Discrete-Domain Model for Hysteresis and its Application on Porous Rocks

Summary Water-alternate-gas injections for improved recovery, slug-wise CO2 injections for permanent storage, and seasonal storage of natural gas and hydrogen are all processes that lead to cyclic flows of drainage and imbibition in the reservoir. Reservoir simulations of these processes need flow functions with hysteresis, which is typically implemented using correlations and hysteresis loop logic that could be inaccurate, especially for the description of higher-order scanning curves. Typically, measuring enough hysteresis-loop data from core-scale experiments is not feasible, and pore-scale simulation of these relations directly on micro-CT images is computationally demanding. The discrete-domain model (DDM) represents an efficient, physics-based, method to describe hysteresis. DDM divides the porous rock into a set of compartments where each compartment is described by a Helmholtz energy and a local saturation. These energy functions describe a rugged energy landscape where hysteresis occurs due to irreversible jumps across energy barriers separating the local energy minima. For pressure-controlled displacement, the evolution equations of the local saturations are uncoupled, while for saturation-controlled (or, rate-controlled) displacement they are coupled by global saturation constraints where capillary pressure is a Lagrange multiplier. The global coupling leads to fluid redistribution among compartments (cooperative behavior) and pressure fluctuations. Thus far, the DDM has only employed simple phenomenological energy functions. The objective of this work is to explore the applicability of the DDM on realistic data from rock samples. For that purpose, we simulate primary drainage, imbibition, and subsequent scanning loops, on segmented micro-CT images of Castlegate sandstone, using a level set model (LSM) for capillary-controlled displacement. The generated data is used to calculate the energy functions for different compartment architectures in the DDM. From the data we also explore how the interfacial area leads to differences in the energy between drainage and imbibition. We find that the DDM reproduces the capillary pressure curves from both the pressure- and saturation-controlled LSM simulations using the energy functions from the saturation-controlled case. A finer compartment division of the rock sample leads to more energy minima and smoother results. Using the same energy landscape for either drainage or imbibition on all processes (including scanning curves) leads to a slight deviation from the LSM results, whereas the case with different energy landscapes for drainage and imbibition shows excellent agreement. Hence, the DDM emerges as a suitable pore-to-core upscaling approach for hysteresis as its compartmental description is based on extensive properties.