##
**Galerkin methods for miscible displacement problems with point sources and sinks - Unit mobility ratio case.**
*(English)*
Zbl 0551.76079

Mathematical methods in energy research, Proc. Spec. Year, Univ. Wyoming 1982-83, 40-58 (1984).

[For the entire collection see Zbl 0546.00029.]

In most cases of reservoir simulation, the size of the well-bore is extremely small in comparison with the size of the reservoir. For this reason point sources and sinks are often used in mathematical models to describe the behavior at the wells. It can be shown that point singularities reduce the regularity of the functions which are determined by the mathematical models. In this paper we shall consider the problem of the miscible displacement of the one incompressible fluid by another in a porous medium using point sources and sinks at the wells. Due to the reduced smoothness of the pressure and concentration caused by the assumption of point singularities, we must use a nonstandard analysis for the semidiscrete Galerkin procedures and obtain convergence rates of lower order than for nonsingular sources and sinks. We treat only the case of unit mobility ratio, when the viscosities of the invading and displaced fluids are equal.

In most cases of reservoir simulation, the size of the well-bore is extremely small in comparison with the size of the reservoir. For this reason point sources and sinks are often used in mathematical models to describe the behavior at the wells. It can be shown that point singularities reduce the regularity of the functions which are determined by the mathematical models. In this paper we shall consider the problem of the miscible displacement of the one incompressible fluid by another in a porous medium using point sources and sinks at the wells. Due to the reduced smoothness of the pressure and concentration caused by the assumption of point singularities, we must use a nonstandard analysis for the semidiscrete Galerkin procedures and obtain convergence rates of lower order than for nonsingular sources and sinks. We treat only the case of unit mobility ratio, when the viscosities of the invading and displaced fluids are equal.

### MSC:

76S05 | Flows in porous media; filtration; seepage |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

76T99 | Multiphase and multicomponent flows |

70-08 | Computational methods for problems pertaining to mechanics of particles and systems |