An Efficient Technique for Inversion of Reservoir Properties Using Iteration Method
Authors: Chu Lifu , Komara M. , Schatzinger R.A. 
Published: December 2007

Abstract
In light of the inverse solution theory, an efficient solution procedure has been developed to generate reservoir descriptions conditioned to statistics for rock properties, hard data, and dynamic data. The technique yields realizations for wellbore skin factors at each active well, porosity, and permeability which honor a priori information and dynamic production data. The technique uses inverse solution theory to construct the objective function and uses a gradient method to generate the maximum a posteriori estimates.
Differing from previous work, we derived and implemented a two-loop iteration method to perform the minimization. By using Krylov space based methods to solve the linear part involved in the minimization, the explicit construction of the sensitivity coefficient matrix is avoided.
Complexity analysis indicates that although the limitations suffered by the procedures using GPST or Carter's methods to construct the sensitivity coefficient matrix are completely removed, the new method is as efficient as those procedures. More significantly, the new method can be easily adapted to multi-phase flow conditions.
We also developed a modified procedure for computing realizations using a Chebyshev approximation of the decomposed a posteriori covariance matrix. In this way, the expensive computational cost of construction and decomposition of an a posteriori covariance matrix are avoided. When estimating multiple categories of parameters, as is the case in this study, our new procedure produces much more accurate results than the conventional Chebyshev method.

Introduction
The most commonly encountered and probably the most challenging work in the management of a mature reservoir is to effectively and efficiently implement various feasible IOR techniques and assess the impact of such implementations on the further performance of the reservoir. To this end, an accurate reservoir description is essential. The description should include at least major localized discontinuities such as faults, fractures, and bounding surfaces, as well as facies distributions, rock property distributions within the facies and rock-fluid properties. Due to the complex nature of the multiple scales of heterogeneity inherent to petroleum reservoirs, different production processes may be sensitive to different scales of heterogeneities, therefore, theoretically, we need an infinite dimensional model space to adequately describe the real reservoir. In reality, what we can do, at best, is to generate an equivalent description of the reservoir at a scale that is suitable to the production process in which we are interested. In other words, we partition the reservoir into a proper gridblock system, and then search the discrete reservoir model corresponding to the partitioned system.
Limited by technology and expense, generating reservoir description based on the direct measurements of the reservoir properties is impractical. Therefore, we are forced to perform the task of transferring to model space, via theoretical correlation, the prior information and the information carried in indirect data sets. By definition, such a process is referred to as an inverse procedure.
The information is usually divided into two classes: prior information and dynamic information. The former includes all the phenomenological information and static data; the later includes production data, pressure transient data, tracer testing data, etc.
Although the prior information plays a critical role in the inverse process, both as a mathematical necessity for stabilizing the ill-posed problems and reducing the dimension of the model space and as a requirements for geological and logical consistencies. The reservoir description based on prior information only will usually provide a unrealistic smooth version of the "true" reservoir model with high uncertainty.
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