The Elimination of the Constraint Equation And Modelling of Problems With a Non-condensable Gas In Steam Simulation

Abstract

Abstract The numerical formulation previously used in CMG's (Computer Modelling Group) steam simulator used an equation set consisting of flow equations and a constraint equation. The effect of analytically eliminating this constraint equation from the basic equation set, thereby reducing the number of primary variables by one, is investigated. It is found that an alternating use of the two equation sets according to certain switching criteria yields the best results. A problem of convergence to non-physical answers, which arises in the simulation of steam displacement with a non-condensable gas, is described, and a method for circumventing this problem is proposed. The results of various test runs are included. INTRODUCTION The equations used to describe the steam or in-situ combustion processes can be divided into two groups. The "flow" equations (energy and mass balance) contain inter-block flow terms; the "constraint" equations (saturations and mole fraction constraint) depend only on variables in a single block(1). It is possible to eliminate the constraint equations analytically from the equation set, and thereby reduce the number of equations per grid block. However, previous in-situ combustion and steam simulators have retained the constraint equations in the basic equation set(2–4). These authors have reported that the addition of constraint equations to the basic equation set results in better stability and convergence rate, and permits larger time steps. As the amount of computation per time step is increased for a larger equation set, it is not clear that the addition of constraint equations will result in a net saving in computing cost. This 4-equation formalism has been discussed because it is a natural subset of CMG"s in-situ combustion model ISCOM(5), which is a fully implicit simulator. When ISCOM is run on combustion problems, the constraint equations are eliminated analytically. No difficulties have been observed with this procedure(5–6). Consequently, it was decided to investigate the effect of eliminating the constraint equation (1) from the steam model. Description of the Model If equation (1) is eliminated from the set (1–4), there are only three "basic" equations left, which express the conservation of oil, water and energy. In order to reduce the equation set, it is necessary to perform variable substitution on the appearance or disappearance of steam, while the equation set remains the same. This is in constrast to the previous method(5), where the independent variables remain the same, but the equation set is switched. For the situation in which there is no steam in a given grid block, the primary variables associated with each equation are: (Equation in full paper) The disappearance of the water phase is avoided using the pseudo k value method(4). The only other fully implicit steam simulator reported in the literature is that developed by Coats(l) as a subset of an in-situ combustion simulator.