Corresponding author: L. C. Murdoch, Department of Environmental Engineering and Earth Science, 340 Brackett Hall, Clemson University, Clemson, SC 29631, USA. (firstname.lastname@example.org)
 The volume of water released from storage per unit head drop per volume of an REV is a basic quantity in groundwater hydrology, but the details of the process of storage change in the vicinity of a well are commonly overlooked. We characterize storage change in a flat-lying fracture or thin sedimentary bed through the apparent hydraulic compliance,Cf, the change in aperture of the fracture or thickness of the layer per unit change in pressure. The results of theoretical analyses and field measurements show that Cf increases with time near the well during pumping, but it drops suddenly and may become negative at the beginning of recovery during a well test. Profiles of Cfincrease with radial distance from a well, but they are marked by a sharp increase and a sharp decrease at the edge of the region affected by the wellbore pressure transient. The conventional view in groundwater hydrology is that storage change at a point is proportional to the local change in pressure, which requires that the hydraulic compliance is uniform and constant. It appears that this conventional view is a simplification of a process that varies in both space and time and can even take on negative values. This simplification may be a source of uncertainty when interpreting well tests and extensometer records or predicting long-term well performance.
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 The volume of water stored in an aquifer decreases during pumping and this is a fundamental process controlling well performance. The typical conceptualization is that the volume of water stored per unit volume of aquifer decreases in response to the pressure change within a representative volume (a historical review of the concept is given by Narasimhan ). This process is typically characterized by the specific storage, Ss, and it results in the equation commonly used to characterize transient hydraulic heads in groundwater hydrology [Bear, 1979; Ingebritsen et al., 2006]
where K is hydraulic conductivity, h is hydraulic head, t is time, and R is a volumetric source. Several types of specific storage, which differ depending on the mechanical constraints on the representative elementary volume (REV), have been introduced in the literature [e.g., Wang, 2000, and references therein]. The unconstrained specific storage, Sσ, assumes water is released from storage while the mean applied stress is constant [Wang, 2000, p. 56], and the uniaxial specific storage applies additional constraints of zero horizontal strain [Green and Wang, 1990]. The unconstrained specific storage is taken as a material property of the aquifer given by
where β is the compressibility of water, φ is porosity, and K is the bulk modulus (1/K is the bulk compressibility) of the aquifer material, and the solid grains of the aquifer are assumed to be incompressible [Wang, 2000]. Equation (2) was initially proposed by [Jacob, 1940] and it is widely used in groundwater hydrology today [Narasimhan, 2006].
 The approach implied by using (1) is that storage change is a local process that occurs in response to a local change in pressure. Pressure loading may deform the aquifer skeleton in a pattern that differs from direct proportionality to the pressure change. This process can be represented by separately considering effects of changes in fluid pressure and changes in strain. Fluid pressure effects can be considered by assuming the volume of the REV is held constant, so external strain is zero. This gives the constrained specific storage [Wang, 2000]
where is the unjacketed bulk modulus, or the modulus of the solid grains for aquifers consisting of a single mineral, and it is assumed that and . is the unjacketed pore bulk modulus. In general, the compressibility of the solid grains is much less than the compressibility of the bulk aquifer material, , so .
 Within the framework of the theory of poroelasticity, the effect of strain on storage can be considered by recognizing that the volumetric strain rate defines the volumetric source in equation (1)
 The use of (1) and (2) in conventional groundwater hydrology and (1), (3), and (4) in poroelasticity results in different perspectives on the process of storage change. In conventional groundwater hydrology, storage change only occurs in response to a local pressure change and the magnitude is proportional to fluid and aquifer compressibilities. In poroelasticity, however, storage change occurs in response to both pressure and stress changes. The latter is particularly significant because it means that storage change may not be proportional to pressure change, as is assumed when using (1) and (2) with R = 0. Despite these differences, there appears to be scant data and few analyses describing the process of storage change in the vicinity of a well. Recently, Berg et al.  evaluated the effect that neglecting mechanical coupling has on the results of well tests. They conducted poroelastic analyses of wells pumping in layered aquifers and then analyzed the pressure time series by inverting the Neuman and Witherspoon  solution to (1). Their findings show that including mechanical coupling can significantly alter the drawdown curve compared to the analytical solution, particularly for observation points that are near the pumping well or observation times that are relatively early. Inverting drawdowns from late times gave parameter estimates that were within the range of uncertainties from typical pumping tests, according to Berg et al. . This provides useful guidelines for practical applications of some well tests, but the processes occurring to cause these effects were not a focus of their study so it is difficult to put these results in a broader context. In addition, the real aquifer material may not necessarily be poroelastic, although poroelasticty provides a useful framework for storage consideration.
