A New Method for Evaluating the Productivity Index of Nonlinear Flows

Abstract

Summary This paper addresses the effects of nonlinearity of flows on the value of the productivity index (PI) of the well. Experimental data show that, during the dynamic process of hydrocarbon recovery, the PI stabilizes to some constant value, which, in general, is a nonlinear function of both the pressure drawdown and the production rate. Linear Darcy flow is well understood, and excellent approximate formulas are available for the PI in various well/reservoir geometries. To handle the more realistic nonlinear situation, the current practice is to solve the nonlinear problem multiple times for different values of production rate and then add ad-hoc corrective parameters in the linear formulas to reproduce the nonlinear nature of the flow. In this paper, we propose a rigorous framework to measure the PI of a well for nonlinear Forchheimer flows. Our approach, based on recent progress in the modeling of transient Forchheimer flows, uses both analytical and numerical techniques. It provides, for a wide class of reservoir geometries, an accurate relation between the PI for nonlinear Forchheimer flows and the PI for linear Darcy flows. The proposed method of building look-up tables and analytical formulas serves as an effective tool for fast PI evaluation in nonlinear cases. Introduction The PI is one of the more basic characteristics of well performance not requiring assumptions about the equations of the flow motion and the state of the fluid. The concept of PI expresses the following: Once the well production is, in some sense, stabilized, then the ratio between the production rate and the pressure drawdown (difference between the reservoir average pressure and the well average pressure) is practically independent from the production history or even from the operating conditions (Muskat 1937; Dietz 1965; Dake 1978; Raghavan 1993; Larsen 2001). The higher the value of the PI is, the better the performance of the well in the reservoir is. We are particularly interested in the asymptotic (late-time) value of the PI. For an isolated reservoir with no-flow condition on the outer boundary and constant production rate, stabilization of the PI means that the pressure drawdown becomes time invariant; this flow regime is called pseudosteady state (PSS). In the case of constant wellbore pressure, both the production and reservoir pressure change in time, but the PI asymptotically stabilizes to a constant value, leading to the flow regime called boundary-dominated (BD) (Dietz 1965; Dake 1978; Raghavan 1993; Larsen 2001).

Long-term dynamics for well productivity index for nonlinear flows in porous media

“…The original PDE system (1)-(3) can be reduced to a scalar linear second order parabolic equation for the pressure only. Namely, substituting (1) and (2) in (3) one gets…”

Structural stability of generalized Forchheimer equations for compressible fluids in porous media

“…Let g 1 = g(s, a (1) ) and g 2 = g(s, a (2) ) be two functions of class FP(N, α). Let p k (k = 1, 2) be the solution of (5.14) ∂p…”

Geometric methods in the analysis of non-linear flows in porous media

“…It represents the production rate versus the pressure drawdown (the difference between averages in the domain and on the boundary Γ i ), and is independent of time. This quantity is called Productivity Index, and it is widely used by engineers to test the performances of a well/reservoir system (see [4,3,15]).…”

“…Following [24,4,1,3] and references herein we have established three types of generalized nonlinear Darcy law, with permeability tensor depending on the gradient of the pressure function. We will show that these constitutive equations correspond to a generalized Forchheimer equation.…”

“…where N is the outward normal vector on the boundary Γ = Γ i ∪ Γ e . Other types of boundary conditions on the exterior boundary are discussed in [4]. The accessible boundary Γ i models the well and defines the regime of filtration inside the domain.…”