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This work consists of two parts. In the first part, we present new methods for generating unstructured polyhedral grids that align to prescribed geometric objects. Control-point alignment of cell centroids is introduced to accurately represent horizontal and multilateral wells, but can also be used to create volumetric representations of fracture n...
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... Although the use of complex-structure wells increase the oil drainage area and have significant benefits in production, the area of working fluid contacting the reservoir is larger and the contact time is longer due to the complex-structure wells' trajectory (Moreno et al., 2006;Sau et al., 2014;Klemetsdal et al., 2017), or multiple branch wells and the long well construction cycle, which leads to greater damage to the reservoir (Joshi, 1994(Joshi, , 2003Jiang et al., 2021). Therefore, it is necessary to study and quantitatively evaluate the reservoir formation damage of complex-structure wells caused during the well construction. ...
Kinds of complex-structure wells can effectively improve production, which are widely used. However, in the process of drilling and completion, complex-structure wells with long drilling cycle and large exposed area of reservoir can lead to the fact that reservoir near wellbore is more vulnerable to the working fluid invasion, resulting in more serious formation damage. In order to quantitatively describe the reservoir formation damage in the construction of complex-structure well, taking the inclined well section as the research object, the coordinate transformation method and conformal transformation method are given according to the flow characteristics of reservoir near wellbore in anisotropic reservoir. Then the local skin factor in orthogonal plane of wellbore is deduced. Considering the uneven distribution of local skin factor along the wellbore, the oscillation decreasing model and empirical equation model of damage zone radius distribution along the wellbore direction are established and then the total skin factor model of the whole well is superimposed to realize the reservoir damage evaluation of complex-structure wells. Combining the skin factor model with the production model, the production of complex-structure wells can be predicted more accurately. The two field application cases show that the accuracy of the model can be more than 90%, which can also fully reflect the invasion characteristics of drilling and completion fluid in any well section of complex-structure wells in anisotropic reservoir, so as to further provide guidance for the scientific establishment of reservoir production system.
... General reservoir geometries are comprised of various features such as faults, fractures, pinch-outs and layered media, with a wide range of variations in porosity and permeability across different layers, e.g., [1][2][3][4][5][6][7][8][9][10] where a range of gridding strategies are presented. In addition reservoirs can have a complex spatial distribution of wells in place [6,11]. ...
Grid generation for reservoir simulation, must honour classical key geological features and multilateral wells. The features to be honoured are classified into two groups; (1) involving layers, faults, pinchouts and fractures, and (2) involving well distributions. In the former, control-volume boundary aligned grids (BAGs) are required, while in the latter, control-point (defined as the centroid of the control-volume) well aligned grids (WAGs) are required. Depending on discretization method type and formulation, a choice of control-point and control-volume type is made, i.e. for a cell-centered method the primal grid cells act as control-volumes, otherwise for a vertex-centered method the dual-grid cells act as control-volumes. Novel three-dimensional unstructured grid generation methods are proposed that automate control-volume boundary alignment to geological features and control point alignment to complex wells, yielding essentially perpendicular bisector (PEBI) meshes either with respect to primal or dual-cells depending on grid type. Both grid types use tetrahedra, pyramids, prisms and hexahedra as grid elements. Primal-cell feature aligned grids are generated using special boundary surface protection techniques together with constrained cell-centered well trajectory alignment. Dual-cell feature aligned grids are generated from underlying primal-meshes, whereby features are protected such that dual-cell control-volume faces are aligned with interior feature boundaries, together with protected vertex-centered (control point) well trajectory alignment. The novel methods of grid generation presented enable practical application of both method types in 3-D for the first time. The primal and dual grids generated here demonstrate the gridding methods, and enable the first comparative performance study of cell-vertex versus cell-centered control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent mesh resolution on challenging problems in 3-D. Pressure fields computed by the cell-centered and vertex-centered CVD-MPFA schemes are compared and contrasted relative to the respective degrees of freedom employed, and demonstrate the relative benefits of each approximation type. Stability limits of the methods are also explored. For a given mesh the cell-vertex method uses approximately a fifth of the unknowns used by a cell-centered method and proves to be the most beneficial with respect to accuracy and efficiency. Numerical results show that vertex-centered CVD-MPFA methods outperform cell-centered CVD-MPFA method.
... Our test cases involve deformed cell geometries and anisotropic permeabilities. The paper can therefore be seen as an update of [14] and [11]. Further 45 comparisons can be found in, e.g., [7]. ...
