Deconvolution of Variable-Rate Reservoir Performance Data Using B-Splines
Authors: Ilk Dilhan , Valko Peter P. , Blasingame Thomas A. 
Published: October 2006

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).