It complements Chapter 3 in the FIRST textbook, but can be read independently of the other chapters here. It uses the state functions from Chapter 5, but can be read independently of the other chapters in the book.
Introduction
The upr module in the MATLAB Reservoir Simulation Toolbox (MRST) can construct unstructured Voronoi meshes that conform to polygonal boundaries and geometric constraints in arbitrary dimensions prescribed within the reservoir volume. The mesh structure of the MATLAB Reservoir Simulation Toolbox (MRST) is highly flexible and allows for completely unstructured topologies and general polyhedral cell geometries. A detailed discussion can be found in Chapter 3 of the MRST textbook [11].) The basic MRST module includes several functions for creating a wide variety of meshes, from simple line meshes, through corner point meshes and unstructured simple meshes to hybrid and multiblock meshes.
Basic Introduction to PEBI Grids
In fact, the center of the circumference around a cell of the Delaunay triangle will be a vertex in the PEBI grid. This function is a wrapper around the voronoin function and will clip the PEBI grid through the convex hull of the specified boundary.
Three Approaches for Optimizing PEBI Grids
To avoid having half-sized cells near the border, you can use clipped Pebi with the face positions offset by [dx,dy]/2 to the centers of the Cartesian grid. The first is based on Delaunay triangulation optimization using the DistMesh1 software [16], discussed in subsection 3.2.4 of the MRST text.
Internal Face Constraints
The output parameter F is a structure containing the sites on opposite sides of each constraint, the circle centers (i.e., the vertices of the 1D tessellation), as well as the point locations, all shown in Figure 1.5. To create a corresponding 2D grid of the circle (right plot), the corresponding 1D intersection lines are first gridded (center).
Adapting Cell Centroids
Restricted sites are shown as black dots and protected sites as red dots. If you prescribe density functions to control cell size based on the distance to cell and plot boundaries, the algorithm uses the smallest value of the two; i.e. distance to nearest.
Worked Examples
The left plot shows a zoom of the grid around the intersection of the cell constraint and the face constraint. As an alternative, we can try to create a grid that conforms to the outline of the high-permeability channels.
Concluding Remarks
Here indexMapmaps from the indexes in the old and new grids that we use to extract the correct subset of the layer, room, and well indicators. These functions are quite flexible and can create almost all of the 2D grids shown in this chapter. Grids that fit simple surfaces can be constructed, but as we've seen, the user has to do more of the work.
Introduction
We outline key functions in the accompanying nfvm module in the MATLAB Reservoir Simulation Toolbox (MRST) and show some examples of how the method is applied. In the following sections, we briefly describe the mathematical problem and delve right into the discretization scheme. The examples demonstrate the effectiveness of nonlinear finite volume (NFV) methods against spurious oscillations that may arise in the solutions of other consistent discretization schemes.
Model Equations
These schemes aim to preserve the monotonicity and positivity of the discrete solutions using a variety of methods, notably positive coefficient interpolation strategies, MPFA continuity or flux interpolation, and inverse distance interpolation.
Nonlinear Finite-Volume Methods
Determining the exact locations of auxiliary points will be explained in the next subsection. The blue arrows start at the cell centerline and end at the harmonic mean points associated with the cell faces. The main idea of the correction algorithm is to "pull back" poorly placed harmonic mean points towards the centers of the faces, while at the same time reducing the errors that arise in the process.
Numerical Examples
The pressure solutions using NTPFA and NMPFA for this case are shown in Figure 2.10 and the convergence histories of Picard's nonlinear solver are shown in Figure 2.11. In contrast, the two nonlinear methods resolve the principal directions of the discontinuous permeability tensors quite well and the pressure solutions remain nonnegative. The pressure solutions using TPFA, MPFA-O, NTPFA and NMPFA are shown in Figure 2.17 and the convergence history of the two nonlinear methods is shown in Figure 2.18.
Concluding Remarks
Monotone finite volume schemes for the diffusion equations on unstructured and regular shaped triangular polygonal meshes. Second-order accurate finite-volume schemes with the maximum discrete principle for solving the Richards equation on unstructured meshes. Interpolation-based second-order monotone finite volume schemes for the anisotropic diffusion equations on general networks.
Introduction
We explain how you can use discontinuous Galerkin methods to formulate implicit higher-order discretizations of the transport equations in stratigraphic and polytopal networks and describe how this is implemented in the dg module of the MATLAB Reservoir Simulation Toolbox (MRST). Higher-order discontinuous Galerkin (dG) methods [5, 16], first introduced by Reed and Hill [15], are an example of mass-conservative methods that are suitable for capturing sharp fronts of displacement without introducing spurious oscillations or excessive numerical smearing. This chapter describes how to formulate implicit discontinuous Galerkin methods for a wide class of meshes, including stratigraphic meshes and general polytopal meshes.
