AN INEQUALITY CONSTRAINED EXTENDED KALMAN FILTER FOR CONTINUAL FORECASTING OF INTERWELL CONNECTIVITIES IN WATERFLOODS

Liu, et al 1 first developed an Extended Kalman Filter (EKF) for a producer‑centric reservoir modeled as a collection of continuous‑time impulse responses that convert injection rates of all contributing injectors into a production rate. This original EKF model estimates, for each producer‑injector pair, two parameters, that can be used to compute an Injector‑Producer‑Relationship (IPR) value. Zhai, et al 4 then modified the EKF model to directly estimate the square‑root of the IPR value for each producer‑injector pair, and developed a set of heuristic strategies for implementing the EKF model on much more complicated and noisy real field data. In both these works, the IPR values of a whole field are obtained by the EKF processing of each producer independently; however, an important type of injector‑centric constraints needs to be imposed across producer‑centric models, namely, the sum of the IPR values of each injector has to be less than or equal to one. Unfortunately, it is not possible to impose such constraints using the current EKF structure. Recently, D. Simon, et al 2, 3 and Yang and Blasch 5 have shown how to incorporate linear equality and inequality constraints into the structure of a Kalman filter. This paper extends their constrained Kalman Filter for our producer‑centric EKF model so as to handle nonlinear inequalities. For the first time in our works, all of the producer‑centric models are coupled together due to the constraint on the sum of the IPRs ‑ constraint coupling. This paper uses a mini‑simulated field to show how unconstrained EKF processing can cause the sum of the IPRs to be greater than one; and, demonstrates how the constrained EKF can be used to couple the whole field thereby obtaining the correct IPR values.