Computation and Interpretation of Miscible Displacement Performance in Heterogeneous Porous Media

Abstract

Summary. Computation of miscible displacement performance requires an estimate of the amount of mixing in the reservoir. Dispersion coefficients measured from displacements in laboratory cores are often used to compute reservoir performance. This paper considers interpretation of effluent flowing concentration data obtained from heterogeneous cores by use of the Coats-Smith, porous-sphere, and transverse-matrix-diffusion heterogeneous models. Analytical solutions to the three heterogeneous models are presented in Laplace space. Approximate solutions for short- and long-time effluent flowing concentrations, developed from simplification of the Laplace transformation solution, are used to develop practical methods of interpreting effluent concentration data to determine model parameters. Criteria for design of laboratory experiments based on these results are suggested. Introduction Design of mixing-sensitive displacement processes, such as miscible flooding and surfactant flooding, requires an estimate of the extent of mixing in the reservoir. Miscible displacement experiments performed on reservoir core samples are designed to non-uniform flow effects. Laboratory miscible displacement experiments may be interpreted in terms of the one-parameter convection-dispersion model to determine the longitudinal dispersion coefficient or in terms of multiparameter models that consider reservoir heterogeneity, such as the Coats-Smith and porous-sphere models. Brigham et al. demonstrated a simple graphical method for interpreting effluent concentration data from homogeneous cores. This paper presents solutions in Laplace space for these models and analytical approximations for short and long times. These approximations are used to develop practical methods for interpreting effluent concentration data from heterogeneous cores. Criteria for design of laboratory miscible displacement experiments based on these results are suggested. In addition, an efficient numerical inverter was developed to invert the model solutions from Laplace to real-time space. A substantial saving in computation time over the conventional finite-difference solution method was possible with the Laplace inversion method. Results from miscible displacement projects indicate that dispersion coefficients measured from field data are often much larger than those measured in laboratory cores. The differences in dispersion coefficients may result from more complex flow phenomena in the reservoir. For example, viscous fingering and gravity segregation are likely to affect displacement in a reservoir. In addition, laboratory-determined dispersion coefficients reflect only the scales of heterogeneity present in the laboratory core. Calculation of miscible displacement performance with a model that considers nonuniform flow effects resulting from flow mechanisms operating in a reservoir is discussed. Problems of the type described in this paper have been considered in both the heat-transmission and pressure-transient literatures. A detailed review of this extensive literature is beyond the purpose of this study. Models This section presents the governing equations and boundary conditions for the convection-dispersion, Coats-Smith, porous-sphere, and transverse-matrix-diffusion models. Convection-Dispersion Model. Viscous and gravity-stable miscible displacement in a uniform linear porous medium may be described by the convection-dispersion equation: (1) The dimensionless form of the equation is (2) where xD = (uld)x, tD=(u2/d)t, and CD(XD, tD)=[C(X, t)-Ci]/ (C1-Ci). Here, Ci=initial concentration in the core and Ci =concentration of the injected fluid. Brighams pointed out that Eq. 2 could describe either the flowing concentration, Cf, or in-situ concentration, C, with the choice of boundary conditions determining which concentration is considered. For continuous fluid injection at concentration C, the in-situ concentration may be determined by solving Eq. 2 with the boundary conditions (3) (4) (5) The effluent concentration flowing out of the core is usually measured in laboratory miscible displacement experiments. The flowing concentration is related to the in-situ concentration by (6) Eq. 2 can be solved by Laplace transformation to give, in Laplace space,(7) (8) The general functional relationship g(s) is used because it is useful for describing solutions for other porous matrix models, as will be shown later. This notation is similar to that used in pressure-transient solutions of dual-porosity systems. From the relationship between flowing and in-situ concentration given in Eq. 6, we can determine the solution for the flowing concentration in Laplace space: (9) The analytical solution for the flowing concentration is determined by inverting Eq. 9 with g(s) = 1 from Laplace space, which yields (10) SPERE P. 69^