##### Publikasjonsdetaljer

**Tidsskrift:**
SPE Reservoir Evaluation and Engineering, vol. 7, p. 107–116, lørdag 1. februar 1992

**Utgivere:**
Society of Petroleum Engineers

**Utgave:**
1

**Internasjonale standardnumre:**

Trykt: 1094-6470

Elektronisk: 1930-0212

**Lenker:**

DOI: doi.org/10.2118/19805-PA

This paper presents a new numerical method for solving saturation equations without stability problems and without smearing saturation fronts. A reservoir simulator based on this numerical method is under development. A set of test problems is used to compare the simulation results of the new simulator with those of an existing finite-difference simulator (FDS).

Introduction

The standard nonlinear, partial-differential equations that describe flow in porous media can be separated into a pressure equation and saturation equations. If the diffusion term is ignored, the saturation equations describe a physical problem where sharp discontinuities in the physical data are possible. Finite-difference methods used to solve these equations typically exhibit numerical dispersion. They also show numerical stability problems so that very short timesteps may be required. Use of an implicit problems so that very short timesteps may be required. Use of an implicit formulation can reduce this limitation of the timestep length, but this will not solve the numerical dispersion problem.

Different methods are suggested in the literature for representing the saturation fronts more accurately. These methods usually center on the method of characteristics and usually are limited to miscible flow problems. Glimm et al. developed a front-tracking simulator (FTS) based problems. Glimm et al. developed a front-tracking simulator (FTS) based on a method that follows the discontinuities separately and solves the saturation equation in the remaining part of the reservoir with a infinite- difference method.

New methods in the field of hyperbolic conservation laws have led to alternative solution procedures for the saturation equations. A simulator based on an implicit pressure, explicit saturation (IMPES) formulation and these methods is under development. The goal of this development work is a 3D, three-phase simulator. In this reservoir simulator, the pressure equation is solved by a infinite-element method (FEM). The grid for the pressure equation can therefore be fitted to the reservoir geometry and the pressure equation can therefore be fitted to the reservoir geometry and the geometry of the sharp discontinuities in the saturations with great flexibility. The linear equation system is solved with a preconditioned conjugate-gradient method.

The method for solving the saturation equations is based on an approximation of the fractional-flow function by a piecewise linear function. This method allows the saturation equations to be solved without stability problems and restrictions on the timestep length caused by the Courant-Friedrichs-Levy (CFL) condition.

A set of test problems has been used to compare the simulation results of the new simulator with those of an existing FDS. The test cases show the FTS to be without the grid and numerical dispersion effects observed in a standard five-point FDS- The FTS is also computationally more efficient than finite-difference methods. The simulator has been used in North Sea field simulation problems.

Front-Tracking Method

Traditional numerical methods have a tendency to smear discontinuities owing to numerical dispersion. This results in a less accurate estimate of the position of the fluids and their motion in the reservoir. By introducing the discontinuities (fronts) as separate entities, their positions can be simulated more accurately. The crucial point is to positions can be simulated more accurately. The crucial point is to represent the physical discontinuities (saturation fronts) correctly. In front-tracking, the sharp change in saturation is replaced by a step function-i.e., a discontinuous function. This saturation jump is called a front, and the method handles the fronts separately.