Data Assimilation Using The Ensemble Kalman Filter With Emphasis On The Inequality Constraints

Abstract

The reliability of reservoir models generally increases as more data are included during their construction. In the recent past, the ensemble Kalman lter (EnKF) technique has established itself as a viable data assimilation method for models with large numbers of variables. The standard implementation of the EnKF and its variants are being used in a number of disciplines including reservoir engineering.

The standard formulation of the EnKF, however, does not take into account the physical constraints on state variables during the data assimilation step so, the constraint violations are often handled heuristically. When the standard implementation of the EnKF is used for history matching, the resulting updates to reservoir properties

sometimes exceed physical bounds, especially when the problem has highly nonlinear model dynamics, the state variables are non-Gaussian, or the ensemble size is small. The physical constraints often contain valuable information about the system which are critical for the valid initialization and the forecast. A successful enforcing of these constraints during the data assimilation step of the EnKF is necessary for valid estimation of reservoir properties.

In the conventional EnKF approach, phase saturations (in case of black-oil models) and molar densities (in case of compositional models) are often included in the state vector. Unfortunately, the analysis step of the EnKF sometimes results in non-physical values of phase saturations or molar densities. In this dissertation, we illustrate the problem of using the standard EnKF with a compositional model in which the updated CO2 molar density in some regions of the model space is observed to take negative values while molar densities of the remaining components are increased. A number of solutions to the problem of constraint violation have been proposed in the past including the iterative EnKF, transformation of state variables, reparameterization, and truncation. The standard truncation schemes avoid negative values of molar densities, but do not address the problem of increased molar densities of other components. The results can include a spurious increase in reservoir

pressure with a subsequent inability to maintain injection. In this dissertation, we present two dierent methods for incorporating the inequality constraints into the EnKF methodology.

In the first part of this dissertation we present a method for the constrained EnKF (CEnKF) which takes into account the physical constraints on the plausible values of state variables during the data assimilation such that the resulting solution is as close as possible to the unconstrained solution obtained from the standard EnKF, and at

the same time, it lies in the feasible region. The proposed method can be implemented in two dierent approaches, both of which convert the inequality constraints to a small number of equality constraints. The first approach uses Lagrange multipliers to enforce the active constraints. In the second approach, the active constraints are used as virtual observations for calibrating the model parameters within plausible ranges. Applying the CEnKF technique in an iterative manner ensures that the resulting solution is within the limits set by the constraints.

The application of the proposed CEnKF method is successfully demonstrated on a synthetic 1D linear problem, on a synthetic 2D compositional model, and on a highly heterogeneous three-phase

flow reservoir model. The effect of the constraints on mass conservation is illustrated using a 1D Buckley-Leverett flow example. Results show

that the CEnKF technique is able to enforce the non-negativity constraints on molar densities and the bound constraints on phase saturations (all phase saturations must be between a lower and an upper bound), and achieve a better estimation of reservoir properties than is obtained using only truncation with EnKF.

An interior-point method for incorporating the inequality constraints into the EnKF methodology (IPCEnKF) is presented in the second part of this dissertation. In this approach, the objective function for data assimilation is reformulated by adding a barrier function to penalize proximity of the state variables to the boundaries of the feasible region and to set a barrier on the state variables against leaving the feasible region. By doing so, the original constrained optimization problem is transformed

into an unconstrained optimization problem. We present the solution of the resulting unconstrained problem in the form of a new iterative EnKF scheme which implicitly contains inequality constraints on the state variables. The proposed IPCEnKF method is efficient compared to the CEnKF as it does not require the identication of the active constraints. Although the method is iterative to reduce the effect of the barrier term at each assimilation step, the iterations do not require running the

simulator.

The IPCEnKF method is successfully tested first on a 1D linear example to illustrate the performance when nonlinearity is not an issue, then on a more realistic 3D, three-phase reservoir flow assimilation problem based on the modied SPE9 model. Results from the reservoir problem show the effectiveness of the newly proposed IPCEnKF method in matching the observations and honoring the inequality constraints

on phase saturations. The proposed method is able to achieve a better estimate of reservoir properties than is obtained using only truncation with the standard EnKF.