A mixed formulation for a modification to Darcy equation based on Picard linearization and numerical solutions to large-scale realistic problems

TL;DR

This paper introduces a stabilized mixed formulation for a modified Darcy equation accounting for pressure-dependent viscosity, demonstrating improved stability and applicability to large-scale, real-world problems like oil recovery and CO2 sequestration.

Contribution

It presents a novel stabilized mixed formulation with equal-order interpolation for pressure and velocity, enabling stable numerical solutions for pressure-dependent viscosity Darcy models.

Findings

The formulation is stable and effective for large-scale problems.

Modified Darcy model predicts different, more accurate flow behaviors.

Application to CO2 sequestration shows significant differences from classical Darcy predictions.

Abstract

In this paper we consider a modification to Darcy equation by taking into account the dependence of viscosity on the pressure. We present a stabilized mixed formulation for the resulting governing equations. Equal-order interpolation for the velocity and pressure is considered, and shown to be stable (which is not the case under the classical mixed formulation). The proposed mixed formulation is tested using a wide variety of numerical examples. The proposed formulation is also implemented in a parallel setting, and the performance of the formulation for large-scale problems is illustrated using a representative problem. Two practical and technologically important problems, one each on enhanced oil recovery and geological carbon-dioxide sequestration, are solved using the proposed formulation. The numerical examples show that the predictions based on Darcy model are qualitatively and quantitatively different from that of the predictions based on the modified Darcy model, which takes into account the dependence of the viscosity on the pressure. In particular, the numerical example on the geological carbon-dioxide sequestration shows that Darcy model over-predicts the leakage into an abandoned well when compared to that of the modified Darcy model. On the other hand, the modified Darcy model predicts higher pressures and higher pressure gradients near the injection well. These predictions have dire consequences in predicting damage and fracture zones, and designing the seal, whose integrity is crucial to the safety of a geological carbon-dioxide sequestration geosystem.