Parameter estimation to infer injector-producer relationships in oil fields: From hybrid constrained nonlinear optimization to inference in probabilistic graphical model

In petroleum community, the oil field optimization, i.e., minimizing the operational cost and maximizing the oil recovery, is a challenging problem. One of the popular oil recovery techniques is waterflooding, which injects water into the oil reservoir to extract oil. Thus, the knowledge about injector-producer relationships (IPRs), i.e., which injectors contribute to the production of which producers, is a key for field optimization. The difficulty associated with field optimization is that the underlying structure of oil fields is unknown and it continuously changes over time. Recently, a capacitance-resistive model (CRM) has been proposed to investigate the IPRs. The CRM is a predictive model that predicts production rates given water injection rates. It consists of two sets of parameters, connectivity and time constant. The connectivity parameter quantifies the contribution of an injector to the production of a producer and the time constant parameter does the degree of fluid storage between the two wells. Three constraints are posed on the two sets of parameters: (1) the sum of the connectivity parameters of an injector to all other producers should be 1; (2) individual connectivity parameters should fall between 0 and 1 (this implies a nonnegativity constraint as well); and (3) nonnegativity constraint that the time constant parameters should be positive real numbers. Estimating parameters is mapped to a parameter estimation problem for continuous nonlinear systems with constraints. The challenge is that an oil reservoir consists of hundreds of injection and production wells, with which the number of CRM parameters increasing quadratically.;We propose two different approaches for the parameter estimation problem. First, we propose a new constrained nonlinear optimization method. The parameter estimation problem for the CRM is mapped to a large-scale constrained nonlinear optimization problem. Nonlinear optimization is challenging due to the nonconvex property of the objective function and/or constraints. Identifying a global optimum is extremely hard even without constraints. The curse of dimensionality makes the problem more difficult to solve: as the number of parameters increases, the number of local optima is likely to increase. Moreover, there is no analytical solution for constrained nonlinear parameter estimation problems due to the constraints. For the large-scale optimization problem, we developed a hybrid constrained nonlinear optimization (HCNO) method, which is based on sequential quadratic programming (SQP) with a quasi-Newton method in line search framework. The constrained nonlinear time constant parameters are estimated by the SQP so that the constrained nonlinear system is converted to a constrained linear system. And then, the constrained linear connectivity parameters are estimated by the constrained linear least squares (CLLS) method using a projected Hessian. The coupled optimization is repeated until convergence. The experimental results show that HCNO outperforms the base SQP method in terms of search time and prediction accuracy, i.e., faster convergence to better solutions.;Second, we developed a factor graph-based probabilistic inference approach. The proposed method is based on greedy forward search and belief propagation in a factor graph for the learning of the graph structure and parameter estimation. To initiate the node and edge search, an initial factor graph is created by introducing an inductive bias, the locality principle. The graph structure is learned on the fly guided by belief discrepancies, estimation error, and correlation analysis iteratively. At each round, the partially constructed graph is the input for the learning process at the next round. The observed data is exploited as evidences by local propagation. To reduce the estimation error caused by the discretization of the continuous search spaces for the CRM parameters, the expected values are computed for the parameters. This learning process is repeated until a stopping criterion is satisfied. Experimental results show that the elapsed time for the message passing and the node and edge search is less than the search time of HCNO for a large number of parameters, with the prediction accuracy comparable to that of the optimization method. It is shown that the messages converge to a stable equilibrium as time goes on in the loopy belief probabilistic system as advocated in [46] and the approximation results in the high prediction accuracy.