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).
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
9 July 2007
- Meeting paper published:
11 November 2007
- Revised manuscript received:
14 July 2008
- Manuscript approved:
29 July 2008
- Published online:
16 July 2009
- Version of record:
22 December 2009
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