Productivity Index for Darcy and Pre-/Post-Darcy Flow (Analytical Approach)

Abstract

We investigate the impact of nonlinearity of high and low velocity flows on the well productivity index (PI). Experimental data shows the departure from the linear Darcy relation for high and low velocities. High-velocity (post-Darcy) flow occurring near wells and fractures is described by Forchheimer equations and is relatively well-studied. While low velocity flow receives much less attention, there is multiple evidence suggesting the existence of pre-Darcy effects for slow flows far away from the well. This flow is modeled via pre-Darcy equation. We combine all three flow regimes, pre-Darcy, Darcy and post-Darcy, under one mathematical formulation dependent on the critical transitional velocities. This allows to use our previously developed framework to obtain the analytical formulas for the PI for the cylindrical reservoir. We study the impact of pre-Darcy effect on the PI of steady-state flow depending on the well-flux and the parameters of the equations.

Non-Darcy Flow in Hydraulic Flushing Hole Enlargement-Enhanced Gas Drainage: Does It Really Matter?

“…where k is the permeability, mD; μ is the dynamic methane viscosity, Pa•s; and v is the gas velocity, m/s. However, as shown in Figure 2(b), the linear relation holds only for 2 Geofluids a certain range of velocities (and pressure gradients, correspondingly) for laminar flow, and for the velocities above or below some critical values, the flow becomes Forchheimer or pre-Laminar [16]. This nonlinear phenomenon is the so-called non-Darcy flow, which is primarily caused by the continuous deceleration and acceleration of fluid molecules traveling along a tortuous flow path through the interconnected pores and also in the proppant pack [17].…”

“…Meanwhile, to incorporate the non-Darcy effect into numerical simulation more expediently, the Forchheimer equation can be reformulated as Figure 2: A conceptual model for gas transport in the coal seam [15,16]. 3…”

A Comprehensive Review of the Simulation Methods for Analysis at the Pore-scale

“…For this reason, use has been limited (pre-and post-Darcy flows) and often has to be modified, extended, or replaced entirely by considering the suitable theoretical foundations. Therefore, numerous scientists (Bloshanskaya et al, 2017;Das et al, 2018;Dukhan et al, 2014;Kovalchuk and Hadjistassou, 2019;Oostrom et al, 2014;Wu, 2016) have sought to work out alternative methods for the determination of porous media flows, along with the prediction of domain onset or an additional zone termination (Das et al, 2018).…”

A Mixed Finite Element Approximation for Fluid Flows of Mixed Regimes in Porous Media

“…When the Reynolds number is high (Re > 10), there is a deviation from the Darcy law and Forchheimer's equations are usually used to account for it [12], see also in [3,32,33]. On the other end of the Reynolds number's range, when it is very small (Re → 0), the pre-Darcy regime is observed but not fully understood, although it contributes to unexpected oil extraction, and improved oil recovery in petroleum reservoirs, see in [4,11,37,40] and references therein.…”