 The purpose of this paper is to describe deformation in the vicinity of a pumping well and show how it affects storage change during slug tests and constant rate pumping. The results could have implications related to interpreting well tests and predicting how wells will perform. The approach will be to use theoretical analyses and field data to characterize deformation and storage change as a function of time and space in the vicinity of an active well. For simplicity, we will assume the well intersects a thin, soft, permeable layer where the variations in strain across the thickness of the layer can be ignored. This will allow the problem to be treated as a 1-D function of radial distance from the well. A flat-lying fracture in low-permeability rock would resemble this geometry, but it could also be applicable to a thin permeable sand bed sandwiched between thick shale layers.
 The volumetric strain rate is affected by changes in thickness of the layer, so it will be convenient to use the apparent hydraulic compliance, defined as
where v is displacement normal to the layer. The name of this term is derived from the similarity to mechanical compliance, dv/dσe, where σe is replaced with p in the derivative because of the focus of this paper on storage effects when pressure changes. The adjective “apparent” is used to distinguish this term from other uses of compliance that refer to a material property. It follows that for a thin layer, the first term in parentheses in (4) is
where δo is the aperture at a reference pressure. The paper begins with a conceptual model that is evaluated using an analytical solution for the deformation of a fracture. This analysis will show that the radial strain is much smaller than the strain normal to the layer, so the term in parentheses in (4) is dominated by (6). This supports the use of hydraulic compliance as a proxy for storage change. Numerical analyses are included to consider additional conditions and coupling. Field data are then presented to support the theoretical results.
2. Conceptual Model
 Most natural fractures in the subsurface are held open by stresses on contacting asperities, pasp, and by fluid pressure, po, in their open space (Figure 1). Changing the fluid pressure by Δp (= p − po), shifts the stresses on the asperities, Δpasp, and may cause a net change in the total stress on the fracture surface. A force balance gives the total stress on the fracture as
where α is the ratio of the open area to the total area of the fracture. The stress on the asperities, averaged over the area of the fracture, is recognized as an effective stress, σeff = (1 − α)pasp. Assuming α remains constant during small changes in load, it follows that small changes in pressure or effective stress (compression is positive) will change the total stress on the fracture:
 Changes in the total stress will result in normal displacement, v, of the walls of the fracture (Figure 1). For simplicity, shear displacements are considered small and are ignored here.
 Normal displacement of the fracture face is assumed to satisfy both local and global constraints [Murdoch and Germanovich, 2006], which correspond to the local response within the fracture and global response within the host material. Normal displacement is assumed to be proportional to changes in effective stress [Rutqvist and Stephansson, 2003]
where Cn is the local compliance of the asperities normal to the plane of the fracture. Equation (9) follows from laboratory experiments and is valid for small changes in σeff. Parameter Cn is equal to the inverse of the fracture normal stiffness [Bandis and Barton, 1983; Rutqvist and Stephansson, 2003]. Nonlinear expressions for displacement resulting from large changes in effective stress are available, but (9) will be adequate for the ranges of small stress change caused by most pumping tests.
 Fracture aperture, δ, changes in response to normal displacement, so the local aperture is
where δo is the initial aperture, or the average separation when two rough surfaces first make contact at zero effective stress [Murdoch and Germanovich, 2006]. In this case, δ is defined based on displacement, rather than on effective hydraulic properties.
 Displacement of the fracture walls at a particular location, however, may be caused by changes in total stress anywhere on the fracture surface. This nonlocal effect can be visualized by conceptualizing the fracture walls as a half space. A point load will indent the half space over a region beyond the location of the load. Displacement caused by a distributed load can be represented by integrating the response from point loads over the half space.
 The change of fracture aperture is represented by the sum of displacements of the two fracture surfaces. Nonlocal displacements will be characterized by following this conceptual model and using the double displacements, v(r, t), of the half-space surface caused by an axially symmetric distribution of total stress [e.g.,Johnson, 1985]
where K(x) is the complete elliptic integral of the first kind, E is Young's modulus, and υ is Poisson's ratio.
 Local and nonlocal responses to changes in loading conditions result in the same displacements, so equating (9) and (11) and substituting (8) while dropping the Δ symbols for convenience gives
 Differentiating (12) with respect to p, using (5), and substituting (8) gives an equation for the dimensionless form of the apparent hydraulic compliance
where Cf has been normalized as .
 The compressibility in the vicinity of a well in a thin, uniform layer with a permeable zone shaped like a circular disk will serve as a useful example. Many fractures are conceptualized as circular disks, and some aquifers or reservoirs are roughly circular and bounded by relatively low permeability materials, so the circular shape will provide a reasonable representation of at least two natural conditions. The pressure distribution p(r, t) caused by a constant line source along the axis of a disk of radius a is given by Carslaw and Jaeger . Then equation (13a) becomes
where ρ = r/a, η = λ/a, τ = tD/a2 with D the hydraulic diffusivity of the layer, and
is obtained by differentiating the analytical expression given by Carslaw and Jaeger . Here pressure has been normalized as P = p/p1 with and q being the volume of liquid injected per unit length of the line source (borehole) per unit time, J0(x) is the Bessel function of the first kind, and xn is the nth positive root of the Bessel function J1(x). When K and Ss in (1) are constant, D = K/Ss.
