A numerical dual‐porosity model with semianalytical treatment of fracture/matrix flow

Abstract

A new dual‐porosity model is developed for single‐phase fluid flow in fractured/porous media. Flow is assumed to take place through the fracture network and between the fractures and matrix blocks. The matrix blocks are treated in a lumped parameter manner, with a single average pressure used for each matrix block. Rather than assuming that fracture/matrix flux is proportional to the difference between the fracture pressure and matrix pressure at each point, as is done in the Warren‐Root model, we use a nonlinear equation which more accurately models the flux over all time regimes, including both early and late times. This flux equation is compared with analytical solutions for spherical blocks with prescribed pressure variations on their boundaries. The nonlinear flux equation is also used as a source/sink term in the numerical simulator TOUGH. The modified code allows more accurate simulations than the conventional Warren‐Root method, with a large savings (about 90%) in computational time compared to methods which explicitly discretize the matrix blocks.

Numerical evaluation of a second‐order water transfer term for variably saturated dual‐permeability models

“…A nonlinear ordinary differential equation accurate for all times was presented by Zimmerman et al [1993]. Their second-order water transfer term, derived by differentiating an analytical solution adapted from Vermeulen [1953], accurately approximates the exact solution for the pressure response of a spherical matrix block to a step function increase in the pressure head at its outer boundary [Zimmerman et al, 1993]. The term was subsequently modified for variably saturated conditions [Zimmerman et al, 1996] and in the notation of this paper is given by…”

“…The approach consisted of two different equations applicable to early-and late-time lateral imbibition. A nonlinear ordinary differential equation accurate for all times was presented by Zimmerman et al [1993]. Their second-order water transfer term, derived by differentiating an analytical solution adapted from Vermeulen [1953], accurately approximates the exact solution for the pressure response of a spherical matrix block to a step function increase in the pressure head at its outer boundary [Zimmerman et al, 1993].…”

“…First, it is theoretically valid only for quasi-steady water transfer [Zimmerman et al, 1993] and therefore underestimates highly transient transfer flux for early times [e.g., Zimmerman et al, 1993;Gerke and van Genuchten, 1993a]. Second, in a lumped formulation such as equation (1), K m needs to be evaluated as some blockaveraged, effective hydraulic conductivity function of the pressure head in the matrix and fractures that best represents the conductivity of the actual pressure head profile at all times.…”

“…A mathematical derivation of equation (1) was given, for instance, by Zimmerman et al [1993Zimmerman et al [ , 1996.…”

“…Linear first-order transfer terms cannot accurately calculate water transfer between the domains during early times of pressure head nonequilibrium. In this study, a second-order equation for water transfer into spherical rock matrix blocks [Zimmerman et al, 1993[Zimmerman et al, , 1996 was adapted and evaluated comprehensively for water transfer into and out of variably saturated soil matrix blocks of different hydraulic properties, geometries, and sizes, for different initial and boundary conditions. Numerical solutions of the second-order term were compared with respective results obtained with a first-order term and a one-dimensional horizontal flow equation.…”