Statistical Scale-Up of Dispersive Transport in Heterogeneous Reservoir

Abstract

Numerical methods are often used to simulate and analyze flow and transport in heterogeneous reservoirs. However, they are limited by computational restrictions including small time steps and fine grid size to avoid numerical dispersion. The ability to perform efficient coarse-scale simulations that capture the uncertainties in reservoir attributes and transport parameters introduced by scale-up remains challenging. A novel method is formulated to properly represent sub-grid variability in coarse-scale models. First, multiple sub-grid realizations depicting detailed fine-scale heterogeneities and of the same physical sizes as the transport modeling grid block are subjected to random walk particle tracking (RWPT) simulation, which is not prone to numerical dispersion. To capture additional unresolved heterogeneities occurring below even the fine scale, the transition time is sampled stochastically in a fashion similar to the continuous time random walk (CTRW) formulation. Coarse-scale effective dispersivities and transition time are estimated by matching the corresponding effluent history for each realization with an equivalent medium consisting of averaged homogeneous rock properties. Probability distributions of scale-up effective parameters conditional to particular averaged rock properties are established by aggregating results from all realizations. Next, to scale-up porosity and permeability, volume variance at the transport modeling scale is computed corresponding to a given spatial correlation model; numerous sets of “conditioning data” are sampled from probability distributions whose mean is the block average of the actual measured values and the variance is the variance of block mean. Multiple realizations at the transport modeling scale are subsequently constructed via stochastic simulations. The method is applied to model the tracer injection process. Results obtained from coarse-scale models where properties are populated with the proposed approach are in good agreement with those obtained from detailed fine-scale models. With the advances in nanoparticle technology and its increasing application in unconventional reservoirs, the method presented in this study has significant potential in analyzing tracer tests for characterization of complex reservoirs and reliable assessment of fluid distribution. The approach can also be employed to study scale-dependent dispersivity and its impacts in miscible displacement processes.