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Hui, Mun-Hong (Chevron Energy Technology Company) | Dufour, Gaelle (Chevron Energy Technology Company) | Vitel, Sarah (Chevron Energy Technology Company) | Muron, Pierre (Chevron Energy Technology Company) | Tavakoli, Reza (Chevron Energy Technology Company) | Rousset, Matthieu (Chevron Energy Technology Company) | Rey, Alvaro (Chevron Energy Technology Company) | Mallison, Bradley (Chevron Energy Technology Company)
Traditionally, fractured reservoir simulations use Dual-Porosity, Dual-Permeability (DPDK) models that can idealize fractures and misrepresent connectivity. The Embedded Discrete Fracture Modeling (EDFM) approach improves flow predictions by integrating a realistic fracture network grid within a structured matrix grid. However, small fracture cells with high conductivity that pose a challenge for simulators can arise and ad hoc strategies to remove them can alter connectivity or fail for field-scale cases. We present a new gridding algorithm that controls the geometry and topology of the fracture network while enforcing a lower bound on the fracture cell sizes. It honors connectivity and systematically removes cells below a chosen fidelity factor. Furthermore, we implemented a flexible grid coarsening framework based on aggregation and flow-based transmissibility upscaling to convert EDFMs to various coarse representations for simulation speedup. Here, we consider pseudo-DPDK (pDPDK) models to evaluate potential DPDK inaccuracies and the impact of strictly honoring EDFM connectivity via Connected Component within Matrix (CCM) models. We combine these components into a practical workflow that can efficiently generate upscaled EDFMs from stochastic realizations of thousands of geologically realistic natural fractures for ensemble applications.
We first consider a simple waterflood example to illustrate our fracture upscaling to obtain coarse (pDPDK and CCM) models. The coarse simulation results show biases consistent with the underlying assumptions (e.g., pDPDK can over-connect fractures). The preservation of fracture connectivity via the CCM aggregation strategy provides better accuracy relative to the fine EDFM forecast while maintaining computational speedup. We then demonstrate the robustness of the proposed EDFM workflow for practical studies through application to an improved oil recovery (IOR) study for a fractured carbonate reservoir. Our automatable workflow enables quick screening of many possibilities since the generation of full-field grids (comprising almost a million cells) and their preprocessing for simulation completes in a few minutes per model. The EDFM simulations, which account for complicated multiphase physics, can be generally performed within hours while coarse simulations are about a few times faster. The comparison of ensemble fine and coarse simulation results shows that on average, a DPDK representation can lead to high upscaling errors in well oil and water production as well as breakthrough time while the use of a more advanced strategy like CCM provides greater accuracy. Finally, we illustrate the use of the Ensemble Smoother with Multiple Data Assimilation (ESMDA) approach to account for field measured data and provide an ensemble of history-matched models with calibrated properties.
Hui, Mun-Hong (Robin) (Chevron Energy Technology Company) | Karimi-Fard, Mohammad (Stanford University) | Mallison, Bradley (Chevron Energy Technology Company) | Durlofsky, Louis J. (Stanford University)
A comprehensive methodology for gridding, discretizing, coarsening, and simulating discrete-fracture-matrix models of naturally fractured reservoirs is described and applied. The model representation considered here can be used to define the grid and transmissibilities, either at the original fine scale or at coarser scales, for any connectivity-list-based finite-volume flow simulator. For our fine-scale mesh, we use a polyhedral-gridding technique to construct a conforming matrix grid with adaptive refinement near fractures, which are represented as faces of grid cells. The algorithm uses a single input parameter to obtain a suitable compromise between fine-grid cell quality and the fidelity of the fracture representation. Discretization using a two-point flux approximation is accomplished with an existing procedure that treats fractures as lower-dimensional entities (i.e., resolution in the transverse direction is not required). The upscaling method is an aggregation-based technique in which coarse control volumes are aggregates of fine-scale cells, and coarse transmissibilities are computed with a general flow-based procedure. Numerical results are presented for waterflood, sour-gas injection, and gas-condensate primary production for fracture models with matrix and fracture heterogeneities. Coarse-model accuracy is shown to generally decrease with increasing levels of coarsening, as would be expected. We demonstrate, however, that with our methodology, two orders of magnitude of speedup can typically be achieved with models that introduce less than approximately 10% error (with error appropriately defined). This suggests that the overall framework may be very useful for the simulation of realistic discrete-fracture-matrix models.
