Stochastic Quantification of Spatial Variability of Flow Fields in Heterogeneous, Non-uniform, Confined Aquifers

Abstract

Stochastic quantification of flow field variability in complex geologic formations under uncertainty is expected to provide valuable information for rational management of regional groundwater resources and analysis of solute transport processes for stochastic environmental risk assessment. Studies of fluid flow behavior in confined aquifers of variable thickness presented in the literature assume that the thickness of the aquifer varies linearly or nonlinearly. However, natural variations, such as the thickness of the aquifer caused by complex natural events, cannot be accurately predicted. Therefore, quantifying the variability of the flow field in heterogeneous, non-uniform, confined aquifers may be done from a stochastic perspective. In this study, the spatial variations in hydraulic conductivity are considered as a stationary random process, while the spatial variations in aquifer thickness are treated as a nonstationary random process with homogeneous (stationary) increments. General expressions for the spatial covariance functions and the evolutionary power spectra of the depth-averaged hydraulic head and integrated specific discharge in the direction of x₁ are derived using the Fourier–Stieltjes spectral representation approach and representation theorem. Closed-form solutions for the evolutionary power spectra of depth-averaged hydraulic head and integrated specific discharge are used to analyze the effect of variation in the thickness of the confined aquifer on the variability of depth-averaged head and integrated discharge. An application of the theory developed here to the case of random aquifer thickness fields exhibiting a power-law semivariogram is given. The results of this study improve the understanding and quantification of flow field variability in natural confined aquifers.

Appendix: Derivation of Eq. (8)

To solve Eq. (7), the random variables in the stochastic differential equation are represented by Fourier–Stieltjes integrals (e.g., Lumley and Panofsky 1964; Priestley 1965; Yaglom 1987; Christakos 1992). From the analysis, a functional relationship is obtained between the variations in depth-averaged head, hydraulic conductivity, and aquifer thickness. Using this spectral relationship, it is then possible to determine the statistics of the integrated flow properties based on the known statistical properties of the random fields of hydraulic conductivity and aquifer thickness.

A second-order random field such as the random field f admits the Fourier–Stieltjes representation as (Lumley and Panofsky 1964)

$$f(x_{1} ,x_{2} ) = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {\exp [i(\mathop R\nolimits_{1} x_{1} + \mathop R\nolimits_{2} x_{2} } } )]\mathop {dZ}\nolimits_{f} (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ),$$

(A1)

where R₁ and R₂ are the components of the wave number vector R (= (R₁, R₂)) and Z_f is a complex-valued distribution with uncorrelated increments on wave number space. According to Yaglom (1987) and Christakos (1992), any nonstationary differentiable process with stationary increments can be represented in the form

$$\beta (x_{1} ,x_{2} ) = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {\frac{{\exp [i(\mathop R\nolimits_{1} x_{1} + \mathop R\nolimits_{2} x_{2} )] - 1}}{iR}} } \mathop {dZ}\nolimits_{Y} (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ),$$

(A2)

where $R = \sqrt {\mathop R\nolimits_{1}^{2} + \mathop R\nolimits_{2}^{2} }$ is the magnitude of the wave number vector and Z_Y denotes a stationary random distribution function with uncorrelated increments, which can be identified by the structure function of the random field β. Following the Priestley (1965) theorem of spectral representation, the nonstationary process h′ can be represented in terms of the following Fourier–Stieltjes integral:

$$h^{\prime}(x_{1} ,x_{2} ) = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {\Lambda (x_{1} ,x_{2} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} )} } \mathop {dZ}\nolimits_{f\beta } (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ),$$

(A3)

where Λ(−) is an oscillatory function in the sense defined by Priestley (1965) and Z_fβ is a complex-valued random function with orthogonal increments.

From the substitution of Eqs. (A1)–(A3) into Eq. (7), it follows that

$$\begin{aligned} & \left[ {\frac{{\mathop \partial \nolimits^{{2}} }}{{{\partial x}_{1}^{2} }}\Lambda (x_{1} ,x_{2} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) + \frac{{\mathop \partial \nolimits^{{2}} }}{{\mathop {\partial x}\nolimits_{2}^{2} }}\Lambda (x_{1} ,x_{2} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} )} \right]\mathop {dZ}\nolimits_{f\beta } (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) \\ & \quad = iJ\mathop R\nolimits_{1} {\text{exp[}}i(\mathop R\nolimits_{1} x_{1} + \mathop R\nolimits_{2} x_{2} ){]}\left[ {\mathop {dZ}\nolimits_{f} (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) - i\frac{2}{{\sqrt {\mathop R\nolimits_{1}^{2} + \mathop R\nolimits_{2}^{2} } }}\mathop {dZ}\nolimits_{Y} (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} )} \right]. \\ \end{aligned}$$

