10-4 10-3 10-2  0  2  4  6  8  10  12  14  16  18 Sensitivity\t\t|J\u03b8m| Time, t (s) (b) K Kskin Ss Sy -1  0  1  0  2  4  6  8  10  12  14  16  18 Sensitivity\t\tJ\u03b8m Time, t (s) (c) L (\u00d7 10-3) Le (\u00d7 10-2) Lobs (\u00d7 10-3) Le,obs (\u00d7 10-4) 10-7 10-6 10-5 10-4 10-3 10-2  0  2  4  6  8  10  12  14  16  18 Sensitivity\t\t|J\u03b8m| Time, t (s) (d) L Le Lobs Le,obs Figure 6: Temporal variation of the sensitivity coe\ufb03cients (linear scale (a & c) and log scale (b & d)) for the indicated parameters at the source-observation pair P13-MC1-5 (3.1 m below watertable). Subplot (a) shows observed response (same scale as response in Figure 5). sensitivity to Le,obs. Whereas those of the parameters L, Le, and Le,obs are linearly independent (not of the same phase), the pair Le and Lobs are only linearly independent at very early time; they oscillate with the same phase after about 4 seconds. This illustrates a long temporal record of observations would not improve the joint estimation of these two parameters. Figure 6 shows the same information as depicted in Figure 5 for a more damped observation location closer to the watertable. Model sensitivity to K is equal to or larger than Kskin for this interval. Sensitivity to Ss is also higher at early-time. Among parameters K, Kskin, Ss, and Sy, sensitivity to Sy is the smallest at early time (Figure 6(b)). The sensitivity to Sy stays approximately constant with time after the \ufb01rst 10 second of the test, while sensitivities to K, Kskin, and Ss continue to decrease. It should be noted however, that the unfavorable SNR (low data quality) makes it very di\ufb03cult to estimate Sy from late-time data. Collecting data at 3.11 m below the watertable did not yield an appreciable improvement in speci\ufb01c yield identi\ufb01ability over the interval at 5.11 m depth in Figure 5. The behavior depicted in Figure 6 also suggests only data collected in the \ufb01rst 12 seconds of the test are needed to estimate model parameters at this depth. The sensitivity coe\ufb03cients for all but K essentially vanish after about 12 seconds and the identi\ufb01ability condition is no longer satis\ufb01ed. Additionally, even for the case where the sensitivity coe\ufb03cients appear to be in phase (linearly dependent) at early-time (compare JK(t) and JKskin(t) for t \u22642 s for test P13-MC1-5), they quickly (in the \ufb01rst 12 s) become linearly independent with time. This again indicates that a temporal record of the response longer than a few seconds is su\ufb03cient for joint estimation of these two parameters. 14 \f4. Conclusions Cross-hole slug test data were analysed with an extended version of the model of Malama et al. (2011). The semi-analytical model was modi\ufb01ed for: 1. predicting heads at observation wells, 2. inclusion of borehole skin e\ufb00ects, 3. use of the \ufb01nite Hankel transform for computation expediency, and 4. inclusion of observation well storage and inertial e\ufb00ects. Estimates were obtained of formation and source/observation well skin hydraulic conductivity, speci\ufb01c storage, speci\ufb01c yield, and well characteristics that control oscillation frequency and degree of damping. The aim of the study was to evaluate the use of cross-hole slug test data to characterize vertical uncon\ufb01ned aquifer heterogeneity and understand identi\ufb01ability and estimability of these parameters, especially speci\ufb01c yield. Estimated values of hydraulic conductivity and speci\ufb01c storage from PEST are indicative of a heterogeneous sand and gravel aquifer. Parameter estimation and sensitivity analysis show the model has e\ufb00ectively linearly independent sensitivity coe\ufb03cients with respect to seven of the eight parameters estimated. These parameters are clearly jointly estimable from the data over the duration of the tests. It should be understood that the model used in this analysis was developed for a homogeneous but anisotropic aquifer and is thus of only limited and approximate applicability to analysis of a heterogeneous system. Of the parameters estimated, model predictions were least sensitive to speci\ufb01c yield even near the watertable, which implies it was the least identi\ufb01able parameter. This is due to a combination of factors, including 1. the short duration of the data record due to rapid signal decay with time (< 20 seconds); 2. the increasing damping observed in monitoring locations near the watertable (resulting in even shorter temporal records), and; 3. the decreasing signal strength near the watertable, resulting in a lower signal-to-noise. The sensitivity function with respect to speci\ufb01c yield shows a relatively modest increase in magnitude with time (model sensitivity to the other model parameters tends to decrease, while that of Sy asymptotically tends to a non-zero constant value), suggesting the importance of late-time data to improve its estimation. The analysis of Malama et al. (2011) also indicated that the largest e\ufb00ect of speci\ufb01c yield on slug test response is at late-time, at which time the amplitude of the signal has decayed signi\ufb01cantly in magnitude and quality. The absence of good quality late-time observations and the relative low sensitivities of speci\ufb01c yield explain the the wide variability of the estimates of Sy. An important shortcoming of using cross-hole slug tests to characterize heterogeneity, as has been suggested in several \ufb01eld (Brauchler et al., 2010, 2011) and synthetic (Paradis et al., 2015) hydraulic tomography studies, is the signi\ufb01cant decay of the signal with distance from the source well and close to the water-table. These lead to low quality observations with low signal-to-noise ratios (SNR), and would require test redesign to improve parameter identi\ufb01ability and estimability. One approach to change test design is to conduct tests with a su\ufb03ciently large initial displacement in the source well to achieve favorable SNR at late-time in the observation wells. This may, however, introduce non-linearities and potentially increase the importance of unsaturated \ufb02ow above the watertable (Mishra and Neuman, 2011). Another approach is to use more sensitive and low-noise pressure sensors, which would increase costs signi\ufb01cantly, especially in the cross-hole multilevel testing set-up where a large network of sensors is used for data acquisition. This would be particularly useful close to the watertable and further from the source well due to signi\ufb01cant signal strength decay decline. This decline in signal strength limits the usefulness of crosshole slug tests for large-scale aquifer characterization using hydraulic tomography. Lastly, conducting multiple test repetitions and stacking the response data, akin to seismic data stacking (Jones and Levy, 1987), can be used to amplify signal and increase the SNR. 15 \fAcknowledgements Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy\u2019s National Nuclear Security Administration under contract DE-AC04-94AL85000. Appendix A: Solution with Linearized Watertable Kinematic Condition The solution can be written in dimensionless form for the intervals above, below, and across from the source well completion interval as sD = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 s(1) D \u2200zD \u2208[0, dD] s(2) D \u2200zD \u2208[dD, lD] s(3) D \u2200zD \u2208[lD, 1], (A-1) where s(n) D solves \u2202s(n) D \u2202tD = 1 r \u2202 \u2202rD   rD \u2202s(n) D \u2202rD ! + \u03ba\u22022s(n) D \u2202z2 D . (A-2) The initial and boundary conditions are s(n) D (tD = 0) = s(n) D (rD = RD) = 0 (A-3) lim rD\u21920 rD \u2202s(1) D \u2202rD = lim rD\u21920 rD \u2202s(3) D \u2202rD = 0 (A-4) \u2202s(1) D \u2202zD \f \f \f \f \f zD=0 = 1 \u03b1D \u2202s(1) D \u2202tD \f \f \f \f \f zD=0 (A-5) \u2202s(3) D \u2202zD \f \f \f \f \f zD=1 = 0 (A-6) rD \u2202s(2) D \u2202rD \f \f \f \f \f rD=rD,w = CD d\u03a6uc dtD , (A-7) \u03a6uc(tD = 0) = 1, (A-8) and \u03b22 d2\u03a6uc dt2 D + \u03b21 d\u03a6uc dtD + \u03a6uc = 1 bD Z lD dD s(2) D (rD,w, zD, tD) dzD. (A-9) Additionally, continuity of head and \ufb02ux is imposed at zD = dD and zD = lD via s(1) D (tD, rD, zD = dD) = s(2) D (zD = dD), (A-10) \u2202s(1) D \u2202zD \f \f \f \f \f zD=dD = \u2202s(2) D \u2202zD \f \f \f \f \f zD=dD , (A-11) s(3) D (tD, rD, zD = lD) = s(2) D (zD = lD), (A-12) and \u2202s(3) D \u2202zD \f \f \f \f \f zD=lD = \u2202s(2) D \u2202zD \f \f \f \f \f zD=lD . (A-13) 16 \fThis \ufb02ow problem is solved using Laplace and Hankel transforms. Taking the Laplace and Hankel transforms of Equation (A-2) for n = 1, 3, and taking into account the initial and boundary conditions in (A-3) and (A-4), gives the ordinary di\ufb00erential equation d2\u02c6 s (n) D dz2 D \u2212\u03b72\u02c6 s (n) D = 0 (A-14) where \u02c6 s (n) D = H{L{s(n) D }} is the double Laplace-Hankel transform