## 1. Introduction

[2] 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* [2006]). This process is typically characterized by the specific storage, *S _{s}*, 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, *S*_{σ}, 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].

[3] 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 .

[4] 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)

where *α _{b}*is the Biot-Willis coefficient,

*ε*is the volumetric strain, and

*σ*is the mean stress. It appears that in the theory of poroelasticity [

*Biot*, 1941, 1955;

*Van der Kamp and Gale*, 1983;

*Detournay and Cheng*, 1993;

*Wang*, 2000],

*S*in (1) represents the storage in conditions of uniaxial strain and constant normal stress in the strain direction, and

_{s}*S*is proportional to the diffusion coefficient in the diffusion equation for the fluid content [

_{s}*Green and Wang*, 1990;

*Detournay and Cheng*, 1993].

[5] 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.* [2011] 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* [1969] 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.* [2011]. 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.

[6] 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.

[7] 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/d**σ*_{e}, where *σ*_{e} 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 *δ _{o}* 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.