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Leung, Wing T. (University of Texas at Austin) | Chung, Eric T. (The Chinese University of Hong Kong) | Efendiev, Yalchin (Texas A&M University) | Vasilyeva, Maria (Texas A&M University) | Wheeler, Mary (University of Texas at Austin)
The objective of this work is to design upscaled model concepts for multi-phase flow and transport. Our approaches are based on recent developments in multiscale simulations and their relations to upscaling.
We propose a novel multi-phase upscaling technique, which employs rigorous multiscale concepts based on the Constraint Energy Minimization (CEM-GMsFEM). CEM-GMsFEM concepts utilize local spectral problems and an energy minimization principle to design multiscale basis functions, which are supported in oversampled regions. A coarse-grid solution defined by these basis functions provides first-order accuracy with respect to the coarse-mesh size and is independent of high contrast of the permeability. The degrees of freedom in multiscale methods represent the coordinates of the solution in the multiscale space. To design an upscaled model, we modify these basis functions such that the degrees of freedom have physical meanings, in particular, the averages of the solution in each continuum. This allows deriving rigorous upscaled models and account for both local and non-local on the effects. The transmissibilities in our upscaled models are non-local and take into account non-neighboring connections.
To extend this approach to nonlinear problems in the context of two-phase flow, we develop non-linear upscaling, where the pressures and saturations are interpolated within an oversampled region based on average values of these quantities. Multicontinua concepts are used to localize the problem to the oversampled regions. Our upscaled model shares some similarities with the pseudo-relative permeability approach with the following differences: (1) the upscaled relative permeabilities depend non-locally on the saturations; and (2) local problems, formulated in oversampled regions, involve constraints and require multi-contiuum concepts.
The numerical results will utilize upscaled methods to predict the solution of single-and two-phase flow dynamics. We will describe upscaled equations, which include the non-local neighborhood connections. Our results demonstrate that the proposed approaches provide a good accuracy and robustness. We consider various types of heterogeneities. The proposed concepts will benefit developing coarse-grid and upscaled models for many applications involving multi-phase flow and transport.
The objective of this work is to design novel multi-layer neural network architectures for simulations of multi-phase flow taking into account the observed data (e.g., production data) and physical modeling concepts. Our approaches use deep learning concepts combined with model reduction methodologies to predict multi-phase flow dynamics. The use of reduced-order model concepts is important for constructing robust deep learning architectures. The reduced-order models provide fewer degrees of freedom and allow handling the cases relevant to reservoir engineering that is limited to production and near-well data.
Multi-phase flow dynamics can be thought as multi-layer networks. More precisely, the solution, pressures and saturations, at the time instant n+1 depends on the solution at the time instant n and input parameters, such as permeability, well rates, and so on. Thus, one can regard the solution as a multi-layer network, where each layer is a nonlinear forward map. The number of time steps is user-defined quantity, which will be treated as an unknown within our deep learning algorithms. We will rely on rigorous model reduction concepts to define unknowns and connections for each layer. Novel proper orthogonal basis functions will be constructed such that the degrees of freedom have physical meanings (e.g., represent the solution values at selected locations) and basis functions have limited support, which will allow localizing the forward dynamics. This will allow writing the forward map for the solution values at selected locations with pre-computed neighborhood structure that will be used in deep learning algorithms.
In each layer, our reduced-order models will provide a forward map, which will be modified (trained) using available data. It is critical to use reduced-order models for this purpose, which will identify the regions of influence and the appropriate number of variables. Because of the lack of available data, the training will be supplemented with computational data as needed and the interpolation between data-rich and data-deficient models. We will also use deep learning algorithms to train the elements of the reduced model discrete system.
In this case, deep learning architectures will be employed to approximate the elements of the discrete system and reduced-order model basis functions.
The numerical results will use deep learning architectures to predict the solution and reduced-order model variables. Trained basis functions will allow interpolating the solution between the observation points. We show how network architecture, which includes the neighborhood connection, number of layers, and neurons, affect the approximation. Our results show that with a fewer number of layers, the multi-phase flow dynamics can be approximated. The proposed approach uses physical model concepts and deep learning methods to design a novel forward map, which combines the available data and physical models. This will benefit to develop a fast and data-based algorithms for reservoir simulations.