... The novelty herein is that we compare a large set of discretizations on the same 140 problem, with access to complete source codes. Our experiments here and in [11] do not clearly point to one preferred method that is significantly better than the others. TPFA is inconsistent and has grid orientation effects, but is monotone and gives sparse matrices with low condition numbers. ...
In this work, we review a set of consistent discretizations for second-order elliptic equations, and compare and contrast them with respect to accuracy, monotonicity, and factors affecting their computational cost (degrees of freedom, sparsity, and condition numbers). Our comparisons include the linear and nonlinear TPFA method, multipoint flux-approximation (MPFA-O), mimetic methods, and virtual element methods. We focus on incompressible flow and study the effects of deformed cell geometries and anisotropic permeability.
... In this example, we study the effect of using higher order dG methods and reordering to simulate a case posed on a 3D, fully unstructured PEBI/Voronoi grid generated using the upr module in MRST [3,14]. The domain is comprised of a rectangular box of dimensions 300 × 100 × 100 m 3 with approximately 20, 10, and 10 cells in each of the axial directions, respectively. ...
The fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-conditioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: (i) a sequential formulation that decouples flow and transport into separate subproblems, and (ii) a highly efficient Gauss–Seidel solver for the transport problems. This solver uses intercell fluxes to reorder the grid cells according to their upstream neighbors, and groups cells that are mutually dependent because of counter-current flow into local clusters. The cells and local clusters can then be solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process. This method has been successfully applied to real-field problems using the standard first-order finite volume discretization. Here, we extend the idea to first-order dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of the open-source OPM Flow simulator.
... Use of polygonal tessellations offers improved resolution control in the lateral direction, but 2.5D PEBI grids have more lateral neighbors that must be accounted for in a spatial reconstruction and pose the same challenges as cornerpoint grids in the vertical direction. Several methods have also been proposed to construct truly 3D PEBI grids adapting to various types of geological objects and curvilinear well paths [43,29]. Such grids will have a fully unstructured topology and general polyhedral cell geometries. ...
Many authors have used higher-order spatial discretizations to reduce numerical diffusion, which can be particularly pronounced when simulating EOR processes involving active chemical substances that are transported by linear or weakly nonlinear waves. Most high-resolution methods reported in the literature are based on explicit temporal discretizations. This imposes severe time-step restrictions when applied to the type of grids seen in industry-standard simulation models of real assets, which usually have orders-of-magnitude variations in porosities and Darcy velocities that necessitate the use of implicit discretization. Herein, we propose a second-order WENO discretization suitable for complex grids with polyhedral cell geometries, unstructured topologies, large aspect ratios, and large variations in interface areas. The WENO scheme is developed as part of a standard, fully implicit formulation that solves for pressure and transported quantities simultaneously. We investigate the accuracy and utility of the WENO scheme for a series of test cases that involve corner-point and 2D/3D Voronoi grids and black-oil and compositional flow models.
... In this example, we study the effect of using higherorder dG methods and reordering to simulate a case posed on a 3D, fully unstructured PEBI/Voronoi grid generated using the upr module in MRST [3,14]. The domain is comprised of a rectangular box of dimensions 300 × 100 × 100 m 3 with approximately 20, 10, and 10 cells in each of the axial directions, respectively. ...
The fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-condi\-tioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: (\textit{i}) a sequential formulation that decouples flow and transport into separate subproblems, and (\textit{ii}) a highly efficient Gauss--Seidel solver for the transport problems. This solver uses intercell fluxes to reorder the grid cells according to their upstream neighbors, and groups cells that are mutually dependent because of counter-current flow into local clusters. The cells and local clusters can then be solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process. This method has been successfully applied to real-field problems using the standard first-order finite volume discretization. Here, we extend the idea to first-order dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of the open-source OPM Flow simulator.
... r et al. 2015], compute gradients along nodal lines connecting neighboring cells, and hence require that these dual edges are orthogonal to the common primal facet [Pruess 2004]. Several heuristic approaches to the generation of Voronoi meshes for such simulators were developed [Bonduà et al. 2017;Freeman et al. 2014;Hu et al. 2016;Kim et al. 2015;S. Klemetsdal et al. 2017]. The situation is further complicated for multi-material domains, where the difficulty of generating conforming meshes necessitates dealing with mixed elements straddling the interface between multiple materials [Dawes 2017;Garimella and Lipnikov 2011;Kikinzon et al. 2017]. In contrast, VoroCrust is a well-principled algorithm for conf ...
Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrary curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably correct algorithm for conforming Voronoi meshing for non-convex and possibly non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while all sharp features are preserved in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.
