Measurement of Streaming Potentials Generated During Laboratory Simulations of Unconfined Aquifer Pumping Tests

Abstract

The streaming potential method has emerged as a promising hydrogeophysical technique for indirect acquisition of spatially dense measurements of the hydraulic response of aquifers to pumping or other system forcings. The method relies on measurements of electric potentials generated by groundwater flow. They arise due to the existence of the electric double layer at the rock–water interface. Mathematical solutions describing the transient electric potentials associated with pumping tests conducted in confined and unconfined aquifers have been recently developed and demonstrated to yield reasonable estimates of hydraulic parameters when applied to tests conducted at the field-scale. We present results of laboratory experiments conducted to investigate the applicability of the unconfined aquifer model under controlled conditions in a sand tank instrumented with pressure transducers for direct measurement of the hydraulic system state, and nonpolarizable electrodes for measurement of the associated electric field. Measurements show unambiguous transient streaming potential responses to groundwater flow in a bounded cylindrical system. Parameters estimated from streaming potential data are compared to those from drawdown data.

7.1 Appendix A: Governing Equations for Coupled Flow and SP

In cylindrical coordinates with no tangential flow, Eq. (7.4) can be written, for flow in a radially infinite aquifer and in nondimensional, form as (Malama et al. 2009a,b)

$$\frac{\partial s_{D}} {\partial t_{D}} = \frac{1} {r_{D}} \frac{\partial } {\partial r_{D}}\left (r_{D}\frac{\partial s_{D}} {\partial r_{D}}\right ) +\delta \kappa \frac{{\partial }^{2}s_{D}} {\partial z_{D}^{2}},$$

(A.1)

where $s_{D} = s/H_{c}$ is dimensionless aquifer drawdown; $s = h(r,z,0) - h(r,z,t)$ is drawdown (m), $r_{D} = r/b_{2}$, $z_{D} = z/b_{2}$, $t_{D} =\alpha _{r}t/b_{2}^{2}$; r is radial distance from the pumping well (m); z is vertical position from the base of the aquifer (m); t is elapsed time since onset of pumping (s); $\kappa = K_{z}/K_{r}$ is the anisotropy ratio; $\alpha _{r} = K_{r}/S_{s}$ is radial hydraulic diffusivity (m²∕s); K _r and K _z are radial and vertical hydraulic conductivities (m∕s), respectively; S _s is specific storage (m^− 1); b ₂ is aquifer or initial saturated thickness (m); $H_{c} = Q/(4\pi b_{2}K_{r})$ is system characteristic head (m); and Q is the volumetric pumping rate (m³∕s). The dimensionless parameter δ is zero for confined aquifers, where flow is entirely radial, and unity for unconfined aquifers, where vertical flow is significant.

For confined aquifer flow, Eq. (999999.1) is solved subject to the initial condition

$$s_{D}(r_{D},t_{D} = 0) = 0,$$

(A.2)

a Dirichlet far-field boundary condition

$$\displaystyle\lim _{r_{D}\rightarrow \infty }s_{D}(r_{D},t_{D}) = 0,$$

(A.3)

and a Neumann boundary line-sink condition at the well

$$\displaystyle\lim _{r_{D}\rightarrow 0}r_{D}\frac{\partial s_{D}} {\partial r_{D}} = -2.$$

(A.4)

For unconfined aquifer flow, two additional boundary conditions are needed because of vertical flow. At the base of the aquifer, a homogeneous (zero-flux) Neumann boundary condition is imposed, namely

$$\left.\frac{\partial s_{D}} {\partial z_{D}}\right \vert _{z_{D}=0} = 0.$$

(A.5)

The linearized kinematic condition of Neuman [1972] is imposed at the water-table, namely,

$$-\left.\frac{\partial s_{D}} {\partial z_{D}}\right \vert _{z_{D}=1} = \left. \frac{1} {\alpha _{D}} \frac{\partial s_{D}} {\partial t_{D}} \right \vert _{z_{D}=1},$$

(A.6)

where $\alpha _{D} =\kappa /\vartheta$, $\vartheta = S_{y}/(b_{2}S_{s})$, and S _y is specific yield or drainable porosity, which is a measure of the fraction of the bulk volume a saturated porous medium would yield when the water is allowed to drain out under the action of gravity.

Solving the flow problem yields s _D, c  = E ₁(x), which is the solution of Theis [1935] for confined aquifer flow. Solving the unconfined flow problem yields the solution of Neuman [1972] given in Eq. (7.12). It should be noted that the flow problem is solved without consideration of the SP problem. This is due to a weaker dependence of fluid flow on electric potential differentials; i.e., the Darcy (pressure gradient) flux term is much greater than the flux term due to the electric field (Ishido and Mizutani 1981, Sill 1983). For the SP problem, however, the current density due pressure differentials (Darcian component) is of comparable magnitude to that due to electric potential differentials (the Ohmic component).