Summary A new-generation compositional reservoir-flow-simulation model is presented for gas-bearing organic-rich source rocks, including convective/diffusive mass-balance equations for each hydrocarbon component in the organic (kerogen), inorganic, and fracture continua. The model accounts for the presence of dispersed kerogen with sorbed-gas corrected dynamic porosity. The Maxwell-Stefan theory is used to predict pressure-and composition-dependence of molecular diffusion in the formation. The equations are discretized and solved numerically by use of the control-volume finite-element method (CVFEM). The simulation is derived from a new multiscale conceptual flow model. We consider that kerogen is dispersed at a fine scale in the inorganic matrix and that it will be the discontinuous component of total porosity at the reservoir-simulation scale, which could be up to six orders of magnitude larger. A simple mass-balance equation is introduced to enable kerogen to transfer gas to the inorganic matrix that is collocated in the same gridblock. The convective/diffusive transport takes place between neighboring gridblocks only in the inorganic matrix. We also observe that although adsorbed fluid could contribute significantly to storage in the shale formation, its contribution to production could be severely limited by the lack of kerogen continuity at the reservoir scale and by a low degree of coupling between the organic and inorganic pores. The contribution of the Maxwell-Stefan diffusion to the overall transport in the shale formation increases as the inorganic matrix permeability is reduced because of pressure decline during production. Introduction Oil and gas production from organic-rich source rocks is characterized by a sharp decline. This decline is a manifestation of flowregime transition from an early time, when the production is mainly caused by flow in fractures, to a later period, when the matrix begins to contribute to the production. The extent of the early transient, which could be ephemeral, depends on the presence of fractures and on the physical qualities (conductivity, spatial distribution) of the fracture network. The later period, on the other hand, could take large production times and is characterized by a long tail in production-history plots. Its extent is controlled by the formation qualities. The latter observation and the desire to manipulate decline rates in shale-gas wells for optimal production have resulted in research interest on the shale matrix, in particular its storage and transport mechanisms.
Wang, Min (Texas A&M University) | Wei, Chenji (Research Institute of Petroleum Exploration & Development, PetroChina) | Song, Hongqing (University of Science and Technology Beijing) | Efendiev, Yalchin (Texas A&M University) | Wang, Yuhe (Texas A&M University)
In this paper, we couple Discrete Fracture Network (DFM) and multi-continuum model with Generalized Multiscale Finite Element Method (GMsFEM) for simulating flow in fractured and vuggy reservoir. Various scales of fractures are treated hierarchically. Fractures that have global effect are modeled by continua while the local ones are embedded as discrete fracture network based on the geologic observation. For independent vugs, a continuum is used to represent their effects with specific configuration that there's no intra-flow of this continua. GMsFEM enables us to systematically develop an approximation space that contains prominent sub-grid scale heterogeneous background information based on the multi-continuum and DFM model. Conforming unstructured mesh is used to surrender the application of random discrete fracture networks. This paper targets on the improvement of the flow simulation performance in complex high-contrast domain by extending the ability of multiscale method to modeling arbitrary discrete fracture network. This advancement by GMsFEM is motivated by the limited capability of Multiscale Finite Element Method (MsFEM) on modeling discrete fractures when multiple fracture networks present in same coarse block. Multiple numerical results are shown to validate the efficiency of our coupled method.
Seismic modeling of fractures can aid in the characterization of fractured reservoirs by providing insights into the effects of fracture properties such as length, spacing, orientation and compliance. We apply new seismic models that explicitly represent fractures, unlike more common effective medium methods, and we can thus incorporate the complex geometry of realistic fracture systems. This also allows the prediction of the influence of a small number of major fractures that may control the reservoir flow behavior or that are generated by hydraulic fracturing. Specifically, we apply new, innovative generalized multiscale finite element methods (GMsFEM) to predict the effect of fracture compliances on scattered seismic waves from natural or hydraulic fractures. This numerical approach represents fractures on a finely discretized mesh; this fine mesh is used to capture fracture properties by generating quantities (basis functions) that are used for modeling wave propagation on a much coarser grid. This methodology reduces the size of the computational problem, allowing faster results while retaining the influence of the original fracture distribution on the fine grid. Another important feature is that the fractures are parameterized by their compliance, the variations of which are correlated with changes in fracture conductivity. Both compliance and conductivity will increase in a propped fracture, but will decrease when in situ stress increases as it closes the fracture. Thus, the inference of changes in compliance using seismic data will also help to identify changes in flow properties, guiding development of predictive models for reservoir management.We have applied the method to model seismic data from both natural fracture systems and hydraulic fracture networks. Simulation of a 2-D seismic survey in a model with multiple swarms of natural fractures illustrates the complex scattering of waves reflected between the fracture sets. Initial application of the method to 2-D models containing multiple hydraulic fracture stages shows how data acquired in microseismic monitoring can be affected by newly formed fracture networks. In particular, by varying fracture compliance, model results show the difference in seismic reflections from propped and unpropped portions of fractures. These models can allow the development of algorithms for using field data to provide new measures to verify hydraulic fracture location and flow properties.
Multiphase flow in carbonate reservoirs has been a hard problem of scientific research for many years. Accurate flow simulation is essential for the efficient exploitation. In this paper, a coupled triple-continuum and discrete fracture network approach is developed for modeling multiphase flow through fractured vuggy porous media. Multiple levels of fractures can be not only modeled as different superimposed continua but also embodied as discrete fracture network based on their geometrical characteristics. We develop a systematic coupling using Multiscale Finite Element Method (MsFEM) as a framework for coarsening and refinement. MsFEM is used to capture subgrid scale heterogeneities and interactions through multiscale basis functions calculated based on the triple-continuum background. Unstructured mesh is applied to model discrete fractures in arbitrary. This paper presents a significant advancement in terms of elevating the limitations of the triple-continuum models in handling complex fractures and extending the model reduction capability of MsFEM. Several numerical examples are carried out to demonstrate the capability of the proposed coupling method.