A comprehensive methodology for gridding, discretizing, coarsening, and simulating discrete-fracture-matrix models of naturally fractured reservoirs is described and applied. The model representation considered here can be used to define the grid and transmissibilities, at either the original fine scale or at coarser scales, for any connectivity-list-based finite-volume flow simulator. For our fine-scale mesh, we use a polyhedral gridding technique to construct a conforming matrix grid with adaptive refinement near fractures, which are represented as faces of grid cells. The algorithm uses a single input parameter to obtain a suitable compromise between fine-grid cell quality and the fidelity of the fracture representation. Discretization using a two-point flux approximation is accomplished with an existing procedure that treats fractures as lower-dimensional entities (i.e., resolution in the transverse direction is not required). The upscaling method is an aggregation-based technique in which coarse control volumes are aggregates of fine-scale cells, and coarse transmissibilities are computed using a general flow-based procedure. Numerical results are presented for waterflood, sour gas injection, and gas condensate primary production. Coarse-model accuracy is shown to generally decrease with increasing levels of coarsening, as would be expected. We demonstrate, however, that by using our methodology, two orders of magnitude of speedup can be achieved with models that introduce less than about 10% error (with error appropriately defined). This suggests that the overall framework may be very useful for the simulation of realistic discrete-fracture-matrix models.
Mallison, Bradley (Chevron Energy Technology Company) | Sword, Charles (Chevron Energy Technology Company) | Viard, Thomas (Chevron Energy Technology Company) | Milliken, William (Chevron Energy Technology Company) | Cheng, Amy (Chevron Energy Technology Company)
Effective workflows for translating Earth models into simulation models require grids that preserve geologic accuracy, offer flexible resolution control, integrate tightly with upscaling, and can be generated easily. Corner-point grids and pillar-based unstructured grids fail to satisfy these objectives; hence, a truly 3D unstructured approach is required. This paper describes unstructured cut-cell gridding tools that address these needs and improve the integration of our overall reservoir-modeling workflows. The construction of simulation grids begins with the geologic model: a numerical representation of the reservoir structure, stratigraphy, and properties. Our gridding utilizes a geochronological (GeoChron) map from physical coordinates to an unfaulted and unfolded depositional coordinate system. The mapping is represented implicitly on a tetrahedral mesh that conforms to faults, and it facilitates accurate geostatistical modeling of static depositional properties. In the simplest use case, we create an explicit representation of the geologic model as an unstructured polyhedral grid. Away from faults and other discontinuities, the cells are hexahedral, highly orthogonal, and arranged in a structured manner. Geometric cutting operations create general polyhedra adjacent to faults and explicit contact polygons across faults. The conversion of implicit models to explicit grids is conceptually straightforward, but the implementation is nontrivial because of the limitations of finite precision arithmetic and the need to remove small cells formed in the cutting process. In practice, simulation grids are often constructed at coarser resolutions than Earth models. Our implementation of local grid coarsening and refinement exploits the flexibility of unstructured grids to minimize upscaling errors and preserve critical geologic features. Because the simulation grid and the geologic model are constructed by use of the same mapping, fine cells can be nested exactly inside coarse cells. Therefore, flow-based upscaling can be applied efficiently without resampling onto temporary local grids. This paper describes algorithms and data structures for constructing, storing, and simulating cut-cell grids. Examples illustrate accurate modeling of normal faults, y-faults, overturned layers, and complex stratigraphy. Flow results, including a field-sector model, show the suitability of cut-cell grids for simulation.
