Data-driven modelling with coarse-grid network models

We propose to use a conventional simulator, formulated on the topology of a coarse volumetric 3D grid, as a data-driven network model that seeks to reproduce observed and predict future well responses. The conceptual difference from standard history matching is that the tunable network parameters are calibrated freely without regard to the physical interpretation of their calibrated values. The simplest version uses a minimal rectilinear mesh covering the assumed map outline and base/top surface of the reservoir. The resulting CGNet models fit immediately in any standard simulator and are very fast to evaluate because of the low cell count. We show that surprisingly accurate network models can be developed using grids with a few tens or hundreds of cells. Compared with similar interwell network models (e.g., Ren et al., 2019, 10.2118/193855-MS), a typical CGNet model has fewer computational cells but a richer connection graph and more tunable parameters. In our experience, CGNet models therefore calibrate better and are simpler to set up to reflect known fluid contacts, etc. For cases with poor vertical connection or internal fluid contacts, it is advantageous if the model has several horizontal layers in the network topology. We also show that starting with a good ballpark estimate of the reservoir volume is a precursor to a good calibration.publishedVersio

Multiscale Mixed Methods on Corner-Point grids: Mimetic versus Mixed Subgrid Solvers

-Multiscale simulation is a promising approach to facilitate direct simulation of large and complex grid-models for highly heterogeneous petroleum reservoirs.  Unlike traditional simulation approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by incorporating sub-scale heterogeneities into local discrete approximation spaces. We consider a multiscale formulation based on a hierarchical grid approach, where basis functions with subgrid resolution are computed numerically to correctly and accurately account for subscale variations from an underlying (fine-scale) geomodel when solving the global flow equations on a coarse grid. By using multiscale basis functions to discretise the global flow equations on a (moderately-sized) coarse grid, one can retain the efficiency of an upscaling method, while at the same time produce detailed and conservative velocity fields on the underlying fine grid.For pressure equations, the multiscale mixed finite-element method (MsMFEM) has shown to be a particularly versatile approach. In this paper we extend the method to corner-point grids, which is the industry standard for modelling complex reservoir geology. We consider two different subsolvers: a mimetic finite difference method on the original corner-point grid and a mixed finite-element method on a tetrahedral subdivision. The versatility and accuracy of the multiscale mixed methodology is demonstrated on two corner-point models: a small Y-shaped sector model and a complex model of a layered sedimentary bed. In particular, we demonstrate how one can avoid the usual difficulties of resampling, when moving from a fine to a coarse grid, and vice versa, since the multiscale mixed formulation allows the cells in the coarse grid to be chosen as an (almost) arbitrary connected collection of cells in the underlying fine grid

Efficient adjoint-based well-placement optimization using flow diagnostics proxies

Model-based optimization of placement and trajectories of wells in petroleum reservoirs by the means of reservoir simulation forecasts is computationally demanding due to the high number of simulations typically required to achieve a local optimum. In this work, we develop an efficient flow-diagnostics proxy for net-present-value (NPV) with adjoint capabilities for efficient computation of well control gradients and approximate sensitivities with respect to placement/trajectory parameters. The suggested flow-diagnostic proxy consists of numerically solving a single pressure equation for the given scenario and the solution of a few inter-well time-of-flight and steady-state tracer equations, typically achieved in a few seconds for a reservoir model of medium size. Although the proxy may not be a particularly good approximation for the full reservoir simulation response, we find that for the cases considered, the correlation is very good and hence the proxy is suitable for use in an optimization loop. The adjoint simulation for the proxy model which provides control gradients and placement sensitivities is of similar computational complexity as the forward proxy model (a few seconds). We employ a version of the generalized reduced gradient for handling individual well constraints (e.g., bottom-hole-pressures and rates). As a result, the individual well constraints are enforced within the flow-diagnostics computations, and hence every parameter update becomes feasible without sacrificing gradient information. We present two numerical experiments illustrating the efficiency and performance of the approach for well placement problems involving trajectories and simulation models of realistic complexity. The suggested placements are evaluated using full simulations. We conclude by discussing limitations and possible enhancements of the methodology

Comparison of two different types of reduced graph-based reservoir models: Interwell networks (GPSNet) versus aggregated coarse-grid networks (CGNet)

Computerized solutions for field management optimization often require reduced-order models to be computationally tractable. The purpose of this paper is to compare two different graph-based approaches for building such models. The first approach represents the reservoir as a graph of 1D numerical flow models that each connects an injector to a producer. One thus builds a network in which the topology is primarily determined by “well nodes” to which “non-well nodes” can be connected if need be. The second approach aims at building richer models so that the connectivity graph mimics the intercell connections in a conventional, coarse 3D grid model. One thus builds a network with topology defined by a mesh-like placement of “non-well nodes”, to which wells can be subsequently connected. The two approaches thus can be seen as graph-based analogues of traditional streamline and finite-volume simulation models. Both model types can be trained to match well responses obtained from underlying fine-scale simulations using standard misfit minimization methods; herein we rely on adjoint-based gradient optimization. Our comparisons show that graph models having a connectivity graph that mimics the intercell connectivity in coarse 3D models can represent a wider range of fluid connections and are generally more robust and easier to train than graph models built upon 1D subgridded interwell connections between injectors and producers only.publishedVersio

