Uniform steady free-surface flow in heterogeneous porous formations

Abstract

The effect of spatial variability of the hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for a sloped mean uniform flow in a randomly heterogeneous porous medium. The model also describes the interface between two fluids of differing densities, e.g. freshwater/saltwater and water/oil with the denser fluid at rest. We develop analytic solutions for the variance and integral scale of free-surface fluctuations and of specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of the head and the specific discharge beneath the free surface. Statistical moments are derived using a first-order approximation and then compared with their counterpart in an unbounded medium. The effect of anisotropy and angle of mean uniform flow on the statistical moments is analysed. The solutions can be used for solving more complex flows, slowly varying in the mean.

The effect of random recharge on uniform steady free-surface flow in heterogeneous porous formations

“…This was done by Amir and Dagan (2002) in the deterministic no-recharge case (c = 0 and C /R ¼ 0), and by Amir (2003) in the case of constant deterministic non-zero recharge (C /R ¼ 0) for a log conductivity with Gaussian covariance structure. The process yields that the Fourier transforms of C /R , C /Y , and C / are given by, …”

“…Tartakovsky and Winter (2001) present a more thorough investigation of free-surface flow in a randomly heterogeneous porous medium. Recently, Amir and Dagan (2002) have studied the problem of three-dimensional gravity driven flow in a semi-infinite randomly heterogeneous phreatic aquifer without recharge, and Amir (2003) has expanded this work to the case of constant deterministic recharge.…”

Uniform steady free surface flow with recharge in heterogeneous porous formations

“…For such applications it is interesting to observe the behavior for the two extreme cases, (1) a = 0, vertical flow beneath the free surface and (2) R n = 0, uniform free surface flow at an angle a. The later case was thoroughly examined by Amir and Dagan [2002]. For the first case, the mixed boundary condition on the free surface, (24), degenerates to a condition of normal/ vertical flux equal to ÀRY 0 e ÀhY i .…”

“…Ignoring the effect of gravity on flow in the saturated zone, Tartakovsky and Winter use a first-order asymptotic expansion to obtain governing equations for the covariance of the free surface position, which they then solve analytically in a one-dimensional case. Most recently, Amir and Dagan [2002] have studied the problem of threedimensional gravity driven flow in a semi-infinite randomly heterogeneous aquifer bounded above by a free surface. They derived analytic and semianalytic solutions for moments of free surface position, head and specific discharge but fail to account for naturally occurring recharge.…”

Modeling Groundwater Flow under Transient Nonlinear Free Surface

“…Common approaches to solving this problem include trial-anderror procedures to approximate the location of the free surface in combination with numerical solutions of the differential equations ͓for example, Neuman and Witherspoon ͑1971͒; Pinder and Gray ͑1977͒; Huyakorn and Pinder ͑1983͒; Liggett and Liu ͑1983͒; Bear and Verruijt ͑1987͒; Gjerde and Tyvand ͑1992͔͒ and linearization of the free-surface boundary condition and the analytical solution of the differential equation ͑Polubarinova- Kochina 1962;Van der Giesen et al 1994͒. Two-dimensional problems can sometimes be solved with the use of the inverse velocity hodograph ͑Harr 1962; Polubarinoba-Kochina 1962; Aravin and Numerov 1965͒; the application of stochastic methods in combination with numerical methods ͑Serrano and Unny 1986; Fenton and Griffith 1996; Dagan and Zeitoun 1998;Tartakovsky 1999;Amir and Dagan 2002;Tartakovsky et al 2002͒; or the adoption of the Dupuit assumptions of essentially horizontal flow that eliminate the vertical coordinate and the free-surface boundary condition. In the latter case, the resulting domain equation, the Boussinesq flow equation, is itself a nonlinear equation that is subsequently linearized or discretized to facilitate its solution ͑Muskat 1937; Charnyi 1951;Dagan 1966;Kirkham 1966;Knight 1981͒. In another approach, an analytical solution of the linear groundwater flow equation subject to an oscillating forcing function is obtained when aquifer compressibility is not neglected ͑Vandenberg 1980͒.…”