Introduction
Most reservoir simulators employ finite-difference methods to solve the appropriate set of equations. Variables that influence the accuracy of the results are the time-step size and the cell dimensions. While the effects of these variables on the results of conventional simulators can be significant, they can be even more important with chemical-flooding models. This is because of the presence of an additional phase, such as a surfactant slug that is moving with time and that can occupy part of, or all of, a cell. Within this slug, fluid saturations and relative permeability relations are different than those ahead or behind it. This causes a mathematical problem that is the subject of this work. problem that is the subject of this work.
DESCRIPTION OF THE PROBLEM
Simulation of the low-tension flooding process involves calculating the three unknown variables, pressure, saturation, and surfactant concentration, pressure, saturation, and surfactant concentration, as a function of time and space. The pressure and saturation distributions can be calculated using the usual finite-difference methods of solution of the equations for immiscible flow. The surfactant concentration distribution can be determined by tracking the surfactant slug boundaries analogous to the scheme proposed by Vela et al. for polymerflood simulation. However, the surfactant slug polymerflood simulation. However, the surfactant slug maintains a fairly sharp boundary as it moves through the reservoir. Therefore, in some finite-difference cells, two distinct parts may exist, one with and one without surfactant. Each part is different from the other in fluid saturations and relative permeability relations. However, the finite-difference method of solution requires that the two parts be represented by one average saturation and by one relative permeability value. Thus, the problem is how to average the two parts and to determine how sensitive the results of simulation are to the averaging scheme, to the cell size, and to the time step.
METHOD OF ATTACK
Ideally, one wants an averaging scheme that (1) gives answers that are not sensitive to the time step or to the cell size, and (2) gives correct answers.
The second criterion is the most difficult to check since no exact solution to the surfactant flood is reported in the literature. The work was started by developing an analytic solution (Buckley-Leverett type) to a one-dimensional surfactant flood. The solution is analogous to solutions for other tertiary recovery projects. It combines the relationship for the normalized motion of a point of constant saturation, with expressions for the
dimensionless velocity of each phase, and vw = (1-fo)/(1-So). (fo is fractional flow of oil; So is oil saturation.) Saturations throughout the one-dimensional reservoir are thereby obtained. Several solution regimes result. (1) For oil-water viscosity ratios greater than unity, oil moves exclusively in front of the surfactant, forming a bank, and all the oil is produced. (2) For slightly unfavorable viscosity ratios, an oil bank is still formed, but the oil gradually invades the surfactant and may result in reduced production. (3) Highly unfavorable viscosity ratios cause all the oil to move through the surfactant, and no bank is formed.
The one-dimensional surfactant flood was then simulated using an incompressible, two-dimensional, polymer-surfactant model that solves for the polymer-surfactant model that solves for the concentration using a point-tracking scheme based on the method of characteristics. This method eliminates numerical dispersion and results in sharp surfactant-slug interfaces. Several sets of runs were made, with each set using a different averaging scheme. The various schemes used are described in the next section.
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A Finite Difference Method for the Variational p-Laplacian
