Injected solvent composition is a key parameter in the design of hydrocarbon miscible floods. Economic optimization of a field flood requires consideration of the effect of solvent composition on flood performance at the reservoir-scale. In this study, a compositional simulation approach was used to study the effects of permeability heterogeneity, fluid crossflow, and phase behavior on displacement performance in two-dimensional, finegrid (x-z) cross-sections. Results of secondary and tertiary solvent flood simulations in various rock property fields indicate that floods with solvent enrichment at or below that required for development of multicontact miscibility in one-dimensional flow can perform as well or better than floods with richer solvents. A significant change to the current practice of solvent enrichment selection is suggested based on analysis of the simulation results.
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^
Th is paper was selected for presentetion by Me Stee ring Commi ttee , fol iowing review of Information contained in en abstract subm itted by the author(s). The paper, as presented hes not been reviewed by the Steering Committee .
SPE Members Abstract This paper concerns the application of frontal advance theory to displacement processes in heterogeneous porous media. The assumptions under which a generalized frontal advance equation can be used to describe a flow process in a heterogeneous porous medium are examined, Material balance equations are derived based on these assumptions, and the theory is illustrated by application to the Dykstra-Parsons model of flow in noncommunicating layers. We show that the resulting formulation can be used to forecast performance and to solve the inverse problem: the determination of a permeability distribution from displacement data. This approach can also be used to determine pseudorelative permeability curves appropriate to a one-dimensional description of a layered system. We also consider the use of pseudorelative permeabilities in simulation studies to represent the effects of reservoir heterogeneities with length scales smaller than a grid block. The dependence of process performance and pseudorelative permeabilities on scale is described. Results show that frontal advance theory may be applied to flow in heterogeneous porous media for processes that exhibit linear characteristics. In such a process, a given saturation travels at a constant velocity through the porous medium. Displacement performance for the two-dimensional Dykstra-Parsons model is exactly duplicated using one-dimensional frontal advance theory for unit mobility ratio displacements. One-dimensional theory reproduces the qualitative performance of the two-dimensional model for nonunit mobility ratios, but exact quantitative agreement is not obtained, because the characteristics are nonlinear. Pseudorelative permeabilities determined for a system of noncommunicating layers apply to any scale as long as the layer structure does not change with system length. However, for processes that are not purely convective, process performance varies with the scale of the displacement. Thus, pseudorelative permeabilities obtained from flow processes that are not purely convective are only applicable to the scale at which the displacement data was measured. Introduction Experiments to determine the flow properties of a porous medium are inevitably interpreted in terms of some model representation of the flow. For example, if a miscible displacement is performed in a laboratory core, or in a field-scale experiment, the resulting effluent composition data may be interpreted by a convection-dispersion equation to determine a dispersion coefficient. That model, with the dispersion coefficient so determined, can be used to compute displacement performance for another miscible displacement in the same porous medium at the same scale, and it may or may not be useful for calculations at different scales. If the model used reflects faithfully the physical mechanisms that dominate the displacement, then calculations at different scales will be accurate. However, more than one model can be adjusted to match displacement data at a given scale. Thus, it may not be clear whether use of parameters so determined is appropriate for displacement in larger flow systems unless measurements have been made at several scales. Another example is use of the Buckley-Leverett equations to determine relative permeability from displacement data, In this case, it is assumed that the measured fractional flow can be interpreted as the result of variations of relative permeabilities with changes in phase saturations. There is, of course, no guarantee that the relative permeability description captures all the detail of the mechanisms that influence the transport of the phases. Thus, any model of a displacement is a compromise between a detailed representation of pertinent physical mechanisms and tractability of the calculation scheme, and availability of data required to support the model. Essentially all calculations of waterflood or other oil recovery process performance are based on the use of relative permeability functions to describe the local movement of phases based on their saturations. The functions used in a particular simulation are average representations of the effects of displacement mechanisms operating at smaller scales. For example, the analyses of Buckley and Leverett, Welge and Johnson, Bossler and Naumann are routinely used to determine relative permeability functions from measurements of the fractional flow of water leaving a core and the pressure drop across the core. These analyses assume that gravity and capillary pressure effects are negligible and that the core is one dimensional, even though most cores do not have strictly uniform permeability. P. 215^
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