Darcy model with optimized permeability distribution (DMOPD) approach for simulating two-phase flow in karst reservoirs

Darcy model fails to accurately model flow in karst reservoirs because the flow profiles in free-flow regions such as vugs, fractures and caves do not conform to Darcy’s law. Flows in karsts are often modelled using the Brinkman model. Recently, the DMOPD approach was introduced to reduce the complexity of modelling single-phase flow in Karst aquifers. Modelling two-phase flow using the Brinkman’s equation requires either a method of tracking the front or introducing the saturation component in the Brinkman’s equation. Both of these methods introduce further complexity to an already complex problem. We propose an alternative approach called the two-phase Darcy’s Model with optimized permeability distribution (TP-DMOPD) to model pressure and saturation distributions in karst reservoirs. The method is a modification to the DMOPD approach. Under the TP-DMOPD model, the caves are initially divided into zones and the permeability of each zone is estimated. During this stage of the TP-DMOPD model, the fluid inside the reservoir is assumed to be in a single-phase. Once the permeability distribution is obtained, the two-phase Darcy model is used to simulate flow in the reservoir. The example applications tested showed that the TP-DMOPD approach was able to model two-phase flow in karst reservoirs.

Structures and Instabilities in Reaction Fronts Separating Fluids of Different Densities

Density gradients across reaction fronts propagating vertically can lead to Rayleigh–Taylor instabilities. Reaction fronts can also become unstable due to diffusive instabilities, regardless the presence of a mass density gradient. In this paper, we study the interaction between density driven convection and fronts with diffusive instabilities. We focus in fluids confined in Hele–Shaw cells or porous media, with the hydrodynamics modeled by Brinkman’s equation. The time evolution of the front is described with a Kuramoto–Sivashinsky (KS) equation coupled to the fluid velocity. A linear stability analysis shows a transition to convection that depends on the density differences between reacted and unreacted fluids. A stabilizing density gradient can surpress the effects of diffusive instabilities. The two-dimensional numerical solutions of the nonlinear equations show an increase of speed due to convection. Brinkman’s equation lead to the same results as Darcy’s laws for narrow gap Hele–Shaw cells. For large gaps, modeling the hydrodynamics using Stokes’ flow lead to the same results.

A New Two Phase Extension of Modified Brinkman Formulation for Fluid Flow through Porous Media

In porous media research, Modified Brinkman’s equation is a very recent development. It is important as it incorporates the concept of viscous effect to inertial effect in a fluid flow system when Darcy’s, Forchheimer’s and Brinkman’s terms are brought all together. So far, researchers have developed the modified equation in its two-dimensional forms; however, limited to only one phase. In reality, petroleum reservoirs experience the multiphase conditions. Therefore, the simulation of a multidimensional, multiphase scenario is mostly desired, the highlight of this paper. The paper presents the formulation of two-dimensional, transient pressure and saturation equations for oil and water phases, one equation for each phase. The difference between phases is noticeable explicitly in their respective saturation, permeability, viscosity and velocity terms. The equations are then solved numerically to generate relative permeability curves. The simultaneous solution of pressure and saturation terms in the governing equations required additional relationships: the phase saturation constraint and capillary pressure as function of saturation. Finally, the numerical results are compared and validated with the experimental results. The implication of this study is manifold. The formulated equations including the solution part for the multiphase conditions are new. The new comprehensive model will describe fluid flow in reservoirs prone to high velocity or fractures more accurately than ever described by Darcy’s or other aforementioned equations.