Multiscale Pressure Solver: Flow Driven by Horizontal and Vertical Well

Compare the fine-grid and the multiscale pressure solver by solving the single-phase pressure equation

for a Cartesian grid with isotropic, homogeneous permeability. This example is built upon the flow-solver tutorial "Using Peacemann well models".

Contents

Define the model and set data

We construct the Cartesian grid, set the permeability to 100 mD, and use the default single-phase fluid with density 1000 kg/m^3 and viscosity 1 cP.

nx = 20; ny = 20; nz = 8;
Nx =  5; Ny =  5; Nz = 2;
verbose   = false;
G         = cartGrid([nx ny nz]);
G         = computeGeometry(G);
rock.perm = repmat(100*milli*darcy, [G.cells.num, 1]);
fluid     = initSingleFluid();

Set two wells, one vertical and one horizontal. Note that a vertical well can be constructed by both addWell and verticalWell, so the appropriate choice depends on whether you know the well cells or the I, J, K position of the well.

W = verticalWell([], G, rock, nx, ny, 1:nz,            ...
                'Type', 'rate', 'Val', 1*meter^3/day, ...
                'Radius', .1, 'Name', 'I');
W = addWell(G, rock, W, 1:nx, 'Type','bhp', ...
           'Val', 1*barsa, 'Radius', .1, 'Dir', 'x', 'Name', 'P');

Set up solution structures

Here we need four solution structures, two for each simulator to hold the solutions on the grid and in the wells, respectively.

xrRef = initResSol( G, 0.0);
xwRef = initWellSol(W, 0.0);
xrMs  = initResSol( G, 0.0);
xwMs  = initWellSol(W, 0.0);

Partition the grid

We partition the fine grid into a regular Nx-by-Ny-by-Nz coarse grid in index space so that each coarse block holds (nx/Nx)-by-(ny/Ny)-by-(nz/Nz) fine cells. The resulting vector p has one entry per fine-grid cell giving the index of the corresponding coarse block. After the grid is partitioned in index space, we postprocess it to make sure that all blocks consist of a connected set of fine cells. This step is superfluous for Cartesian grids, but is required for grids that are only logically Cartesian (e.g., corner-point and other mapped grids that may contain inactive or degenerate cells).

p = partitionUI(G, [Nx, Ny, Nz]);
p = processPartition(G, p, 'Verbose', verbose);

% Generate the coarse-grid structure
CG = generateCoarseGrid(G, p, 'Verbose', verbose);

Plot the partition and the well placement

clf
plotCellData(G, mod(p,2),'EdgeColor','k','FaceAlpha',0.5,'EdgeAlpha', 0.5);
plotWell(G, W, 'radius', 0.1, 'color', 'k');
view(3), axis equal tight off

Assemble linear systems

First we compute the inner product to be used in the fine-scale and coarse-scale linear systems. Then we generate the coarse-scale system

gravity off
S  = computeMimeticIP(G, rock, 'Verbose', verbose);

mob = fluid.mob(struct('s', ones([G.cells.num, 1])));
CS = generateCoarseSystem(G, rock, S, CG, mob, 'Verbose', verbose);

Then, we assemble the well systems for the fine and the coarse scale.

W = assembleWellSystem(G, W);
W = generateCoarseWellSystem(G, S, CG, CS, mob, rock, W);
disp('W(1):'); display(W(1));
disp('W(2):'); display(W(2));

W(1):
    cells: [8x1 double]
     type: 'rate'
      val: 1.1574e-05
        r: 0.1000
      dir: 'z'
       WI: [8x1 double]
       dZ: [8x1 double]
     name: 'I'
    compi: [1 0 0]
        S: [1x1 struct]
       CS: [1x1 struct]

W(2):
    cells: [20x1 double]
     type: 'bhp'
      val: 100000
        r: 0.1000
      dir: 'x'
       WI: [20x1 double]
       dZ: [20x1 double]
     name: 'P'
    compi: [1 0 0]
        S: [1x1 struct]
       CS: [1x1 struct]

Solve the global flow problems

[xrRef, xwRef] = solveIncompFlow  (xrRef, xwRef, G, S,                  ...
                                   fluid, 'wells', W, 'Solver', S.type, ...
                                   'MatrixOutput', true);
[xrMs , xwMs ] = solveIncompFlowMS(xrMs, xwMs, G, CG, p, S, CS,         ...
                                   fluid, 'wells', W, 'Solver', S.type, ...
                                   'MatrixOutput', true);

% Report pressure in wells.
dp = @(ws) num2str(convertTo(ws(1).pressure, barsa));
disp(['DeltaP, Fine: ', dp(xwRef)])
disp(['DeltaP, Ms:   ', dp(xwMs )])

DeltaP, Fine: 1.4965
DeltaP, Ms:   1.5009

Plot Schur complement matrices

clf
   subplot(1,2,1); spy(xrRef.A); title('Schur complement matrix, fine scale');
   subplot(1,2,2); spy(xrMs.A);  title('Schur complement matrix, coarse scale');

Plot solution

clf
subplot('Position', [0.02 0.52 0.46 0.42]),
   plotGrid(G, 'FaceColor', 'none', 'EdgeColor', [.6, .6, .6]);
   plotGrid(G, W(1).cells, 'FaceColor', 'b', 'EdgeColor', 'b');
   plotGrid(G, W(2).cells, 'FaceColor', 'r', 'EdgeColor', 'r');
   title('Well cells')
   view(3), camproj perspective, axis tight equal off, camlight headlight

subplot('Position', [0.54 0.52 0.42 0.40]),
   plot(convertTo(xwRef(2).flux, meter^3/day), '-ob'); hold on
   plot(convertTo(xwMs (2).flux, meter^3/day), '-xr');
   legend('Fine', 'Multiscale')
   title('Producer inflow profile')

subplot('Position', [0.02 0.02 0.46 0.42]),
   plotCellData(G, convertTo(xrRef.cellPressure, barsa))
   title('Pressure, fine [bar]')
   view(3), camproj perspective, axis tight equal off, camlight headlight
   cax = caxis; colorbar

subplot('Position', [0.52 0.02 0.46 0.42]),
   plotCellData(G, convertTo(xrMs.cellPressure, barsa));
   title('Pressure, coarse [bar]')
   view(3), camproj perspective, axis tight equal off, camlight headlight
   caxis(cax); colorbar

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Last time modified: $Id: simpleWellExampleMS.m 1945 2009-03-31 09:22:39Z bska $

Published with MATLAB® 7.6