A Numerical Solution with Second-Order Accuracy for Multicomponent Compressible Stable Miscible Flow
- Narayan M. Chaudhari (Gulf Research and Development Co.)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- April 1973
- Document Type
- Journal Paper
- 84 - 92
- 1973. Society of Petroleum Engineers
- 5.4.9 Miscible methods
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This paper presents an extension of an earlier one, which presented an improved numerical technique for solving multicomponent stable miscible flow equations. Diffusion-convection equations for each component together with a pressure equation are solved. Second-order correct approximations for individual component flows, resulting in numerical dispersion correction terms, are employed in solving the diffusion-convection equations. Backward-in-time implicit and Crank-Nicolson approximation techniques are used to solve the pressure equation. Pressure and saturation distributions using first- and second-order correct finite-difference approximations for individual component flows are compared. One- and two-dimensional systems are considered for these comparisons. The results show that in a compressible miscible displacement process, first-order correct finite-difference scheme process, first-order correct finite-difference scheme for individual component flows results in considerable numerical smearing in the saturation distribution. The pressure distribution, which depends upon the saturation distribution, is very sensitive to numerical smearing. Therefore, high-order correct approximations for individual component flours are necessary. The results also demonstrate That the earlier approach employed to reduce numerical smearing in the numerical solution of incompressible miscible flow processes can be extended to compressible miscible flow systems.
Many investigators have tried to solve in some fashion the system of equations governing the miscible displacement process. In most cases, even though the system was compressible, an incompressible system was considered. One may state several reasons for this. Since the system of equations governing compressible miscible flow is nonlinear, analytical solution of such a system, even in a one-dimensional case, is not known In a multidimensional case it may be a practical impossibility. The only methods known to solve the above problem are the numerical methods. One of such methods is presented here.
Basic flow equations for the individual components present in the compressible miscible flow system present in the compressible miscible flow system are assumed. These flow equations together with the individual component continuity equations are used to arrive at the pressure equation. Crank-Nicolson and backward-in-time approximation schemes are used in an iterative procedure to solve the pressure equation. Flows of individual components are calculated with first- and second-order correct finite-difference approximations.
It is also shown that a semirigorous technique, which eliminates most of the numerical smearing from the numerical solution of an incompressible miscible system, can also be used for a compressible miscible flow system. It has been observed that pressure distributions in a one-dimensional case are pressure distributions in a one-dimensional case are significantly different when the individual component flow terms are approximated with first- and second-order correct finite-difference approximations. Also, the order of approximation of the flow equations has more effect on the pressure distribution than the order of approximation (time approximation) used to solve the pressure equation.
FORMULATION OF THE PROBLEM
Consider a flow system where one fluid is displacing another in a porous medium under the following conditions: (1) The two fluids are completely miscible, and the fluid densities and viscosities are functions of pressure. (2) The miscible flow is stable, and the dispersion of a component is independent of its concentration. (3) The two fluids, when mixed, satisfy the law of additive volumes.
The assumption of stable flow implies that volumetric flux of a component n from an element of a porous medium is given by USn, where U is be volumetric flux of the mixture of the two fluids and Sn is the volumetric fraction or saturation of the nth component in the mixture. It is important to note that the fluids have different viscosities. Therefore, the displacement may take place under unfavorable mobility conditions. Stable flow assumption here means that finger smaller than the size of an element in the porous medium is neglected.
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