of the function s(n) D , \u03b72 = (p + a2 i )/\u03ba, and p and ai are the Laplace and \ufb01nite Hankel transform parameters, respectively. Equation (A-14) has the general solution \u02c6 s (n) D = Ane\u03b7zD + Bne\u2212\u03b7zD. (A-15) The boundary condition at the watertable, Equation (A-16), in Laplace\u2013Hankel transform space, becomes \u2202\u02c6 s (1) D \u2202zD \f \f \f \f \f zD=0 = p \u03b1D \u02c6 s (1) D (zD = 0). (A-16) Applying this boundary condition leads to (1 \u2212\u03b5)A1 \u2212(1 + \u03b5)B1 = 0, (A-17) where \u03b5 = p/(\u03b7\u03b1D). Applying the continuity conditions at zD = dD (Equations (A-10) and (A-11)), lead to A1e\u03b7dD + B1e\u2212\u03b7dD = \u02c6 s (2) D (zD = dD), (A-18) and \u03b7 \u0000A1e\u03b7dD \u2212B1e\u2212\u03b7dD\u0001 = d\u02c6 s (2) D dzD \f \f \f \f \f zD=dD . (A-19) Similarly, applying the no \ufb02ow boundary condition at zD = 1 (Equation A-6), leads to \u02c6 s (3) D = 2B3e\u2212\u03b7 cosh (\u03b7z\u2217 D) , (A-20) where z\u2217 D = 1 \u2212zD. Continuity conditions at zD = lD lead to 2B3e\u2212\u03b7 cosh (\u03b7l\u2217 D) = \u02c6 s (2) D (zD = lD), (A-21) \u22122\u03b7B3e\u2212\u03b7 sinh (\u03b7l\u2217 D) = d\u02c6 s (2) D dzD \f \f \f \f \f zD=lD , (A-22) where l\u2217 D = 1 \u2212lD and d\u2217 D = 1 \u2212dD. For n = 2, solving Equation (A-2) in Laplace-Hankel transform space yields \u02c6 s (2) D = \u02c6 uD + \u02c6 vD, (A-23) where \u02c6 uD = CD(1 \u2212p\u03a6uc) \u03ba\u03b72\u03bewK1(\u03bew) , (A-24) and \u02c6 vD = A2e\u03b7zD + B2e\u2212\u03b7zD. (A-25) 17 \fThe \ufb01ve equations (A-17)\u2013(A-19), (A-21) and (A-22), together with Equation (A-23) can be used to determine the \ufb01ve unknown coe\ufb03cients A1, A2, and B1\u2013B3. It can then be shown that \u02c6 vD = \u2212 \u02c6 uD \u22060 {\u22061 cosh (\u03b7z\u2217 D) + sinh (\u03b7l\u2217 D) [cosh (\u03b7zD) + \u03b5 sinh (\u03b7zD)]} . (A-26) The integral in Equation (A-9) is 1 bD Z lD dD \u02c6 s (2) D dzD = \u02c6 uD + 1 bD Z lD dD \u02c6 vD dzD = \u02c6 uD +  \u02c6 vD \u000b . (A-27) Substituting Equation (A-26) into (A-27) leads to 1 bD Z lD dD \u02c6 s (2) D dzD = \u02c6 uD \u00001 \u2212  \u02c6 wD \u000b\u0001 , (A-28) where \u27e8\u02c6 wD\u27e9= 1 bD\u03b7\u22060 [\u22061 sinh (\u03b7d\u2217 D) + (\u22062 \u22122\u22061) sinh (\u03b7l\u2217 D)] , \u22060 = sinh(\u03b7) + \u03b5 cosh(\u03b7), \u22061 = sinh(\u03b7dD) + \u03b5 cosh(\u03b7dD), \u22062 = sinh(\u03b7lD) + \u03b5 cosh(\u03b7lD). (A-29) Taking the Laplace transform of (A-9) and replacing the integral on the left-hand-side with (A-28), gives (p2 + \u03b21p + \u03b22)\u03a6uc \u2212p \u2212\u03b21 = 1 2 \u00001 \u2212p\u03a6uc \u0001 \u2126 (A-30) where \u02c6 \u2126is de\ufb01ned in (11). Solving (A-30) for \u03a6uc yields the required source well response in Laplace space. 18 \fNotation ai \ufb01nite Hankel transform parameter \u2212 B Aquifer initial thickness L bs Length of source well test interval L Cw Coe\ufb03cient of wellbore storage L2 d/do Depth of top of source/observation well test interval below watertable L g Acceleration due to gravity L \u00b7 T\u22122 H Hydraulic head change from equilibrium position in source well L H0 Initial slug input L K Formation hydraulic conductivity L \u00b7 T\u22121 Kr Radial formation hydraulic conductivity L \u00b7 T\u22121 Kz Vertical formation hydraulic conductivity L \u00b7 T\u22121 Kskin Skin hydraulic conductivity L \u00b7 T\u22121 l/lo Depth of bottom of source/observation well test interval below watertable L L/Lobs Characteristic length for source/observation well damping term L Le/Le,obs Characteristic length for source/observation well oscillatory term L p Laplace transform parameter \u2212 r Radial coordinate, out from center of source well L R Domain radius, out from center of source well L rc Radius of source well tubing at water-table L rw Radius of source well at test interval L s Hydraulic head change from initial conditions L Ss Speci\ufb01c storage L\u22121 Sy Speci\ufb01c yield \u2212 t Time since slug initiation T Tc Characteristic time (Tc = B2/\u03b1r,1) T z Vertical coordinate, down from water-table L \u03b1r,i Hydraulic di\ufb00usivity of ith zone L2 \u00b7 T\u22121 \u03b3s Source well damping coe\ufb03cient T\u22121 \u03bd Kinematic viscosity of water L2 \u00b7 T\u22121"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1304.6738v1",
+                    "title": "Core-scale solute transport model selection using Monte Carlo analysis",
+                    "abstract": "Model applicability to core-scale solute transport is evaluated using\nbreakthrough data from column experiments conducted with conservative tracers\ntritium (H-3) and sodium-22, and the retarding solute uranium-232. The three\nmodels considered are single-porosity, double-porosity with single-rate\nmobile-immobile mass-exchange, and the multirate model, which is a\ndeterministic model that admits the statistics of a random mobile-immobile\nmass-exchange rate coefficient. The experiments were conducted on intact\nCulebra Dolomite core samples. Previously, data were analyzed using single- and\ndouble-porosity models although the Culebra Dolomite is known to possess\nmultiple types and scales of porosity, and to exhibit multirate\nmobile-immobile-domain mass transfer characteristics at field scale. The data\nare reanalyzed here and null-space Monte Carlo analysis is used to facilitate\nobjective model selection. Prediction (or residual) bias is adopted as a\nmeasure of the model structural error. The analysis clearly shows single- and\ndouble-porosity models are structurally deficient, yielding late-time residual\nbias that grows with time. On the other hand, the multirate model yields\nunbiased predictions consistent with the late-time -5/2 slope diagnostic of\nmultirate mass transfer. The analysis indicates the multirate model is better\nsuited to describing core-scale solute breakthrough in the Culebra Dolomite\nthan the other two models.",
+                    "authors": "Bwalya Malama, Kristopher L. Kuhlman, Scott C. James",
+                    "published": "2013-04-24",
+                    "updated": "2013-04-24",
+                    "primary_cat": "physics.data-an",
+                    "cats": [
+                        "physics.data-an"
+                    ],
+                    "main_content": "Introduction During the last 30 years, signi\ufb01cant e\ufb00ort has been expended to understand contaminant transport in fractured rock [Huyakorn et al., 1983, Sun and Buscheck, 2003] due in part to the necessity to evaluate site suitability for nuclear waste disposal. Contaminant transport in fractured rock is of common concern to regulators and stakeholders at nuclear waste disposal sites because o\ufb00-site contaminant migration could impact groundwater resources. Modeling contaminant transport in fractured rock is challenging due to the complex and inherently heterogeneous nature of the transport domain, and the multitude of physical and chemical processes controlling contaminant interaction with the host rock. This has led to 1 \fa development of several potentially competing conceptualizations of the transport environment [van Genuchten and Wagenet, 1989, Zheng et al., 2010]. Model selection is typically based on subjective expert judgment. Hence, there is a need for objective criteria for selecting physically-based models that best describe observed transport behavior and provide minimal predictive uncertainty. In this work, we present a criterion for selecting between competing models for describing transport at the core scale. Three models are considered: the single-porosity model; the traditional double-porosity model with single-rate mobile-immobile domain mass exchange [van Genuchten and Wagenet, 1989, Gamerdinger et al., 1990], and; a double-porosity model with multiple rates of mobile-immobile-domain mass exchange controlled by a random mass transfer coe\ufb03cient [Haggerty and Gorelick, 1995, 1998]. We refer to the traditional doubleporosity model as simply the double-porosity model, and to the model with multiple rates of mass exchange as the multirate model following Haggerty and Gorelick [1995], Haggerty et al. [2000] and Meigs et al. [2000]. In the multirate model, the mass transfer coe\ufb03cient is a random variable, not a single deterministic parameter. This conceptualization re\ufb02ects spatial, not temporal, variability (due to heterogeneity, i.e., multiple types and scales of porosity). The probability density function of the transfer coe\ufb03cient gives the probability that a mobileimmobile interface (assumed to be randomly distributed in space), encountered by a particle along its trajectory through the transport domain, has a particular mass transfer coe\ufb03cient value. The three models are used to analyze breakthrough data collected in core-scale laboratory