Reservoir Engineering: Multi-Phase Fluid Flow

1.  Whether the generalization of Darcy’s law to the simultaneous flow of two immiscible fluids by Wyckoff and Botset (eight years following 1856’s Darcy’s law) remains sufficient enough to bring in capillary effects, although, each immiscible fluid obeys Darcy’s law within the diminished pore-space (where, each immiscible fluid occupies an available space in which it can flow, which consists of the pore space minus the space the other immiscible fluid occupies)?

2.  Even though, mass conservation equation considers an explicit flow rate for each fluid phase (water, oil & gas), how does “the combined flow” react to a given pressure drop @ field-scale?

In such cases, whether, the entire reservoir’s flow regime will uniformly obey the linear relation between flow rate and pressure drop?

What is the guarantee that both the immiscible fluids will tend to percolate @ low flow rates (where, the capillary forces remain to be too strong for the viscous forces to move the fluid interfaces), so that linear Darcy’s law would still prevail?

Won’t the enhanced flow rates lead to a non-linearity between flow rate and pressure drop, resulting from the gradual increase in mobilized interfaces?

If not, then, the two different (say, oil and water) (Newtonian) immiscible fluids flowing @ pore-scale - do no more behave as a (single) Newtonian fluid @ continuum-scale (if the effective viscosities of oil and water remain to depend on shear rate)?

In essence, how do we have a control over the following four different scenarios in an oil reservoir @ field-scale?

(a)          Low flow rates (where, the capillary forces remain to be too strong for the viscous forces to move the fluid interfaces, and thereby, still obeying linear Darcy’s law)

(b)         Moderately higher flow rates (appearance of strong pressure fluctuations, but, still obeying linear Darcy’s law)

(c)          Significantly higher flow rates (where, non-linearity set in, and a power law relation between flow rate and pressure drop gets developed)

(d)         Very high flow rates (where, capillary forces become negligible compared to viscous forces, and, eventually, comfortably following linear Darcy’s law)

3.   Feasible to capture the main mechanism responsible for the non-linearity in the flow-pressure relationship @ laboratory-scale?

If so, then, how could we deduce the threshold pressure (the pressure drop at which the interfaces start getting mobilized), associated with the correlation between time-averaged flow rate and excess pressure drop?

And also, would it remain feasible to capture the movement of the interfaces, which consumes energy, and which, eventually, leads to the reduction in the effective permeability @ lab-scale?

Suresh Kumar Govindarajan

Professor (HAG)

IIT Madras

22-Sep-2024