We present a semianalytic method for modeling the productivity testing of
vertical, horizontal, slanted, or multilateral wells. The method is applicable
to both oil and gas reservoirs and automatically accounts for well
interference. The use of analytic expressions ensures that short-time transient
behavior and long-time semisteady-state behavior are handled appropriately,
whether close to the well or further into the reservoir. Calculation times are
still very limited—on the order of a few minutes to a few seconds when all
wells are vertical. This makes the tool suitable for evaluating well testing
and determining well productivity.
We based the approach on an earlier derived productivity prediction tool, in
which the steady-state equations were solved. It has now been extended to solve
the time-dependent diffusion equation. In our current method, the equations
have first been transformed using the Laplace transformation. The expressions
for the producing wells are combined with auxiliary sources outside the
reservoir. The crux of the semianalytic method involves an adjustment of the
positions and strengths of these sources in order to approximate the boundary
conditions at the reservoir boundaries. The solution obtained is transformed
back into the time domain by use of a Stehfest algorithm.
The new approach has been validated with numeric tools, including both
reservoir simulators and well-test interpretation software. Validations were
performed with artificial cases and with field production data, using both
single-well and multiple-well production tests. The results of these tests were
In a previous paper, we presented a novel semianalytic method to calculate
productivities of complex wells in steady-state or pseudosteady-state
conditions (Fokker et al. 2005b). The method is very flexible in that it is
able to combine finite-conductivity wells, well interference, nonhomogeneous
reservoirs, and finite-conductivity natural or hydraulic fractures. Innovative
well geometries and completion, including variably spaced producing intervals,
can be taken into account to estimate the well productivity under any flow
conditions. Despite this, calculations with the new approach are fast.
The methodology that we propose can be classified as an example of the
method of fundamental solutions, as reviewed by Fairweather and Karageorghis
(1998). A similar method has been developed to predict subsidence caused by
exploiting hydrocarbon reservoirs in a layered subsurface (Fokker and Orlic
2006). For a review of other available fast models for predicting productivity,
see Fokker et al. (2005b) and the references provided therein.
Our new method is suitable for incorporation into a reservoir simulator. It
can also be applied as a fast model to evaluate completion scenarios for a
given reservoir (e.g., in an automated decision support environment). In this
paper, we present a refinement of the method, for describing the pressure—and,
thus, the evolution—of well productivity under transient conditions.
The available literature on well testing is extensive, and any summary
within the context of this paper is bound to be incomplete. The interested
reader is therefore referred to Matthews and Russell (1967), Earlougher (1977),
Lee (1982), Bourdet (2002), Horne (1995), Zheng and Corbett (2005), and
Gringarten (2008) for an overview of the currently available interpretation
methodology. For the testing of horizontal wells or multilaterals (including
interference between wells or well branches), we also refer the reader to
select papers: Ozkan et al. (1989), Kuchuk (1995), Basquet et al. (1998), Ding
(1999), Al-Khamis et al. (2001, 2005), Kuchuk and Onur (2003), Yildiz (2003),
Shimamoto (2006), and Medeiros et al. (2006). Most of the approaches in this
field use expressions that are, in principle, exact solutions of the
boundary-value problem. Shimamoto (2006) uses a totally different approach, by
combining a streamline simulation and a mapping of the physical space on a
radially symmetric domain to obtain simple, approximate "S-functions"
that represent the transient response of the well. Our approach is more like
the approach of Furman and Neuman (2003), who use "analytic elements"
(i.e., analytic functions), in such a way that the boundary or interface
conditions are approximated.
The method we propose can be applied to determine the well productivity of
any type of well in the short or long term; in the latter case, the method’s
reliability can be improved by fine-tuning the model with the transient
pressure response obtained from a well test.
The diffusion equation is transformed with the Laplace transform. In Laplace
space, the solution is obtained similarly to the original, pseudosteady-state
solution. Because no analytic solution to the transformed diffusion equation of
a finite well segment in 3D is available, a numeric integration has to be used.
A back transformation with the Stehfest algorithm completes the approach.
We will start with a brief mathematical description of the new approach.
Then we will present a validation of the method for oil and gas wells of simple
geometry by comparison with the results obtained with a reservoir simulator and
with well-testing software. The usefulness of the code will be demonstrated
with some applications to data obtained in real well tests and a comparison
with a well-testing software package. We complete the paper with a discussion
© 2008. Society of Petroleum Engineers
View full textPDF
- Original manuscript received:
21 March 2005
- Meeting paper published:
13 June 2005
- Revised manuscript received:
11 December 2007
- Manuscript approved:
27 January 2008
- Version of record:
20 June 2008