Anisotropic dispersion and upscaling for miscible displacement

Dispersion is the irreversible mixing that occurs during miscible displacements. Dispersion can reduce local displacement efficiency by lessening solvent peak concentration or increase volumetric sweep efficiency by spreading of the injected solvent to more of the reservoir. Dispersion is therefore an important parameter in predicting and simulating miscible displacements. The difficulty of simulating miscible displacement and understanding the effect of dispersion is also compounded by numerical dispersion, which increases the apparent dispersion in finite difference simulation models. This paper presents an approach to estimate the total longitudinal and transverse dispersion in large-scale media using continuous solvent injection in a medium of finite thickness. The simulations are based on the experimental arrangement of Blackwell (1962) to estimate transverse dispersion, whose experimental consisted of co-injecting two miscible fluids into different sections of the medium at similar rates. Coupled with analytical solutions for two-dimensional convection dispersion equation for a continuously injected solvent we can determine longitudinal and transverse dispersivity simultaneously for the flow medium. In this manner we investigate the effects of stochastic permeability distributions and other scaling groups affecting first contact miscible simulations on dispersion. Sensitivity analysis of dispersion in stochastic permeability fields confirms that both longitudinal and transverse dispersion are scale-dependent. Results also show that the effect of increasing autocorrelation of cell permeability in the longitudinal direction (parallel to flow) is to increase longitudinal dispersion, as solvent travels through more continuous layers, while reducing transverse dispersion. Increasing autocorrelation in the transverse direction is to reduce dispersion in both directions. This reduction is caused by equilibration of solvent concentrations in continuous sections of the reservoir resulting in reduced solute fingering and channeling. Finally, we developed a simple procedure to use the estimated dispersivities to determine a-priori the maximum grid-block size that will maintain equivalent level of dispersion between fine-scale and upscaled coarse models. Large grid block sizes can be used for highly heterogeneous and layered reservoir models. Non-uniform coarsening (upscaling) methods were also recommended and validated for reservoir models with sets of sequential but different permeability distributions. The procedure was extended to multi-contact miscible simulations. The sweep and recovery from upscaled multi-contact miscible simulations were comparable to those of fine scale models.