Summary
In this work, we present a simple analytical formula for the stabilized
productivity index (PI) of an arbitrary well in an arbitrary closed, naturally
fractured reservoir that can be modeled as a double-porosity reservoir. The
formula relates the PI to the PI of a well in a single-porosity reservoir with
permeability equal to the effective fracture permeability of the
double-porosity reservoir. We have elaborated this formula for a vertical and a
horizontal well in a rectangular reservoir made up of orthogonal matrix blocks.
It appears that the double-porosity characteristics, along with reservoir flow
capacity and reservoir volume, combine into a single factor, which plays the
same role as the familiar skin factor. For small values of this double-porosity
skin, well productivity is dominated by fracture permeability; for large
values, the controlling parameter is matrix permeability. Inflow-performance
relationship (IPR) curves of wells in double-porosity gas reservoirs can be
created readily with the help of the new PI formula. An example is included
showing the production profiles of a vertical and a horizontal well in a
double-porosity gas reservoir with cubes and square pillars as matrix blocks.
The results of this work are particularly relevant for the reservoir
engineering of naturally fractured gas reservoirs.
Introduction
The productivity of a well is commonly indicated by a PI, which is defined
as the ratio of the production rate to the difference between average reservoir
pressure and well pressure (i.e., pressure drawdown). In the case of
compressible liquid flow in a bounded reservoir, the PI starts out with a
relatively high value but declines rather rapidly down to a constant value
reflecting flow that is dominated by the outer reservoir boundary. This is true
regardless of whether production takes place at a constant rate or at a
constant well pressure. The constant PI value is known as the stabilized PI.
Most, if not all, of the analytical work on stabilized PIs concerned wells in
nonfractured, single-porosity reservoirs.
In this work, we address the stabilized PI of wells in naturally fractured
reservoirs that can be modeled as double-porosity reservoirs. A double-porosity
reservoir comprises two distinct and coupled flow systems: (1) a highly
conductive, continuous fracture network with little storage capacity that
surrounds (2) a system of poorly conductive matrix blocks with a high storage
capacity. Fluid transport to wells occurs exclusively through the fractures,
while the matrix blocks act as sources of fluid that feed into the
fractures.
In the early 1960s, Warren and Root (1963) set down the theoretical basis
for the description of compressible-fluid flow in double-porosity reservoirs.
Their work and later work by others focused on the interpretation of
transient-pressure well tests and as a consequence were restricted mainly to
the early-time transient-flow regime. To the best of our knowledge, no formulas
exist in the open petroleum-engineering literature for the PI of wells in
double-porosity reservoirs in the stabilized-flow regime.
The outline of this paper is as follows. We begin with the general flow
equations for stabilized flow of a slightly compressible fluid with constant
viscosity in double-porosity reservoirs. This leads to a general formula for
the stabilized PI of an arbitrary well in an arbitrary double-porosity
reservoir. We then elaborate this formula for a vertical and a horizontal well
in a rectangular-box-shaped reservoir that comprises orthogonal matrix blocks
with arbitrary aspect ratios. We further show how the PI formula is to be
modified to include wells in fractured gas reservoirs. Finally, we present an
example calculation of the production profiles of a vertical and a horizontal
well in a double-porosity gas reservoir with cubes and square vertical pillars
as matrix blocks.
The results of this work are particularly relevant for the reservoir
engineering of naturally fractured gas reservoirs where stabilized-flow
conditions often prevail during most of their producing lives. The formulas
presented enable quick-and-easy estimations of well productivities and related
sensitivities to the double-porosity characteristics. In addition, the results
may serve as an analytical benchmark for numerical simulations of complex
naturally fractured reservoirs.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
8 March 2007
- Revised manuscript received:
18 February 2008
- Manuscript approved:
23 February 2008
- Version of record:
25 October 2008
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