Abstract

This work presents the development, validation and applica-tion of a novel deconvolution method based on B-splines for analyzing variable-rate reservoir performance data. Variable-rate deconvolution is a mathematically unstable problem which has been under investigation by many researchers over the last 35 years.We believe that this work is an important addition to existing well test/production data analysis methods where few deconvolution methods are practically applicable.

We use B-splines for representing the derivative of unknown unit-rate drawdown pressure and numerical inversion of the Laplace transform is utilized in our formulation.When significant errors and inconsistencies are present in the data functions, the direct and indirect regularization methods (i.e., mathematical "uniformity" processes) are used.We provide examples of under and over-regularization, and we discuss procedures for ensuring proper regularization.

We validate our method with synthetic examples generated with and without errors (for this work we provide cases with 10 percent error, but we have considered cases with as high as 40 percent error).Upon validation, we then demonstrate our deconvolution method using a variety of field cases — including traditional well tests, permanent downhole gauge data as well as 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.

Objectives

The following objectives are proposed for this work:

  • To develop and validate a new deconvolution method based on B-spline representations of the derivative of unknown constant rate drawdown pressure response (i.e., the undistorted pressure response).

  • To create a practical and robust deconvolution tool that can tolerate relatively large (random) errors in the input rate and pressure functions. The proposed process should also be capable of tolerating small systematic errors in the input functions (via calibrated regularization).

  • To apply this new method to traditional variable-rate/ pressure problems, such as wellbore storage distortion, long-term production data, permanent downhole (pres-sure) gauge data, and well tests having multiple flow sequences.

Introduction

The constant-rate drawdown pressure behavior of a well/reser-voir system is the primary signature used to classify/establish the characteristic reservoir model.Transient well test proce-dures are typically 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 via 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 been long 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 po-tential advantage of variable-rate deconvolution has become particularly obvious with the appearance of permanent down-hole instrumentation.

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