In this study, we present a history matching framework for oil production forecast based on synthetic and real production data developed using the stochastic Discrete Well Affinity (DiWA) model. With the increase in the complexity of the geological model and the uncertainty in the geological data, it becomes more difficult (sometimes infeasible) to conduct model inversion and production based on conventional technique. To address this problem, we proposed a stochastic DiWA model with unstructured low-resolution mesh to represent the location of wells and reservoir fluid dynamics. With this method, we can efficiently train the forward model based on production data and a stochastic ensemble of property realization. The performance of forward evaluation benefits from the Operator-Based Linearization (OBL) technique and the adjoint method for gradient calculation. Before the model training, a large ensemble size of stochastic DiWA models is generated based on the permeability statistics of the real reservoir, and those models are then filtered using the misfit between the true production data and the DiWA model response. The filtered models have the best fit with the production history of the real reservoir, while they also contain the basic geological information of the real field. The proposed method is tested first on a synthetic data ensemble for production forecast and then applied to a real field. Based on real observations, we use the DiWA model for data quality diagnostic and identify certain flaws in the collected data and model assumptions. Based on these findings, the original assumptions and data observations have been adjusted and the resulting DiWA model was successfully trained. The prediction quality of the trained DiWA model is comparable to conventional simulation techniques based on detailed geological models and has the advantage of a much more efficient and faster ability to update and maintain the subsurface model when continuous updates in production data become available. This study shows that the proposed method can provide the history matching results with high accuracy and low computational costs. Furthermore, the performance of the stochastic DiWA model can be further improved using more comprehensive and geologically constrained initial and boundary conditions.

In realistic field history-matching problems uncertainty parameters are subject to upper- and lower bounds which must be satisfied. Violation of bounds (e.g., using a negative porosity or permeability in a grid-block) may result in unphysical solutions or the failure of simulations. The Gauss-Newton (GN) optimizer using a trust-region (TR) search method performs more efficiently and robustly than using a line-search method. The GN trust-region search optimizer requires solving a trust-region subproblem (GNTRS) iteratively. Given gradient and Hessian evaluated at the current best solution, the objective function can be approximated by a quadratic model of the search step. The global minimum of the quadratic model within a ball-shaped trust-region, which is the solution of the GNTRS, is used as the new search step for the next iteration. However, available methods to solve a GNTRS cannot correctly handle bound constraints.

This paper introduces an iterative dimension-reduction procedure to solve the GNTRS with bound constraints, which involves the following three steps. First, an unconstrained GNTRS with n variables is solved and the solution is accepted if no bound is violated. Otherwise, at least one bound is violated, and the dimension of the problem is reduced to m by activating one or more violated bounds, according to the Karush-Kuhn-Tucker (KKT) conditions. Second, the gradient, Hessian, and trust region size are updated in the reduced subspace accordingly. Third, an unconstrained GNTRS with m variables is solved in the reduced subspace. We repeat the last two steps until no bound is violated. To achieve better performance, we devised several algorithms to update gradient, sensitivity matrix and Hessian in the reduced subspace adapted to the problem type: (1) using the full Hessian expression to solve the GNTRS directly for problems with more observed data, (2) applying the matrix inversion lemma for problems with the regularization term and with fewer observed data, and (3) applying the linear transformation approach for problems without the regularization term and with fewer observed data.

The proposed new GNTRS solver is first validated on different synthetic problems with known solutions and then tested on a suite of realistic field history matching problems. Our numerical tests confirm that the newly proposed GNTRS solver outperforms other methods for handling bound constraints. In our testing the new solver finds the correct solutions in all cases – with the least CPU time – while other methods failed for some test problems.

Advances in applications of reservoir management workflows have shown the value of closed-loop optimization in leveraging the learning from measurements gathered during operations to improve subsequent operational decisions. Closed-loop workflows rely on the combination of optimization and history matching procedures, which both can involve a very large number of reservoir simulations when employing ensemble-based methods for uncertainty quantification. This renders such approach unfeasible in many real-life applications which involve large-scale models. The problem is amplified in more advanced workflows where the closed-loop calculations must be repeated several times (e.g., value of information approaches to assess and optimize the effectiveness of monitoring strategies).

Recent developments in direct forecasting techniques such as data-space inversion (DSI) have shown promising results to alleviate the computational burden associated with the generation of ensemble of simulated forecasts conditioned to measurement data and their use in optimization workflows. In this work, we present an implementation of the DSI framework using the ES-MDA method available within a mature open-source data assimilation tool suitable for large-scale reservoir applications. The developed workflow utilizes machine learning techniques to better handle the presence of non-linearities typical of real-life applications (e.g., well shut-ins) and also accounts for the variability of well controls to enable the use of the forecasts for well control optimization purposes.

We demonstrate the workflow with two realistic synthetic case studies. In the first case, we illustrate the forecasting of the extent of the plume of CO₂ injected in an aquifer reservoir. In the second example we couple the developed DSI framework to an ensemble-based optimization framework to optimize water injection rates in an oil-water reservoir based on the forecasts of cumulative production conditioned to production data. The outcome achieved with DSI is verified against the response of synthetic truth models to assess the validity of the approach. The results obtained confirm the potential of DSI as a suitable technique to enable the acceleration of closed-loop and monitoring design optimization workflows. Moreover, the coupling with an ensemble-based optimization framework opens up opportunities to extend its use to optimize other types of reservoir management and field development decisions.

Gaussian Process models have been proposed as statistical models that allow interpolation between existing data points. One advantage of this approach is that the Gaussian process model includes an estimate of the accuracy of the predicted expected value at any point within the parameter space, unlike direct interpolators often used as proxy models. When used as part of an optimisation process we can use the Gaussian process model to eliminate those areas where we have high confidence that the optimal solution will not be found. This allows the efficient targeting of resources on those areas of parameter space that could yield the optimal solution, and also facilitates a more global analysis of the parameter space. Gaussian Processes naturally provide clear visualisation of the objective surface at various stages of the optimisation, which generates insight into the optimisation process sometimes lacking in alternative approaches, and thus facilitate human validation/intervention if desired.

To understand the theoretical basis of the approach requires a level of statistical knowledge that is not commonly found outside of the statistics community, which may have inhibited uptake. However, the approach can be easily implemented from first principles in python, using a recipe by Rasmussen and Williams, without needing a deep understand of the theoretical underpinning. The recipe has a small number of controls that need to be set by the user. In this paper we construct empirical models of the effect of these controls on the interpolation and explain their limitations from a theoretical perspective. We explore how dynamic adjustment of the controls might be used as part of an optimisation scheme.

We apply our approach to the well placement optimisation problem. The reservoir model used for the exercise is the PUNQ Complex Model, which is a 2.4 million cell representation of a BRENT sequence reservoir. A combination of producers and injectors are sequentially placed in the model using a greedy algorithm with the optimal position at each iteration being selected using the Gaussian Process model as a proxy for the true objective surface. The result is compared to a manually derived solution by an experienced reservoir engineer which required 22 wells. The result obtained by this approach reaches the same level of performance using only 18 wells.