Simulation of EOR Processes in Stochastically Generated Permeable Media

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Summary This paper deals with the competition between effects during miscible displacements through heterogeneous permeability media. Displacements with an even front are dispersive, others are a type of bypassing: fingering, caused by an adverse mobility ratio, or channeling, caused by heterogeneity and autocorrelation. Under most flow circumstances, channeling is the predominant type. Introduction With the advent of computerized imaging and the minipermeameter to infer small-scale inhomogeneities quantitatively, many rocks once thought to be homogeneous, such as Berea sandstone, have been shown to contain significant heterogeneity.1 Even sandpacks and other synthetic media may suffer from packing problems that can create permeability channels, especially along the boundaries of the medium. In nonunit-mobility ratio displacements in heterogeneous media, it is difficult to separate the effects of the mobility ratio (called fingering if the mobility ratio is adverse) and heterogeneity, or channeling. The terms fingering and channeling are often used synonymously because they both result in uneven displacement fronts, poor oil recovery, and premature breakthrough of the displacing fluid. The terms represent separate effects in this paper because they have different causes. We examine channeling through numerical simulation of unit-mobility of unit-mobility-ratio displacements and then show how a mobility change can affect the flow pattern. Heterogeneity. A permeability field is heterogeneous if it is spatially varying. The simplest representation of heterogeneity is a collection of continuous layers, each with a single value for permeability and porosity. One of the goals of reservoir characterization is to emphasize the actual geology of the reservoir. With such a minute fraction of the reservoir actually sampled by a well, however, a complete description of the reservoir is nearly hopeless. Geostatistics addresses this problem by using statistical techniques to create an image of the reservoir from information observed at the wells. Statistical models are not unique; the same set of input parameters will produce different results, or realizations. The choice of the correct realization(s) to use for a reservoir study depends on the fluid-flow behavior through the model compared with the field production history. Statistical models that are based on conditional simulation generate realistic fields, and their flexibility implies that they can be used to represent a wide variety of geologically realistic media. Such flexibility and realism suggests that the results we describe here will apply generally to what is occurring in displacements through naturally occurring permeable media. Generation Technique. We used longitudinal (x-direction) permeability fields, kx(x,z), generated by the turning-bands method (TBM),2 but the results should apply equally well to synthetic permeable media generated by similar statistical techniques. All kx(x,z) fields are log-normally distributed with a spherical variogram model of autocorrelation. Thus, the fields are completely described statistically by a measure of heterogeneity, which we take to be the Dykstra-Parsons3 coefficient, VDP, and the range ?R. The range in kx is oriented longitudinally and is expressed as a fraction of the total medium length, L. Spatial correlation of kx(x,z) in the direction transverse to bulk flow is held constant at 0.2 times the medium thickness, h; the effect of 2D correlation variability is not treated in this work. Different realizations are generated by choosing a different random number seed to enter into the generation program. Porosity was held constant in all runs. The degree of heterogeneity, VDP, measures the variability of permeability values. When VDP=0, all permeabilities are equal, representing a homogeneous medium. As VDP increases, permeability values deviate further from a mean. Typical field values for undifferentiated core data range from 0.6 to 0.8, but values as large as 0.9 have been observed.4 The dimensionless range, ?R, measures how well neighboring permeability values are related to each other. Qualitatively, ?R relates to the depositional environment, ranging from such high-energy environments as valley-fills, which have little correlation, to such low-energy environments as marine bars, which have extensive correlation.5 When ?R=0, there is no correlation between neighbors; as ?R?8, the depositional environment becomes strictly layered (homogeneous in the direction of correlation). Simulation at Vertical Equilibrium (VE) Conditions. Much of this study is based on the VE concept described elsewhere.6-8 Most simulations had very good transverse communication to ensure that VE was a good assumption. Unless noted otherwise, we forced VE by making the effective aspect ratio, RL=(L/h) kz/kx, large. For linear flow (the only kind considered here) RL is a good indicator of the approach to VE.* We performed a few runs at the other extreme of RL=0, but these proved to be unenlightening because good transverse communication is required to propagate fingers.9 We show one such case below; however, RL is generally large for field-scale displacements. A black-oil finite-difference simulator run in a miscible fluids mode8 was used to simulate first-contact-miscible displacements at the viscosity ratio of interest. A simulation grid of 120 x-direction blocks and 20 z-direction blocks was found to be relatively insensitive to further...