Model-reduced gradient-based history matching

Abstract

Since the world's energy demand increases every year, the oil & gas industry makes a continuous effort to improve fossil fuel recovery. Physics-based petroleum reservoir modeling and closed-loop model-based reservoir management concept can play an important role here. In this concept measured data are used to improve the geological model, while the improved model is used to increase the recovery from a field. Both problems can be formulated as optimization problem, i.e. history matching identifies the parameter values that minimize an objective function that represents the mismatch between modeled and observed data while production optimization identifies wells controls that maximize the total oil recovery or monetary profit. One of the most efficient class of methods to solve history matching and production optimization problems are gradient-based methods where the gradients are calculated with the use of an adjoint method. The implementation of the adjoint method for parameter estimation and control optimization is, however, very difficult if no Jacobians of the model are available. This implies that there is a need for gradient-based, but adjoint-free optimization methods. A requirement becomes even more pressing if reservoir simulation is combined with another simulation, e.g. simulation of geomechanics or rock physics, with a code for which no Jacobians are available. The research objective of this thesis was to evaluate the performance of a model-reduced gradient-based history matching routine that does not require a difficult implementation and involves the reduction of the reservoir system. Additionally, the use of model-reduced method for production optimization of a reservoir operating under induced fracturing conditions was considered. In history matching problems one deals with a large number of uncertain parameters and very sparse observations, while in the production optimization one controls a large dimensional system by adjusting a limited number of controls. Consequently, the values of many model parameters cannot be verified with measurements due to a relatively few information content present in them, while in the production optimization only a limited part of the system can be indeed controlled. In this thesis we proposed a new method inspired by the results in reduced order modeling (ROM) and system-theoretical concepts of controllability and observability of the reservoir system. The new approach assumes that the reservoir dynamics relevant for history matching or production optimization can be represented accurately by a much smaller number of variables than the number of grid cells used in the simulation model. Consequently, the original (nonlinear and high-order) forward model is replaced by a linear reduced-order forward model and the adjoint of the tangent linear approximation of the original forward model is replaced by the adjoint of a linear reduced-order forward model. The reduced-order model is constructed by means of the Proper Orthogonal Decomposition (POD) method or Balanced Proper Orthogonal Decomposition (BPOD) method. The reduced-order model is not, however, obtained by the projection of the nonlinear system of equations as in the conventional projection-based ROM techniques, but instead it is approximated in the reduced subspace. The conventional POD method requires the availability of the high-order tangent model, i.e. of the Jacobians with respect to the states which are not available. The model-reduced method obtains a reduced-order approximation of the tangent linear model directly by computing approximate derivatives of the reduced-order model. Then due to the linear character of the reduced model, the corresponding adjoint model is easily obtained. The gradient of the objective function is approximated and the minimization problem is solved in the reduced space; the procedure is iterated with the updated estimate of the parameters if necessary. The POD-based approach is adjoint-free and can be used with any reservoir simulator, while the BPOD-based approach requires an adjoint model but does not require the Jacobians of the model with respect to uncertain parameters or controls. At first the model-reduced method was applied to history matching problems and was evaluated based on its computational efficiency and robustness. In order to make a valuable judgment this approach was compared to the classical adjoint-based method, which was available for the estimation of the permeability field. Permeabilities are described at each cell of the model, and therefore they need to be re-parameterized. The KL-expansion was used to reduce the parameters space. The significant reduction of the dimension of the dynamic reservoir model and parameter space made the approximation of the reduced-order system feasible in acceptable computation time. The pressure field required relatively low number of patterns which modeled mostly the changes around the wells. The saturation field required much more patterns and they modeled mostly the moving front of the saturation field. In the first studies simplistic reservoir models were used, for which the model-reduced approach showed to perform very well. The obtained estimates of the permeability field significantly improved compared to the prior fields and gave the acceptable history-matches; the quality of the prediction capabilities of the estimated models were very high and comparable to those obtained by the classical adjoint-based approach. The POD-based method was approximately twice as expensive as the classical approach, but the BPOD-based method was comparable to the adjoint-based method. Moreover, both methods were considerably cheaper than the finite difference approach. These preliminary results were the first applications of the model-order reduction to history matching problems. After this proof of concept, further studies were carried on more complex and larger models. The proposed method was capable to obtain satisfactory match with a computational efficiency about five times lower than the adjoint-based method. Similarly, an improvement in the prediction was obtained. The second problem considered in this research was to apply the adjoint-free methods to production optimization of the reservoir operating under special conditions that required coupling of two simulators and for which the adjoint code is not available. The model-reduced method could not be applied because of a low accuracy of the simulation solution which in case of long time simulations resulted in large approximation errors. Therefore, simultaneous perturbation stochastic algorithm (SPSA) was applied together with the finite difference gradient-based method to solve the production optimization problem. SPSA is a gradient-based method where the gradients are approximated by random perturbations of all controls in once, while the finite difference method approximates the gradients by perturbation of each control separately. Both approaches were very simple to implement, they resulted in the improvement of the production, but they were computationally relatively expensive.