Streamline-based methods can be used as effective post-processing tools for assessing flow patterns and well allocation factors in reservoir simulation. This type of diagnostic information can be useful for a number of applications, including visualization, model ranking, upscaling validation, and optimization of well placement or injection allocation. In this paper, we investigate finite-volume methods as an alternative to streamlines for obtaining flow diagnostic information. Given a computed flux field, we solve the stationary transport equations for tracer and time of flight by use of either single-point upstream (SPU) weighting or a truly multidimensional upstream (MDU) weighting scheme. We use tracer solutions to partition the reservoir into volumes associated with injector/producer pairs and to calculate fluxes (well allocation factors) associated with each volume. The heterogeneity of the reservoir is assessed with time of flight to construct flow-capacity/storage-capacity (F-vs.-F) diagrams that can be used to estimate sweep efficiency. We compare the results of our approach with streamline-based calculations for several numerical examples, and we demonstrate that finite-volume methods are a viable alternative. The primary advantages of finite volume methods are the applicability to unstructured grids and the ease of implementation for general-purpose simulation formulations. The main disadvantage is numerical diffusion, but we show that a MDU weighting scheme is able to reduce these errors.
Multidimensional transport for reservoir simulation is typically solved by applying 1D numerical methods in each spatial-coordinate direction. This approach is simple, but the disadvantage is that numerical errors become highly correlated with the underlying computational grid. In many real-field applications, this can result in strong sensitivity to grid design not only for the computed saturation/composition fields but also for critical integrated data such as breakthrough times. Therefore, to increase robustness of simulators, especially for adverse-mobility-ratio flows that arise in a variety of enhanced-oil-recovery (EOR) processes, it is of much interest to design truly multidimensional schemes for transport that remove, or at least strongly reduce, the sensitivity to grid design.
We present a new upstream-biased truly multidimensional family of schemes for multiphase transport capable of handling countercurrent flow arising from gravity. The proposed family of schemes has four attractive properties: applicability within a variety of simulation formulations with varying levels of implicitness, extensibility to general grid topologies, compatibility with any finite-volume flow discretization, and provable stability (monotonicity) for multiphase transport. The family is sufficiently expressive to include several previously developed multidimensional schemes, such as the narrow scheme, in a manner appropriate for general-purpose reservoir simulation.
A number of waterflooding problems in homogeneous and heterogeneous media demonstrate the robustness of the method as well as reduced transverse (cross-wind) diffusion and grid-orientation effects.
Aavatsmark, Ivar (Center for Integrated Petroleum Research) | Eigestad, Geir T. (University of Bergen) | Heimsund, Bjorn-ove (Center for Integrated Petroleum Research) | Mallison, Bradley (Chevron) | Nordbotten, Jan M. (University of Bergen) | Øian, Erlend (Center for Integrated Petroleum Research)
Multipoint-flux-approximation (MPFA) methods were introduced to solve control-volume formulations on general simulation grids for porous-media flow. While these methods are general in the sense that they may be applied to any matching grid, their convergence properties vary.
An important property for multiphase flow is the monotonicity of the numerical elliptic operator. In a recent paper (Nordbotten et al. 2007), conditions for monotonicity on quadrilateral grids have been developed. These conditions indicate that MPFA formulations that lead to smaller flux stencils are desirable for grids with high aspect ratios or severe skewness and for media with strong anisotropy or strong heterogeneity. The ideas were pursued recently in Aavatsmark et al. (2008), where the L-method was introduced for general media in 2D. For homogeneous media and uniform grids, this method has four-point flux stencils and seven-point cell stencils in two dimensions. The reduced stencils appear as a consequence of adapting the method to the closest neighboring cells.
Here, we extend the ideas for discretization on 3D grids, and ideas and results are shown for both conforming and nonconforming grids. The ideas are particularly desirable for simulation grids that contain faults and local grid refinement.
We present numerical results herein that include convergence results for single-phase flow on challenging grids in 2D and 3D and for some simple two-phase results. Also, we compare the L-method with the O-method.