experiments [Lucero et al., 1998] using conservative tracers tritium (3H) and sodium-22 (22Na), and the retarding tracer uranium-232 (232U). The experiments analyzed herein were performed on a rock core collected from a formation known to exhibit multiple types and scales of rock-matrix porosity. Previous analysis of the experimental data with singleand double-porosity models by Lucero et al. [1998] yielded poor model \ufb01ts to these data due to the inability of the two models to describe the long-tailing behavior of conservative solutes. The multirate model has been shown to properly describe this behavior in breakthrough data obtained in \ufb01eld-scale tracer tests [Meigs and Beauheim, 2001, Haggerty et al., 2001, McKenna et al., 2001]. It is applied herein for the \ufb01rst time to core-scale breakthrough data to demonstrate multirate mass-transfer e\ufb00ects are observable at this scale. Null-space Monte Carlo analysis (NSMC) is used to evaluate model prediction uncertainty for each of the three model based on breakthrough data. It yields multiple sets of parameters that calibrate the model [Tonkin et al., 2007, Tonkin and Doherty, 2009, James et al., 2009, Gallagher and Doherty, 2007], leading to multiple realizations of model \ufb01ts to data at parameter estimation optimality. By prediction uncertainty we mean the variance and bias of the ensemble of these model-prediction realizations relative to observed behavior. Variance describes the scatter of realizations about mean behavior, while the residuals bias associated with each data point at optimality over all NSMC realizations provides a measure of the systematic departure of predicted from observed behavior. This work presents the \ufb01rst use of residual bias in the solute transport literature as a criterion for model selection. 2 \f2 The multirate transport model The multirate model is based on the traditional double porosity model where the transport domain is conceptualized as comprising two overlapping continua, namely the mobile (advective or fracture porosity) and immobile (di\ufb00usion-dominated matrix porosity) domains. Unlike the traditional double porosity model where a single deterministic constant is used to characterize mobile-immobile-domain mass exchange, a random variable is used in the multirate model. Using this conceptual approach, the governing equation for transport of a sorbing radionuclide in the mobile domain [Haggerty and Gorelick, 1995, 1998] is given in nondimensional form \u2202C \u2202T + Z \u221e 0 \u03b2(\u03c9D) \u0012\u2202Cim \u2202T + \u03bbDCim \u0013 d\u03c9D = 1 Pe \u22022C \u2202X2 \u2212\u2202C \u2202X \u2212\u03bbDC, (1) where C = c/Cc, Cim = cim/Cc, X = x/Lc, T = t/Tc, c and cim are mobileand immobilephase solute concentrations [M L\u22123], x and t are space-time coordinates, Cc, Lc, and Tc are characteristic concentration, length, and time, \u03bbD = \u03bbTc, \u03bb is the \ufb01rst-order radioactive decay constant [T\u22121], \u03c9D = \u03c9Tc is the dimensionless \ufb01rst-order mass-transfer rate coe\ufb03cient (Damk\u00a8 ohler-I number), \u03b2(\u03c9D) = \u03b2Tp(\u03c9D) is the rock matrix point capacity ratio, \u03b2T = \u03c6imRim/\u03c6mRm is the dimensionless rock-matrix total capacity ratio, p(\u03c9D) is the probability density function (pdf) of \u03c9D, Pe = Lc/\u03b1L is the P\u00b4 eclet number, \u03b1L [L] is the longitudinal dispersivity, \u03c6m and \u03c6im are the mobileand immobile-domain porosities, and Rm and Rim are the mobileand immobile-domain retardation factors. The dimensionless governing equation for immobile domain transport is \u2202Cim \u2202T = \u03c9D(C \u2212Cim) \u2212\u03bbDCim, (2) the lumped-parameter formulation of immobile-domain mass transport. The transport equations are solved subject to the initial condition C(X, T = 0) = Cim(T = 0) = C0, (3) indicating initial equilibrium between mobile and immobile-domain concentrations. The boundary condition at X = 0 is \u0012 A Lc \u2202C \u2202X + BC \u0013\f \f \f \f X=0 = Cinj(T), (4) where Cinj is a normalized injection concentration and A [L] and B are parameters to specify the X = 0 boundary condition type (A = 0 and B = 1 correspond to a Dirichlet boundary condition, while A = \u2212D/v and B = 1 correspond to a Robin boundary condition). The downstream boundary condition is lim X\u2192\u221e \u0012 \u22121 Pe \u2202C \u2202X + C \u0013 = 0, (5) 3 \findicating zero solute \ufb02ux in\ufb01nitely far downstream. The solution to (1)\u2013(5) is obtained on a semi-in\ufb01nite domain 0 \u2264X < \u221eas a simpli\ufb01cation and limiting case of the \ufb01nite domain considered by Haggerty and Gorelick [1995, 1998]. It is given by \u00af C(X) = \u0012 \u00af Cinj \u2212B \u00af Cp B + uA/Lc \u0013 euX + \u00af Cp, (6) where the overbar indicates the Laplace transform, s is the Laplace transform parameter, u = \u0010 1 \u2212 p 1 + 4 \u00af f1/P \u0011 P/2, \u00af Cp = C0/(s+\u03bbD), \u00af f1(\u03bbD) = (s+\u03bbD) \u00af f0(\u03bbD), \u00af f0(\u03bbD) = 1+\u03b2T \u00af g(\u03bbD), and \u00af g(\u03bbD) = Z \u221e 0 \u03c9Dp(\u03c9D) s + \u03bbD + \u03c9D d\u03c9D. (7) The function \u00af g(\u03bbD) is the Laplace transformed memory function of Haggerty et al. [2000]. For single porosity \u00af g(\u03bbD) \u22610, whereas for double porosity with single-rate mass transfer \u00af g(\u03bbD) = \u03c9D/(s + \u03bbD + \u03c9D). The inverse Laplace transform of (6) is obtained using the de Hoog et al. [1982] algorithm. For all results reported herein, Cc = cinj is the injection concentration, Lc is core length, and Tc = Lc/vR, where vR = v/Rm and v is the average linear velocity [L T\u22121]. 2.1 Mass-Transfer Coe\ufb03cient Distribution To evaluate the memory kernel \u00af g(\u03bbD) numerically, the probability density function p(\u03c9D) must be speci\ufb01ed. All valid probability density functions are admissible in the computation of the memory function, including single-parameter distributions such as the power-law used by Haggerty et al. [2000] and Schumer et al. [2003]. However, single-parameter distributions may not lead to improved multirate model predictions of breakthrough behavior compared to the single-rate mass transfer model. Here, without loss of generality, we use the lognormal distribution because several key geological properties appear to approximately follow this distribution [Haggerty and Gorelick, 1998], including hydraulic conductivity [Neuman, 1982, Hoeksema and Kitanidis, 1985] and grain size [Buchan et al., 1993]. Other equally valid examples of distributions that have been used in the literature to characterize the mobileimmobile mass transfer coe\ufb03cients are summarized in Haggerty et al. [2000]. Using any of these models with two or more parameters, would likely yield multirate models that outperform the single-porosity and single-rate double-porosity models. The standard two-parameter lognormal distribution for \u03c9D \u2208[0, \u221e) was used by Haggerty and Gorelick [1995, 1998]. For the case where physical bounds exist \u03c9D \u2208[\u03c9D,min, \u03c9D,max], it may be more appropriate to use the random variable \u02c6 \u03c9D = (1/\u03c9D \u22121/\u03c9D,max)\u22121 \u2212\u03c9D,min, where \u03c9D,min and \u03c9D,max are the minimum and maximum physically allowable \u03c9D values. The pdf of \u02c6 \u03c9D is p(\u02c6 \u03c9D) = 1 \u02c6 \u03c9D\u02c6 \u03c3 \u221a 2\u03c0 exp \" \u2212 \u0012log(\u02c6 \u03c9D) \u2212\u02c6 \u00b5 \u02c6 \u03c3 \u221a 2 \u00132# , (8) where \u02c6 \u00b5 and \u02c6 \u03c3 are the mean and standard deviation of log(\u02c6 \u03c9D). This is the lowerand uppertail-truncated lognormal distribution, which is a more plausible distribution when there are 4 \fphysical limits on permissible Damk\u00a8 ohler-I numbers. These physical limits may be estimated from data. In the limit as \u03c9D,min \u21920 and \u03c9D,max \u2192\u221e, \u02c6 \u03c9D \u2192\u03c9D, and p(\u02c6 \u03c9D) degenerates to the standard two-parameter lognormal distribution. We set \u03c9D \u2208[0, 1000], loosely based on the work of Haggerty and Gorelick [1995], where \u03c9D = 100 was suggested as the limit for signi\ufb01cant multirate mass transfer. 