The research presented in this paper was partly funded by the Integrated Systems Approach to Petroleum Production (ISAPP) and Recovery Factory (RF) projects.The 'Egg Model' is a synthetic reservoir model consisting of an ensemble of 101 relatively small three-dimensional realizations of a channelized oil reservoir produced under water flooding conditions with eight water injectors and four oil producers. It has been used in numerous publications to demonstrate a variety of aspects related to computer-assisted flooding optimization and history matching. Unfortunately the details of the parameter settings are not always identical and not always fully documented in several of these publications. We present a 'standard version' of the Egg Model which is meant to serve as a test case in future publications, and a dataset of 100 permeability realizations in addition to the permeability field used for the standard model. We implemented and tested the model in four reservoir simulators: Dynamo/Mores (Shell), Eclipse (Schlumberger), AD-GPRS (Stanford University) and MRST (Sintef), which produced near-identical output. This article describes the input parameters of the standard model. Together with the input files for the various simulators, it has been be uploaded in the 3TU.Datacentrum repository with free access to external users.
Foam can be used for gas mobility control in different subsurface applications. The success of foam-injection process depends on foam-generation and propagation rate inside the porous medium. In some cases, foam properties depend on the history of the flow or concentration of the surfactant, i.e., the hysteresis effect. Foam may show hysteresis behavior by exhibiting multiple states at the same injection conditions, where coarse-textured foam is converted into strong foam with fine texture at a critical injection velocity or pressure gradient. This study aims to investigate the effects of injection velocity and surfactant concentration on foam generation and hysteresis behavior as a function of foam quality. We find that the transition from coarse-foam to strong-foam (i.e., the minimum pressure gradient for foam generation) is almost independent of flowrate, surfactant concentration, and foam quality. Moreover, the hysteresis behavior in foam generation occurs only at high-quality regimes and when the pressure gradient is below a certain value regardless of the total flow rate and surfactant concentration. We also observe that the rheological behavior of foam is strongly dependent on liquid velocity.
With an increase in the number of applications of ensemble optimization (EnOpt) for production optimization, the theoretical understanding of the gradient quality has received little attention. An important factor that influences the quality of the gradient estimate is the number of samples. In this study we use principles from statistical hypothesis testing to quantify the number of samples needed to estimate an ensemble gradient that is comparable in quality to an accurate adjoint gradient. We develop a methodology to estimate the necessary ensemble size to obtain an approximate gradient that is within a predefined angle compared to the adjoint gradient, with a predefined statistical confidence. The method is first applied to the Rosenbrock function (a standard optimization test problem), for a single realization, and subsequently for a case with uncertainty, represented by multiple realizations (robust optimization). The maximum allowed error applied in both experiments is a 10° angle between the directions of the EnOpt gradient and the exact gradient. For the single-realization case we need, depending on the perturbation size, 900, 5 and 3 samples to estimate a "good" gradient with 95% confidence at 50 points in the optimization space for 50 different random sequences. For the robust case, the conventional EnOpt approach is to couple one model realization with one control sample, which leads to a computationally efficient technique to estimate a mean gradient. However, our results show that in order to be 95% confident the original one-to-one model realization to control sample ratio formulation is not sufficient. To achieve the required confidence requires a ratio of 1:1100, i.e. each model realization is paired with 1100 control samples using the original formulation. However, using a modified formulation we need a ratio of 1:10 to stay within the maximum allowed error for 95% of the points in space, though a 1:1 ratio is sufficient for 85% of the points. We also tested our methodology on a reservoir case for deterministic and robust cases, where we observe similar trends in the results. Our results provide insight into the necessary number of samples required for EnOpt, in particular for robust optimization, to achieve a gradient comparable to an adjoint gradient.
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