Equation (13b) is a nonhomogeneous Fredholm integral equation of the second kind with respect to the unknown function . When its kernel is square integrable it has a unique solution [Rudin, 1991; Kolmogorov and Fomin, 1999], which is the case for the kernel defined in (11) because
Note that while transitioning from (13a) to (13b), we changed the upper limit in the integral from infinity to 1 because = 0 outside this radius, so (13b) is automatically satisfied for ρ > 1. This means that equation (13b) has a unique solution only for ρ < 1, which is sufficient for the purpose of this work.
 In general, the solution of (13b)can be obtained numerically, but a closed-form solution is possible when parameterε is sufficiently small. Specifically, the fixed point theorem [Kolmogorov and Fomin, 1999], suggests that for ε < 1/G, the converging Neumann series
represents the unique solution of (13b). Hereafter,
and E(ρ) is the complete elliptic integral of the second kind. As follows from (13d), in the case under consideration, 1/G ≈ 0.572, and (14a) converges for ε < 0.572.
 The integrals in equations (14b) and (14c) are evaluated numerically. Because the terms in converging series (14a) alternate their sign, the error of the truncated series does not exceed the first omitted term [e.g., Hardy, 2008]. Keeping the first term in (14a) defines the compressibility to the first order in ε
 The radial displacement in the half space is given by [Johnson, 1985]
where we assumed that the shear tractions on the half-space surface are absent due to the problem symmetry and that the slip at the interface between fracture and wall rock can be ignored. Then the volumetric strain inside the layer can be expressed as [e.g.,Timoshenko and Goodier, 1970]
where we used (14e) in (14f) and that . Because scales with pressure, p, the second term in (14f) is roughly 3 to 6 orders of magnitude smaller than the normal strain using ranges of parameters typical of problems in fractured rock. As a result, the last term in (14f) can be ignored, so εkk = v/do. Therefore, although radial strain can be important in thick aquifers [Burbey, 2001b], it appears to be negligible in soft, thin fractures or layers. Differentiating (12) with respect to time, we see, after some simplifications, that . This expression is not correct in general, and cannot be obtained directly from the poroelastic relation (4). It holds, however, for the fracture (layer) model considered in this work, and it follows, using εkk = v/do, that
It appears that the contribution of volumetric strain rate to storage change is proportional to the hydraulic compliance of the fracture, or thin layer.
 The analysis of equation (14d) shows that hydraulic compliance changes with both time and location (Figure 2). At early time, increases with radial distance by more than an order of magnitude. This effect diminishes with increasing time, although the general trend of with distance persists (Figure 2a). Hydraulic compliance also varies temporally, increasing by nearly an order of magnitude over 0 < τ < 0.1 for ρ = 0.1 (Figure 2b). The fracture is essentially becoming softer with time, and it is stiffer near the well than further away from it, according to these results.
 The extreme case of large times, τ → ∞ results in in (13c). Consequently, (14d) becomes
which represents the first-order solution inε for an infinite fracture with a uniformly pressurized circular area (p = τ). Hydraulic compliance still depends upon the distance from the borehole at late times.
 The findings from the analytical solution identify a fundamental behavior where hydraulic compliance increases with both distance and time during a well test (Figure 2). The example given in Figure 2 is for ε = 0.1, but the general behavior described in Figure 2 occurs for a range of ε < 0.572 in the obtained perturbation solution.
 Many field conditions, including those at the field site used for this research, are characterized by parameters where ε > 0.572. Moreover, the analytical solution was obtained by assuming that the fluid pressure is coupled to the deformation of the fracture walls while ignoring the reverse coupling (equation (13c)) where the deformation influences the fluid pressure. A numerical solution was used to evaluate cases where ε > 0.572 and to include the effects of full coupling between deformation and fluid flow.
3. Numerical Analyses
 Well tests in deformable fractures have been evaluated by coupling analyses of fluid flow through discrete fractures to simulations of elastic deformation of the enveloping host rock [Gale, 1975; Guglielmi et al., 2003; Cappa, 2005; Cappa et al., 2005, 2006; Murdoch and Germanovich, 2006; Svenson et al., 2007; Schweisinger et al., 2011]. This type of fully coupled analysis has been developed using several numerical techniques for a variety of scenarios, including geometrically realistic field conditions [Cappa et al., 2005, 2006]. Here we are interested in evaluating a fundamental storage process rather than results from a specific site, so we intentionally avoid geometrical complexities in favor of an analysis with axial symmetry. The model assumes the fracture is circular with radius, a, and that it is embedded in an infinite material; effects from mechanical boundaries, like the ground surface, are ignored. Slug or pumping tests are simulated by applying appropriate boundary conditions at r = rw. The model used for the analyses is described in detail by Murdoch and Germanovich .
 Both the analytical model outlined above and the numerical model assume a no-flow boundary occurs atr = a. The numerical model assumes displacement is zero at r = a, whereas the analytical model does not require zero displacement at r = a because it assumes the fracture face displaces as a half space. This difference is negligible when the area affected by pressure change is small compared to a.