Hence, Eq. (7.1) can be written in nondimensional cylindrical form as (Malama et al. 2009a,b)

$$\frac{1} {r_{D}} \frac{\partial } {\partial r_{D}}\left (r_{D}\frac{\partial \phi _{D,2}} {\partial r_{D}} \right ) + \frac{{\partial }^{2}\phi _{D,2}} {\partial z_{D}^{2}} -\left [ \frac{1} {r_{D}} \frac{\partial } {\partial r_{D}}\left (r_{D}\frac{\partial s_{D}} {\partial r_{D}}\right ) +\delta \frac{{\partial }^{2}s_{D}} {\partial z_{D}^{2}}\right ] = 0,$$

(A.7)

for the aquifer, with δ = 0 for confined and δ = 1 for unconfined flow. For the upper unit, which is confining in the confined aquifer case, but corresponds to the vadose (unsaturated) zone for the unconfined case, the SP problem is given by

$$\frac{1} {r_{D}} \frac{\partial } {\partial r_{D}}\left (r_{D}\frac{\partial \phi _{D,i}} {\partial r_{D}}\right ) + \frac{{\partial }^{2}\phi _{D,i}} {\partial z_{D}^{2}} = 0,$$

(A.8)

where i = 1 for the upper unit and i = 3 for the lower unit, $\phi _{D,i} =\phi _{i}/\Phi _{c}$ for i = 1, 2, 3, and $\Phi _{c} = (\gamma \ell_{2}/\sigma _{2})H_{c}$. Note that the Darcian component of the current density (term is square brackets in Eq. (999999.7)) does not appear in Eq. (999999.8) because it is assumed (Malama et al. 2009a,b) that there is no fluid flow in layers 1 and 3. Additionally, it should be noted that only vertical fluid flow is neglected in the confined case; the electric field, on the other hand, has vertical components, which make it possible for the SP signal to be measured at land surface.

The SP problem is solved subject to the initial condition

$$\left.\phi _{D,i}\right \vert _{t_{D}=0} = 0,$$

(A.9)

the homogeneous Dirichlet boundary condition at the far-field in all layers,

$$\displaystyle\lim _{r_{D}\rightarrow \infty }\phi _{D,i} = 0,$$

(A.10)

the homogeneous Neumann (insulating condition) boundary conditions at ground surface

$$\left.\frac{\partial \phi _{D,1}} {\partial z_{D}} \right \vert _{z_{D}=1+b_{D,1}} = 0$$

(A.11)

and at the base of layer 3

$$\left.\frac{\partial \phi _{D,3}} {\partial z_{D}} \right \vert _{z_{D}=-b_{D,3}} = 0,$$

(A.12)

where $b_{D,1} = b_{1}/b_{2}$ and $b_{D,3} = b_{3}/b_{2}$ are the normalized thicknesses of layers 1 and 3. Note that the insulating condition in Eq. (999999.12) is applied at the contact between layer 3 and an underlying highly resistive layer ($z_{D} = 1 + b_{D,3}$). It has been shown in Malama et al. [2009a] that the line-sink condition at the pumping well can be written as

$$\displaystyle\lim _{r_{D}\rightarrow 0}r_{D}\frac{\partial \phi _{D,i}} {\partial r_{D}} = \left \{\begin{array}{ll} 0 &\quad i = 1,3\\ - 2 &\quad i = 2 \end{array} \right..$$

(A.13)

Additionally, continuity conditions are imposed at the upper and lower boundaries of the aquifer, namely,

$$\left.\phi _{D,1}\right \vert _{z_{D}=1} = \left.\phi _{D,2}\right \vert _{z_{D}=1},$$

(A.14)

$$\left.\phi _{D,3}\right \vert _{z_{D}=0} = \left.\phi _{D,2}\right \vert _{z_{D}=0},$$

(A.15)

$$\sigma _{D,1}\left.\frac{\partial \phi _{D,1}} {\partial z_{D}} \right \vert _{z_{D}=1} = \left.\frac{\partial \phi _{D,2}} {\partial z_{D}} \right \vert _{z_{D}=1},$$

(A.16)

$$\sigma _{D,3}\left.\frac{\partial \phi _{D,3}} {\partial z_{D}} \right \vert _{z_{D}=0} = \left.\frac{\partial \phi _{D,2}} {\partial z_{D}} \right \vert _{z_{D}=0}$$

(A.17)

where $\sigma _{D,i} =\sigma _{i}/\sigma _{2}$, the electrical conductivity of the ith layer normalized by that of layer i = 2. Malama et al. [2009a,b] developed the analytical solutions to the confined and unconfined SP problems. These are repeated here for completeness (and in a notation that is consistent for the confined and unconfined aquifer problems) in Appendix B.

