Summary
Based on a linearized model for the isothermal flow of a single,
compressible phase through a reservoir of arbitrary shape with impermeable or
constant-pressure boundaries and spatially varying, anisotropic rock
properties, we develop a multiwell extension of the superposition principle and
re-examine the question of reciprocity between wells that may be modeled as
point sinks or as extended sinks. In the latter case, we find that the answer
depends on the wellbore boundary conditions: Reciprocity holds for infinite
conductivity wells but fails to hold for spatially uniform sink strength. We
also derive a multiwell generalization of the fractional transformation in the
Laplace domain, which adds skin and wellbore storage to a reservoir model, and
find that its impact on reciprocity is neutral: It preserves reciprocity if it
holds for the reservoir model.
Introduction
Data from interfering wells have been a long-standing challenge for well
test analysis. The challenge is more acute than ever as multiple active wells
per reservoir compartment are now the norm in optimized production plans; it is
compounded by the trend toward more complex well trajectories (see Fig. 1 for
an example). The central signal processing task remains to estimate the
rate-normalized pressure drop and its time derivative (Bourdet et al. 1983,
1989) for each well in response to its own production as well as to that of the
other wells. As in the case of a single well (van Everdingen and Hurst 1949),
this is a deconvolution problem.
A recent study (Levitan 2007) showed how the same principles that proved
successful in single-well deconvolution (von Schroeter et al. 2004; Levitan
2005) can be extended to multiple wells. This study assumed reciprocity between
wells (i.e., that the rate-normalized pressure drop at one well in response to
production at another is the same as vice versa), which halves the number of
interference signals to be estimated. For wells modeled as point sinks (such as
fully penetrating vertical wells in a conducting layer of constant thickness),
reciprocity follows from the symmetry of Green's function in its spatial
arguments, a fact established by several authors (McKinley et al. 1968; Deng
and Horne 1993) under various physical assumptions.
However, for extended sinks (such as horizontal, inclined, curved, and
fractured wells) the picture is more complicated, as we show in this paper.
Based on a linearized model for the flow of a single, compressible phase
through a reservoir of arbitrary shape with spatially varying permeability
tensor, we derive multiwell extensions of the superposition principle and
deduce the symmetry of Green's function, which establishes reciprocity at the
level of point sinks for a wider class of reservoir models than hitherto
considered, and by a simplified mathematical route. For extended wells, we find
that reciprocity depends on the boundary conditions: It holds for infinite
conductivity wells but fails to hold for spatially uniform sink strength.
Moreover, the fractional transformations applied in the Laplace domain to add
skin and wellbore storage to a reservoir model preserve reciprocity if it holds
for the reservoir model.
As our investigation relies heavily on Green's functions and related
mathematical concepts, we illustrate the methodology with a simple yet
instructive analytic example.
© 2009. Society of Petroleum Engineers
View full textPDF
(
314 KB
)
History
- Original manuscript received:
8 August 2007
- Meeting paper published:
11 November 2007
- Revised manuscript received:
30 June 2008
- Manuscript approved:
7 July 2008
- Published online:
23 July 2009
- Version of record:
28 September 2009
View Cart