3 Application to Core-scale Breakthrough Data The data considered here were collected in a series of column experiments conducted on \ufb01ve intact cores (denoted A through E) of the Culebra Dolomite as reported by Lucero et al. [1998]. The Culebra Dolomite member of the Rustler formation of the Permian Basin in southeastern New Mexico is known to exhibit several categories and scales of porosity [Holt, 1997] including inter-crystalline, inter-particle, fracture, and vuggy porosities (Figure 1). The multiple types and scales of porosity are also clearly observable in Culebra Dolomite cores (Figure 2). The only breakthrough data analyzed in this work were collected on the B core for the conservative tracers 3H and 22Na, and the retarding tracer 232U. Core B, pictured in Figure 2, was selected because its length-to-diameter ratio (50.9 cm to 14.5 cm) ratio was such that boundary e\ufb00ects can be neglected, thus permitting the use of the analytical solution developed for a 1D semi-in\ufb01nite (0 \u2264x < \u221e) transport domain. Dry bulk density \u03c1bulk = 2400 kg/m3 and total porosity \u03c6T = 0.14 were determined by standard laboratory methods [Lucero et al., 1998]. Additional details on experiment setup, solute injection, \ufb02ow rates, and e\ufb04uent analysis, are available in Lucero et al. [1998] and are not repeated here. Figure 3 shows normalized concentrations plotted against pore volume (PV) computed using \u03c6T. Solute injection pulses were longer in duration for tests shown in Figure 3(b) than for those in Figure 3(a). Plotting data on a log-log scale as in Figure 3(b) clearly shows that the e\ufb04uent was not collected for a su\ufb03ciently long time to completely reveal the late-time tracer behavior. A long breakthrough tail is characteristic of mobile-immobile-domain mass transfer for conservative tracers. Despite this shortcoming, the data can be used to assess the performance of the three models. The data in Figure 3(a) show early breakthrough for both conservative tracers [Lucero et al., 1998], suggesting the occurrence of preferential \ufb02ow in an advective porosity that is signi\ufb01cantly smaller than the total core porosity \u03c6T. Breakthrough data for 232U are shown in Figure 4 (22Na data from the same test are included for comparison). 232U breakthrough clearly occurs much later than 22Na because the former sorbs onto the Culebra Dolomite. Peak 232U concentration arrival occurs around 1 PV, about four times later than 22Na. Using the single-porosity model, Lucero et al. [1998] estimated the 232U retardation factor to be 4.5 and 3.7, from B3 and B7 data, respectively. For the dual-porosity model, they obtained mobileand immobile-zone retardation factor values of {Rm = 1.14, Rim = 65.4} and {Rm = 4.35, Rim = 1.00}, from B3 and B7 data, respectively. The value of Rim = 65.4 appears to be an error in recording the estimated value. 5 \f3.1 Parameter Estimation To estimate model parameters we let cobs be the breakthrough data vector, ccal(\u03b8) the model-calculated concentrations vector, and \u03b8 the vector of estimated model parameters. For 3H and 22Na, \u03b8 = (\u03c6m, \u03b1L, \u00b5, \u03c3, tinj), whereas for 232U, \u03b8 = (Rm, Rim, \u00b5, \u03c3). Injection pulse concentration (cinj) was \ufb01xed for tests B1, B2, B3, and B7, but was estimated for tests B4, B5, and B8. Increased test durations for B4, B5, and B8 made it more di\ufb03cult to maintain constant injection concentrations over prolonged test periods, resulting in injection concentrations that varied appreciably with time [Lucero et al., 1998]. Since this temporal variability is not incorporated explicitly into the solution, and its functional form in unknown, the injection concentrations for tests B4, B5, and B8 are treated as unknown constants and are estimated from breakthrough data. Initial concentration (c0) was \ufb01xed for all tests and was determined from e\ufb04uent concentration values measured prior to solute injection. The truncated lognormal distribution (\u03c9D \u2208[0, 1000]) was used to describe the mass-transfer coe\ufb03cient distribution. The advective porosity (\u03c6m), dispersivity, and the injected pulse (tinj) duration were estimated with the multirate model for 22Na data and used as \ufb01xed input parameters when estimating the retardation factor and \u03c9D distribution parameters from 232U data. Distribution parameters were also estimated for 232U because \u03c9D is a function of the tracer-speci\ufb01c molecular di\ufb00usion coe\ufb03cient. We examine model sensitivity coe\ufb03cients to determine whether all model parameters are estimable from available data. Sensitivity coe\ufb03cients are derivatives of model-predicted ef\ufb02uent concentrations with respect to model parameters, which are elements of the Jacobian matrix (J). They provide a measure of parameter identi\ufb01ability, because the determinant of JTJ must be su\ufb03ciently larger than zero to be estimable from data [\u00a8 Ozi\u00b8 sik and Orlande, 2000]. Small sensitivity coe\ufb03cients imply |JTJ| \u22480 and the inverse problem is ill conditioned. Here, sensitivity coe\ufb03cients were estimated with PEST using central di\ufb00erences, and their variation with time is shown in Figure 5 for (a) short (B2) and (b) long (B4) solute injection pulses. The sensitivities are su\ufb03ciently larger than zero to permit estimation of all parameters from breakthrough data. The coe\ufb03cients are also linearly independent for much of the time data were collected. Apparent linear dependence is restricted to late-time data, implying parameters cannot be uniquely estimated solely from late-time data. The parameter sensitivity curves obtained in both shortand long-pulse injection tests show a weak symmetry between two opposite-sign branches associated with arrival and elution tracer breakthrough waves. Absolute values of sensitivity coe\ufb03cients are largest when measured concentrations are changing most rapidly. Variation of sensitivity coe\ufb03cients with time for retarding tracer 232U in test B3 are shown in Figure 6. These are also su\ufb03ciently larger than zero indicating that parameters, including Rm and Rim, are estimable from breakthrough data. Parameter estimation was performed using PEST [Doherty, 2010]. The optimal vector of model parameters (\u03b8opt) was obtained by minimizing the sum of squared residuals, \u03a6(\u03b8) = e(\u03b8)Te(\u03b8), (9) where e = cobs \u2212ccal(\u03b8) is the vector of residuals. PEST uses the Levenberg-Marquardt non6 \flinear optimization algorithm [Marquardt, 1963]. Parameter estimates and multirate model \ufb01ts to data are compared to those obtained using singleand double-porosity models. Parameter values obtained by inverting 3H and 22Na breakthrough data with the all three models are summarized in Table 1; parameters estimated from 232U data are in Table 2. Because tinj was not reported in the original study [Lucero et al., 1998], it was estimated from data. The \u03c9D column also includes the mean (\u27e8\u03c9D\u27e9) and variance (\u03c32 \u2126D) of the Damk\u00a8 ohler-I number determined from the estimated values of \u02c6 \u00b5 and \u02c6 \u03c3. The last row of Table 1 shows estimated model parameters from simultaneous inversion of B4, B5, and B8 tracer-test breakthrough data. Parameter estimates are comparable to those from individual tests, even though the three tests were conducted with \ufb02ow rates ranging over an order of magnitude (0.05, 0.1, and 0.5 ml/min). This indicates minimal model structural error with regard to simulating average pore-water velocity. Model \ufb01ts to data for parameter values listed in Table 1 are shown in Figure 7 (B1\u2013 B3, B7) and Figure 8 (B4, B5, B8) for 3H and 22Na. Figures are in pairs of (a) linear or semi-log (concentration on linear scale) and (b) log-log plots, to illustrate how models match data over multiple time scales and over several concentration orders of magnitude. The two plotting scales are complementary because an apparently good model \ufb01t on a semi-log or linear plot may reveal a poor \ufb01t on log-log scale, and vice versa. Model-\ufb01t results for 232U data are shown in Figure 9. Lucero et al. [1998] parameter estimates are comparable to those obtained here using singleand double-porosity models, but they did not estimate tinj. Parameter estimation using the multirate model yielded improved model \ufb01ts to breakthrough data compared to those obtained using singleand double-porosity models (see R2 values in Table 1). Mobile-domain porosity values (\u03c6m) estimated with singleand doubleporosity models were comparable (means of 0.069 and 0.065, respectively), but were appreciably larger than those obtained using the multirate model (mean of 0.045). Dispersivity (\u03b1L) values were consistently largest for the single-porosity model (mean of 12.1 cm) and smallest for the multirate model (mean of 3.76 cm) for all tests. Table 1 shows there is significantly more variability in \u03b1L estimated using the single-porosity model than those obtained using the double-porosity and multirate models (standard deviations of 4.2 cm, 2.4 cm, and 2.3 cm, respectively). The Damk\u00a8 ohler-I numbers estimated with the double-porosity model appear closer (though not equal) to the geometric mean (\u27e8\u03c9D\u27e9g = e\u00b5) of the multirate model than to the mean (\u27e8\u03c9D\u27e9= e\u00b5+\u03c32/2). Results show absolute values of \u00b5 and \u03c3 for the 3H tracer test (B1) are