Summary
We propose a novel well-test for in-situ estimation of relative
permeabilities under two-phase (oil/water) flow conditions. The test consists
of three periods: injection of water into an oil reservoir operating above
bubblepoint pressure, a falloff test, and a producing period. The producing
period is critical because it yields production data that reflect changes in
sandface mobility and thus is highly sensitive to the parameters used to model
relative permeability curves, whereas our results indicate that
injection/falloff pressure data by themselves are not as reliable for defining
relative permeability curves. We have developed optimization code based on the
Levenberg-Marquardt algorithm and coupled it with a commercial reservoir
simulator to obtain a procedure for data analysis wherein the reservoir
simulator is used as the forward model. By matching data by minimization of a
weighted least-squares objective function, we generate estimates of absolute
permeability, relative permeability, and the well skin factor. We show the
method can be applied with either power-law models or B-splines. We introduce a
variable transformation that can be used to ensure that the estimated relative
permeabilities are monotonic and concave up when B-splines are used.
Introduction
Numerous papers from the 1970s and 1980s discuss techniques for the
estimation of relative permeability curves by matching pressure and rate (or
displaced volume) data from laboratory corefloods using an optimization
algorithm to minimize a least-squares objective function (Archer and Wong 1973;
Sigmund and McCaffery 1979; Kerig and Watson 1986; Watson et al. 1988).
Assuming power-law relative permeability curves yields a small number of
parameters to be estimated and a well-conditioned optimization problem.
However, relative permeability curves may not be well represented by power law
models. Because of this, various authors used splines, especially B-spines, to
parameterize relative permeability curves, but then special techniques were
required to regularize the optimization process and ensure that monotonic
relative permeability curves are obtained (Watson et al. 1988).
Although many papers have also proposed using similar techniques to adjust
relative permeability curves by matching long-time production data, our
objective is to generate a procedure to estimate relative permeability curves
from a well test which has a duration of a few hours to a very few days.
Intuitively, to do this successfully, we should generate data that are
sensitive to a wide range of saturations (i.e., generate data similar to those
which can be obtained in a laboratory coreflood). To do so, we propose an
injection/falloff test followed by a production period. The idea is that,
during the flowback period, the sandface will be exposed to a wide range of
water saturations and the associated pressure and phase-rate data or water-cut
data should be sufficient to obtain good estimates of relative permeability
curves.
We compare results obtained from the proposed test with those obtained from
a more standard injection/falloff test. Even though accurate approximate
analytical solutions for the injection and falloff pressure response for radial
flow problems have been presented previously in the petroleum engineering
literature (Abbaszadeh and Kamal 1989; Bratvold and Horne 1990; Levitan 2003;
Peres and Reynolds 2003; Peres et al. 2006), all data considered in this paper
are analyzed by nonlinear regression using a commercial reservoir simulator as
the forward model (IMEX 2000). Of the analytical solutions for the
injection/falloff pressure, the one provided by Levitan (2003) is most general
and most useful as it applies for a multirate injectivity test where the rate,
during one or more of the periods, can be zero to simulate shutin periods for
falloff testing. His solution, however, does not apply during a subsequent
production period and thus cannot be used to analyze the test proposed
here.
We consider both the case in which relative permeabilities are modelled by
power-law formulas as well as a more flexible parameterization based on
B-splines. Parameters defining relative permeability curves are estimated
simultaneously with reservoir absolute permeability and skin factor.
© 2008. Society of Petroleum Engineers
View full textPDF
(
2,605 KB
)
History
- Original manuscript received:
26 July 2005
- Meeting paper published:
9 October 2005
- Revised manuscript received:
19 May 2007
- Manuscript approved:
19 October 2007
- Version of record:
25 February 2008
View Cart
