Contents

Consistent Discretization

The two-point flux-approximation (TPFA) method, which is the current industry standard discretization method, is not consistent and only convergent on K-orthogonal grids. In this example, we will compare this method to two consistent methods that are also available in two different modules in MRST: a mimetic method with two-point inner product, and a MPFA-O multi-point flux approximation method.

To compare the methods, we condiser the single-phase pressure equation

$$\nabla\cdot v = q, \qquad v=\textbf{--}\frac{K}{\mu}\nabla p,$$

for a two-dimensional Cartesian grid with anisotropic, homogeneous permeability which will violate the K-orthogonality condition. To drive the flow, we impose a single well and zero Dirichlet boundary conditions.

The main idea of the TPFA method is to approximate the flux v over a face f by the difference of the cell centered pressures in the neighboring cells (sharing the face f) weigthed by a face transmissibility T:

$$ v_{ij} = T_{ij}(p_i \textbf{--} p_j).$$

The pressure in each cell is approximated by solving a linear system Ap = b. When ignoring wells, sources, and bc, A and b are given by

$$ a_{ik} = \left\{\begin{array}{cc}
             \sum_j t_{ij}  & \textrm{if } i=k, \\
             -t_{ij} &  \textrm{if }\, i\neq k,
             \end{array} \right. \quad
             \textrm{and} \quad b_i = \int_{i} q \, dx. $$

Once the pressure is known, the flux is calculated using the expression given above.

In the same manner, the MPFA method approximates the flux v over a face f as a linear combination of the cell pressure and cell pressures in neighbor cells sharing at least one vertex with the face f.

The mimetic method approximates the face flux as a linear combination of cell pressures and face pressures. Only in special cases is it possible to make a local stencil for the face flux in terms of cell pressures, while the stencil for the flux in terms of face pressures is always local.

In this example we show non-monotone solutions to the pressure equation that arise from both the MPFA-method and the Mimetic method.

verbose = false;
MODS = mrstModule;
mrstModule add mimetic mpfa

Define and process 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 formate in which cells, faces, nodes, etc. are given explicitly.

nx = 11; ny = 11;
G = cartGrid([nx, ny]);
G = computeGeometry(G);

Set rock and fluid data

The only parameters in the single-phase pressure equation are the permeability $K$, which here is homogeneous, isotropic and equal 100 mD. The fluid has density 1000 kg/m^3 and viscosity 1 cP. We make a non diagonal rock tensor

theta=30*pi/180;
U=[cos(theta),sin(theta);-sin(theta),cos(theta)];
rocktensor = U'*diag([0.1,100])*U;
rocktensor =[rocktensor(1,1),rocktensor(1,2),rocktensor(2,2)];
rock.perm = repmat(rocktensor, [G.cells.num, 1]) .* 1e-3*darcy();
fluid     = initSingleFluid('mu' ,    1*centi*poise     , ...
                            'rho', 1014*kilogram/meter^3);

gravity off

Introduce wells

We will include two wells, one rate-controlled vertical well and one horizontal well controlled by bottom-hole pressure. Wells are described using a Peacemann model, giving an extra set of equations that need to be assembled. We need to specify ('InnerProduct', 'ip_tpf') to get the correct well model for TPFA.

The first well is vertical well (vertical is default):

cellsWell1 =  sub2ind(G.cartDims,floor(nx/2)+1,floor(ny/2)+1);
radius     = .1;
% well with wellindex calculated for TPFA
bhp=1;
W = addWell([], G, rock, cellsWell1,          ...
            'Type', 'bhp', 'Val', bhp*barsa(), ...
            'Radius', radius, 'InnerProduct', 'ip_tpf');
% well with wellindex calculated for MIMETIC
W_mim = addWell([], G, rock, cellsWell1,          ...
            'Type', 'bhp', 'Val', bhp*barsa(), ...
            'Radius', radius, 'InnerProduct', 'ip_simple');
The second well is horizontal in the 'y' direction:

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 (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 = pside([], G, 'LEFT',  0);
bc = pside(bc, G, 'RIGHT', 0);
bc = pside(bc, G, 'BACK', 0);
bc = pside(bc, G, 'FRONT', 0);

APPROACH 1: Direct/Classic TPFA

Initialize solution structure with reservoir pressure equal 0. Compute one-sided transmissibilities for each face of the grid from input grid and rock properties. The harmonic averages of ones-sided transmissibilities are computed in the solver incompTPFA.

