Modeling impacts of subscale heterogeneities on dispersive solute transport in subsurface systems

Previous works in the literature demonstrated that dispersion increases with heterogeneities and travel distance in heterogeneous reservoirs. However, it remains challenging to quantify the effects of subscale heterogeneities on dispersion. Scale-up of input dispersivity and other reservoir attributes to the transport modeling scale should account for subscale heterogeneity and its variability. A method is proposed to quantify the uncertainties in reservoir attributes and dispersivity introduced by scale-up. A random walk particle tracking (RWPT) method, which is not prone to numerical dispersion, is used for transport modeling. First, to scale-up rock properties including 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, multiple sub-grid geostatistical realizations depicting detailed fine-scale heterogeneities and of the same physical sizes as the transport modeling grid block are subjected to RWPT simulation. Effective longitudinal and transverse (horizontal) dispersivities in two-dimensional models are determined simultaneously by matching the corresponding breakthrough concentration history for each realization with an equivalent medium consisting of averaged homogeneous rock properties. Aggregating results derived with all realizations, we generate probability distributions of scaled-up dispersivities conditional to particular averaged rock properties, from which values representative of the transport modeling scale are randomly drawn. The method is applied to model a tracer injection process. Results obtained from coarse-scale models, where reservoir properties and dispersivities are populated with the proposed approach, are compared to those obtained from fine-scale models. Our results verify that dispersivity increases with scale and demonstrate that (1) uncertainty distributions in recovery obtained by accounting for variability owing to scale-up capture the actual fine-scale behavior; and (2) ignoring sub-scale uncertainties would underestimate the ensuing uncertainty in recovery performance. An important contribution of this work is that it presents a quantitative and systematic procedure to scale-up both rock and flow-related properties. It reinforces the notion that deterministic conditioning data does not exist in reservoir modeling.