20. Statistical Scale-up of Reservoir Properties and Dispersivities in Heterogeneous Reservoirs

Previous works demonstrated that dispersion increases with heterogeneities and travel distance in heterogeneous reservoirs. Therefore, scale-up of input dispersivity and reservoir attributes to the transport modeling scale should account for subscale heterogeneity and its variability. A quantitative procedure to scale-up both rock and flow-related properties is presented. First, 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 measure values and the variance is the variance of block mean. Stochastic simulations are subsequently performed to generate multiple realizations at the transport modeling scale. Next, a stochastic random walk particle tracking (RWPT) method, which is not prone to numerical dispersion, is applied. Multiple sub-grid realizations illustrating fine-scale heterogeneities and of the same sizes as the transport modelling mesh size are subjected to RWPT simulation. Effective longitudinal and transverse dispersivities are computed simultaneously by matching the corresponding breakthrough concentration history for each realization with an equivalent medium consisting of averaged homogeneous properties. Probability distributions of scaled-up dispersivities conditioned to average porosity and permeability values are established. Tracer injection responses obtained with the coarse-scale models and fine-scale models are compared. Anomalous behavior is observed, and the interplay between large-scale heterogeneity and sub-scale variability is studied. Scaled-up dispersivity is shown to increase with scale. This study demonstrates that (1) accounting for variability owing to scale-up could capture the actual fine-scale behavior; and (2) the ensuing uncertainty in transport response is underestimated when sub-scale variability is ignored. It underlines the significance of quantification and integration of uncertainty in subsurface “hard” data.