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Many difference equations used to approximate reservoir flow problems treat the phase pressures implicitly but not the mobility-density coefficients. Such difference equations are neither wholly explicit nor implicit, but might be described as mixed. Mixed equations are relatively easy to apply. But the associated time truncation error is relatively large, and when used to solve problems characterized by high flow rates, these equations may be unstable for practical size time steps. This paper outlines the development of a completely implicit difference analogue for reservoir simulation, along with a Newtonian iterative method for solving the resulting nonlinear set of algebraic equations that arise at each time step. While this implicit equation requires two to three times more work than does the mixed equation, it is shown to markedly decrease the time truncation error and to yield a stable solution for much larger time steps than does the mixed equation. Introduction Calculation of multiphase, multidimensional flow in porous media is generally accomplished by numerical methods that involve approximating systems of particle differential equations by systems of partial difference equations. The early development of difference equation techniques was directed toward the solution of linear differential systems. However, the equations of multiphase fluid flow through porous media are highly nonlinear in that mobility and density often are strong functions of pressure and saturation. Thus, solution of the flow pressure and saturation. Thus, solution of the flow problems by difference equations involves solving problems by difference equations involves solving sets of algebraic equations whose coefficients change from step to step of the calculations. Current practice is to evaluate most coefficients at the beginning of a time step and then to apply difference methods well suited for solving linear problems. At least two major difficulties arise from problems. At least two major difficulties arise from the use of this practice. First, evaluation of coefficients at the old time level and evaluation of pressures at the new time level results in larger pressures at the new time level results in larger time truncation errors near displacement fronts than if all quantities in the distance derivative were evaluated at the same time level. Second, evaluation of mobility coefficients at old time levels results in an unstable difference equation in regions of high flow rate. The first of these difficulties often results in optimistic recovery behind the displacement front, and the second requires the use of small time steps or special techniques in solving coning problems or problems of gas percolation. Evaluation of all quantities in the distance differences at the new time level results in a completely implicit difference system. Such a difference system has a lower time truncation error than equations in which mobilities are evaluated at the old level and pressures evaluated at the net, level. The algebraic equations resulting from the fully implicit difference equation are, in this case, nonlinear and require some iterative method for their solution. This paper develops fully implicit difference equations for two-phase flow in porous media, describes their solution by Newtonian iteration, and gives examples of problems more easily solved by the new method than by previous methods. EVOLUTION OF DIFFERENCE EQUATIONS FOR NONLINEAR PROCESSES The completely implicit difference equation used in these calculations results from evolution beginning with explicit difference equations, proceeding through what we will call "mixed" equations, and arriving now at a completely implicit equation. SPEJ P. 417

A mathematical model describing isothermal, two-phase flow in porous media has been developed. The model, which consists of describing differential equations and algorithms for their numerical solution, was applied to the problem of vertical groundwater movement in unsaturated soils in the absence of evaporation and transpiration. The equations describing water-air flow through porous media are second order, nonlinear partial differential equations. These equations were converted to finite difference form and were solved with the aid of a digital computer using an iterative implicit procedure. The model includes effective permeabilities of each phase and capillary pressure as functions of liq.uid saturation. The properties of the porous media may be varied in the model as functions of position. A comparison was made between computed results and experimental field data on moisture movement beneath a shallow surface pond. Water was added to the pond at controlled rates to maintain an approximately constant head for a set time period. Following this wetting period the pond was kept d.ry, but covered to reduce evaporation. At different times during the wetting and dwing periods, neutron logs were run to measure water saturation versus depth at depths of up to 22 feet. The experiment was simulated with the computer model and excellent agreement between calculated results and the data was obtained; thus the mathematical model could be used to describe soil moisture movement during wetting and drying periods. Pw • Pwboundary • X---Xboundary (6) Pa • Paboandary 864 GREEN ET AL.

A STUDY of the behavior of retrograde condensation from gas mixtures was made in the presence and absence of sand in order to determine if the condensed liquid would revaporize in the presence of sand. Methanebutane and methane-pentane mixtures that would form liquids by retrograde condensation when produced from a constant volume cell were used.The methane-butane mixtures of similar composition were produced by three different methods. The first two were charged into an empty cell and were produced in one case within a period of 9 hr and in the other within a period of 3 days. The third mixture was charged to a sand-packed cell and produced within a period of 9 hr. The curves relating composition of the produced gas to pressure obtained from these experiments show that equilibrium was maintained as long as liquid was condensing. However, during the portion of the pressure decline where the condensed liquid should be revaporizing, equilibrium was maintained only when the mixture was produced from the sand-packed cell.The methane-pentane mixture is produced only in the presence of sand. The data obtained for this system also show that equilibrium is maintained at all times during the pressure decline.These results indicate that revaporization is aided rather than prevented by the fact that the condensate" wets the sand."

A phase-behavior approach to the prediction of the performance characteristics of a dissolved-gas-drive reservoir is unique in that the problem of choosing flash, differential, or composite-solution gas-oil ratios and formation-volume factors has been circumvented. Data required are a compositional analysis of the reservoir fluid, the bubble point of this fluid, and the relative-permeability curves for the reservoir rock.Gas-oil ratios and formation-volume factors were'calculated under conditions duplicating the performance of the reservoir. A comparison was then made between these results and those obtained by calculations involving a differential, a flash, and a composite process. A vital factor in the solution of the problem is the accuracy of the calculated equilibrium constant. Agreement within 3% was obtained when a calculated differential formation-volume curve was compared with an experimentally determined curve.This assumed Fg/ro is compared with the average V J V , calculated from-the two instantaneous formations (V,/Vo) (i) associated with the liquid saturations at P2 and PI. This latter value is calculated by the use of Equation (8).