smaller than those obtained with the tracer 22Na. With exception of B7, the 22Na tests yielded consistent values of \u00b5 and \u03c3 with |\u00b5| > 1.0 and \u03c3 \u22481.9. Those obtained from 232U data (Table 2) are signi\ufb01cantly di\ufb00erent. For the non-conservative tracer 232U, \u03c6m and \u03b1L were estimated with the 22Na tracer from the same experiment, because these parameters are intrinsic transport medium properties. Estimated retardation factors from tests B3 and B7 are listed in Table 2. For test B3, \ufb01tting the multirate and double-porosity models to data yields Rm values appreciably smaller than the value obtained with single-porosity model. This is because retardation is distributed between the mobile and immobile domains in the former two models. It is surprising to \ufb01nd the multirate model Rim in test B3 is signi\ufb01cantly larger than the double-porosity model 7 \fRim. Intuitively, one would expect results similar to those obtained from test B7, because delayed breakthrough is partly due to matrix mass transfer and partly due to solid-phase sorption. In addition, the retardation factors, Rm and Rim, estimated with the doubleporosity and multirate models showed signi\ufb01cant di\ufb00erences between test B3 and B7. These two results may be attributable to interplay between multirate mass-transfer and nonlinear sorption kinetics, where retardation is concentration dependent. The models all assume linear instantaneous sorption, variability in retardation factors between tests B3 and B7 may be an artifact of inherent model de\ufb01ciency to account for nonlinear sorption kinetics. 232U column tests tests B3, B6 (not discussed here) and B7 were performed serially on the same core. B3 had the lowest initial relative 232U concentration with c0/cinj \u22432 \u00d7 10\u22125, while for B7 c0/cinj \u224310\u22123. B7 was performed after the core had already been conditioned with 232U from the previous two tests. These initial concentration di\ufb00erences are expected to a\ufb00ect the estimated retardation factors in the presence of nonlinear sorption kinetics. 3.2 Predictive Analysis All models approximate a complex reality, and the discrepancy between reality and mathematical models is commonly referred to as model structural error. It is a measure of model de\ufb01ciencies that lead to prediction errors even when the models are supplied with optimal input parameters. Structural error cannot be attributed to measurement errors inherent in observations [Doherty and Welter, 2010] and typically decreases as models become more realistic with increased understanding of underlying causal mechanisms of processes. A measure of structural error would thus provide an objective criterion for model selection. Predictive uncertainty analysis presented here is used to demonstrate the structural de\ufb01ciency of the singleand double-porosity models, and how this de\ufb01ciency leads to increased model prediction error. The analysis was undertaken with PEST for test B8. Details for conducting a PEST predictive uncertainty analysis can be found elsewhere [James et al., 2009, Tonkin and Doherty, 2009, Tonkin et al., 2007, Gallagher and Doherty, 2007]. Using parameter values at optimality (Table 1) and the associated covariance matrix, 500 random parameter sets were generated and projected onto the Jacobian matrix null space. No clear null space was found from the singular value decomposition of the Jacobian matrix, therefore we assumed the null space to be a single dimension in these low-dimensional (\u22646) models. Model predictions computed beyond the last observation based on the 500 parameter sets generated in this manner are shown in the left column of Figure 10 for (a) single-porosity, (c) double-porosity, and (e) multirate models. They show signi\ufb01cant model prediction uncertainty for the single-porosity model, and only moderate uncertainty for the other two models. Using these parameter sets projected onto the null space as initial guesses, further minimization of \u03a6 was undertaken, using the Jacobian matrix associated with the calibrated state. Using the value of \u03a6 at optimality (\u03a6opt), the 500 null-space-projected parameter sets were processed with PEST to minimize the objective function such that \u03a6 \u22642\u03a6opt. Predictions associated with the re-calibrated parameter sets are shown in the second column of Figure 10 for (b) single-porosity, (d) double-porosity, and (f) multirate models. As would be expected, post re-calibration model predictions for all three models show a marked decrease 8 \fin model prediction uncertainty from the pre-calibration predictions. The late-time \u22123/2 and \u22125/2 slope lines are included, which are diagnostic of double-porosity and multirate models [Haggerty et al., 2000]. Clearly, the model behavior projected beyond the time of the last observation follows the \u22123/2 slope for the dual-porosity model, and the \u22125/2 slope for the multirate mass transfer model. Re-calibration single-porosity model projections show signi\ufb01cant underestimation of latetime observations. Dual-porosity model predictions are skewed toward overestimating the late-time observations. Multirate model projections are uniformly centered about the data and are consistent with the observed trend of the elution curve. Figure 11 shows histograms of residuals associated with the three models plotted at (a) t = 4.1 and (b) t = 4.7 days. Whereas the residuals computed at t = 4.7 days with the multirate model have zero bias, those of the doubleand single-porosity models show clear bias to negative (concentration overestimation) and positive (underestimation) values. Only the multirate model shows minimal bias about the observed late-time data, even though its ensemble of predictions has comparable spread (variance) to those of the double-porosity model beyond the last observation. The residual bias signi\ufb01es model structural error associated with singleand dual-porosity models. Comparing results in Figure 11 (a) and (b) shows residual bias and singleand double-porosity model structural error increase with time, while bias for the multirate model does not show appreciable change. At time t = 4.1 days, the dual-porosity model residuals have zero mean and are nearly coincident with the multirate model. However, at t = 4.7 days there is a growth in double-porosity model prediction bias. Prediction error due to model structural error increases with time. Figure 12 shows histograms of 500 calibrated multirate model parameter sets obtained from the posterior null-space Monte Carlo analysis described above. These distributions provide a measure of parameter estimation uncertainty. However, as indicated by Keating et al. [2010], parameter sets obtained using null-space Monte Carlo analysis do not necessarily constitute a sample of the posterior density function of the parameters in the strict Bayesian sense. This is especially true with low-dimensional models (at most 6 parameters for the present case) for which a proper null space may not exist. This can be seen by comparing the posterior distribution obtained with the null-space Monte Carlo analysis with those obtained to a formal Bayesian approach using the Di\ufb00eRential Evolution Adaptive Metropolis (DREAM) algorithm [Vrugt et al., 2008, 2009a,b]. For the problem considered here with 6 parameters to be estimated from log-transformed concentrations, DREAM ran 6 di\ufb00erent Markov chains, and after a burn-in period of about 35,000 model runs per chain, we obtained the parameter posterior distributions shown in Figure 13. DREAM required 300,000 total model runs. Clearly, the computational demands of formal Bayesian analysis with DREAM can be prohibitively high [Keating et al., 2010]. The parameter posterior distributions shown in Figure 13 show the \ufb01nal 10,000 model runs. Normal distributions are included in the \ufb01gure for comparison. The results show that posterior distributions obtained with DREAM have smaller variances and are more Gaussian than those obtained with the PEST posterior nullspace Monte Carlo analysis. Whilst PEST results indicate greater variability in estimated parameter values that calibrate the model, DREAM results indicate that parameter estima9 \ftion uncertainty is actually smaller. The low-dimensionality of the parameter space leads to an overestimation of parameter estimation uncertainty using null-space Monte-Carlo analysis. Thus, PEST-based parameter estimation uncertainty, obtained with null-space Monte Carlo analysis for a signi\ufb01cantly lower computational cost, may be viewed as the upper bound of the true uncertainty computed with DREAM, for cases like the low-dimensional models used here. 3.3 Statistical Model Selection For a given number of obervations, as models become more realistic, the