Use of Dispersion Relationships To Model Adverse-Mobility-Ratio Miscible Displacements

Abstract

Summary. Improved predictions of petroleum reservoir performance are needed to reduce the risks associated with field-scale miscible flooding. A major shortcoming of established prediction techniques is their failure to model accurately the effects of mobility ratio and viscous fingering on volumetric sweep and displacement efficiency. This paper establishes the feasibility of using physical dispersion relationships to account for fluid mixing associated with viscous fingering. The dispersion relationships use an additional parameter to incorporate a dependence on the local viscosity gradient. Adverse viscosity gradients increase mixing, while favorable viscosity gradients decrease mixing. The dispersion relationships reduce to the normal form when no viscosity gradients are present. The dispersion parameter needed is determined with published data for flow in a long, sandstone core system. Once established, the dispersion relationships are used to predict the performance of a much more complex displacement of reservoir oil by CO,. The simulations reproduce all the salient features observed experimentally and establish the validity of the approach. In particular, the effect of system length is modeled correctly. Introduction A strong interest in miscible flooding has existed since the early 1950's. The driving force behind this interest is the realization that even after thorough waterflooding, about 50 to 60% of a reservoir's original oil is unrecovered. Despite the target's magnitude, major projects have been initiated only recently. This slow development is a result of the large expenditures involved and the uncertainty of oil recovery performance. One of the principal causes of this uncertainty is the incomplete understanding of the effects of mobility ratio and viscous fingering on volumetric sweep and displacement efficiency. This paper addresses one aspect of this problem. Numerous laboratory studies of viscous fingering have used contact-miscible fluids; however, scaling of these results to field conditions has been the subject of extensive debate. In systems that are basically, viscous fingering has been observed to cause an initial rapid growth of the mixing zone followed by a slower "stabilized" growth rate. Two modeling approaches have been used for these systems. The first method uses a fine 2D grid and the convective-dispersion equation to model the viscous fingers explicitty. No practical way currently exists to extend this approach to field-scale systems. The second approach uses a ID grid and neglects dispersion altogether. This method alters the fractional-flow function so that a large transition zone is predicted. Methods of this type cannot predict the stabilization effect observation experimentally. In the present study, we investigate the feasibility of using the convective-dispersion equation and a ID grid to model these processes. The dispersion coefficient is treated as a function of the viscosity gradient, so that a larger "effective" dispersion coefficient results for adverse-mobility-ratio displacements. Fluids used in field applications are not contact miscible. Instead, they develop a miscible-like character because of component transfer between the phases and are called multiple contact miscible (MCM). Laboratory corefloods using CO2 and reservoir oils exhibit a significant length effect. Substantially immiscible behavior has been observed in short cores (6 to 8 ft), while longer cores have behaved miscibly. The mixing caused by viscous fingering is the most likely reason for this behavior; however, the fluid-phase equilibrium is also important. Two different modeling approaches have been used for MCM processes. One-dimensional compositional models based on upstream-weighted finite differences are used most frequently. Normally, the grid size must be adjusted to match performance. This approach cannot model the length effect observed experimentally. Finely gridded 2D models also have been used. Unforlunately, this approach is not practical for field-scale simulation. In the present study, we successfully use a ID model with physical dispersion to predict the performance of MCM experiments. These simulations use dispersion relationships determined with contactmiscible data. The rigorous simulation of physical dispersion effects for MCM processes is difficult because of the phase transition that takes place. 13 A numerical technique capable of simulating these effects is developed and tested. In this paper, the simulation of contact-miscible systems is discussed. We review pertinent experimental work and establish a dis persion relationship that adequately simulates these systems. Then, the fluid properties used for simulating CO2/reservoir-oil systems are described. Next, numerical methods for simulating physical dispersion effects in MCM systems are discussed. Then, the predictions Of CO2 displacing reservoir oil are presented and compared with experimental observations. Finally, a brief discussion of scaling criteria is used to place these results in perspective. Contact-Miscible Systems Contact-miscible fluids have been used in numerous experimental studies of viscous fingering. Fig. 1 shows a typical displacement front observed in a basically ID system. The differential equations governing the flow of miscible fluids were derived for constant-viscosity systems by use of volume-averaging concepts. With volume averaging, variations of a given quantity that are small relative to the size of the system are averaged over a suitable volume. The dispersion term arises in the averaged equations and is represented with experimentally determined constitutive relationships. For a ID system with constant density and viscosity, the governing equation is ..........................................(1) ..........................................(2) The volume-averaging approach has not been extended to systems with variable viscosity. We propose to use Eq. 1 for systems lie that shown in Fig. 1, where C represents the concentration averaged across the system. Eq. 1 will not yield results consistent with experimental observations unless the constitutive relationship (Eq. 2) is generalized. Before we proceed with this idea, some justification for this approach is needed. One view within the petroleum industry has been that the convective-dispersion equation cannot be used for adverse-mobility-ratio displacements unless the detailed structure of the concentration variations are explicitly modeled. Fine 2D grids have been used to model small laboratory systems successfully. Simulations with such fine detail, however, are not practical for fieldscale problems. Some claim that if the viscous fingers are not explicitly modeled, the convective-dispersion equation will overpredict oil recovery. Published results, however, show that this is not true. SPERE P. 309^