History matching can play a key role in improving geological characterization and reducing the uncertainty of reservoir model predictions. Application of reservoir history matching is restricted by the huge computational cost by amongst others the many runs of the full model. Surrogate models with a reduced complexity are therefore used to reduce the computational demands. This paper presents an efficient surrogate-assisted deterministic inversion framework to primarily explore the possibility of applying deep neural network (DNN) surrogate to approximate the gradient of large-scale history matching by using auto-differentiation (AD). In combination with the deep neural network model, the AD enables us to evaluate the gradients efficiently in a parallel manner. Furthermore, the benefits of using stochastic gradient optimizers in the deep learning practice, instead of full gradient optimizers in conventional deterministic inversions, is investigated as well. Numerical experiments are conducted on a 3D benchmark reservoir model in the context of a water-flooding production scenario. The quantity of interest, e.g., dynamic saturation for an ensemble of test models, can be accurately predicted. The proposed surrogate-assisted inversion with stochastic gradient optimizer obtains a very quick convergence rate against the model and data noise for the high-dimensional history matching problem with a large number of data and parameters. In addition, we also conduct several comparisons and evaluations with our previously proposed projection-based subdomain POD-TPWL approach in terms of computational efficiency and accuracy. The subdomain POD-TPWL constructs a local surrogate model, which is repeatedly reconstructed a number of times for maintaining a satisfactory accuracy, while DNN constructs a global surrogate model based on the entire training data and generally does not require additional reconstructions. The subdomain POD-TPWL is very sensitive to how the domain is decomposed, increasing the training samples does not infinitely improve the history matching results by a fixed decomposition. Overall, these two kinds of surrogate models have demonstrated great potential in solving large-scale history matching problem. The DNN surrogate is particularly useful to generate multiple posteriors for model uncertainty quantification.

Imaging-type monitoring techniques are used in monitoring dynamic processes in many domains, including medicine, engineering, and geophysics. This paper aims to propose an efficient workflow for application of such data for the conditioning of simulation models. Such applications are very common in e.g. the geosciences, where large-scale simulation models and measured data are used to monitor the state of e.g. energy and water systems, predict their future behavior and optimize actions to achieve desired behavior of the system. In order to reduce the high computational cost and complexity of data assimilation workflows for high-dimensional parameter estimation, a residual-in-residual dense block extension of the U-Net convolutional network architecture is proposed, to predict time-evolving features in high-dimensional grids. The network is trained using high-fidelity model simulations. We present two examples of application of the trained network as a surrogate within an iterative ensemble-based workflow to estimate the static parameters of geological reservoirs based on binary-type image data, which represent fluid facies as obtained from time-lapse seismic surveys. The differences between binary images are parameterized in terms of distances between the fluid-facies boundaries, or fronts. We discuss the impact of the choice of network architecture, loss function, and number of training samples on the accuracy of results and on overall computational cost. From comparisons with conventional workflows based entirely on high-fidelity simulation models, we conclude that the proposed surrogate-supported hybrid workflow is able to deliver results with an accuracy equal to or better than the conventional workflow, and at significantly lower cost. Cost reductions are shown to increase with the number of samples of the uncertain parameter fields. The hybrid workflow is generic and should be applicable in addressing inverse problems in many geophysical applications as well as other engineering domains.

A reduced order modeling algorithm for the estimation of space varying parameter patterns in numerical models is proposed. In this approach domain decomposition is applied to construct separate approximations to the numerical model in every subdomain. We introduce a new local parameterization that decouples the computational cost of the algorithm from the number of global principal components and therefore provides attractive scaling for models with a very large number of uncertain parameter patterns. By defining uncertain parameter patterns only in the various subdomains the number of full order simulation required for the derivation of the reduced order models can be reduced drastically. To avoid non-smoothness at the boundaries of the subdomains, the optimal local parameters patterns are projected onto global parameter patterns. The computational effort of the new methodology hardly increases when the number of parameter patterns increases. The number of training models depends primarily on the maximum number of local parameters in a subdomain, which can be decreased by refining the domain decomposition. We apply the new algorithm to a large-scale reservoir model parameter estimation problem. In this application 282 parameters could be estimated using only 90 full order model runs.