increase in model complexity and the number of parameters leads to increased parameter estimation uncertainty because the number of observations available per estimated parameter decreases. In the present case, model complexity and the number of parameters increase from the singleporosity to the multirate model, but the respective model parameters are estimated with the same number of observations. Hence, statistical criteria that account for decreased information content due to increased model complexity may be used to augment model selection based on structural error evaluation. The corrected Akaike Information Criterion, AICc [Hurvich and Tsai, 1997, Anderson and Burnham, 1999, Poeter and Anderson, 2005] is used here for this purpose AICc = 2n \u0014 log(\u03c3e) + k n \u2212k \u22121 \u0015 , (10) where n is the number of observations, k is the number of estimated parameters, and \u03c3e is the standard deviation of residuals at optimality. The \ufb01rst term typically decreases as model complexity increases, representing improved model \ufb01t to data, while the second penalty term increases. Because AICc is a relative measure, it is preferable to use di\ufb00erentials of AICc [Posada and Buckley, 2004], denoted \u2206AICc, over all the three models under consideration. For the ith model, \u2206AICc,i = AICc,i \u2212min AICc, where min AICc is the smallest AICc value among all models for this dataset. The AICc are computed using PEST and \u2206AICc are listed in Table 1. The minimum AICc corresponds to the multirate model, except in test B5, where the it corresponds to the double-porosity model. Clearly the relative AICc values con\ufb01rm the results of predictive analysis that the multirate model is better suited than the other two models to describing transport in the Culebra Dolomite core. For time series data with high autocorrelation, the penalty for model complexity is vanishingly small when n \u226bk and the AICc reduces to a ranking of the models by residual variance. This is only a problem however, when the increased number of observations does not singin\ufb01cantly increase the information content of the observation about the estimated parameters. Hence, a separate optimization with PEST using only 30 of the original 269 data in test B8 to determine whether the ranking of the three models with the AICc would change appreciably. The resulting model \ufb01ts are shown in Figure 14. Basically the same results were obtained, with the multirate model outperforming the other two models. This is because the estimation variance is always smallest for the multirate model, and arti\ufb01cially reducing n only has a modest e\ufb00ect on the \ufb01nal outcome. It should also be noted that a large 10 \fn allows one to better capture the variability in the data due to random measurement error, which are assumed to be Gaussian in minimization of the sum of squared residuals. Further, the number of parameters to be estimated increased only by 2 from the single-porosity to the multirate model, whereas the estimation variance changes by a factor of about 2 (7.6 \u00d7 10\u22126 to 3.2 \u00d7 10\u22127). The temporal structure of the residuals was examined to determine whether they show strong temporal autocorrelation. They are plotted in Figure 15. It can be seen in the Figure that moderate autocorrelation is limited to very early-time. Additionally, in this early-time period, it can be seen that only the single-porosity model residuals show appreciable temporal autocorrelation, which decreases as one moves to the multirate model. The computed responses of the single-porosity model show strong departure from observed behavior. As can be seen in Figure 15, the residuals obtained with the multirate model for the long tests (B4, B5 & B8) show only moderate temporal autocorrelation (at early-time) and are mostly randomly distributed about zero. It should also be noted that the statistical rigor of DREAM does not depend on the distribution of the residuals but on the sampling of the parameter space for parameters that minimize the sum of squared residuals. 4 Discussion and Conclusions We reanalyzed core-scale 3H and 22Na breakthrough data from experiments conducted by Lucero et al. [1998] on a Culebra Dolomite core using the single-porosity, double-porosity, and the multirate model of [Haggerty and Gorelick, 1995, 1998] on a semi-in\ufb01nite domain to determine which of the models best describes the observed breakthrough behavior. Previous analysis of these data by Lucero et al. [1998] had suggested that the single-porosity model was su\ufb03cient to describe core-scale Culebra transport, a \ufb01nding that was at odds with \ufb01ndings based on \ufb01eld-scale tests conducted in the Culebra Dolomite formation [Meigs et al., 2000, McKenna et al., 2001]. In the results presented herein, the multirate model yielded better model \ufb01ts to the data and parameter values that di\ufb00ered signi\ufb01cantly from those obtained with the singleand double-porosity models. The mobile-domain porosity and dispersivity values obtained with the multirate model were consistently lower than those obtained with the other two models because solute dispersion in the core is also accounted for by porosity variability encapsulated in the distribution parameters of the mobile/immobile domain masstransfer coe\ufb03cient. The smaller dispersivity obtained with the multirate model is more plausible than those obtained with the other models, considering the length scale of the experiments. Model-prediction uncertainty was evaluated using breakthrough data from test B8 and post-calibration null-space Monte Carlo analysis as implemented with PEST. The prediction uncertainty analysis revealed the presence of model structural error in the singleand double-porosity models as demonstrated by signi\ufb01cant bias in the residuals of model predictions made with these models with optimal parameter values. The residual bias increased with time over the span of the elution curve where breakthrough data are available, showing increased departure of model predictions from the observed trend (\u22125/2 slope line) of 11 \fbreakthrough data. The parameters associated with the null-space Monte Carlo predictive analysis may be viewed as samples from the posterior parameter distributions and were used to evaluate parameter estimation uncertainty. The posterior distributions estimated using null-space Monte Carlo analysis were compared to those obtained with the more rigorous Bayesian analysis in the DREAM algorithm. The comparison suggests that measures of parameter estimation uncertainty obtained with null-space Monte Carlo may be treated as upper bounds of the true posterior distributions, particularly for low dimensional models where a true null space may not exist. The analysis presented herein clearly shows the residual bias associated with the singleand double-porosity models increases with time indicating increasing systematic departure of predicted from observed behavior due to the inherent structural de\ufb01ciencies of these models. The multirate model residuals, however, maintain minimal bias with time, indicating low model structural error. Although the predictions with the double-porosity and multirate models beyond the last observation have comparable variance, only the residuals of the multirate model have zero bias. These results show that the multirate model is the most appropriate of the three models for describing solute breakthrough behavior in Culebra core even though the three models yield parameters with comparable variances of posterior distributions. This \ufb01nding was further con\ufb01rmed using statistical model selection using the di\ufb00erential AICc where the AICc value was typically smallest for the multirate model. The one test where the double-porosity model yielded the smallest di\ufb00erential AICc value, the value associated with the multirate model was only marginally larger (0.5%). More elution data would be needed to resolve this minor departure from the norm given that the two models predict disparate long-term tailing behaviors. Acknowledgements Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy\u2019s National Nuclear Security Administration under contract DE-AC0494AL85000. This research is funded by WIPP programs administrated by the O\ufb03ce of Environmental Management (EM) of the US Department of Energy."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1301.0772v2",
+                    "title": "Unsaturated Hydraulic Conductivity Models Based on Truncated Lognormal Pore-size Distributions",