Efficient adjoint-based well-placement optimization using flow diagnostics proxies

Model-based optimization of placement and trajectories of wells in petroleum reservoirs by the means of reservoir simulation forecasts is computationally demanding due to the high number of simulations typically required to achieve a local optimum. In this work, we develop an efficient flow-diagnostics proxy for net-present-value (NPV) with adjoint capabilities for efficient computation of well control gradients and approximate sensitivities with respect to placement/trajectory parameters. The suggested flow-diagnostic proxy consists of numerically solving a single pressure equation for the given scenario and the solution of a few inter-well time-of-flight and steady-state tracer equations, typically achieved in a few seconds for a reservoir model of medium size. Although the proxy may not be a particularly good approximation for the full reservoir simulation response, we find that for the cases considered, the correlation is very good and hence the proxy is suitable for use in an optimization loop. The adjoint simulation for the proxy model which provides control gradients and placement sensitivities is of similar computational complexity as the forward proxy model (a few seconds). We employ a version of the generalized reduced gradient for handling individual well constraints (e.g., bottom-hole-pressures and rates). As a result, the individual well constraints are enforced within the flow-diagnostics computations, and hence every parameter update becomes feasible without sacrificing gradient information. We present two numerical experiments illustrating the efficiency and performance of the approach for well placement problems involving trajectories and simulation models of realistic complexity. The suggested placements are evaluated using full simulations. We conclude by discussing limitations and possible enhancements of the methodology.publishedVersio

User Guide to Flow Diagnostics in MRST - Flow Diagnostics Preprocessors for Model Ensembles

Flow diagnostics are simple quantities that can be derived from basic flow simulations to probe a reservoir model, establish connections and basic volume estimates, and measure heterogeneity in flow paths. This user guide introduces various types of flow diagnostics, followed by an overview of two graphical user interfaces (GUIs) developed in the MATLAB Reservoir Simulation Toolbox (MRST) that can be used to quickly interrogate an ensemble of model realizations and investigate relative differences in flow patterns between them, prior to running computationally expensive, multiphase flow simulations. The first GUI enables you to inspect and cross-plot various measures of dynamic heterogeneity as well as simplified estimates of (economic) objectivity functions such as recovery factor and netpresent value for the whole ensemble. The second GUI focuses more on volumetric connections, communication patterns, and timelines for fluid transport within individual models or selected subsets of the full ensemble. It offers much of the same visualization capabilities as the GUI developed for flow-diagnostic postprocessing of multiphase flow simulations.publishedVersio

The use of flow diagnostics to rank model ensembles

Ensembles of geomodels provide an opportunity to investigate a range of parameters and possible operational outcomes for a reservoir. Full-featured dynamic modelling of all ensemble members is often computationally unfeasible, however some form of modelling, allowing us to discriminate between ensemble members based on their flow characteristics, is required. Flow diagnostics (based on a single-phase, steady-state simulation) can provide tools for analysing flow patterns in reservoir models but can be calculated in a much shorter time than a full-physics simulation. Heterogeneity measures derived from flow diagnostics can be used as proxies for oil recovery. More advanced flow diagnostic techniques can also be used to estimate recovery. With these tools we can rank ensemble members and choose a subset of models, representing a range of possible outcomes, which can then be simulated further. We demonstrate two types of flow diagnostics. The first are based on volume-averaged travel times, calculated on a cell by cell basis from a given flow field. The second use residence time distributions, which take longer to calculate but are more accurate and allow for direct estimation of recovery volumes. Additionally we have developed new metrics which work better for situations where we have a non-uniform initial saturation, e.g., a reservoir with an oil cap. Three different ensembles are analysed: Egg, Norne, and Brugge. Very good correlation, in terms of model ranking and recovery estimates, is found between flow diagnostics and full simulations for all three ensembles using both the cell-averaged and residence time based diagnostics.publishedVersio

The use of flow diagnostics to rank model ensembles

Ensembles of geomodels provide an opportunity to investigate a range of parameters and possible operational outcomes for a reservoir. Full-featured dynamic modelling of all ensemble members is often computationally unfeasible, however some form of modelling, allowing us to discriminate between ensemble members based on their flow characteristics, is required. Flow diagnostics (based on a single-phase, steady-state simulation) can provide tools for analysing flow patterns in reservoir models but can be calculated in a much shorter time than a full-physics simulation. Heterogeneity measures derived from flow diagnostics can be used as proxies for oil recovery. More advanced flow diagnostic techniques can also be used to estimate recovery. With these tools we can rank ensemble members and choose a subset of models, representing a range of possible outcomes, which can then be simulated further. We demonstrate two types of flow diagnostics. The first are based on volume-averaged travel times, calculated on a cell by cell basis from a given flow field. The second use residence time distributions, which take longer to calculate but are more accurate and allow for direct estimation of recovery volumes. Additionally we have developed new metrics which work better for situations where we have a non-uniform initial saturation, e.g., a reservoir with an oil cap. Three different ensembles are analysed: Egg, Norne, and Brugge. Very good correlation, in terms of model ranking and recovery estimates, is found between flow diagnostics and full simulations for all three ensembles using both the cell-averaged and residence time based diagnostics