Reliability of Horizontal Well Performance On a Field Scale Through Automatic History Matching
Authors: Kalogerakis N. , Tomas C. 
Published: March 2010

Abstract
An efficient procedure for the determination of the error bounds (95% confidence intervals) of forecasted horizontal well production rates is presented. In particular, we advocate the use of automatic history matching (based on non-linear regression) to arrive at the best reservoir parameter values (typically porosities and permeabilities) for a given reservoir structure (i.e., for a given grid block representation). If the Gauss-Newton-method is used for the parameter search, the sensitivity coefficients required for the estimation of the standard error of the parameters and the well production rates are computed as part of the parameter search algorithm. Using typical techniques from statistical analysis (and the statistical properties of multivariable non-linear regression estimates), one can readily compute the confidence intervals of the predicted production rates. These confidence intervals around the expected reservoir behaviour incorporate the uncertainty in all reservoir parameters. As a result, a realistic measure of the uncertainty in the model predictions of future reservoir performance is obtained. The proposed procedure is illustrated with a case study.

Introduction
The final task of a reservoir simulation study is often the computation of future production levels of the history matched reservoir under alternative depletion plans. Moreover, the reservoir engineer often examines the sensitivity of the forecasted performance to different reservoir descriptions in an effort to establish the risks associated with a particular depletion plan. Typically, this is accomplished by varying key reservoir parameters such as porosities and permeabilities and determining how much the performance of a particular well or of the whole reservoir is affected. With the continuous development of powerful workstations and desktop personal computers, a renewed interest in automatic history marching procedures can be observed in the past few years. For the minimization of the resulting least squares (LS) objective function, two different approaches have been widely used: nonlinear regression(1–6) and optimal control methods(7–11). Recently, the simulated annealing(12) method has also proven to be a reliable procedure. In practical applications some form of reguralization(13,14) is always required to overcome the very serious problem of ill-conditioning, particularly as the number of parameters increases to more than ten or so. In our approach to automatic history matching, we minimize the resulting nonlinear LS objective function using an efficient implementation of the Gauss-Newton method(1).
The necessary computation of the sensitivity coefficients when he Gauss-Newton method is used enables us to readily determine the uncertainty in the estimated parameter values(2). Furthermore, if we compute the sensitivity coefficients at future times where the performance of the reservoir is forecasted, we can generate 95% confidence intervals of the model predictions. The confidence intervals around the expected reservoir behaviour incorporate the uncertainty in all reservoir parameters and hence this approach provides a more realistic measure of the uncertainty in the model predictions of future reservoir behaviour(15).
When several different grid representations of the reservoir are available with a known probability of being correct, the above described approach yields conditional expectations of future production rates which can subsequently be combined to arrive at an overall probability distribution of key variables, like cumulative oil production, WOR, GOR, etc.