3.1. Time Series at the Well
 A constant-rate pumping test followed by a recovery period was simulated using parameters inTable 1. The results show that the head h in the well drops by roughly 12 m and the aperture closes by v = −30 μm during pumping for 825 s (Figure 3a). Then the hydraulic head recovers to several centimeters and v = −5 μm at t = 2500 s. Displacement, v, is a hysteretic function of head at the wellbore, with closing displacements during pumping as much as 8 μm less than they are at a similar head during recovery (Figure 4). The fracture is assumed to be enveloped within impermeable rock, so the water removed from the fracture by pumping results in a permanent negative Δh and v when the system equilibrates after recovery. This is why Δh and v do not return to ambient conditions at the end of the test (Figures 3 and 4).
Table 1. Values of Parameters Used in Numerical Simulations of Pumping and Slug Tests
10−4 m3 s−1
−10−3 m3 s−1
Fracture radius a
Contact aperture at well δow
6.25 × 10−4 m
6.25 × 10−4 m
Initial aperture at well δw
2.5× 10−4 m
2.5 × 10−4 m
Plane strain modulus E′
Asperity compliance Cn
10−9 m Pa−1
10−9 m Pa−1
Casing radius rc
Well radius rw
 The results indicate Cf at the wellbore is 3 × 10−12 m Pa−1 at early time and it increases continuously during pumping. However, Cf decreases abruptly and becomes slightly negative when pumping stops at t = 825 s (Figure 3a). The portion of the curve in Figure 3 with negative slope is readily apparent. Hydraulic compliance, Cf, then increases throughout the recovery period and when the simulation is terminated, Cf = 5 × 10−10 m Pa−1, which is more than 2 orders of magnitude greater than at the start of the simulation.
 A slug test was simulated by injecting water into the well rapidly for 4 s to elevate the head in the wellbore, and then letting this perturbation recover. This is a reasonable approximation to a real slug-in test where the head is raised rapidly by applying air pressure or dropping a weight to displace water in a wellbore. The results from this analysis show the head increasing and the aperture opening during the first 4 s. The head falls duringt > 5 s, but aperture continues to open until it reaches a maximum at t = 45 s. Then the fracture closes. A small elevated head and positive aperture persist after several thousand seconds (Figure 3b).
 Hydraulic compliance, Cf, changes during a slug test in much the same way that it does during and following pumping: it increases initially, abruptly decreases and becomes negative when the slope of the wellbore head changes sign, and then it increases for the remainder of the test (Figure 3b). One difference is that Cf decreases much more during the slug test than it does during the pumping test: it drops to −1.3 × 10−9 m Pa−1 during the slug test, which is off the scale of Figure 3, whereas it drops to roughly −10−12 m Pa−1 during the transition from pumping to recovery.
 The appearance of negative compliance is initially counterintuitive; it means that the fracture is opening while the wellbore head is falling, or the other way around. Nevertheless, this behavior is evident by comparing h and v in Figure 5 during the slug test for (4 s < t < 45 s). The explanation for this behavior is that the displacement results from the total stress, or driving pressure, integrated over the fracture surface (e.g., equation (14)). The rapid pressure increase that occurs early in a slug test is confined to a relatively small radial distribution, so the displacement is relatively small. However, the pressure increase spreads with time and this allows the displacement to increase even though the pressure is decreasing at the well. This explanation suggests that the dynamics of pressure and displacements within the fracture itself are controlling what occurs at the wellbore.
3.2. Radial Profiles
 Profiles of hydraulic head, displacement of the fracture face and Cf were calculated as functions of time to evaluate how storage changes within the fracture. The results show that head and displacement decrease with r during both tests, so that h and v are negligible beyond a certain value of r (Figure 5). Even though profiles of h and v are generally similar, by no means are they proportional. At early times during both the slug and pumping tests, for example, v in μm is less than h in m at the wellbore, but h is greater than v several meters from the well (upper plots in Figure 5). With time, however, the relative magnitudes change so that v (in μm) > h (in m) for all values of r.
 Profiles show that Cf generally increases with r, but the pattern of change is complex near the leading edge of the zone affected by head change and displacement (Figure 5). In this region at early times, Cf increases and then decreases with increasing r. This indicates that near the leading edge of the pressurized region, the aperture is opening unusually rapidly in response to a unit increase in the local pressure. But at slightly greater distances, the aperture is closing in response to a local increase in pressure. This zone of sharply changing Cf moves away from the well with time (Figure 5).
 The abrupt changes in Cf are accompanied by anomalies in Δh and v, which occur in Figure 5, but that can be seen more readily by changing the scale (Figure 6). The example in Figure 6 shows that the head rises slightly over a band 15 m < r < 50 m, even though it drops in the vicinity of the well. Similarly, closing displacements characterize most of the fracture, but slight opening displacements occur over 24 m < r < 50 m. This effect occurs in all the panels in Figure 5, although the signs of the changes during the slug test are opposite those shown in Figure 6 for a pumping test.
