Gradients Method Constrained by Geological Bodies for History Matching
Authors: Rahon D. , Blanc G. , Guerillot D. 
Published: December 2007

Abstract
Conventional gradient methods have already been applied to reservoir engineering for matching the history of former field performances. The key point of these methods is to select the best areal reservoir zoning for reduction of the amount of reservoir parameters to be identified. In this paper we propose a zoning based on reservoir lithofacies, thus making a more natural than geographical choice.
The gradients of the bottom hole well pressure with respect to the reservoir characteristics to be identified are required by all history matching processes. These gradients are calculated using either a perturbation method or a so-called analytical method. Without any reservoir zoning both methods require as many problem solving jobs as grid blocks.
In this article we propose computing gradients relating to lithofacies starting with a geological model in lithofacies that is generally from a seismic interpretation and pixel or object-based geostatistical simulation we try to quantify the effect of a lithofacies on well pressure response or production history. The influence of a lithofacies is measured through its petrophysical parameters. This method drastically reduces the number of parameters to be identified and allows the use of the natural partition of the reservoir into geological bodies.
The gradient calculation relating to lithofacies has been successfully implemented in an implicit single-phase fluid flow model. This model can be used for well-tests simulations.
By introducing gradients to minimize an objective function that measures the difference between observed and simulated well pressure responses, we can effectively achieve the inversion of petrophysical lithofacies parameters.
Several examples of inversion are given at the end of the article to illustrate the effectiveness of this gradient method.

Introduction
The characterization of a reservoir is a delicate problem for geophysicists, geologists and reservoir engineers. Interpreting production results, interference tests and well-tests plays a very important role in attempting to specify the description of the geologic model and to reduce the attenuating uncertainties.
Some of these interpretation techniques are based on inversion procedures in which, by using a meshed geologic model, an attempt is made to calibrate numerical simulations of field measurements by varying the parameters of this model. The variable parameters are both geometric (position and size of geologic bodies) and petrophysical (value of porosity, absolute permeabilities, and points of relative-permeability curves, etc.).
One way of varying the petrophysical characteristics automatically is to use a method of gradients with respect to these parameters. In a meshed model used by a flow simulator in porous media, the number of parameters is often a multiple of the number of meshes. To reduce the number of these parameters, the reservoir is broken down into influence zones in which they vary uniformly.
This article describes an new gradient method using the description of the reservoir model in lithofacies. This method is particularly well-suited for an approach that is used more and more often. It consists first of all in building a geologic model in terms of lithofacies before quantifying it in petrophysical data.
This approach does away with breaking down the reservoir into zones of the meshed model, by using a natural zoning in geologic bodies and by directly measuring the influence of a lithofacies, through its petrophysical characteristics, on the pressure response of a well-test or on production curves.
A complete description of the gradient method in relation to the lithofacies was presented at ECMOR. We will present here a brief description of this method and more particularly extensions that have been developed for gaussian and log-normal petrophysical distributions associated with the lithofacies. Next application cases are presented to illustrate the efficiency of the algorithm.
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