A Highly Implicit Steamflood Model
Authors: Coats K.H. 
Published: December 2007

Abstract
This paper describes a three-dimensional, highly implicit numerical model for simulating steamflooding with distillation or solution gas. The model uses direct solution to solve simultaneously three and four equations for the dead oil and two-component oil cases, respectively. The model is compared in stability and computing time with a model reported earlier. The paper includes comparative discussion of alternate steamflood model formulations, one of which we have adopted as a highly stable, isothermal, black-oil model formulation.

Introduction
A brief review of published descriptions of steamflood models is given in an earlier paper. That paper described a partially compositional, three-dimensional model that solves first a single-variable pressure equation, then two simultaneous saturation equations. In our experience with dead-oil steamflood problems, that model exhibits adequate stability in most cases and marginal stability in some cases. In some compositional problems, the formulation of that model leads to problems, the formulation of that model leads to deteriorating material balances for light hydrocarbon components.
The model described here was developed to gain improved stability for all types of steamflood problems and to eliminate the material balance problems and to eliminate the material balance shortcoming of the earlier model formulation in compositional problems. This highly implicit, three-dimensional model treats oil as a two-component mixture to accommodate problems involving solution or inert gas or distillation. The model simultaneously solves three equations for the dead-oil case and four equations for the compositional case. Transmissibilities, capillary pressures, and production terms are treated pressures, and production terms are treated implicitly in saturations and composition; they also are treated implicitly in temperature in grid blocks where no free gas is present.
The term "implicit" refers to evaluation of interblock flow terms and production rates at the new time level, n + 1. We have found insensitivity to explicit or implicit dating of molar densities and viscosities in these terms and therefore simply evaluate them explicitly. We evaluate relative permeabilities at time level n + 1 by the first-order permeabilities at time level n + 1 by the first-order approximation, which ignores second- and higher-order Taylor series terms . Temperature dependence of relative permeability, if present, is treated explicitly.
We present the model equations, and describe the highly implicit formulation and method for solution. This model is compared with the earlier steamflood model in stability and efficiency through discussion and example field problems.

MODEL DESCRIPTION

BASIC EQUATIONS
The model consists of five equations expressing conservation of energy, conservation of mass, and phase equilibrium. The mass conservation equations phase equilibrium. The mass conservation equations apply to water and to two hydrocarbon components. In finite-difference form, these equations are
Energy Balance
(1)
Mass Balance on H2O
(2)
SPEJ
P. 369