+                    "abstract": "We develop a closed-form three-parameter model for unsaturated hydraulic\nconductivity associated with the Kosugi three-parameter lognormal moisture\nretention model. The model derivation uses a slight modification to Mualem's\ntheory, which is nearly exact for non-clay soils. Kosugi's three-parameter\nlognormal moisture retention model uses physically meaningful parameters, but a\ncorresponding closed-form relative hydraulic conductivity model has never been\ndeveloped. The model is further extended to a four -parameter model by\ntruncating the underlying pore size distribution at physically permissible\nminimum and maximum pore radii. The proposed closed-form models are fitted to\nwell-known experimental data, to illustrate their utility. They have the same\nphysical basis as Kosugi's two-parameter model, but are more general.",
+                    "authors": "Bwalya Malama, Kristopher L. Kuhlman",
+                    "published": "2013-01-04",
+                    "updated": "2014-04-04",
+                    "primary_cat": "physics.geo-ph",
+                    "cats": [
+                        "physics.geo-ph",
+                        "physics.flu-dyn"
+                    ],
+                    "main_content": "Introduction Understanding and predicting in\ufb01ltration of water into unsaturated soil is critical to both agricultural and groundwater-hydrology applications. Precipitation in\ufb01ltrates through the vadose zone, often carrying contaminants from the surface to regional aquifers. Numerically simulating moisture redistribution in the vadose zone using Richards\u2019 equation requires functions relating soil water content and relative hydraulic conductivity to capillary pressure head (e.g., Warrick (2003, \u00a72.5)). Durner and Lipsius (2006) summarize techniques to collect data characterizing these unsaturated \ufb02ow relationships in soils. Many di\ufb00erent functional forms have been proposed to capture the essential characteristics of observed soil behavior, allowing for e\ufb03cient and accurate predictive simulation. Although any arbitrary function can be adopted to represent a soil\u2019s behavior during \u2217bnmalam@sandia.gov \u2020klkuhlm@sandia.gov 1 arXiv:1301.0772v2  [physics.geo-ph]  4 Apr 2014 \fin\ufb01ltration, simpler mathematical forms often allow closed-form representation of the moisture retention and relative hydraulic conductivity curves. Common closed-form functions include models by Gardner (1958), Brooks and Corey (1964), Van Genuchten (1980), and Kosugi (1994). Closedform expressions for unsaturated hydraulic conductivity allow straightforward implementation in numerical models, avoiding costly and error-prone numerical integration. Closed-form expressions can provide more insight into the relationship between parameters in the moisture retention and hydraulic conductivity models than purely numerical schemes (e.g., Priesack and Durner (2006)). Kosugi (1994) assumed soil pore size is a lognormal random variable and derived a physically based three-parameter model for moisture retention, the three parameters being the mean and variance of the pore-size distribution and the maximum pore radius. In the limiting case where the maximum pore radius becomes in\ufb01nite the three-parameter model simpli\ufb01es to a two-parameter model for which Kosugi (1996) developed the closed-form expression for unsaturated hydraulic conductivity using the theory of Mualem (1976). Kosugi (1996) did not develop a three-parameter closed-form equation for hydraulic conductivity, but reverted to the two-parameter form, owing to di\ufb03culty in analytically integrating the expression of Mualem (1976). We extend the work of Kosugi (1994, 1996), developing closed-form expressions for unsaturated hydraulic conductivity associated with the three-parameter lognormal moisture retention model. The derivation of the closed-form equation for unsaturated hydraulic conductivity is made possible by an approximation to the theory of Mualem (1976). Further, we modify the pore-size probability density function (PDF) of Kosugi (1994) by incorporating a nonzero minimum pore radius, as suggested by Brutsaert (1966). This modi\ufb01cation results in a four-parameter moisture retention model and a corresponding four-parameter closed-form equation for unsaturated hydraulic conductivity, again obtained using the modi\ufb01ed theory of Mualem (1976). The four parameters are all based on porous medium properties like Kosugi (1996); they are not \ufb01tting parameters without physical signi\ufb01cance. The four-parameter model is a generalization of the two-parameter model of Kosugi (1996) and simpli\ufb01es to it when the minimum and maximum pore sizes tend to zero and in\ufb01nity. 2 Theory The lognormal distribution is commonly used to statistically characterize pore size in granular porous media. Brutsaert (1966) and Kosugi (1994, 1996) considered lowerand upper-tail truncated lognormal PDFs for pore-size distributions. Brutsaert (1966) considered the log-transformed random pore radius R \u2212r0, where r0 is the radius at which the e\ufb00ective moisture content vanishes (associated with residual saturation). 2.1 Three-parameter lognormal model The classical (non-truncated) lognormal pore-size distribution (Brutsaert, 1966) is measured here by the random pore radius R \u2208[0, \u221e]. In a physically realistic porous medium R \u2208[0, rmax], where rmax is some \ufb01nite maximum pore radius. To account for the \ufb01nite interval, Kosugi (1994) used the random variable Re = (1/R \u22121/rmax)\u22121 to rescale the classical PDF. The PDF of R is related to the PDF of Re as fR(r) = fRe h (1/r \u22121/rmax)\u22121i (1 \u2212r/rmax)2 . (1) 2 \fAccording to Young-Laplace theory, capillary pressure head h and pore radius r are related by h = \u03ba/r, where \u03ba = 2\u03b3 cos \u03b1/(\u03c1g), \u03b3 is interface surface tension, \u03b1 is the interface contact angle, \u03c1 is \ufb02uid density, and g is gravitational acceleration. For water in a glass tube, \u03ba \u22480.149 cm2. It follows the PDF of the random capillary pressure head H is fH(h) = fRe h (h/\u03ba \u22121/rmax)\u22121i (h/\u03ba \u22121/rmax)2 . (2) Further, if Re is lognormally distributed, the PDF for the capillary pressure head H can be written as fH(h) = 1 \u221a 2\u03c0\u03c3Z(h \u2212hc) exp \" \u2212 \u0012ln (h \u2212hc) \u2212\u00b5\u03b7 \u221a 2\u03c3Z \u00132# , (3) for all h > hc, where hc = \u03ba/rmax is the bubbling pressure head, \u00b5\u03b7 = ln(\u03ba) \u2212\u00b5Z is the mean of ln(H), \u03c32 Z is the variance of Z, \u00b5Z is the mean of Z, and Z = ln (Re). Kosugi (1994) used the dimensionless random variable R\u2032 e = Re/rmax to obtain the PDF in (3) with the parameters \u00b5Z and \u03c32 Z scaled appropriately. Kosugi (1994) showed the three-parameter moisture retention curve related to (3) is \u03b8\u2217(h) = ( 1 2erfc \u0010 ln(h\u2212hc)\u2212\u00b5\u03b7 \u221a 2\u03c3Z \u0011 h > hc, 1 h \u2264hc, (4) where \u03b8\u2217(h) = (\u03b8(h) \u2212\u03b8r)/(\u03b8s \u2212\u03b8r) is moisture capacity, \u03b8(h) is volumetric moisture content, \u03b8r is residual moisture content, \u03b8s is saturated moisture content, and erfc is the complementary error function (e.g., Abramowitz and Stegun (1964, \u00a77)). Kosugi (1994) did not develop a corresponding closed-form equation for unsaturated hydraulic conductivity from 3. Mualem (1976) developed a widely used functional relation between unsaturated hydraulic conductivity K(\u03b8\u2217) = KsKr and h Kr(\u03b8\u2217) = \u221a \u03b8\u2217 \" Z \u03b8\u2217 0 dx h(x) ! \u001e \u0012Z 1 0 dx h(x) \u0013#2 , (5) where Kr and Ks are relative and saturated hydraulic conductivity and x is an integration variable. Equation (5) can be rewritten in terms of the capillary pressure head PDF as Kr(\u03b8\u2217) = \u221a \u03b8\u2217 \u0014\u0012Z \u221e h fH(x) x dx \u0013 \u001e \u0012Z \u221e 0 fH(x) x dx \u0013\u00152 . (6) Using the theory of Mualem (1976), Kosugi (1996) developed the two-parameter closed-form equation for unsaturated hydraulic conductivity by setting rmax \u2192\u221e. Kosugi (1996) made this simpli\ufb01cation because the theory of Mualem (1976) as given in (6) is not readily amenable to integration when rmax is \ufb01nite. We obtain an approximate closed-form expression for unsaturated hydraulic conductivity for the lognormal PDF with \ufb01nite values of rmax by modifying (5) due to Mualem (1976) into Kr(\u03b8\u2217) = \u221a \u03b8\u2217 \" Z \u03b8\u2217 0 dx h(x) \u2212hc ! \u001e \u0012Z 1 0 dx h(x) \u2212hc \u0013#2 , (7) 3 \fbased on the assumption fH(h)/h \u2248fH(h)/(h\u2212hc). Using this approximation the relative hydraulic conductivity for the three-parameter lognormal model is Kr(h) \u22cd ( \u221a \u03b8\u2217 n 1 2erfc h ln(h\u2212hc)\u2212\u00b5\u03b7+\u03c32 Z \u221a 2\u03c3Z io2 h > hc, 1 h \u2264hc. (8) In the limit as rmax \u2192\u221e, hc \u21920, the assumption given above is exact, and (8) reduces to the two-parameter unsaturated hydraulic conductivity expression derived by Kosugi (1996). The approximation introducted by the assumption leading to (7) is best when rmax is relatively large (hc is small). Figure 1 shows that the truncated lognormal pore-size distribution (8) shifts the Kr curves (Kr = 1 at h = hc \u2013 dashed curves) compared to the solid curves corresponding to the two-parameter model of Kosugi (1996), where Kr = 1 is reached at h = 0. 