7.2 A.5 Appendix B: Analytical SP Solutions

For a confined aquifer, Malama et al. [2009a] derived the exact solution for SP response in the aquifer and the confining units, in double Hankel–Laplace transform space. The details of the derivation, including the appropriate governing equations and initial and boundary conditions, can be found in that work. When derived in the same coordinate system as the unconfined SP solution, the confined SP solution becomes

$$\overline{\phi }_{D,i}^{{\ast}}(p,a,z_{ D}) = \overline{s}_{D,\mathrm{c}}^{{\ast}}(p,a)v_{ D,i}^{{\ast}}(a,z_{ D})$$

(B.1)

where i = 1 for the upper confining unit, i = 2 for the aquifer, i = 3 for the lower confining unit, a and p are the Hankel and Laplace transform parameters, respectively and $\overline{s}_{D,c}^{{\ast}}(p,a)$ is the Hankel–Laplace transform of the Theis solution

$$\displaystyle\begin{array}{rcl} v_{D,i}^{{\ast}}(p,a,z_{ D}) = \left \{\begin{array}{@{}l@{\quad }l@{}} \frac{1} {\xi _{1,1}} \cosh [a(b_{D,1} + z_{D}^{\prime})]v_{D,2}^{{\ast}}\vert _{z_{D}=1}\quad &i = 1, \\ 1 + w_{D}^{{\ast}}(a,z_{D}) \quad &i = 2, \\ \frac{1} {\xi _{3,1}} \cosh [a(b_{D,3} + z_{D})]v_{D,2}^{{\ast}}\vert _{z_{D}=0}\quad &i = 3,\\ \quad \end{array} \right.& &\end{array}$$

(B.2)

$$\displaystyle\begin{array}{rcl} w_{D}^{{\ast}}(a,z_{ D})& = \frac{1} {\Delta _{2}} \left [\xi _{3,2}g_{1}(z_{D}) +\xi _{1,2}g_{3}(z_{D})\right ]&\end{array}$$

(B.3)

where $z_{D}^{\prime} = 1 - z_{D}$,

$$\displaystyle\begin{array}{rcl} \xi _{1,1}& =\cosh (ab_{D,1})&\end{array}$$

(B.4)

$$\displaystyle\begin{array}{rcl} \xi _{3,1}& =\cosh (ab_{D,3})&\end{array}$$

(B.5)

$$\displaystyle\begin{array}{rcl} \xi _{1,2}& =\sigma _{D,1}\sinh (ab_{D,1})&\end{array}$$

(B.6)

$$\displaystyle\begin{array}{rcl} \xi _{3,2}& =\sigma _{D,3}\sinh (ab_{D,3})&\end{array}$$

(B.7)

$$\displaystyle\begin{array}{rcl} \Delta _{2}& = \vert A_{1}\sinh (a) + A_{2}\cosh (a)\vert &\end{array}$$

(B.8)

$$\displaystyle\begin{array}{rcl} A_{1}& =\xi _{1,2}\xi _{3,2} +\xi _{1,1}\xi _{3,1}&\end{array}$$

(B.9)

$$\displaystyle\begin{array}{rcl} A_{2}& =\xi _{1,2}\xi _{3,1} +\xi _{1,1}\xi _{3,2}&\end{array}$$

(B.10)

$$\displaystyle\begin{array}{rcl} g_{1}& =\xi _{1,1}\cosh (az_{D}^{\prime}) +\xi _{1,2}\sinh (az_{D}^{\prime})&\end{array}$$

(B.11)

$$\displaystyle\begin{array}{rcl} g_{3}& =\xi _{3,1}\cosh (az_{D}) +\xi _{3,2}\sinh (az_{D}).&\end{array}$$

(B.12)

The unconfined aquifer solution obtained by Malama et al. [2009b] is given by

$$\displaystyle\begin{array}{rcl} \overline{\phi }_{D,i}^{{\ast}}(p,a,z_{ D}) = \left \{\begin{array}{@{}l@{\quad }l@{}} \frac{1} {\xi _{1,1}} \cosh [a(b_{D,1} + z_{D}^{\prime})]\overline{\phi }_{D,2}^{{\ast}}\vert _{z_{D}=1}\quad &i = 1 \\ \overline{s}_{D,\mathrm{u}}^{{\ast}} + \overline{w}_{D,\mathrm{u}}^{{\ast}}(p,a,z_{D}) \quad &i = 2 \\ \frac{1} {\xi _{3,1}} \cosh [a(b_{D,3} + z_{D})]\overline{\phi }_{D,2}^{{\ast}}\vert _{z_{D}=0}\quad &i = 3\\ \quad \end{array} \right.& &\end{array}$$

(B.13)

$$\displaystyle\begin{array}{rcl} \overline{w}_{D,\mathrm{u}}^{{\ast}}(p,a,z_{ D})& = \frac{1} {\Delta _{2}} \left [\left (\xi _{1,2} -\frac{p\theta } {a\kappa }\xi _{1,1}\right )g_{3}(z_{D})\overline{s}_{D,\mathrm{u}}^{{\ast}}\vert _{ z_{D}=1} +\xi _{3,2}g_{1}(z_{D})\overline{s}_{D,\mathrm{u}}^{{\ast}}\vert _{ z_{D}=0})\right ]&\end{array}$$

(B.14)

where the various functions and parameters are the same as those given above and $\overline{s}_{D,\mathrm{u}}^{{\ast}}$ is the Laplace–Hankel transform of unconfined aquifer drawdown. The inverse transforms were obtained numerically.