Basic Flow-Solver Tutorial
The purpose of this example is to give an overview of how to set up and use the single-phase mimetic pressure solver to solve the single-phase pressure equation
for a flow driven by Dirichlet and Neumann boundary conditions. Our geological model will be simple a Cartesian grid with anisotropic, homogeneous permeability.
In this tutorial example, you will learn about:
- the grid structure,
- how to specify rock and fluid data,
- the structure of the data-objects used to hold solution,
- how to assemble and solve linear systems,
- the structure of the mimetic linear systems,
- useful routines for visualizing and interacting with the grids and simulation results.
Contents
Define geometry
Construct a Cartesian grid of size 10-by-10-by-4 cells, where each cell has dimension 1-by-1-by-1. Because our flow solvers are applicable for general unstructured grids, the Cartesian grid is here represented using an unstructured format, in which cells, faces, nodes, etc. are given explicitly.
nx = 10; ny = 10; nz = 4; G = cartGrid([nx, ny, nz]); display(G);
G = cells: [1x1 struct] faces: [1x1 struct] nodes: [1x1 struct] cartDims: [10 10 4] type: {'tensorGrid' 'cartGrid'}
After the grid structure is generated, we plot the geometry.
plotGrid(G); view(3), camproj orthographic, axis tight, camlight headlight
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Process geometry
Having set up the basic structure, we continue to compute centroids and volumes of the cells and centroids, normals, and areas for the faces. For a Cartesian grid, this information can trivially be computed, but is given explicitly so that the flow solver is compatible with fully unstructured grids.
G = computeGeometry(G);
Set rock and fluid data
The only parameters in the single-phase pressure equation are the permeability and the fluid viscosity
. We set the permeability to be homogeneous and anisotropic
The viscosity is specified by saying that the reservoir is filled with a single fluid, for which de default viscosity value equals unity. Our flow solver is written for a general incompressible flow and requires the evaluation of a total mobility, which is provided by the fluid object.
rock.perm = repmat([1000, 100, 10].* milli*darcy(), [G.cells.num, 1]);
mrstModule add incomp fluid = initSingleFluid('mu' , 1*centi*poise , ... 'rho', 1014*kilogram/meter^3);
Initialize reservoir simulator
To simplify communication among different flow and transport solvers, all unknowns are collected in a structure. Here this structure is initialized with uniform initial reservoir pressure equal 0 and (single-phase) saturation equal 0.0 (using the default behavior of initResSol)
resSol = initResSol(G, 0.0); display(resSol);
resSol = pressure: [400x1 double] flux: [1380x1 double] s: [400x1 double]
Impose Dirichlet boundary conditions
Our flow solvers automatically assume no-flow conditions on all outer (and inner) boundaries; other type of boundary conditions need to be specified explicitly.
Here, we impose Neumann conditions (flux of 1 m^3/day) on the global left-hand side. The fluxes must be given in units of m^3/s, and thus we need to divide by the number of seconds in a day. Similarly, we set Dirichlet boundary conditions p = 0 on the global right-hand side of the grid, respectively. For a single-phase flow, we need not specify the saturation at inflow boundaries. Similarly, fluid composition over outflow faces (here, right) is ignored by pside.
bc = fluxside([], G, 'LEFT', 1*meter^3/day()); bc = pside (bc, G, 'RIGHT', 0); display(bc);
bc = face: [80x1 int32] type: {1x80 cell} value: [80x1 double] sat: []
Construct linear system
Construct mimetic pressure linear system components for the system Ax = b
based on input grid and rock properties for the case with no gravity.
gravity off;
mrstModule add mimetic S = computeMimeticIP(G, rock); % Plot the structure of the matrix (here we use BI, the inverse of B, % rather than B because the two have exactly the same structure) clf, subplot(1,2,1) cellNo = rldecode(1:G.cells.num, diff(G.cells.facePos), 2) .'; C = sparse(1:numel(cellNo), cellNo, 1); D = sparse(1:numel(cellNo), double(G.cells.faces(:,1)), 1, ... numel(cellNo), G.faces.num); spy([S.BI , C , D ; ... C', zeros(size(C,2), size(C,2) + size(D,2)); ... D', zeros(size(D,2), size(C,2) + size(D,2))]); title('Hybrid pressure system matrix')
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The block structure can clearly be seen in the sparse matrix A, which is never formed in full. Indeed, rather than storing B, we store its inverse B^-1. Similarly, the C and D blocks are not represented in the S structure; they can easily be formed explicitly whenever needed, or their action can easily be computed.
display(S);
S = BI: [2400x2400 double] ip: 'ip_simple' type: 'hybrid'
Solve the linear system
Solve linear system construced from S and bc to obtain solution for flow and pressure in the reservoir. Function incompMimetic demands that we pass a well solution structure even if the reservoir has no wells, so we initialize an empty wellSol structure. The option 'MatrixOutput=true' adds the system matrix A to resSol to enable inspection of the matrix.
resSol = incompMimetic(resSol, G, S, fluid, ... 'bc', bc, 'MatrixOutput', true); display(resSol);
resSol = pressure: [400x1 double] flux: [1380x1 double] s: [400x1 double] facePressure: [1380x1 double] A: [1340x1340 double]
Inspect results
The resSol object contains the Schur complement matrix used to solve the hybrid system.
subplot(1,2,2), spy(resSol.A);
title('Schur complement system matrix');
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We then plot convert the computed pressure to unit 'bar' before plotting result.
clf plotCellData(G, convertTo(resSol.pressure(1:G.cells.num), barsa()), ... 'EdgeColor', 'k'); title('Cell Pressure [bar]') xlabel('x'), ylabel('y'), zlabel('Depth'); view(3); shading faceted; camproj perspective; axis tight; colorbar
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