... The generated grid conforms to the tracked planes and has fairly uniform cells. Recently, we extended the method to also generate conforming cells at intersections [20], thereby giving a robust method for generating Voronoi grids with control-point alignment of cell centroids and boundary alignment of cell faces in 2D. In 3D, the method only guarantees full alignment away from intersections. ...
... This method is promising, even though one often needs to treat the grid manually after the optimization. Especially cells at fracture intersections can have undesirable geometries, and fracture planes are not reproduced exactly like in [20]. Another optimization method was proposed by [36] for discrete fracture-network models to reduce highly skewed cells and ensure good grid quality around fracture tips and intersections and in regions of high fracture density. ...
... By building the grid this way, we ensure that it preserves intersections between lower dimensional constraints and that the cells from a (d − 1)-dimensional constraint will be faces in the d-dimensional grid. Our new method and its predecessors described in [20] have been implemented as a separate module called upr in the Matlab Reservoir Simulation Toolbox (MRST), which is an open-source community code designed for rapid prototyping and validation of new models and computational methods for simulating flow in porous media [21]. ...
We present a novel mixed-dimensional method for generating unstructured polyhedral grids that conform to prescribed geometric objects in arbitrary dimensions. Two types of conformity are introduced: (i) control-point alignment of cell centroids to accurately represent horizontal and multilateral wells or create volumetric representations of fracture networks, and ii) boundary alignment of cell faces to accurately preserve lower dimensional geological objects such as layers, fractures, faults, and/or pinchouts. The prescribed objects are in this case assumed to be lower dimensional, and we create a grid hierarchy in which each lower dimensional object is associated with a lower dimensional grid. Further, the intersection of two objects is associated with a grid one dimension lower than the objects. Each grid is generated as a clipped Voronoi diagram, also called a perpendicular bisector (PEBI) grid, for a carefully chosen set of generating points. Moreover, each grid is generated in such a way that the cell faces of a higher dimensional grid conform to the cells of all lower dimensional grids. We also introduce a sufficient and necessary condition which makes it easy to check if the sites for a given perpendicular bisector grid will conform to the set of prescribed geometric objects.
... Structured grids are naturally orthogonal, but the discontinuous nature of geological bodies makes it almost impossible to align structured grids with arbitrarily distributed geological objects, e.g., horizons, pinch-outs, faults, fractures, bore holes, etc. Although the improved finite difference (FD) schemes, such as MPFA [1][2][3], mimetic finite difference (MFD) [4][5][6][7], and nonlinear two-point flux approximation(NTPFA) [8,9], can diminish the effect of non-orthogonality, these schemes complicate the flux stencil, introducing additional difficulties to the solution of the discretized equations. To preserve the geometrical details and keep the flux stencil simple, building a high-quality grid is always the optimal choice. ...
Grid generation is critical to numerical reservoir simulations. High-quality grids guarantee the fidelity of a reservoir model and keep the flow calculations simple. In this study, we propose a 3D unstructured grid, the generalized prism grid (GPG), to model reservoirs with complicated geological geometries, including horizons, pinch-outs, faults, fractures, and bore holes. GPG is a layered, pillar-based grid. The location of a face node is specified by its elevation, and the pillar to which it is attached. Compared with the hexahedral corner point grid (CPG), GPG is a polygon prism and therefore more flexible; whereas, compared with the 2.5D perpendicular bisection (PEBI) grid, GPG allows polygons morphing through the stratum. We built a gridding algorithm to fulfil the features of GPG. The algorithm first constructs a 2D triangular mesh for one layer by setting up control points and grid densities for geological objects, such as fractures, faults, and wells, distributing triangular grid points with the “advancing front method,” and performing Delaunay optimization to the points. The polygon mesh is the dual grid of the triangular mesh. Taking the polygon mesh as a reference, the mesh for each layer of the strata is a morphing of it, with edges being stretched and points being assigned with heights. We also designed a compact file format to store GPG data and implemented the flux calculation method for GPG in a reservoir simulator. The attractive features of GPG are demonstrated through four examples. The conciseness and flexibility of GPG make it a potential new standard grid format replacing CPG.
... In this example, we study the effect of using higherorder dG methods and reordering to simulate a case posed on a 3D, fully unstructured PEBI/Voronoi grid generated using the upr module in MRST [3,14]. The domain is comprised of a rectangular box of dimensions 300 × 100 × 100 m 3 with approximately 20, 10, and 10 cells in each of the axial directions, respectively. ...
