Uniform steady free surface flow with recharge in heterogeneous porous formations

Abstract

[1] Free surface flow in a randomly heterogeneous medium with recharge is studied in a stochastic setting. A three-dimensional gravitational flow model with sloped mean uniform flow is used to describe flow in a phreatic aquifer experiencing natural gradient flow, far from wells. The model seeks to investigate the effects of medium heterogeneity and recharge on flow on and below the free surface. Statistical moments of free surface position, head, and specific discharge are derived using a first-order approximation and compared with their parallel in an unbounded medium. Analytic solutions for the variance of free surface fluctuations and of the specific discharge on the free surface are developed as well as semianalytic solutions for the statistical moments of head and specific discharge beneath the free surface. The effect of recharge, angle of mean free surface and anisotropy on the statistical moments is analyzed.

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

“…In the case of deterministic constant recharge (R 0 ¼ 0; c ¼ R n cos aÞ, Amir (2003) showed thatĈ u ðk;z; z 0 Þ can be derived analytically by lettingĈ R ðkÞ ¼ 0 in (10). The resulting solution for Cuðr;z; z 0 Þ is the covariance of / conditioned on the value of R. Now considering R to be a random variable, we can assess the effect of recharge measurement error via simple probability distribution theory to show how the variance of head and free-surface position depend on r 2 R the variance of R. The joint density function of / and R; qð/RÞ is equal to qð/jRqRÞ, the product of the conditional density function of / given R and the marginal density function of R. As the marginal density of / is the integral of the joint density qð/; RÞ over all realizations of R, qðuÞ ¼ R qðu RÞ j qðRÞdR, and the variance of head is therefore given by,…”

“…Since r 2 /c ðzjR n Þ can be reduced to one quadrature when z < 0 and can be calculated analytically on the free surface i.e. when z ¼ 0 as in Amir (2003), we can easily calculate r 2 / (z) on the free-surface yielding,…”

“…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, …”

“…where / 0 is the mean zero fluctuation of /; r 2 /c ðzjRÞ is the conditional variance of / given R [the quantity derived in Amir (2003)], and hi is the expected value. Likewise, we can work with the dimensionless/normalized recharge R n s.t.…”

“…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.…”