 The pressure and displacement change sign at the leading edge of the pressure perturbation caused by a well test because they are influenced by the loading near the well. The fracture closes in response to the head drop induced by pumping, for example, but closing displacements occur beyond the limit of the region where head is directly affected by pumping. In the example shown in Figure 6, the head is affected by pumping to roughly r = 15 m, but the closing displacements caused by the change in head from pumping to extend beyond this point. Closing displacements compress the water in the fracture and increase its pressure, causing elevated head in the range r > 15 m. This elevated head causes slight opening displacements at r > 24 m. Thus, it is the coupling between displacement and fluid pressure that causes sharp changes in Cf that move progressively away from the well in Figure 5.
 Sharp changes in Cf with radial distance accompany the initiation of the well test, but they also occur when the pressure change at the wellbore reverses sign. This occurs in Figure 5a between panels t = 717 s and t = 840 s as the pump is turned off, and in Figure 5bbetween panels 4 s and 9 s as pressurization during slug-in is completed. As a result, it appears that radial profiles are characterized by two zones of sharply changingCf during the recovery period following pumping, and during most of a slug test.
3.3. Comparison to Analytical Model
 The numerical results are similar to equations (12) when the assumptions used to derive (12) are applicable. For example, the shape of the curve representing compressibility with time (Figure 2b) resembles the compliance as a function of time during pumping prior to 825 s in Figure 3a and generally during the brief period of pressure increase at the beginning of a slug test (Figure 3b). Likewise, the compressibility profile resembles the profile of compliance in the vicinity of the well in the numerical analyses (Figure 5). The compliance or compressibility increases and the slope diminishes with distance in both analyses.
 It seems reasonable to expect the numerical analyses to differ from the analytical results when the flow rate at the well changes because the analytical solution assumes constant flow rate. Reducing the flow rate is what causes the sharp drop in compliance at 825 s in the numerical results shown in Figure 5a, for example. The radial profiles of compliance from the numerical analysis also differ significantly from the analytical solution beyond a critical radius. This is because the numerical analysis includes full coupling between pressure and displacement, and this coupling causes the displacement to change sign beyond a critical radial distance. The compliance also changes sign, and this is the reason for the sharp drop in compliance in the numerical analyses. The analytical solution only includes one-way coupling, so it is unable to produce this effect without additional modifications.
4. Field Experiments
 Fracture compliance can be measured in the field by deploying a borehole extensometer during well testing [Gale, 1975; Martin et al., 1990; Hesler et al., 1990; Thompson and Kozak, 1991]. The extensometer used for this work consists of two anchors that attach to the borehole wall above and below a fracture. A reference rod extends from one anchor to the other, and a displacement sensor attached to the rod is used to measure the displacement of the anchors [Schweisinger and Murdoch, 2002; Schweisinger et al., 2007; Murdoch et al., 2008b]. In an ideal situation where a single, flat-lying fracture occurs in fresh crystalline rock between the anchors, the displacement measured by the transducer is assumed to be equal to the normal displacement of the fracture walls. This assumes displacement due to poroelastic effects in the enveloping rock matrix [Murdoch et al., 2009] can be neglected. Fracture compliance is determined as the derivative of displacement as a function of water pressure in the well.
 Field data show that Cf generally increases during slug and pumping tests; for example, Cf increases from 0.1 × 10−10 to 8 × 10−10 m Pa−1 during the pumping test shown in Figure 7a. Cf is calculated using finite differences applied to displacement data. However, when the pump is turned off at the start of the recovery period, Cf decreases and becomes negative. It then increases throughout the recovery period and eventually approaches a value similar to that at the end of pumping (Figure 7a).
 Fracture compliance also increases during the rapid pressure rise at the beginning of a slug test (the first 5 to 8 s in Figure 7b). Compliance decreases and becomes negative when the pressure begins to decrease. The period of negative Cf is brief, however, and after 20 s it changes sign once again and increases to a value of roughly 4 × 10−10 m Pa−1.
 The transient behavior of Cf during the slug test is qualitatively similar to that during the pumping test. Cf increases initially, but then it decreases and becomes negative when the trend of the wellbore head reverses. This is followed by values that increase and approach 4 × 10−10 to 7 × 10−10 m Pa−1 for the fracture zone at 22.3 m depth. Similar behavior occurred at the deeper fracture zones, although they appear to be slightly stiffer than the fracture zone at 22.3 m (Figure 7).
 Field data from pumping and slug tests in Figure 7 show a complex signal that is remarkably similar to the numerical results in Figure 3, and the initial stage of the pumping and slug tests resemble the analytical solution in Figure 2. Both slug and pumping tests involve a reversal in the sign of pressure change at the wellbore (the pressure increases and then decreases, or the other way around). This reversal decreases the apparent compliance and it may even cause the sign to become negative at the wellbore (Figure 7). This occurs because pressure distributed over a broad region of the fracture causes it to continue to open even when the pressure head at the wellbore is falling.