2.2 Four-parameter lognormal model We incorporate the lower-tail truncation of Brutsaert (1966) into the distribution of Kosugi (1994), by introducing the random variable \u02c6 Re = \u0012 1 R \u2212r0 \u2212 1 rmax \u0013\u22121 , (9) which yields the following lognormal PDF for capillary pressure head, fH(h) = 1 \u221a 2\u03c0u\u03c3Z exp \" \u2212 \u0012ln(u) \u2212\u00b5\u03b7 \u221a 2\u03c3Z \u00132# , (10) for all h \u2208[hc, hmax] where u = (1/h \u22121/hmax)\u22121 \u2212hc and hmax = \u03ba/r0 is the pressure head associated with the smallest undrainable pores. Figure 2 shows the three lognormal PDFs for capillary pressure head: the classical (non-truncated) lognormal distribution, the upper-truncated lognormal distribution (3), and the doubly truncated lognormal distribution (10). It can be seen the three-parameter model of Kosugi (1994) departs from the classical lognormal distribution only at small head values (large pore radii) whereas the proposed four-parameter distribution departs from the classical distribution at both the lower and upper limbs of the function. The four-parameter model can be considered most physically realistic, while the twoand three-parameter models are simpli\ufb01cations. For some soils the simpli\ufb01ed PDFs may be adequate. A moisture retention model is derived from (10) in a similar manner, and is given by \u03b8\u2217(h) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 2erfc h ln(u)\u2212\u00b5\u03b7 \u221a 2\u03c3Z i hc < h < hmax, 1 h \u2264hc, 0 h \u2265hmax. (11) Finally, it can be shown that the closed-form expression for unsaturated hydraulic conductivity using the doubly truncated PDF (10) and the modi\ufb01ed equation of Mualem (1976) (6) is Kr(h) \u22cd \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 \u221a \u03b8\u2217 n 1 2erfc h ln(u)\u2212\u00b5\u03b7\u2212\u03c32 Z \u221a 2\u03c3Z io2 hc < h < hmax, 1 h \u2264hc, 0 h \u2265hmax. (12) In the limit as both rmax \u2192\u221eand r0 \u21920, (11) and (12) simplify to corresponding two-parameter expressions from Kosugi (1994, 1996). 4 \f3 Fitting lognormal models to experimental data The threeand four-parameter lognormal models for moisture retention (4) and (11) were \ufb01tted to experimental data (also used by Van Genuchten (1980) and Kosugi (1996)) via a quasi-Newton optimization algorithm (Zhu et al., 1997) from scipy (Oliphant, 2007). The parameters rmax, \u00b5Z, \u03c32 Z, \u03b8r, and \u03b8s were estimated for the three-parameter model, while r0 was additional estimated for the four-parameter model. Model-predicted unsaturated hydraulic conductivity (8) and (12) were compared to measured values. Figures 3 and 4 show data (dots) and best-\ufb01t models (lines) for Hygiene sandstone (Mualem (1976) soil 4130) and Silt Loam G.E. 3 (Mualem (1976) soil 3310). The 3-parameter model is plotted in red and the four-parameter model is plotted in black. Table 1 provides a summary of estimated parameters for the three soils and two models. Figures 3b, 4b, and 5b show solid curves representing the approximate, but closed-form expression for Kr (7), and dashed curves representing the numerically integrated Mualem relationship (5) for comparison. The numerically integrated form takes several orders of magnitude more time to evaluate than the closed-form expression, making its direct use in numerical models impractical. The numerical and analytically derived curves are nearly identical for the sandstone (Figure 3) and silt (Figure 4). The \ufb01ts are comparable to those of Kosugi (1996) and Van Genuchten (1980) for these same soils. The threeand four-parameter lognormal models were simultaneously \ufb01tted to moisture retention and unsaturated hydraulic conductivity data for Beit Netofa clay (Mualem (1976) soil 1006). The four-parameter model (black line in Figure 5) simultaneously \ufb01tted both unsaturated conductivity and moisture retention data. The three-parameter model only either \ufb01t hydraulic conductivity data (Figure 5) or moisture retention data (not shown \u2013 see examples in Kosugi (1996)); we could not get the three-parameter model to \ufb01t both data sets simultaneously. Figure 5b shows the di\ufb00erence between the closed-form solutions (solid lines) and the numerically integrated form (dashed lines). The closed-form and numerical solutions for the 3-parameter model are identical (a single red line). The numerically integrated 4-parameter model deviates more signi\ufb01cantly from its closed-form counterpart. 4 Discussion We present a three-parameter approximate closed-form expression for the hydraulic conductivity associated with the three-parameter moisture retention curve of Kosugi (1994, 1996). We use this approach to develop an analogous four-parameter lognormal model, which provides \ufb01ts to moisture retention data from three soils, similar to the \ufb01ts with the widely used models of Van Genuchten (1980) and Kosugi (1996). Predictions of unsaturated hydraulic conductivity from model \ufb01ts to moisture retention data are comparable to those of Van Genuchten (1980) and Kosugi (1996) for Hygiene sandstone and Silt Loam G.E. 3. The four-parameter model yielded a di\ufb00erent estimate of rmax than the three-parameter model did (Table 1), and it predicted the unsaturated permeability curve better (Figure 4b). Estimating model parameters for Beit Netofa clay using only moisture retention data did not yield good predictions of unsaturated hydraulic conductivity (similar to the models of Van Genuchten (1980) and Kosugi (1996)). It was essential to use both moisture retention and hydraulic conductivity data to estimate the parameters and improve the \ufb01t of the proposed approximate analytical model over the models of Kosugi (1994) and Van Genuchten (1980). Moisture retention data alone are not su\ufb03cient to estimate all four (or six, when including \u03b8r and \u03b8s) model parameters. If conductivity measurements are available for clays, they should be 5 \fused with moisture retention data to arrive at a more realistic closed-form model for unsaturated soil moisture retention and hydraulic conductivity. The di\ufb00erence between the proposed approximate analytical expressions for 3and 4-parameter lognormal unsaturated conductivity and their much more computationally expensive numerically integrated counterparts is minimal for the two non-clay soils. The models provide useful alternative formulations to the only closed-form lognormal model (with 2 parameters) given by Kosugi (1996). To rigorously verify the validity of the approximate closed-form expressions for a given set of parameters, we suggest comparing them to the results of numerically integrating (5), as was done in Figures 3\u20135. One would expect a model with a larger number of adjustable parameters to \ufb01t observed data better than similar models with fewer parameters, but improved \ufb01t is often at the expense of parameter physical signi\ufb01cance. The proposed four-parameter model is more physically plausible model than the simpler twoand three-parameter forms, which make the simplifying assumption of in\ufb01nite minimum or maximum pore sizes. Models with too few parameters to adequately explain observed data (e.g., the three-parameter model in the case of the Beit Netofa Clay) are structurally de\ufb01cient; other parameters may take on physically unrealistic values to compensate for the structural de\ufb01ciency of the model. Likewise, a model with too many parameters will have high uncertainty associated with the unnecessary parameters. For any given soil, the simplest appropriate model should be used. All parameters follow the philosophy Kosugi (1994) used in deriving his physically based model: they are related to pore-size distribution statistics and limits, rather than being exponents or powers (i.e., \ufb01tting parameters). The modi\ufb01ed model of Mualem (1976) enables approximate but closed-form expressions for unsaturated hydraulic conductivity for both the threeand four-parameter lognormal pore size distributions. The model derived here is a generalization of the lognormal model of Kosugi (1996) for non-zero minimum pore radius and non-in\ufb01nite maximum pore radius. Acknowledgments Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy\u2019s National Nuclear Security Administration under contract DE-AC04-94AL85000."
+                }
+            ]
+        },
+        "edge_feat": {}
+    }
+}
\ No newline at end of file