Summary
We use B-splines for representing the derivative of the unknown unit-rate
drawdown pressure and numerical inversion of the Laplace transform to formulate
a new deconvolution algorithm. When significant errors and inconsistencies are
present in the data functions, direct and indirect regularization methods are
incorporated. We provide examples of under- and over-regularization, and we
discuss procedures for ensuring proper regularization.
We validate our method using synthetic examples generated without and with
errors (up to 10%). Upon validation, we then demonstrate our deconvolution
method using a variety of field cases, including traditional well tests,
permanent downhole gauge data, and production data. Our work suggests that the
new deconvolution method has broad applicability in variable rate/pressure
problems and can be implemented in typical well-test and
production-data-analysis applications.
Introduction
The constant-rate drawdown pressure behavior of a well/reservoir system is
the primary signature used to classify/establish the characteristic reservoir
model. Transient-well-test procedures typically are designed to create a pair
of controlled flow periods (a pressure-drawdown/-buildup sequence) and to
convert the last part of the response (the pressure buildup) to an equivalent
constant-rate drawdown by means of special time transforms. However, the
presence of wellbore storage, previous flow history, and rate variations may
mask or distort characteristic features in the pressure and rate responses.
With the ever-increasing ability to observe downhole rates, it has long been
recognized that variable-rate deconvolution should be a viable option to
traditional well-testing methods because deconvolution can provide an
equivalent constant-rate response for the entire time span of observation. This
potential advantage of variable-rate deconvolution has become particularly
obvious with the appearance of permanent downhole instrumentation.
First and foremost, variable-rate deconvolution is mathematically
ill-conditioned; while numerous methods have been developed and applied to
deconvolve “ideal” data, very few deconvolution methods perform well in
practice. The ill-conditioned nature of the deconvolution problem means that
small changes in the input data cause large variations in the deconvolved
constant-rate pressures. Mathematically, we are attempting to solve a
first-kind Volterra equation [see Lamm (2000)] that is ill-posed. However, in
our case the kernel of the Volterra-type equation is the flow-rate function
(i.e., the generating function); this function is not known analytically but,
rather, is approximated from the observed flow rates. In practical terms, this
issue adds to the complexity of the problem (Stewart et al. 1983).
In the literature related to variable-rate deconvolution, we find the
development of two basic concepts. One concept is to incorporate an a priori
knowledge regarding the properties of the deconvolved constant-rate response.
The observations of Coats et al. (1964) on the strict monotonicity of the
solution led Kuchuk et al. (1990) to impose a “nonpositive second derivative”
constraint on pressure response. In some respects, this tradition is maintained
in the work given by von Schroeter et al. (2004), Levitan (2003), and
Gringarten et al. (2003) when they incorporate non-negativity in the “encoding
of the solution.” We note that in the examples given, this concept
(non-negativity/monotonicity of the solution) requires less-straightforward
numerical methods (e.g., nonlinear least-squares minimization).
The second concept is to use a certain level of regularization (von
Schroeter et al. 2004; Levitan 2003; Gringarten et al. 2003), where
“regularization” is defined as the act or process of making a system regular or
standard (smoothing or eliminating nonstandard or irregular response features).
Regularization can be performed indirectly, by representing the desired
solution with a restricted number of “elements,” or directly, by penalizing the
nonsmoothness of the solution. In either case, the additional degree of freedom
(the regularization parameter) has to be established, where this is facilitated
by the discrepancy principle (effectively tuning the regularization parameter
to a maximum value while not causing intolerable deviation between the model
and the observations). In some fashion, each deconvolution algorithm developed
to date combines these two concepts (non-negativity/monotonicity of the
solution or regularization).
© 2006. Society of Petroleum Engineers
View full textPDF
(
7,027 KB
)
History
- Original manuscript received:
17 July 2005
- Revised manuscript received:
10 July 2006
- Manuscript approved:
18 July 2006
- Version of record:
20 October 2006
View Cart
