6.2 Development of Equations Describing Aquifer Responses to Pumping
Equations developed to relate the hydrogeologic properties of earth materials in a groundwater system to the changes in hydraulic head when the aquifer is pumped, are founded on Darcy’s Law and the conservation of mass (water). These general governing equations are presented in the Groundwater Project book Hydrogeologic Properties of Earth Materials and Groundwater Flow↗ (Woessner & Poeter, 2020). The equations describe factors controlling the movement of groundwater in confined and unconfined water bearing units under steady‑state and transient conditions, and in isotropic, anisotropic, homogeneous, and heterogeneous settings.
The flow of groundwater to wells under a variety of conditions can be represented by these same governing equations (Woessner & Poeter, 2020). When conditions are defined by linear mathematical differential equations, and specific mathematical boundary and initial conditions are applied, calculus solution techniques result in algebraic equations referred to as analytical solutions or analytical models.
The governing equations for confined and unconfined aquifers form the basis for development of analytical solutions describing the flow of water to a pumping well. A wide variety of analytical solutions that predict the corresponding water level response to pumping in both steady‑state and transient conditions are documented in the literature. However, in almost all cases, basic formulations are limited to radial flow in an isotropic and homogeneous hydrogeologic unit that is considered unbounded (i.e., of infinite lateral extent). Other boundary conditions can be specified in some cases. These simplifying conditions reduce the mathematical challenges encountered when solving and formulating an analytical solution.
Confined Aquifers
The development of analytical solutions for pumping conditions starts with the governing equations that describe steady‑state and transient flow in two dimensions (i.e., in horizontal x‑ and y‑cartesian coordinates) in a homogeneous and isotropic confined groundwater system. Transmissivity, T, describes the ease with which water moves through an aquifer and is defined as the product of hydraulic conductivity and aquifer thickness. Given their two‑dimensional nature, they are applicable to essentially horizontal flow. Equation (1) describes steady conditions and Equation (2) describes transient conditions.
| [latex]0\ = \ \left( \frac{\partial^{2}h}{\partial x} + \frac{\partial^{2}h}{\partial y} \right)[/latex] | (1) | |
|---|---|---|
| [latex]S\frac{\partial h}{\partial t}\ = \ T\ \left( \frac{\partial^{2}h}{\partial x} + \frac{\partial^{2}h}{\partial y} \right)[/latex] | (2) |
where:
| S | = | storativity (dimensionless) |
|---|---|---|
| h | = | hydraulic head (L) |
| t | = | time (T) |
| T | = | transmissivity (L2T‑1) |
| x,y | = | coordinates (L) |
These two‑dimensional equations in cartesian coordinates (x,y) can be converted into polar coordinates by recognizing r = (x2 +y2)0.5 as shown in Equations (3) and (4).
| [latex]0\ = \ \left( \frac{\partial^{2}h}{\partial r^{2}} + \frac{1}{r}\frac{\partial h}{\partial r} \right)[/latex] | (3) | |
|---|---|---|
| [latex]S\frac{\partial h}{\partial t}\ = \ T\ \left( \frac{\partial^{2}h}{\partial r^{2}} + \frac{1}{r}\frac{\partial h}{\partial r} \right)[/latex] | (4) |
where:
| S | = | storativity (dimensionless) |
|---|---|---|
| h | = | hydraulic head (L) |
| t | = | time (T) |
| T | = | transmissivity (L2T‑1) |
| r | = | radial distance from the origin (the pumping well) (L) |
Additional documentation on the derivation of Equations (3) and (4) is provided in in the work of Lohman (1972, in the section presenting partial differential equations for radial flow).
Formulating the steady‑state and transient equations, Equations (5) and (6) respectively, in terms of drawdown (s) for any r value is presented in many documents (e.g., Loaiciga, 2009).
| [latex]0 = \frac{1}{r}\frac{\partial s}{\partial r} + \left( \frac{\partial^{2}s}{\partial r^{2}} \right)[/latex] | (5) | |
|---|---|---|
| [latex]\frac{S}{T}\frac{\partial s}{\partial t} = \frac{1}{r}\frac{\partial s}{\partial r} + \left( \frac{\partial^{2}s}{\partial r^{2}} \right)[/latex] | (6) |
where:
| s | = | drawdown = hcorrected-static‑ht (L) |
|---|---|---|
| hcorrected-static | = | initial water level prior to pumping corrected for background variations (L) |
| ht | = | water level at time t since the pumping started (L) |
| r | = | radial distance from the center of the pumped well to the observation well (L) |
| t | = | elapsed time since the start of pumping (T) |
| T | = | transmissivity (L2T‑1) |
| S | = | storativity (dimensionless) |
Equations (3), (4), (5), and (6) are referred to mathematically as linear differential equations and can be solved using calculus techniques. They are the basis for the development of an analytical equation describing the response of confined aquifers to pumping. Additional factors can be added to these equations to account for other groundwater sources and sinks such as leakage at a constant rate or the constant discharge of water from a pumping well (negative value of w) as in Equations (7) and (8).
