Page 157 - gas transport in porous media

Sahimi et al.
150
Finite-Difference Methods
8.7.2
Conventional finite-difference (FD) discretizations have also been used to obtain
improved physical understanding of miscible displacements; see, for example,
Christie and Bond (1987), Christie (1989), Bradtvedt et al. (1992), Fayers et al.
(1992), and Christie et al. (1993), as well as Sorbie et al. (1995) and Zhang et al.
(1996). The overall accuracy of these methods are typically of second order, but often
they still suffer from significant amount of numerical dispersion. In terms of formal
accuracy, an important seminal work was that of Leventhal (1980), who applied
the fourth-order Operator Compact Implicit (OCI) family of FD schemes (see, e.g.,
Berger et al., 1980; Morton, 1996) to one-dimensional, two-phase immiscible water-
flood problems, and demonstrated a significant reduction of the adverse effect of
numerical diffusion. Tchelepi and Orr (1993a,b) minimized the amount of numerical
diffusion by employing a particle tracking technique (Araktingi and Orr, 1990; see
below), although the solution of the accompanying pressure equation still resulted in
some artificial smoothing.

8.7.3 Streamline Method

Inthisapproach, theflowproblemisdecoupledintoasetof1Dproblems, solvedalong
streamlines, which reduces the simulation time and suppresses the numerical disper-
sion. The method has also been used as a scale-up technique. The obvious advantages
of the streamline method have increased its wide application and fast commercializa-
tion. However, compared with the conventional FD simulations, the streamline-based
simulations usually account only for relatively simple physics. These simulators are
limited to production scenarios where the effect of gravity can be neglected. Given
that compositional simulations are much more time consuming, it is of great interest
to speed up the compositional simulators using a streamline method.
Thiele et al. (1995) proposed a compositional simulation approach to inves-
tigate compositional displacements in two-dimensional heterogeneous reservoirs.
Thiele et al. (1997) combined a three-dimensional streamline method with a
one-dimensional two-phase numerical solver and developed a streamline-based three-
dimensional compositional reservoir simulator. Jessen and Orr (2002) developed a
three-dimensional compositional reservoir simulator based on the analytical map-
ping approach. By mapping the analytical solution to one-dimensional two-phase
multicomponent gas injection along each streamline, the simulator can be orders of
magnitude faster and completely free of numerical dispersion. A limitation of the
analytical mapping method is, however, that it is only applicable to uniform initial
condition.
More recently, Yan et al. (2004) developed a three-phase compositional streamline
simulator by integrating a 1D numerical solver. The numerical solver is optimized
for calculating the efficiency, and various descriptions of phase equilibrium between
hydrocarbon and water, as well as the gravity effect, are included by the operator-
splitting method. Operator splitting (Batycky et al., 1996; Bradtvedt et al., 1996;