Critical review on challenges in the application of Darcy's law for multi-phase fluid flow in petroleum reservoirs

The physical description of the transient flow of multi-phase compressible fluid through a petroleum formation should ideally be built on conservation principles of mass and momentum. In practice, however, the extended or modified version of Darcy's law is employed as the foundation of the multi-directional, multi-phase flow of reservoir fluids. Darcy's equation is purely based on experimental studies that describe the flow of single-phase incompressible fluid through a one-dimensional porous medium under steady-state conditions. The original Darcy's law had a single constant of proportionality that remained to depend on both rock and fluid properties. However, petroleum reservoirs are associated with the flow of multi-phase fluid systems. Not only was the capillary pressure concept introduced at the pore scale, but permeability was also assumed to depend on both fluid and rock properties. The original coefficient in Darcy's law has not only been allowed to vary as a function of time and space but also has been allowed to vary non-linearly as a function of rock and fluid properties in petroleum reservoirs. In addition, multi-phases through water, oil and gas make these coefficients more complex. Thus, in petroleum reservoir application, the simple constant of proportionality introduced by Darcy has become a complex function of reservoir fluid, rock and rock-fluid interaction characteristics. This Darcian approach causes mismatches in rock and fluid parameters while up-scaling the sub-pore-state to the Darcian scale. Reservoir simulation in the absence of conceptual and mathematical modelling and machine learning data handling capability has taken precedence over any attempt that employs conceptual and mathematical principles for deducing explicit conservation principles of flow through a petroleum reservoir associated with multiple scales of interest. This article has attempted to deduce some essential assumptions and challenges required in preliminary discussions, which could pave the way for constructing more comprehensive physical models.