 Compliance varies spatially as well as temporally. In particular, Cf generally increases with radial distance, but there are one or two zones where it increases and decreases sharply. These zones of sharp change in Cfoccur because displacement is induced in a region beyond that affected by the head change at the well. This causes the pressure to drop at the leading edge of a pressurized zone. This effect is well known in the mechanics of hydraulic fracturing, where a low-pressure region at the crack tip causes fluid to lag behind the leading edge of a propagating pressurized fracture [Germanovich and Murdoch, 2010; Murdoch, 1993, and references therein]. Hart and Wang  show that this effect also occurs during pulse tests used to measure permeability of tight rocks, and Wang describes this as a case of the Mandel-Cryer effect. The pressure decrease at the leading edge of a slug-in test has been observed in the field [Murdoch et al., 2008a]. It appears, therefore, that the pressure reversal shown in Figure 5 has been recognized in a variety of scenarios in the lab and field, and this is further verification of spatial variability of Cf.
 Reversals in piezometric head have been described during pumping tests as Nordbergum or Rhade effects [Verruijt, 1969; Kim and Parizek, 1997, 2005; Hsieh, 1996]. These reverse responses in piezometric head occur in low-permeability confining units and are a result of the deformation of those units in response to pressure change in the aquifer. The effect described in this study occurs in the permeable layer undergoing pumping, so it differs from the Noordbergum or Rhade effects, even though all of these effects involve a pressure response that is opposite from the one expected.
 The models presented here assume an infinite medium and so they do not consider effects of the ground surface. In a practical sense, this means either a or are smaller than the depth. This constraint is satisfied if the well tests are sufficiently short (e.g., some slug tests). If the tests are longer, then the kernel in equation (13b) will differ from that given equation (11), but the solution of (13b) for hydraulic compliance will still be a function of time and space.
5.1. Specific Storage and Well Tests
 Specific storage is typically regarded as a material property that remains constant during a well test (e.g., (2)), and indeed, one of the common objectives of well tests is to estimate Ss. Specific storage can be used as a material property in poroelastic analyses [Green and Wang, 1990], but when this is done a term that couples the fluid flow analysis to a mechanical analysis of the strain rate is also used [Wang, 2000]. Equations (3), (4), and (7) can be combined to develop a term that enters the diffusion equation as Ss in (1)
 The last term in (17) accomplishes mechanical coupling to strain rate, so using (17) in (1) is equivalent to the diffusion equation from poroelasticity [e.g., Wang, 2000, equation (4.65); Detournay and Cheng, 1992, equation (82); Rutqvist and Stephansson, 2003, equation (54)]. In general, the first two terms in (17) are much smaller than the third, so the apparent hydraulic compliance is roughly proportional to an effective specific storage that lumps the mechanical coupling from poroelasticity into a single storage term, as in (1) with R = 0.
 The implication is that using Ss as a constant and uniform property in (1) overlooks the spatial and temporal changes of Cf outlined in the previous pages. Presumably values of Ss estimated during well tests by inverting equation (1) will be some average of a spatially and temporally variable quantity, although the details of the averaging process are currently unclear. In a uniform fracture of infinite extent, Cf approaches Cn at late time [Svenson et al. 2007], so there is some evidence that the hydraulic compliance is related to a material property in some cases.
 One implication is that the results shown here indicate that specific storage measured during a short-term well test should be stiffer than the response during prolonged pumping. This has implications in evaluating the long-term sustainability of aquifers, and assessing the risks associated with pressurizing saline aquifers by injecting fluids containing CO2 [Zhou et al., 2008; Zhou and Birkholzer, 2011].
 It appears that assuming Cf is constant in (17) will introduce errors when interpreting well tests using (1). The magnitude of these errors is unclear and they appear to depend on the duration and rate of pressure change.
Berg et al.  generated synthetic drawdown curves using a poroelastic analysis and then used parameter estimation techniques with conventional analyses based on equation (1) to analyze the results. Their findings show that errors in estimates of K and Sscan appear when deformation-induced effects are ignored, but these errors diminish as the duration of the test increases. This result is consistent with the findings of this study showing that effective storage changes markedly at the onset of pumping, but the rate of change diminishes with time. Even though the effective storage never becomes constant or uniform, the results ofBerg et al.  indicate that the rate of change may be small enough for conventional well test analyses to yield acceptable results at late times. The effects of poroelastic deformation on slug test interpretation have not been evaluated to our knowledge, but large changes in compliance during slug tests (Figures 5b and 7) may contribute to the uncertainty that slug tests have in estimating specific storage [Butler, 1997].
