Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures

Abstract

Sudicky and Frind [1982] considered the physically important process of contaminant transport in a fractured porous medium. They developed analytic expressions for contaminant concentrations within equally spaced parallel fractures as well as within the matrix of the porous rock material. However, their expressions for the contaminant concentration in the porous matrix are incorrect, both for the general case with dispersion in the fracture and the simplified instance where dispersion is ignored. (Also note the error in their (35b); the term should be exp (-R2z/v) instead of exp (R2z/v)). We present the correct solutions in this paper. The steady state expressions given by Sudicky and Frind for concentrations in the porous matrix are correct. It must be emphasised that the main results illustrated and discussed by Sudicky and Frind are not affected by these errors, since they did not evaluate the expressions obtained for the porous matrix. The notation and numbering of equations used by Sudicky and Frind is retained in the working here. Where any differences occur, a definition will be given. Equations developed here are identified with numbers of the form (1'). Two errors can be identified in the expressions given by Sudicky and Frind for the concentration in the porous matrix. The first, an omission, relates to the concentration at the fracture wall, while the second stems from an error in integration. Any solution to a boundary value problem must, by definition , satisfy the imposed boundary conditions. The condition imposed at the interface between the fracture and the porous matrix is stated as c'(b, z, t) = c(z, t) (1') However, the solution given for the porous matrix (equation (31)) satisfies c'(b, z, t)= 0 (2') for all times. This indicates an omission in the solution given by Sudicky and Frind. The Laplace inversion in (29) is incomplete , since for x = b the result incorrectly gives L-•{1} =0 (Y) where L-• is the inverse Laplace transform operator. Equation (29) is therefore only correct for x > b. For x-b the inversion yields a delta function •5(t) and must be included in the overall solution (31). Sudicky and Frind make no covering statement, claiming (29) or (31) to be valid only in the region x > b. This immediately resolves the discrepancy between (1') and (2'), since the integrand in (31) is discontinuous at x = b. However, while both equations (29) and (30) are correct for x > b, a second error is apparent in expressions (31) and (36b). The time integration, in proceeding from (30) to (31), was performed incorrectly. Although they noted it earlier, Sudicky and Frind ignored the presence of u[T(0] in the kernel of the time integration of (30) by retaining the lower integral limit as zero. Here we have used the notation u[T0:)] = 0 •: < A r (4') u[T0:)] = 1 z > A Y where TOO = •:-A Y. Clearly the inclusion of (4') necessitates integration from A Y to t, not zero to t. This error by Sudicky and Frind required the introduction of the parameter fl'. In the correct solution it does not appear. Upon c...

Connectivity of joint networks with power law length distributions

“…The model of Snow [1969], in which a fracture is represented by two infinite parallel plates, remains, perhaps, the most popular fracture connectivity model [e.g., Gibson and ToksOz, 1990;Marrett, 1996;Sidle et al, 1998;Sudicky and Frind, 1982], despite its uncertain applicability to natural fracture systems. More recently, a limited number of alternative models have been developed to account for the finite lengths of fractures.…”

An analytical solution for slug tracer tests in fractured reservoirs

“…[11] Sudicky and Frind [1982] gave a general solution for the above problem under a constant boundary concentration that is different from (6a). We may apply their solution to derive the solution for the tracer slug test using the superposition principle.…”

“…Reversible sorption in the matrix is accounted for by a retardation factor. On the basis of these assumptions, we use the following governing equations simplified from Sudicky and Frind [1982]:…”

“…Analytical solutions for contaminant transport in fractured porous media have been available since the early 1980s [Tang et al, 1981;Sudicky and Frind, 1982]. The former paper is for a single fracture where the matrix is assumed to extend to infinity away from the fracture, while the latter is for the case of a set of parallel fractures with uniform fracture spacing and identical solute concentration at the entrance of the fractures.…”

Semianalytic solution for transport of a two-member decay chain in discrete parallel fractures

“…A good introduction to currently existing solutions may be found in chapter 6 of Rumynin [2011]. The decay of a single species has been modeled in a single fracture by Tang et al [1981] and in a set of parallel fractures by Sudicky and Frind [1982]. A two-species decay chain was also later considered in a system with a single isolated fracture in Sudicky and Frind [1984], modified in a special case by Cormenzana [2000], and reconsidered for longer chains by Sun and Buscheck [2003].…”