Summary
Determination of the influence function of a well/reservoir system from thedeconvolution of wellbore flow rate and pressure is presented. Deconvolution isfundamental and is particularly applicable to system identification. A varietyof different deconvolution algorithms are presented. The simplest algorithm isa direct method that works well for presented. The simplest algorithm is adirect method that works well for data without measurement noise but that failsin the presence of even small amounts of noise. We show, however, that amodified algorithm that imposes constraints on the solution set works verywell, even with significant measurement errors.
Introduction
In reservoir testing, we generally know the characteristic features of thesystem from its constant-flowrate and constant-pressure behavior. Thus it isimportant to determine the constant-rate or -pressure behavior of the systemfor the identification of its characteristic features. For instance, identification of a one-half on a log-log plot of the pressure data mayindicate a vertically fractured well, as two parallel straight lines on a Homergraph may indicate a fractured reservoir. The presence of eitherwellbore-storage or flow-rate variations, however, usually masks characteristicsystem behavior, particularly at early times. For many systems, it is desirableto have a wellbore pressure that is free of wellbore-storage and/orvariable-flowrate effects to obtain information about the well/reservoirgeometry and its parameters. For example, the effects of partial penetration, hydraulic fractures, solution gas within the vicinity of the wellbore, gas cap, etc., on the wellbore pressure can be masked entirely by wellbore storage, flowrate variations, or both.
Although deconvolution of pressure and flow rate has not been commonly usedfor reservoir engineering problems, one can still find a few works ondeconvolution (computing influence function) in the petroleum engineeringliterature. Hutchinson and Sikora, Jones et al., and Coats et al. presentedmethods for determining the influence function directly from field data.
Jargon and van Poollen were perhaps first to use the deconvolution ofwellbore-flowrate and pressure data to compute the constant-rate behavior (theinfluence function) of the formation in well testing. Bostic et al. used adeconvolution technique to obtain a constant-rate solution from a variable-rateand -pressure history. They also extended the deconvolution technique tocombine production and buildup data as a single test. Pascals also usedproduction and buildup data as a single test. Pascals also used deconvolutiontechniques to obtain a constant-rate solution from variable-rate (measured atthe surface) and -pressure measurementof a drawdown test. Kucuk and Ayestaranpresented several deconvolution methods including the Laplace transform andcurve fit. Thompson et al. and Thompson and Reynolds presented a stableintegration procedure for deconvolution.
This paper focuses on deconvolution methods. Mathematically, thedeconvolution operation can be defined as obtaining solutions forconvolution-type, linear, Volterra integral equations. In reservoir testing, itis defined as determining the pressure behavior (in-fluence function orunit-response behavior) of a system fro simultaneously measured downholepressure and flow rate. In other words, deconvolution computes the pressurebehavior of a well/reservoir system as if the well were producing at a constantrate. We call the computed pressure behavior of the system "deconvolvedpressure."
Convolution Integral
The convolution integral, which is a special case of the Volterra integralequations, is widely known for providing techniques for solving time-dependentboundary-value problems. It is also known as the superposition theorem (i.e., Duhamel's principle) and has played an important part in transient well-testanalysis. In recent played an important part in transient well-test analysis. In recent years, there has been more interest in the solution of theconvolution integral in connection with analysis of simultaneously measuredwellbore pressure and flow rate. Although we restrict our discussion mostly todetermining influence functions for the constant-rate case, we do treat theconvolution integral in a general manner. In other words, the influencefunction can also be the solution of the constant-pressure case.
In a linear causal system (reservoir), the relationship between input (thetime-dependent boundary condition that can be either the flow rate or pressure)and output (the system response measured as either the flow rate or pressure)at the wellbore can be described as a convolution operation. We let thequantities measured at the wellbore, above the sandface, be p =wellborepressure, Q =cumulative wellbore production, and q =wellbore flow rate.
The convolution integral is (1)
where
The functions delta p (t) and Q (t) or q (t) are the solutions of thediffusivity equation for the constant-flowrate or -pressure case with orwithout wellbore-storage and skin effects. Although it is usually small, thedeconvolved pressure will always be affected by the wellbore volume between themeasurement point and the sandface because the sandface flow rate is differentfrom the wellbore flow rate, q, when it is measured by the flowmeter at somewellbore location above the perforations.
As shown by Coats et al., the general solutions of the diffusivity equationwith the first and second kind of internal boundary conditions and nonperiodicinitial and outer-boundary conditions, satisfy the constraints
(2)
(3)
(4)
and (5)
when the real time is greater than 1 second and if the diffusivity constant, k/phi mu c, is not very small. Coats et al. used linear programming with theabove constraints to compute K(t) from measured g(t) programming with the aboveconstraints to compute K(t) from measured g(t) and f(t). Here we use theseconstraints to compute the system influence function.
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