 Techniques for simultaneously interpreting pressure signals from multiple observation wells using hydraulic tomography are being developed to estimate permeability distributions in aquifers. Many applications [Bohling, 1993; Yeh and Liu, 2000; Bohling and Butler, 2010b; Cardiff et al., 2009] use steady state analyses, but some studies [Zhu and Yeh, 2005; Liu et al., 2007; Xiang et al., 2009; Illman et al., 2009; Berg and Illman, 2011; Brauchler et al., 2011] have started to employ transient forward analyses based on equations (1) and (2) where Ss is assumed constant in time. Inverting transient data sets requires forward models that represent subsurface processes using techniques that are computationally tractable. Equation (1) can be solved more rapidly than a fully coupled poroelastic analysis, and this computational advantage may currently justify the simplification resulting from assuming Cf is constant. However, using equations (1) and (2) may cause temporal changes in Cf to be misinterpreted as spatial distributions of permeability, further complicating a problem that already has difficulties with uniqueness [Bohling and Butler, 2010a]. The results shown here suggest that this problem will be most noticeable for short well tests because volumetric strain is changing rapidly during these tests. Including measurements of deformation from extensometers, tiltmeters, GPS sensors, InSAR or other methods [Burbey, 2008; Burbey et al., 2012] should help to further constrain the inversion process and may help to offset the additional computational burden needed to include deformation in the forward model.
5.2. Interpreting Extensometer Data
 Extensometers have long been used to measure deformation of sedimentary aquifers during pumping [Lofgren, 1961; Davis et al., 1968; Riley, 1969], and plotting displacement as a function of head at the wellbore commonly produces a hysteretic plot similar to Figure 4. The hysteresis in extensometer data is typically attributed to the time lag between the response of confining beds and aquifers cut by the well housing the extensometer [Riley, 1969]. The slope of the plot at early time is attributed to elastic storage, whereas the flattening of the slope (when plotted as in Figure 4) is attributed to the increasing importance of deformation of confining beds with time. Deformation of the confining beds is assumed to be inelastic when the effective stress exceeds a previous maximum, so the late time slope in the hysteresis loop is proportional to the “inelastic” or “nonrecoverable” storage.
 The loop-like hysteresis in extensometer records has been observed and analyzed at many locations, from Riley's early work in California to more recent applications in Colorado [Harmon, 2002]. Cleveland et al.  applied this approach to extensometer data from Houston, TX. They demonstrated that hysteresis occurs in these data, and they used the hysteresis loop to identify parameters associated with elastic and inelastic deformation. Burbey [2001a] evaluated Riley's approach with a detailed theoretical analysis, which shows that the shape of the hysteresis loop depends on properties and geometries of the aquifer and confining beds. This conclusion is similar to findings by Cappa et al. [2005, 2006], and Svenson et al.  showing the shape of the hysteresis loop during pulse or slug tests depends on aquifer properties and geometries. Cappa et al. extend this conclusion to 3-D.
 The analysis described in the preceding pages and shown in Figures 3–5 is for a single fracture or thin permeable layer, such as a thin confined aquifer. Our theoretical analysis ignores compaction of a confining unit, and the permeable layer is only allowed to deform vertically–no horizontal deformation occurs in the layer. Nevertheless, the displacement is a hysteretic function of head (Figure 4). As a result, it seems that the hysteretic behavior of displacement as a function of head can occur solely due to the changing pressure distribution within the aquifer. We agree that compaction of confining units and horizontal stresses in aquifers may also contribute to hysteresis, but they are not necessary to produce this effect.
5.3. Parameters Controlling Storage Change
 The analyses and field data outlined in the previous pages show that the apparent hydraulic compliance increases during pumping, so a fracture or layer becomes effectively softer with time. This occurs because of the change in fluid pressure distribution, which in turn shifts the parameters that control release from storage. An illustration of this effect is given by returning to the conceptual model used in the numerical model [Murdoch and Germanovich, 2006] where a fracture is represented as a circular disk of radius a in an infinite medium and loaded by an axially symmetric distribution of total stress, σ(r, t). The average aperture in the fracture as a result of global loading of stress on the fracture surface is
where the compliance of the wall rock
can be obtained from the solution in Sneddon . Substituting (8) for σw in (18), and then equating the average aperture from (18) with the local aperture defined by (10) gives the average aperture as a function of the fluid pressure at the well
Taking the derivative of this expression and assuming the aperture at the well is approximately equal to the average aperture gives the average hydraulic compliance as
and it follows that
 The distribution of total stress on the face of the fracture varies with time depending on the type of well test being conducted, heterogeneities within the fracture, leakage out of the fracture, and other factors. This function
illustrates a distribution of total stress shaped like a sharp spike confined to small values of r when τ′ is small, and it transitions to a gradually varying function of r when , and to a uniform distribution when τ′ = 1 (Figure 8). The various shapes of (23) created by changing τ′ resemble the general distribution of pressure with time during many well tests. To our knowledge, equation (23) is not the solution to the total stress on a fracture during any particular well test, but it has the characteristics of total stress from early to late times and so it will be used to represent a generic suite of stress distributions that could occur during well tests.