| [latex]0\ = \ \left( \left( \ \frac{\partial^{2}h}{\partial r^{2}} + \frac{1}{r}\frac{\partial h}{\partial r} \right) + w \right)[/latex] | (7) | |
|---|---|---|
| [latex]S\frac{\partial h}{\partial t}\ = \ T\ \left( \left( \ \frac{\partial^{2}h}{\partial r^{2}} + \frac{1}{r}\frac{\partial h}{\partial r} \right) + \frac{w}{T} \right)[/latex] | (8) |
where:
| S | = | storativity (dimensionless) |
|---|---|---|
| h | = | hydraulic head (L) |
| t | = | time (T) |
| T | = | transmissivity (L2T‑1) |
| r | = | radial distance from origin (pumping well) (L) |
| w | = | source or sink term (LT‑1) |
Unconfined Aquifers
The two‑dimensional governing equation representing essentially horizontal steady‑state and transient flow in a homogeneous and isotropic unconfined groundwater system are shown in Equations (9) and (10). These equations are based on the assumptions that the water table is the top of the aquifer (i.e., drainage of pores in the capillary fringe is negligible) and elastic storage is negligible (i.e., water comes from drainage of pores below the water table and not from expansion of water or compression of the aquifer). Head values are measured from the horizontal base of the unconfined unit to the water table thus represent the saturated thickness as well as head.
| [latex]0\ = \frac{\partial^{2}h^{2}}{\partial x^{2}} + \frac{\partial^{2}h^{2}}{\partial y^{2}}[/latex] | (9) | |
|---|---|---|
| [latex]S_{y}\frac{\partial h}{\partial t}\ = \frac{K}{2}\left( \frac{\partial^{2}h^{2}}{\partial x^{2}} + \frac{\partial^{2}h^{2}}{\partial y^{2}} \right)[/latex] | (10) |
where:
| S y | = | specific yield (dimensionless) |
|---|---|---|
| h | = | hydraulic head measured from the base of the aquifer (L) |
| K | = | hydraulic conductivity (LT‑1) |
| t | = | time (T) |
| x,y | = | Cartesian coordinates (L) |
| [latex]0 = \left( \ \frac{\partial^{2}h^{2}}{\partial r^{2}} + \frac{1}{r}\frac{\partial h^{2}}{\partial r} \right)[/latex] | (11) | |
|---|---|---|
| [latex]S_{y}\frac{\partial h\ }{\partial t} = \frac{K}{2}\left( \ \frac{\partial^{2}h^{2}}{\partial r^{2}} + \frac{1}{r}\frac{\partial h^{2}}{\partial r} \right) = \frac{K}{r}\frac{\partial}{\partial r}\left( h\ r\ \frac{\partial h\ }{\partial r}\ \right)[/latex] | (12) |
where:
| Sy | = | specific yield (dimensionless) |
|---|---|---|
| h | = | hydraulic head measured from the base of the aquifer (L) |
| K | = | hydraulic conductivity (LT‑1) |
| t | = | time (T) |
| r | = | distance from origin (pumping well) (L) |
Additional factors can be added to these equations to account for other groundwater sources and sinks such as leakage at a constant rate into an unconfined aquifer under steady‑state or transient conditions as shown in Equations (13) and (14).
| [latex]0 = \left( \ \frac{\partial^{2}h^{2}}{\partial r^{2}} + \frac{1}{r}\frac{\partial h^{2}}{\partial r} \right) + \frac{2}{K}w[/latex] | (13) | |
|---|---|---|
| [latex]S_{y}\frac{\partial h\ }{\partial t} = \frac{K}{2}\left( \left( \ \frac{\partial^{2}h^{2}}{\partial r^{2}} + \frac{1}{r}\frac{\partial h^{2}}{\partial r} \right) + \frac{2w}{K} \right)[/latex] | (14) |
where:
| S y | = | specific yield (dimensionless) |
|---|---|---|
| h | = | hydraulic head measured from the base of the aquifer (L) |
| K | = | hydraulic conductivity (LT‑1) |
| t | = | time (T) |
| r | = | distance from origin (pumping well) (L) |
| w | = | source or sink term (LT‑1) |
Though these differential equations describing radial flow in an unconfined system are properly stated, they are also mathematically nonlinear. These governing equations cannot be easily converted to analytical solutions using standard calculus methods (Fetter, 2001). As a result, numerical methods are commonly used to simulate groundwater flow to wells in unconfined aquifers (e.g., Anderson et al., 2015). Transient analytical equations associated with basic methods to graphically solve unconfined hydraulic testing results are formulated using an approximation of Equation (12) that is linearized with the confined formulation, and a condition that the drawdown in the unconfined system is small at observations points such that hydraulic properties are constant. This is accomplished by taking h outside of the derivative of the right‑hand side of Equation (12) and substituting b for h to derive Equation (15).
| [latex]S_{y}\frac{\partial h}{\partial t}\ \cong \frac{Kb}{r}\frac{\partial}{\partial r}\left( r\ \frac{\partial h\ }{\partial r}\ \right) = \ T\ \left( \frac{\partial^{2}h}{\partial r^{2}} + \frac{1}{r}\frac{\partial h}{\partial r} \right)[/latex] | (15) |
|---|
where:
| Sy | = | unconfined storativity (specific yield) (dimensionless) |
|---|---|---|
| h | = | head measured as the difference between the water table elevation and the elevation of the horizontal base of the unconfined aquifer (L) |
| t | = | elapsed time since the start of pumping (T) |
| K | = | hydraulic conductivity (LT‑1) |
| b | = | saturated thickness measured from the base of the aquifer to the water table at the pumping well (L) |
| r | = | radial distance from the center of the pumped well to the observation well (L) |
| T | = | Kb, unconfined aquifer transmissivity (L‑2T‑1) |
Section 10 discusses this approach where specific analytical models describing pumping unconfined systems are presented.