Reservoir Engineering

Non-Darcy flow; Reynolds Number; Chemical EOR

1. Whether non-Darcy flow behavior in an Oil Reservoir results from the increase of the microscopic viscous force @ high velocity?

2. What is the critical range of Reynolds Number that describe the onset of non-Darcian fluid flow associated with an Oil Reservoir and a Gas Reservoir?

And, what would be the associated errors, in the above cases, if ‘non-Darcy effect’ (the ratio of pressure gradient consumed in overcoming liquid-solid interactions to the total pressure gradient) remains to be ignored?

3. Whether, non-Darcy flow in an Oil Reservoir would remain to be similar to that of a turbulent flow in a pipe (where, Reynolds Number plays a critical role)?

If so, then, what would be (a) the characteristic length (given the complexity of a reservoir’s pore-structure); and (b) the characteristic velocity, associated with Re?

Feasible to deduce particle diameter, in such cases, towards estimating Re?

If not, then, how could we bring in Reynolds Number to distinguish non-Darcy flow from that of a Darcy-flow regime?

If so, then, how to deduce non-Darcy coefficient (and permeability) towards estimating Forchheimer Number (the ratio of the liquid-solid interaction pressure gradient to that by viscous resistance; and that remains directly connected to the non-Darcy error)?

4. What is the fundamental limitation associated with ‘volumetric averaging theory’ – applied by Whitaker (1969) – towards deducing the permeability tensor for the Darcy’s equation (which is an empirical relationship based on experimental observations of one-dimensional water flow through packed sands @ low velocity; and which states that the pressure gradient remains linearly proportional to the fluid velocity in a porous medium)?

📷

Does the term (mew*v/k) on RHS precisely represent the pressure gradient required to overcome viscous resistance in a gas reservoir?

To what extent, the concept of volumetric averaging theory remains related with pore-scale details of capillary pressure, IFT, contact angle and pore-geometry – in the context of chemical EOR applications?

5. How does the work by Hassanizadeh and Gray (1980) on the development of a mathematical model that describes the macroscopic behavior of fluid flow through a porous medium (and, which yields Darcy’s equation @ low velocities upon linearization of their proposed equations) remain to differ from that of Darcy’s equation?

6. When do we require the addition of a second order of the velocity term to take into account the microscopic inertial effect in an EOR application, where, Darcy’s equation gets translated into Forchheimer (1901) equation (that includes a non-Darcy coefficient,β)?

📷

Does the 2nd term on RHS precisely represent the pressure gradient required to overcome fluid-solid interactions?

7. Do we have the requirement of considering the second-order derivatives of the velocity to the Darcy equation in an EOR application, as proposed by Brinkman (1947), which essentially considers the macroscopic shearing effect between the fluid and the pore walls?

Do we have a scenario, in an EOR application, where, the change of velocity across the pore throat remaining to be significant?

📷

8. How about the significance of non-Darcy behavior on well performance?

Whether the presence of non-Darcy effect @ near-wellbore region, would try to mitigate the gas production significantly?

Dr Suresh Kumar Govindarajan,

Professor [HAG]

IIT-Madras 18-March-2025