A dual-porosity model for gas reservoir flow incorporating adsorption behaviour—Part II. Numerical algorithm and example applications

Abstract

In the present study, an algorithm is presented for the dual-porosity model formulated in Part I of this series. The resultant flow equation with the dual-porosity formulation is of an integro-(partial) differential equation involving differential terms for the Darcy flow in large fractures and integrals in time for diffusion within matrix blocks. The algorithm developed here to solve this equation involves a step-by-step finite difference procedure combined with a quadrature scheme. The quadrature scheme, used for the integral terms, is based on the trapezoidal method which is of second-order precision. This order of accuracy is consistent with the first-and second-order finite difference approximations used here to solve the differential terms in the derived flow equation. In an approach consistent with many petroleum reservoir and groundwater numerical flow models, the example formulation presented uses a first-order implicit algorithm. A two-dimensional example is also demonstrated, with the proposed model and numerical scheme being directly incorporated into the commercial gas reservoir simulator SIMED II that is based on a fully implicit finite difference approach. The solution procedure is applied to several problems to demonstrate its performance. Results from the derived dual-porosity formulation are also compared to the classic Warren-Root model. Whilst some of this work confirmed previous findings regarding Warren-Root inaccuracies at early times, it was also found that inaccuracy can re-enter the Warren-Root results whenever there are changes in boundary conditions leading to transient variation within the domain.

A Statistical Representation of the Matrix–Fracture Transfer Function for Porous Media

“…Thus, discussion of the errors for larger times, say,t > 1 would not be meaningful. However, as demonstrated in Connell and Lu (2007), the conclusion that the mass exchange rate approaches zero at larger times may not always true in general. For example, when transients on the boundary are present the mass exchange rate would respond to the changes of boundary conditions accordingly.…”

“…However, except for this, the two expressions are basically of the same form. Numerical algorithms for this transfer function were proposed in Connell and Lu (2007) and Lu et al (2008). These publications present application examples of these numerical schemes through their incorporation into a commercial gas reservoir simulator (SIMED II).…”

“…15 for the mass transfer function, although more sophisticated in comparison with the single-block-based representation described by Eq. 7 or 8, is of a more basic form when compared with that used in the numerical algorithms presented by Connell and Lu (2007). However, the schemes proposed by Connell and Lu (2007) and Lu et al (2008) still apply to it.…”

“…10, as can be understood, for sufficiently large times, say t > [(m 2 /l 2 X + n 2 /l 2 Y + p 2 /l 2 Z )π 2D ] −1 , all terms in the series beyond the first are negligible under the boundary condition addressed above. However, on the other hand, for t → 0 the series is slow to converge and a large number of terms may be needed (Zimmerman et al 1993;Connell and Lu 2007). In practice, one can set a sufficiently small time, say,t = 10 −5 to be the lower bound of the time concerned.…”

“…This is because they are generally simpler and may provide quantitative flow predictions for volume averages and in a tractable fashion. Traditionally, the transfer functions are derived from either a fully analytical or a quasi-analytical treatment often based on a single matrix block with regular geometry (e.g., Warren and Root 1963;de Swann 1977;Najurieta 1980;Moench 1984;Zimmerman and Bodvarsson 1989;Zimmerman et al 1996;Dykhuizen 1990;Chen 1989Chen , 1990Zimmerman et al 1993;Zimmerman and Bodvarsson 1995;Lim and Aziz 1995;Choi et al 1997;Rasmussen 2001, 2005;Babadagli and Zeidain 2004;Rangel-German and Kovscek 2006;Lu and Connell 2007a;Connell and Lu 2007). The transfer functions derived in this way are sometimes referred to as single rate mass transfer relations, as they merely involve the geometric properties of a single block such as the matrix block size and the shape.…”