T = computeTrans(G, rock);

Initialize well solution structure (with correct bhp). No need to assemble well system (wells are added to the linear system inside the incompTPFA-solver).

resSol1 = initState(G, W, 0);

% Solve linear system construced from T and W to obtain solution for flow
% and pressure in the reservoir and the wells. Notice that the TPFA solver
% is different from the one used for mimetic systems.
resSol1 = incompTPFA(resSol1, G, T, fluid, 'wells', W, 'bc',bc);

APPROACH 2: Mimetic with TPFA-inner product

Initialize solution structure with reservoir pressure equal 0. Compute the mimetic inner product from input grid and rock properties.

IP = computeMimeticIP(G, rock, 'InnerProduct', 'ip_tpf');

Generate the components of the mimetic linear system corresponding to the two wells and initialize the solution structure (with correct bhp)

resSol2 = initState(G, W, 0);

Solve mimetic linear hybrid system

resSol2 = solveIncompFlow(resSol2, G, IP, fluid, 'wells', W_mim,'bc',bc);

APPROACH 3: MPFA method

Initialize solution structure with reservoir pressure equal 0. Compute the transmisibility matrix for mpfa

T_mpfa = computeMultiPointTrans(G, rock);

Generate the components of the mimetic linear system corresponding to the two wells and initialize the solution structure (with correct bhp) We can use the same well structure as for TPFA

resSol3 = initState(G, W, 0);

Solve mimetic linear hybrid system

resSol3 = incompMPFA(resSol3, G, T_mpfa, fluid, 'wells', W,'bc',bc);

Plot solutions

Plot the pressure and producer inflow profile make Caresian grid

X=reshape(G.cells.centroids(:,1),G.cartDims);
Y=reshape(G.cells.centroids(:,2),G.cartDims);
clf
p = get(gcf,'Position'); set(gcf,'Position', [p(1:2) 900 500]);
subplot(2,3,1)
   plotCellData(G, resSol1.pressure(1:G.cells.num) ./ barsa());
   title('Pressure: direct TPFA'); view(2), axis tight off
   colorbar('Location','SouthOutside');
subplot(2,3,4)
   mesh(X,Y,reshape(resSol1.pressure(1:G.cells.num) ./ barsa(),G.cartDims));
   axis tight, box on, view(30,60);

subplot(2,3,2)
   plotCellData(G, resSol2.pressure(1:G.cells.num) ./ barsa());
   title('Pressure: mimetic'), view(2), axis tight off
   colorbar('Location','SouthOutside');
subplot(2,3,5)
   mesh(X,Y,reshape(resSol2.pressure(1:G.cells.num) ./ barsa(),G.cartDims));
   axis tight, box on, view(30,60);

subplot(2,3,3)
   plotCellData(G, resSol3.pressure(1:G.cells.num) ./ barsa());
   title('Pressure: mpfa'); view(2), axis tight off
   colorbar('Location','SouthOutside');
subplot(2,3,6)
   mesh(X,Y,reshape(resSol3.pressure(1:G.cells.num) ./ barsa(),G.cartDims));
   axis tight, box on, view(30,60);

% display the flux in the well for tpfa, mimetic and mpfa
disp('');
disp('Flux in the well for the three different methods:');
disp(['     TPFA   : ',num2str(resSol1.wellSol(1).flux .* day())]);
disp(['     Mimetic: ',num2str(resSol2.wellSol(1).flux .* day())]);
disp(['     MPFA-O : ',num2str(resSol3.wellSol(1).flux .* day())]);
Flux in the well for the three different methods:
     TPFA   : 0.59397
     Mimetic: 0.50002
     MPFA-O : 0.17445
mrstModule clear
mrstModule('add', MODS{:})