We propose a quantitative model-based workflow for conformance verification of CO₂ storage projects. Bayesian inference is applied to update an ensemble of simulation models that capture prior uncertainty based on mismatches with measured data. Conformance assessments are derived by comparison of updated model predictions with storage permit requirements and confidence criteria. Two examples, one conceptual and one based on a real candidate storage site, are provided in which the quantitative workflow is applied to the a priori assessment of candidate monitoring strategies. The examples illustrate the limitations of pressure monitoring in the presence of realistic subsurface uncertainties, and the potential for cost saving by informed design of geophysical monitoring surveys. Approximate methods are discussed that could make the workflow also applicable for (quasi) real-time conformance monitoring.

We are concerned with the efficiency of stochastic gradient estimation methods for large-scale nonlinear optimization in the presence of uncertainty. These methods aim to estimate an approximate gradient from a limited number of random input vector samples and corresponding objective function values. Ensemble methods usually employ Gaussian sampling to generate the input samples. It is known from the optimal design theory that the quality of sample-based approximations is affected by the distribution of the samples. We therefore evaluate six different sampling strategies to optimization of a high-dimensional analytical benchmark optimization problem, and, in a second example, to optimization of oil reservoir management strategies with and without geological uncertainty. The effectiveness of the sampling strategies is analyzed based on the quality of the estimated gradient, the final objective function value, the rate of the convergence, and the robustness of the gradient estimate. Based on the results, an improved version of the stochastic simplex approximate gradient method is proposed based on UE(s²) sampling designs for supersaturated cases that outperforms all alternative approaches. We additionally introduce two new strategies that outperform the UE(s²) designs previously suggested in the literature.

The general field development optimization problem is complex due to the potentially large number of controls of mixed type and discontinuities in the objective function related to varying numbers and types of wells being placed in a discretized grid. This may make the problem challenging or even unsuitable for certain types of optimization methods that rely on, e.g., the availability of (adjoint) gradients. It is not yet clear which alternative approaches will be most useful. Here we investigate the application of stochastic gradient-based optimization techniques to field development optimization. Since their initial application to large-scale well rate and pressure control problems, such techniques have been shown to produce useful results of practical value also for other types of reservoir optimization problems such vertical well placement, well drilling scheduling, and water-alternating-gas strategy optimization. Here we introduce an efficient parameterization for well trajectory optimization and discuss a simple way to handle the number of wells that is placed. The full field development problem is split into subproblems that are addressed sequentially. The sequential workflow is applied to the Olympus benchmark model which represents a complex green field development optimization challenge. Initial experiments show that the proposed approach based on stochastic gradient methods is able to find much improved development strategies, as defined by the number and trajectories of wells, a platform location and a drilling sequence, at relatively low computational cost. We additionally identify a number of possible improvements to the applied workflow that are expected to make it applicable to other field cases of intermediate complexity.

We describe and evaluate a physics-based proxy model approach for reservoir prediction and optimization. It builds on the recent development of so-called flow-network models which represent flow paths between wells by discrete 1D grids with permeability and pore volume properties. These types of models represent an alternative to capacitance resistance and correlation-based models and have the benefit of allowing for all physics supported by regular 3D grid-based commercial simulators. The new model is different from a previously proposed model in that we include additional nodes in the network that allow for more and indirect flow paths between wells, as well as extra nodes to represent an aquifer. We describe the structure of our flow network and investigate the impact of design and training parameters on the performance of the network, both in history matching and prediction mode. Examples include the number and placement of network nodes, the treatment of aquifers, and the size and sampling of prior model property values. We distinguish between the accuracy of the history match and the generalizability by cross-validating the flow network performance on future well control strategies that are different from that encountered during the history period. Using this procedure, we aim to prevent overfitting of the model while ensuring sufficient predictive power. Results are presented for experiments based on phase rate and bottom hole pressure measurements and predictions generated with the Brugge benchmark model which is used as a synthetic truth. We subsequently present a first application of flow network models for well control optimization under uncertainty. To this end we employ a stochastic simplex gradient-based optimization approach and demonstrate that strategies that are expected to deliver improved NPV can be identified at much lower computational cost and within a much shorter time frame than would be required otherwise.

