Aggregating Petroleum Reservoir Interwell Connectivities Using the Generalized Chouqet Integral

Abstract

Abstract In waterflood management, there exists several models to describe a petroleum reservoir for predicting the future production rates using scheduled injection rates. Most of them have the ability to estimate how much the injectors impact some specific producers, namely, the interwell connectivities between the injectors and the producers. Knowing these values not only reduces the cost of water injection, but can also increase the oil production. In this paper, we construct four different models for the interaction between a group of injectors and a producer and then dynamically estimate the parameters of these models, along with the interwell connectivities using an Iterated Extended Kalman Filter (IEKF) and Smoother (EKS). We then use the Generalized Choquet Integral (GCI) to aggregate the estimated interwell connectivities. The GCI is optimized to minimize mean-square errors in future forecasted production rates. This is done by using Quantum Particle Swarm Optimization (QPSO) to search for the optimal set of fuzzy densities which are required by the GCI. Several experiments are conducted to show the improved average performance of our approach on a set of data from a real reservoir. I. Introduction Flooding an oil field with extraneous water has been a widely accepted method for increasing a reservoir's oil recovery since the 1950's. Water is injected into dedicated injection wells strategically located throughout the reservoir, in order to displace the remaining oil towards the producing wells. If properly designed and operated, a waterflood can double the reservoir's oil recovery. In almost all waterflood operations, measured injection and production rates are the most abundant available data. They are considered to be correlated to each other in some very complicated way, and many methods have been previously proposed to infer the interwell connectivities (referred as the "Injector-Producer-Relationship (IPR)?? in this paper) between each producer and its surrounding contributing injectors using only these data. In all those works, the reservoir is modeled as a dynamical system in which the injection rates act as the system's inputs and the production rates are the system's outputs. Heffer et al. [7] used Spearman rank correlations to relate injector-producer pairs and associated these relations with geomechanics. Panda and Chopra [18] used artificial neural networks to determine the interactions between injection and production rates. Albertoni and Lake [1] estimated the interwell connectivity based on a linear model using a multiple linear regression (MLR) method. Yousef et al. [28], [29] improved this work by building a more complex model, named "capacitance model," to describe the relationship between injection and production rates. Lee [12] further generalized the capacitance model to the Distributed Capacitance Model (DCM) by taking into account that the reservoir between some injector-producer pairs is highly heterogeneous or includes some high permeability channels, fractures or faults. Liu and Mendel [13] modeled the reservoir using continuous impulse responses which were characterized as a two-parameter auto-regressive model between each single injector and a single producer. All of these approaches not only can estimate the IPR values using only measured injection and production data, but can also predict future production rates given future scheduled injection rates; however, different approaches focus on different aspects of modeling the reservoir and utilize different estimation methods to estimate the model's parameters. For example, Lee's DCM basically was derived from a total mass balance with compressibility (the same as the capacitance model) and then its parameters were estimated using constrained quadratic programming. Liu and Mendel's model, referred as the LMM in this paper, was developed based on domain expert knowledge, and its parameters were estimated using the Extended Kalman Filter (EKF).