Generalized Approach for Modeling the Nonlinear Hysteretic Response of Reservoir Media

Significant progress has been made recently in the numerical simulation of heterogeneous reservoir media. One of the fundamental reasons for the hysteretic nonlinear behavior of porous reservoir media is that heterogeneous or damaged materials contain an enormous number of mesoscopic features such as microcracks and macrocracks, joints, and grain to grain contacts containing multiple phases. Each of these mesoscopic units exhibits a hysteretic behavior which dominates the macroscopic reservoir response. Based on the work of Preisach and Mayergoyz (P-M space model), Coleman and Hodgdon, a generalized phenomenological model has been developed to describe the hysteretic nonlinear response of capillary pressure and relative permeability. The proposed approach enables the description of active hysteresis by the solution of either a differential or an integral equation. The model focuses on the correct representation of the primary drainage (forced), the imbibition (spontaneous and forced), the secondary drainage (spontaneous and forced) curves and scanning curves. The functions and parameters used in the model can be fine-tuned to match different experimental data or can be used as history matching parameters. It is shown that the proposed model incorporates the Killough type hysteresis as one analytical solution. The differential form of the proposed model allows a smooth transition of both relative permeability and capillary pressures from drainage-to-imbibition or imbibition-to-drainage states and requires minimum storage of parameters during the simulation. Validation of the model indicates that the proposed hysteresis model is stable and robust. The results are presented and discussed and future studies outlined.