3D Field-Scale Automatic History Matching Using Adjoint Sensitivities and Generalized Travel-Time Inversion

Abstract

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractConditioning geologic models to production data is generally done in a Bayesian framework. The commonly used Bayesian formulation and its implementation have difficulties in three major areas, particularly for large scale field applications. First, the CPU time increases quadratically with increasing model size, thus making it computationally expensive for field applications with large number of parameters; second, the sensitivity coefficients that define the relationship between reservoir properties and the production response typically depend on either the number of model parameters or the number of data points; and third, the calculation of the prior covariance matrix (or its inverse) can be time consuming and memory intensive.We propose a fast and robust adaptation of the Bayesian formulation for inverse modeling that overcomes much of the current limitations and is well suited for large-scale field applications. Our approach is based on a generalized travel time inversion and utilizes the adjoint method for computing the sensitivity of the travel time with respect to reservoir parameters such as porosity and permeability. The sensitivity calculations depend on the number of wells integrated which can be orders of magnitude less than the number of data points or the model parameters. The adjoint sensitivities can be computed from the pressure and water saturation distribution that is readily available from the numerical simulator. For solving the inverse problem, we utilize an iterative minimization algorithm based on efficient singular value decomposition. Prior information is incorporated using an approximation of the square root of the inverse of the prior covariance calculated using a numerically-derived stencil applicable to a wide class of covariance models. Our proposed approach is computationally efficient and more importantly the CPU time scales linearly with respect to model size making it particularly well-suited for large-scale field applications.We demonstrate the power and utility of our approach using synthetic and field examples. The synthetic examples show the robustness and efficiency of this algorithm. The field example is from the Goldsmith San Andres Unit (GSAU) in West Texas and includes multiple patterns consisting of 11 injectors and 31 producers. Using well log data and water-cut history from producing wells; we characterize the permeability distribution, thus demonstrating the feasibility of our approach for large-scale field applications. m …………….….………..(7) Substituting Eq. 7 in Eq. 6, e J O T = ∇ ) (m m ………………...…….(8) The Hessian is obtained by taking the gradient of Eq. 8 with respect to the model parameter (m): m T T H J J e J = + ∇ …………….(9) For quasilinear problems or near the solution, we can approximate Eq. 9 as follows J J H T ≅ ……………………..(10)This approximation for the Hessian, Eq. 10, is the same as that of the Gauss-Newton algorithm. Now, substituting Eqs.8 and 10 in Eq. 5; e J J J

Estimation of the water–oil–gas relative permeability curve from immiscible WAG coreflood experiments using the cubic B-spline model

“…In contrast to the existing analytical methods, when the numerical inversion methods are adopted to interpret laboratory coreflood data, production performance prior to and after water breakthrough can be utilized comprehensively. The estimated result is not only complete but also highly precise (Daoud and Velasquez 2006;Barroeta and Thompson 2010). In recent decades, a variety of numerical inversion methods have been developed to implicitly estimate the relative permeability curve for water-oil or oilgas systems (Chen et al 2008;Li et al 2009;Eydinov et al 2009;Wang et al 2010;Wang and Li 2011;Li and Yang 2011;Abdollahzadeh et al 2011;Zhang and Yang 2013;Xu et al 2013;Miao et al 2014).…”

Look‐ahead vertical seismic profiling inversion approach for density and compressional wave velocity in Bayesian framework

“…For the gradientbased algorithms, it is classified according to its search direction (Nocedel and Wright 1999;Daoud and Vega 2006) into steepest descent: Newton, quasi-Newton and conjugate gradient. The gradient-free algorithms like simulated annealing or genetic algorithms can be computationally prohibitive particularly when many parameters are involved.…”

“…The application of Bayesian inversion was widely used in the context of the geophysical seismic inversion applications (Duijndam 1987;Gouveia and Scales 1998) and in the context of reservoir engineering and simulation (Ogele et al 2006(Ogele et al , 2012Gavalas, Shah, and Seinfeld 1976;Oliver 1994;Reynolds, He, and Oliver 1997;Wu, Reynolds, and Oliver 1999;Li, Reynolds, and Oliver 2001;Zhang et al 2003;Wu, and Datta-Gupta 2002;Daoud, and Vega 2006). In the context of geophysical seismic inversion, the inversion was done for both density and compressional wave velocity between the wells based on three-dimensional (3D) seismic data.…”

Dynamic Penalty Function Evolution Algorithms for History Matching of Oil and Gas Reservoir Models

“…Automatic history matching techniques, however, have evolved significantly over the past decade in the academic literature. Methods that increase the efficiency of history matching can be categorized in four categories: (1) optimization algorithms that find the optimal reservoir parameters such that the mismatch between the measurements and the simulated measurements is minimum [5,32,21,3,20], (2) algorithms that aid history matching through reducing the number of observed data [16,17,30,6,7], (3) algorithms that reduce the number of reservoir parameters (reservoir parameterization) [11,2,12,15], and (4) the use of simple and faster simulators [28,29,14].…”