Model Equations
3.2) As an abbreviation, we use the (slightly imprecise) notation u to represent the unknown variables, which for an immiscible polyphase transport problem consist of the phase saturations Sα. The source volumetric terms, qα, are assumed to be known temporal functions for injection wells and functions of the unknown phase saturations, Sα, for production wells. Higher-order temporal discretizations involve explicit terms that constrain the time step, and so to allow for very large time steps, we accept the reduced accuracy of the unconditionally stable backward Euler method.
Discontinuous Galerkin Methods
To obtain a dG method with lower-order polynomial basis functions, we instead define the basis as the set of all functions of the form. Basic functions are conveniently implemented in the Polynomial class, which defines associated exponents and coefficients. However, since all linear basis functions are rectangular by construction, this can be avoided by replacing u0i with the cell meanu¯i in (3.20) and (3.22).
Numerical Examples
Readers familiar with explicit G methods may be somewhat disappointed by the solution of the higher order DG schemes in this example. This introduces significant numerical smearing that counteracts the effect of the high-resolution spatial discretization. The setup for dG(1) is completely analogous. Figure 3.11 shows 3D plots of the water saturation at the end of each simulation.
Concluding Remarks
Since dG(1) introduces less smearing of water fingers, it also predicts earlier water breakthrough. For compositional flow and other transport equations with many primary unknowns, this will quickly increase the size of linear systems. The multilevel restriction smoothed basis (MsRSB) method is currently the state-of-the-art method in multilevel methods.
Introduction and Background Discussion
Likewise, key parts of the strategy for generating primal-double partitions in the MsFV method were first developed in MRST. The multiscale nature of macroscale models comes primarily from heterogeneity in the petrophysical characteristics of the porous rock. Over the years, research focus has shifted towards accelerating the solution of the original fine-scale problem, yielding iterative multiscale methods [34,56] that challenge algebraic multilevel methods [12,52,54] .
Multiscale Finite-Volume Methods
4.9) To ensure that the support of the basis function does not extend beyond the support region (defined as all cells i ∈ S(j )), we reset dj to zero in all support boundary cells B(j ) of S(j. In contrast, the reconstructed flux field reproduces the fine-scale transport properties of the solution with good accuracy, as can be seen in the streamlined graphs in Figure 4.12. To motivate the iterative method, we start with the plotting of the residue of the multi-scale fine-scale approximation in the top left of Figure 4.14.
Numerical Examples
The left display in the figure clearly shows how the edges of the rough layers fall close to the plateaus in the cumulative cell distribution (these plateaus are more pronounced in the cumulative pore volume distribution). Let us begin by considering the fully implicit (FI) formulation, which can be considered the industry standard approach in recent decades. Chapters 11 and 12 of the MRST textbook [25] give more details.) A natural way to use multi-step methods in an FI setting would be to use them as solvers for an elliptic pressure equation formulated in a constrained pressure residual (CPR) preconditioner (see e.g. . , [25, subsection 12.3.4]), which represents a state-of-the-art solution for solving the linearized black oil equations. The problem is difficult to simulate because of the orders of magnitude differences in flow velocities in the background sand and highly permeable fractures.
Concluding Remarks
Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media. Multiscale finite-volume method for compressible multiphase flow in porous media.Journal of Computational Physics. An operator formulation of the multiscale finite-volume method with correction function.Multiscale Modeling & Simulation.
Introduction
Collectively, the techniques in this chapter represent the next generation of the AD-OO framework, demonstrating new levels of flexibility and modularity for MRST as a prototyping platform. Further improvements to AD-OO are presented in Chapter 6, where we describe how you can increase simulation efficiency by using high-performance automatic differentiation backends, faster linear solvers, and simulation case management. In addition, Chapter 7 on enhanced water-based oil recovery, Chapter 8 on compositional simulation, and Chapter 11 on unified fractured media modeling make extensive use of the new state function framework and show how it is used in practice.
Numerical Models in MRST
Takes the current and previous state, together with the driving forces and the corresponding time step, and produces the linearized residual equations of the model, differentiated with respect to the canonical set of primary variables (unless the optional 'resOnly' argument is enabled to only the residuals are calculated). As an alternative, it is possible to instead implement the getModelEquations interface that computes the residual equations directly from the state without any awareness of the global set of primary variables. Referring to the green box in Figure 5.2, the model can initialize any of these primary variables with the initVariablesAD function from the AD backend (see Chapter 6) and return the primary variables to the model through initStateAD, the function that provides the state is appropriate for fitting discrete residual equations.
StateFunctions : Framework for AD Functions
We first define a state function class, called TutorialNumber, which simply retrieves any of the numbers x,y,a,b from the corresponding named field in the state structure. This is an external dependency, because the values are obtained from outside the current group of functions, in this case from the state. In the listing, we have also shown the plot of the state function group as it appears when visualized from within MATLAB.