In the paper, we propose an online adaptive POD-DEIM model reduction method for fast multiscale reservoir simulations in highly heterogeneous porous media. The approach uses Proper Orthogonal Decomposition (POD) Galerkin projection to construct a global reduced system. The nonlinear terms are approximated by the Discrete Empirical Interpolation Method (DEIM). To adapt at the online stage the states (velocity, pressure and saturation) of the system, we incorporate new data, as it becomes available. Once the criterion for updates is satisfied, we adapt the reduced system online by updating the POD subspace and the DEIM approximation of the nonlinear functions. These global online basis function updates improve the accuracy of snapshot approximation. Since the adaption is performed infrequently, the new methodology does not add a significant computational overhead due to the adaptation of the reduced bases. Our approach is particularly useful for situations where one needs to solve the reduced system for inputs or controls that result in a solution outside the span of the snapshots generated in the offline stage. Our method is able to construct a robust reduced system even if an initial poor choice of snapshots is used. We demonstrate with a numerical experiment to demonstrate the efficiency of our method.
Yang, Yanfang (Texas A&M University) | Ghasemi, Mohammadreza (Texas A&M University) | Gildin, Eduardo (Texas A&M University) | Efendiev, Yalchin (Texas A&M University) | Calo, Victor (King Abdullah University of Science and Technology)
We present a global/local model reduction for fast multiscale reservoir simulations in highly heterogeneous porous media. Our approach identifies a low-dimensional structure in the solution space. We introduce an auxiliary variable (the velocity field) in our model reduction that achieves a high compression of the model. This compression is achieved because the velocity field is conservative for any low-order reduced model in our framework, whereas a typical global model reduction that is based on proper-orthogonal-decomposition (POD) Galerkin projection cannot guarantee local mass conservation. The lack of mass conservation can be observed in numerical simulations that use finite-volume-based approaches. The discrete empirical interpolation method (DEIM) approximates fine-grid nonlinear functions in Newton iterations. This approach delivers an online computational cost that is independent of the fine-grid dimension. POD snapshots are inexpensively computed with local model-reduction techniques that are based on the generalized multiscale finite-element method (GMsFEM) that provides (1) a hierarchical approximation of the snapshot vectors, (2) adaptive computations with coarse grids, and (3) inexpensive global POD operations in small dimensional spaces on a coarse grid. By balancing the errors of the global and local reduced-order models, our new methodology provides an error bound in simulations. Our numerical results, by use of a two-phase immiscible flow, show a substantial speedup, and we compare our results with the standard POD-DEIM in a finite-volume setup.
We apply the Generalized Multi-scale Finite Element Method (GMsFEM) to simulate seismic wave propagation in fractured media. Fractures are represented explicitly on a fine-scale triangular mesh, and they are incorporated using the linear-slip model. The motivation for applying GMsFEM is that it can reduce computational costs by utilizing basis functions computed from the fine-scale fracture model to simulate propagation on a coarse grid. We first apply the method to a simple model that has a uniform distribution of parallel fractures. At low frequencies, the results could be predicted using a homogeneous, effective medium, but at higher frequencies GMs-FEM results allow simulation of more complex, scattered wave-fields generated by the fractures. The second, complex model has two fracture corridors in addition to a few sparsely distributed fractures. Simulations compare scattered wavefields for different acquisition geometries. GMsFEM allows a reduction of computation of about 90% compared to a conventional finite element result computed directly from the fine-scale grid.
Presentation Date: Thursday, October 20, 2016
Start Time: 10:10:00 AM
Presentation Type: ORAL
Ghasemi, Mohammadreza (Texas A & M University and KAUST) | Yang, Yanfang (Texas A & M University and KAUST) | Gildin, Eduardo (Texas A & M University and KAUST) | Efendiev, Yalchin (Texas A & M University and KAUST) | Calo, Victor (KAUST)
In this paper, we present a global-local model reduction for fast multiscale reservoir simulations in highly heterogeneous porous media with applications to optimization and history matching. Our proposed approach identifies a low dimensional structure of the solution space. We introduce an auxiliary variable (the velocity field) in our model reduction that allows achieving a high degree of model reduction. The latter is due to the fact that the velocity field is conservative for any low-order reduced model in our framework. Because a typical global model reduction based on POD is a Galerkin finite element method, and thus it can not guarantee local mass conservation. This can be observed in numerical simulations that use finite volume based approaches. Discrete Empirical Interpolation Method (DEIM) is used to approximate the nonlinear functions of fine-grid functions in Newton iterations. This approach allows achieving the computational cost that is independent of the fine grid dimension. POD snapshots are inexpensively computed using local model reduction techniques based on Generalized Multiscale Finite Element Method (GMsFEM) which provides (1) a hierarchical approximation of snapshot vectors (2) adaptive computations by using coarse grids (3) inexpensive global POD operations in a small dimensional spaces on a coarse grid. By balancing the errors of the global and local reduced-order models, our new methodology can provide an error bound in simulations. Our numerical results, utilizing a two-phase immiscible flow, show a substantial speed-up and we compare our results to the standard POD-DEIM in finite volume setup.