Physical Interpretation of the Associated Mathematics on Extended Form of Darcy's Law towards Characterizing Multi-Phase Fluid Flow through Petroleum Reservoirs

Darcy’s Law (Part 01)

1.Applicable for a two-phase immiscible fluid flow through a deformable porous medium?

2.Whether the internal boundaries between pore-spaces and solid-grains would remain to be rigid (or fixed) upon oil production?

3.Whether the entry and exit of a given pore (@ pore-scale) would always remain to be ‘percolating’?

4.Feasible to correlate the magnitude of Darcy-flux with the magnitude of phase-velocity of the interstitial-fluid just by connecting them with average res porosity?

5.How exactly to correlate ‘macroscopic pressure drop’ associated with ‘average res pressure’ with that of ‘Pc @ pore-scale’?

6.Does Darcy’s law hold good for fluid flow with significant flow rate in the absence of unidirectional fluid flow?

7.Whether extended form of Darcy’s law results from volume averaging assumptions? If not, whether, the concept of ‘freezing out phases’ would remain to be relevant in PE applications?

How could we freeze, say, brine flow, in order to merge brine with solid rigid matrix, during oil production so that Darcy’s law for single-phase flow remains applicable to the non-frozen oil flow?

Is it fine to apply Darcy’s law twice, once for brine with frozen oil, and then, once for oil with frozen brine?

Darcy’s Law (Part 02)

1.Whether the extended form of Darcy’s law that correlates Darcy flux as a function of pressure gradient (k & μ) would remain to be valid, if we have significant viscous forces? In other words, whether the extended form of Darcy’s equation applied in Petroleum Reservoir applications remains to be valid, only when, capillary forces remains to be dominant with insignificant viscous forces in a given REV?

2.If so, then, what exactly we mean by the extended form of Darcy’s law, where, the estimation of water & oil velocities becoming a function of macroscopic viscous pressure drops (Δpo & Δpw)?

3.If viscous forces need to be negligible with reference to capillary forces, then, what is the very purpose of using Capillary Number in EOR applications?

4.Further, Ca depends only on velocity, viscosity & IFT. However, in addition to these parameters, the wettability characteristics of the reservoir and the local-scale pore-network geometry also dictates the resulting mobilization of the residual oil by EOR. If so, then, what exactly is the purpose of using Ca in EOR applications?

Further, how could we assume both oil & brine flow rate would remain to be so small (for larger oil/brine velocities, viscous stresses begin to mobilize the interfaces) so that the brine-oil interface could be kept fixed by capillary forces?

Won’t viscous friction disrupt and subsequently destroy the coherence of the fluid-fluid interface, when the fluid phase velocities remain to be larger or significant?

Suresh Kumar Govindarajan

Professor (HAG)

IIT Madras

25-Aug-2024