Multi-Phase Fluid Flow through Petroleum Reservoirs

Extension of Darcy's Law

1. Inherent Instability of Gas-Oil-Water Flows?

2. Lack of Producibility of Controlled Experiments?

3. Challenging to Model Virtually @ Any Scale?

In EOR applications,

{where, we already have lost 'connected pathway flow' (where, oil/gas & water use to flow through separate and uninterrupted pathways)}'

are we trying to focus on all of the below mentioned pore-scale dynamics?

(a) small-ganglion dynamics (brine remains continuous, while oil remains discontinuous and oil flows in the form of ganglia, with ganglion-size being smaller than the average pore-size)

(b) large-ganglion dynamics (brine remains continuous, while oil remains discontinuous and oil flows in the form of ganglia, with ganglion-size being larger than the average pore-size)

(c) drop traffic flow (oil still remains discontinuous and flows in the form of drops whose size remains comparable to the average pore-size)

Now, the interesting aspect is, when the fluid flow (particularly oil flow) remains discontinuous, can we make use of PDEs?

If so, then, are we not violating the fundamental assumptions associated with the application of basic Differential Calculus?

In other words, having known the cause for the hysteric behavior of the Relative Permeability curves {that corresponds to the transition from connected oil pathway regimes at least until the end of water-flooding to an intermittent regime (ganglion-dynamics) at the beginning of an EOR application}, how could we apply PDEs in the absence of continuous fluid (oil) flow (where, we not only have significant changes in oil-phase connectivity but also, the generation of intermittent structures of the oil-phase ganglia)?