We investigate the potential for improved recovery of subsurface energy resources (hydrocarbons or heat) through in-depth diversion technology. A number of pilot studies in the North Sea have demonstrated in recent years that sodium silicate can be used to block preferential flow paths and divert water to previously unswept areas of a reservoir. Accompanying simulation studies based on an explicit weak coupling of a reservoir flow simulator and an external chemical module have attempted to replicate the observed behaviour. Since the development of silicate gels and the accompanying permeability reduction is essentially a coupled flow-chemical process, we first will present a fully implicit compositional-reactive flow and transport implementation and investigate the impact of the grid and time-stepping resolution on simulation performance in 2D subsurface reservoirs mimicking petroleum and geothermal applications. We proceed to investigate the sensitivity of the recovery to design parameters of the in-depth diversion strategy. Since adjoint gradients are not typically available for these parameters and uncertainties associated with an application of in-depth divergence are large, we use an ensemble-based methodology to perform an optimization study. This study aims to find optimal strategies for combined waterflooding and design of in-depth diversion under geological uncertainty. It is demonstrated that in-depth diversion can significantly extend the life-time of hydrocarbon or geothermal fields when the timing of injection and the size of the sodium silicate batch is optimized. Finally, we discuss methods that help to address an issue of computational cost associated with the high resolution required for accurate simulation of the coupled process.

Ensemble-based optimization has recently received great attention as a potentially powerful technique for life-cycle production optimization, which is a crucial element of reservoir management. Recent publications have increased both the number of applications and the theoretical understanding of the algorithm. However, there is still ample room for further development since most of the theory is based on strong assumptions. Here, the mathematics (or statistics) of Ensemble Optimization is studied, and it is shown that the algorithm is a special case of an already well-defined natural evolution strategy known as Gaussian Mutation. A natural description of uncertainty in reservoir management arises from the use of an ensemble of history-matched geological realizations. A logical step is therefore to incorporate this uncertainty description in robust life-cycle production optimization through the expected objective function value. The expected value is approximated with the mean over all geological realizations. It is shown that the frequently advocated strategy of applying a different control sample to each reservoir realization delivers an unbiased estimate of the gradient of the expected objective function. However, this procedure is more variance prone than the deterministic strategy of applying the entire ensemble of perturbed control samples to each reservoir model realization. In order to reduce the variance of the gradient estimate, an importance sampling algorithm is proposed and tested on a toy problem with increasing dimensionality.

We consider robust ensemble-based (EnOpt) multiobjective production optimization of on/off inflow-control devices (ICDs) for a sector model inspired by a real-field case. The use of on/off valves as optimization variables leads to a discrete control problem. We propose a reparameterization of such discrete controls in terms of switching times (i.e., we optimize the time at which a particular valve is either open or closed). This transforms the discrete control problem into a continuous control problem that can be efficiently handled with the EnOpt method. In addition, this leads to a significant reduction in the number of controls that is expected to be beneficial for gradient quality when using approximate gradients. We consider an ensemble of sector models where the uncertainty is described by different permeability, porosity, net/gross ratios, and initial water-saturation fields. The controls are the ICD settings over time in the three horizontal injection wells, with approximately 15 ICDs per well. Different optimized strategies resulting from different initial strategies were compared. We achieved a mean 4.2% increase in expected net present value (NPV) at a 10% discount rate compared with a traditional pressure-maintenance strategy. Next, we performed a sequential biobjective optimization and achieved an increase of 9.2% in the secondary objective (25% discounted NPV to emphasize shortterm production gains) for a minimal decrease of 1% in the primary objective (0% discounted NPV to emphasize long-term recovery gains), as averaged over the 100 geological realizations. The work flow was repeated for alternative numbers of ICDs, showing that having fewer control options lowers the expected value for this particular case. The results demonstrate that ensemble-based optimization work flows are able to produce improved robust recovery strategies for realistic field-sector models against acceptable computational cost.