Image-based simulations of absolute permeability with massively parallel pseudo-compressible stabilised finite element solver

Abstract

We apply an accurate parallel stabilised finite element method to solve for Navier-Stokes equations directly on a binarised three-dimensional rock image, obtained by micro-CT imaging. The proposed algorithm has several advantages. First, the linear equal-order finite element space for velocity and pressure is ideal for presenting the pixel images. Second, the algorithm is fully explicit and versatile for describing complex boundary conditions. Third, the fully explicit matrix–free finite element implementation is ideal for parallelism on high-performance computers, similar to lattice Boltzmann. In the last, the memory usage is low compared with lattice Boltzmann or implicit finite volume. We compute the permeability of a range of rock images. The stabilisation parameter may affect the velocity, and an optimal parameter is chosen from the numerical tests. The steady state results are comparable with lattice Boltzmann method and implicit finite volume. The transient behaviour of pseudo-compressible stabilised finite element and lattice Boltzmann method is very similar. Our analysis shows that the stabilised finite element is an accurate and efficient method with low memory cost for the image- based simulations of flow in the pore scale up to 1 billion voxels on 128-GB ram workstation and on distributed clusters.

Appendix: The single phase lattice Boltzmann method

We use the multi-relaxation-time scheme D3Q19 lattice Boltzmann method for Stokes flow calculation [17]. D3 represents space dimension and Q19 indicates the number of discretised microscopic velocities.

The LBM equation is:

$$ f_{i}(t+{\Delta} t, x+e_{i}{\Delta} t)=f_{i}(t,x) + {\Omega}_{i},\quad i = 0,...,18, $$

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where e_i is the discretised microscopic velocities:

$$ e_{i} = \left\{ \begin{array}{cccccccccccccccccccc} 0 & c & -c & 0 & 0 & 0 & 0 & c & -c & c & -c & c & -c & c & -c & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & c & -c & 0 & 0 & c & -c & -c & c & 0 & 0 & 0 & 0 & c & -c & c &-c\\ 0 & 0 & 0 & 0 & 0 & c & -c & 0 & 0 & 0 & 0 & c & -c & -c & c & c & -c & -c & c \end{array} \right\} $$

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f_i are mass fractions and e_i are discretised micro velocities. The collision operator for MRT model :

$$ {\Omega}=M^{-1}S((Mf)-m^{eq}) $$

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M_ij is the transformation matrix defined as:

$$ M= \left[ \begin{array}{cccccccccccccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -2 & -2 & -2 & -2 & -2 & -2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 0 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0\\ 0 & -2 & 2 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & 1 &-1\\ 0 & 0 & 0 & -2 & 2 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1\\ 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\ 0 & 0 & 0 & 0 & 0 & -2 & 2 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\ 0 & 2 & 2 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -2 & -2 & -2 &-2\\ 0 & -2 & -2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -2 & -2 & -2 &-2\\ 0 & 0 & 0 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 &0\\ 0 & 0 & 0 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 1 &-1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \end{array} \right] $$

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The equilibrium moments is calculated by m^eq = Mf^eq. The collision matrix S is diagonal comprised of relaxation rates:

$$ \begin{array}{@{}rcl@{}} &&s_{1,1} = -s_{e},\\ &&s_{2,2} = -s_{\xi},\\ &&s_{4,4} = s_{6,6} = s_{8,8} = -s_{q},\\ &&s_{10,10}=s_{12,12}=-s_{\pi},\\ &&s_{9,9} = s_{11,11} = s_{13,13} = s_{14,14} = s_{15,15} = -s_{\nu},\\ &&s_{16,16}=s_{17,17}=s_{18,18}=-s_{m}. \end{array} $$

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where s_ν is the relaxation rate and is defined by viscosity ν as:

$$ \frac{1}{s_{\nu}}=3\frac{\nu}{c^{2} {\Delta} t}+\frac{1}{2}. $$

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c is a constant microscopic reference velocity and it is set as 1 in this work. For other relaxation rates, values suggested by [11] is used to achieve improved accuracy and stability:

$$ s_{e}=s_{\xi}=s_{\pi}=s_{\nu}, s_{q}=s_{m}=8\frac{2-s_{\nu}}{8-s_{\nu}}. $$

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To save memory, the calculation is only performed on fluid nodes, and no memory is allocated for any solid node. MPI is used for code parallelisation. A unstructured graphs partition library ParMetis is used to optimise the geometry partition. ParMetis is a multilevel partitioning method, which is able to divide the geometry into partitions of which the mesh sizes and communication sizes between different domains are as homogeneous as possible.