Ensemble-based method with combined fractional flow model for waterflooding optimization
Authors: Santos Oliveira Dilayne , Horowitz Bernardo , Rojas Tueros Juan Alberto 
Published: January 2021

Proxy models are widely used to estimate parameters such as interwell connectivity in the development and management of petroleum fields due to their low computational cost and not require prior knowledge of reservoir properties. In this work, we propose a proxy model to determine both oil and water production to maximize reservoir profitability. The approach uses production history and the Capacitance and Resistance Model based on Producer wells (CRMP), together with the combination of two fractional flow models, Koval [Cao (2014) Development of a Two-phase Flow Coupled Capacitance Resistance Model. PhD Dissertation, The University of Texas at Austin, USA] and Gentil [(2005) The use of Multilinear Regression Models in patterned waterfloods: physical meaning of the regression coefficient. Master’s Thesis, The University of Texas at Austin, USA]. The proposed combined fractional flow model is called Kogen. The combined fractional flow model can be formulated as a constrained nonlinear function fitting. The objective function to be minimized is a measure of the difference between calculated and observed Water cut (Wcut) values or Net Present Values (NPV). The constraint limits the difference in water cuts of the Koval and Gentil models at the time of transition between the two. The problem can be solved using the Sequential Quadratic Programming (SQP) algorithm. The parameters of the CRMP model are the connectivity between wells, time constant and productivity index. These parameters can be found using a Nonlinear Least Squares (NLS) algorithm. With these parameters, it is possible to predict the liquid rate of the wells. The Koval and Gentil models are used to calculate the Wcut in each producer well over the concession period which in turn allows to determine the accumulated oil and water productions. To verify the quality of Kogen model to forecast oil and water productions, we formulated an optimization problem to maximize the reservoir profitability where the objective function is the NPV. The design variables are the injector and producer well controls (liquid rate or bottom hole pressure). In this work the optimization problem is solved using a gradient-based method, SQP. Gradients are approximated using an ensemble-based method. To validate the proposed workflow, we used two realistic reservoirs models, Brush Canyon Outcrop and Brugge field. The results are shown into three stages. In the first stage, we analyze the ensemble size for the gradient computation. Second, we compare the solutions obtained with the three fractional flow models (Koval, Gentil and Kogen) with results achieved directly from the simulator. Third, we use the solutions calculated with the proxy models as starting points for a new high-fidelity optimization process, using exclusively the simulator to calculate the functions involved. This study shows that the proposed combined model, Kogen, consistently generated more accurate results. Also, CRMP/Kogen proxy model has demonstrated its applicability, especially when the available data for model construction is limited, always producing satisfactory results for production forecasting with low computational cost. In addition, it generates a good warm start for high fidelity optimization processes, decreasing the number of simulations by approximately 65%.