 The average aperture of a crack loaded by (23) is given by Tada et al.  and can be used to show that
 Specific storage depends on the distribution of pressure through τ′ in (26). Assuming τ′ can be used as a surrogate for time allows the parameters controlling (26) to be inferred (Table 2). This is consistent with three basic mechanisms controlling change in storage in a fracture: (1) fluid compression, (2) wall rock deformation, and (3) asperity deformation. The importance of these mechanisms changes as a consequence of the evolving pressure distribution. Wall rock and asperity deformation will be negligible at the outset of a well test when the pressure change is limited to close proximity to the well, and storage change will be controlled by the fluid or mineral grain compressibility (initial time in Table 2).
Table 2. Values of Ss at Different Times as Inferred From Different Values of τ′
Conditions on τ′
 After the initial influence of fluid compressibility is over, the Ssof a water-saturated fracture will be controlled by elastic deformation that alters the aperture of the fracture itself. To understand this process it is important to recognize that both the wall rock enveloping the fracture and the asperities on the fracture face must deform when the aperture changes. The wall rock can bulge in or out in response to a change in the total stress distributed over the fracture face, whereas the asperities can extend or compress in response to a change in the local effective stress. The mechanism that causes the most resistance to deformation will controlSs.
 At early times the fluid pressure anomaly caused by a well test is limited to a small radial distance from the well. The local pressure can be large enough to deform asperities, but the small extent of the pressure distribution limits the magnitude of deformation of the wall rock. The elastic deformation of the wall rock controls Ss at early time, and this is why Ss is proportional to 1/E′ at early time in Table 2.
 The pressure anomaly caused by a well test spreads radially with time, so the pressure distribution becomes progressively more uniform and this controls the late time behavior. This change in pressure distribution may occur with prolonged pumping, but it is particularly applicable as pressures return to ambient conditions after a slug test or during the recovery following pumping. The displacement caused by a unit pressure change when the fluid pressure anomaly is distributed uniformly is much greater than it is when the pressure anomaly is radially restricted. As a result, the compliance of the wall rock increases as the pressure anomaly spreads with time and eventually it can exceed the compliance of the asperities. The limiting case is for a fracture where the fluid pressure anomaly is uniform. In this scenario, Ss is proportional to Cn, a property of the fracture (Table 2), a result that is consistent with the findings of Svenson et al. , who used other methods.
 Similar effects are also recognized by Yin et al. [2007a, 2007b], who present poroelastic analyses showing that the elastic modulus and depth of a confining unit plays important roles in the pressure response to pumping. Increasing the modulus or the depth of the confining unit reduces the amount the confining unit deforms, which increases the drawdown [Yin et al., 2007a, their Figure 7]. This effect is detailed in an analysis by Rothenburg et al. , which shows that the deformation at a point in an aquifer depends on both the pressure at that point and the integral of the pressure response over the entire aquifer [Rothenburg et al., 1994, their equations (17) and (18)]. This is similar to the approach in equation (12). The overlying confining unit in the analyses by Yin et al. [2007b] is analogous to the deforming wall rock described in the analysis presented here.
 The apparent hydraulic compliance (dv/dp) of a deformable uniform fracture, or other thin, soft permeable layer, changes with space and time when a well intersecting the fracture is used for pumping or injection, according to results from a closed form analytical solution, a numerical analyses, and field data. Hydraulic compliance is relatively small at early times because the pressure disturbance is localized and deformation is resisted by the stiffness of the wall rock. Hydraulic compliance near the well increases with time, however, as the distribution of pressure on the fracture face changes, and it may approach the asperity compliance at late pumping times when the pressure distribution is relatively uniform. Turning off the pump causes an abrupt decrease in Cf, and it may even cause Cf to become negative at the well because the fracture continues to contract after the pressure starts to increase during recovery. Radial profiles also indicate significant variability, with Cf generally increasing with r until the leading edge of pressure change from the well test is approached and then Cf abruptly decreases and becomes negative (Figures 5 and 6).
 Conventional analyses of pressure transients in aquifers use the diffusion equation (1)and assume storage change occurs solely in proportion to the local change in pressure. This assumption implies that the hydraulic compliance of a uniform fracture is constant and uniform, a contrast to the results shown here. The consequences of making this simplifying assumption will depend on the application for the well. One example of a potential problem caused by this effect is when parameters derived from short-term well tests are used to predict long-term performance of wells. Including deformation measurements during well tests can help identify these effects.
radius of fracture.
compliance of wall rock, dv/dsw.
dimensionless hydraulic compliance.
apparent hydraulic compliance of a fracture, dv/dp.
normal compliance of fracture asperities, dv/dσe.
complete elliptic integral of the second kind.
E/(1 − υ2).
Bessel function of the first kind.
complete elliptic integral of the first kind.
stress on asperities.
flow rate from well per unit length.
radius of well.
effective specific storage.
displacement normal to fracture.
fracture aperture at zero effective stress.
fracture aperture at wellbore at zero effective stress.
change in fluid pressure.
ratio of open area to total area of fracture.
compressibility of aquifer.
compressibility of fracture.
compliance scaling parameter.
unit weight of fluid.
dimensionless radial distance.
total confining stress.
total stress on fracture face.
total driving stress.
 This research was supported by National Science Foundation grants EAR0609950 and 0944354.