(A4)

Equation (A4) is known as the two-dimensional Poisson equation, which can be solved under the given boundary conditions. Consider a rectangular region with given values for the dead-averaged head at the locations x₁ = 0 and x₁ = L₁ (i.e., the variations are zero at the boundaries) and no flow at the locations x₂ = 0 and x₂ = L₂. As such,

$$h^{\prime}(0,x_{2} ) = 0,$$

(A5a)

$$h^{\prime}(\mathop L\nolimits_{{1}} ,x_{2} ) = 0,$$

(A5b)

$$\frac{\partial }{{\mathop {\partial x}\nolimits_{2} }}h^{\prime}(x_{1} ,0) = 0,$$

(A5c)

$$\frac{\partial }{{\mathop {\partial x}\nolimits_{2} }}h^{\prime}(x_{1} ,\mathop L\nolimits_{{2}} ) = 0,$$

(A5d)

with the corresponding boundary conditions in the wave number space as

$$\Lambda (0,x_{2} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) = 0,$$

(A6a)

$$\Lambda (\mathop L\nolimits_{{1}} ,x_{2} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) = 0,$$

(A6b)

$$\frac{\partial }{{\mathop {\partial x}\nolimits_{2} }}\Lambda (x_{1} ,0;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) = 0,$$

(A6c)

$$\frac{\partial }{{\mathop {\partial x}\nolimits_{2} }}\Lambda (x_{1} ,\mathop L\nolimits_{{2}} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) = 0.$$

(A6d)

The solution of Eqs. (A4) and (A6a)–(A6d) for the oscillatory function Λ is as follows:

$$\Lambda (x_{1} ,x_{2} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} )\mathop {dZ}\nolimits_{f\beta } (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) = - J\mathop \Lambda \nolimits_{h} (x_{1} ,x_{2} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} )\left[ {\mathop {dZ}\nolimits_{f} (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) - i\frac{2}{R}\mathop {dZ}\nolimits_{Y} (\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} )} \right],$$

(A7a)

where

$$\begin{aligned} \mathop \Lambda \nolimits_{h} (x_{1} ,x_{2} ;\mathop R\nolimits_{1} ,\mathop R\nolimits_{2} ) & = 2\mathop L\nolimits_{1} \frac{{\mathop \mu \nolimits_{1} }}{{\mathop \mu \nolimits_{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{(\mathop \phi \nolimits_{2} - 1)[\mathop \phi \nolimits_{1} \cos (n\pi ) - 1]}}{{n\pi (\mathop \mu \nolimits_{1}^{2} - \mathop n\nolimits^{2} \mathop \pi \nolimits^{2} )}}} \sin (n\pi \mathop \xi \nolimits_{1} )] \\ & \quad + \frac{4}{{\mathop L\nolimits_{1} }}\mathop \mu \nolimits_{1} \mathop \mu \nolimits_{2} \sum\limits_{n = 1}^{\infty } {\sum\limits_{m = 1}^{\infty } {\frac{n}{\pi }\frac{1}{{\frac{{\mathop n\nolimits^{2} }}{{\mathop L\nolimits_{1}^{2} }}{ + }\frac{{\mathop m\nolimits^{2} }}{{\mathop L\nolimits_{2}^{2} }}}}\frac{{[\mathop \phi \nolimits_{1} \cos (n\pi ) - 1][\mathop \phi \nolimits_{2} \cos (m\pi ) - 1]}}{{(\mathop \mu \nolimits_{1}^{2} - \mathop n\nolimits^{2} \mathop \pi \nolimits^{2} )(\mathop \mu \nolimits_{2}^{2} - \mathop m\nolimits^{2} \mathop \pi \nolimits^{2} )}}} } \sin (n\pi \mathop \xi \nolimits_{1} )\cos (m\pi \mathop \xi \nolimits_{2} ), \\ \end{aligned}$$

(A7b)

where μ₁ = R₁L₁, μ₂ = R₂L₂, ϕ₁ = exp(iμ₁), ϕ₂ = exp(iμ₂), ξ₁ = x₁/L₁, and ξ₂ = x₂/L₂.

Combining Eqs. (A7a), (A7b) with Eq. (A3), the perturbation of nonstationary depth-averaged head fields takes the form of Eq. (8).