A tractable, predictive modeling approach for Cold Heavy Oil Production with Sand (CHOPS) remains a significant practical challenge. In CHOPS, instabilities of oil-sand interactions lead to formation of meso-scale structures, such as wormholes or other defects. The sand stress concentration around these structures often leads to sand failure, and production of sand in the process. The failure and motion of sand alter the permeability, and hence affect the oil production. Because these structures result from the instability of the oil-sand interactions, sizes, shapes and locations of these structures are unpredictable. One can only describe them using the statistical methods. Furthermore, in typical well-drainage scale or multi-well-scale numerical simulations these mesoscale structures are typically below the level of resolution and must be modeled as sub-grid-scale effects. Rigorous well-drainage-scale averaged governing equations can be easily derived, but closure models for the sand stress and sand production cannot be adequately developed without proper description of the sub-grid effects.
The sub-grid momentum balance equation for sand demands that the divergence of the sand stress balances the gravity, pressure gradient and oil drag. Since gravity is a constant, and for a given mesoscale structure, the oil flow rate and hence the drag is determined by the imposed macroscopic pressure gradient, without information from sub-grid-scale pressure variation, we assert that the pressure gradient is the only independent variable affecting the stress and failure in sand. This leads to the proposition of closing the well-drainage-scale averaged governing equations in terms of the pressure gradient without explicitly involving the sand stress. Based on this proposition we derive a well-drainage-scale pressure diffusion equation without explicit representation of the sand stress.
To develop the closure models, we can extract information from both explicit full-physics simulations of the sub-grid-scale effects with resolved sand stress and from pilot scale production data. Full physics simulations using the multiphase material point method are shown to adequately model lab-scale wormhole experiments. History matched simulations at the well-drainage scale are shown to track transients in pilot data and are a measure of the success of the modeling approach.