Selection of Domain Size for Lattice Boltzmann Simulation to Calculate Flow Properties of Porous Media
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The Lattice Boltzmann method (LBM) is commonly used to simulate fluid flow through porous media because of its ease of implementation in computer programming. The discretization required for LBM is inherent and readily available for the segmented micro-CT images in the form of pixels. LBM is an alternative way to solve the Navier-Stokes equation that describes the motion of fluids. LBM considers a definite number of discrete nodes or positions in the domain in which fluid can only move from one node to the other along the particular set of velocity directions. In the D3Q19 stencil system of LBM, 19 velocity directions are considered in the three dimensional domain. For LBM implementation, for each node in space needs to hold 19 velocity values along with other variables. To keep the account of all these variables at every time step and to update their values for the next time step, a large amount of computational memory is required. This computational memory requirement limits the maximum domain size possible to simulate in a computer with a finite amount of memory. For handling large domain sizes, computer memory requirements for the simulation can be met by using a distributed memory system. The question is how big the domain should be to simulate fluid flow to approximate the core level results from the micro-CT images. This paper proposes some new insights that help select the domain size for the LBM simulation of micro-CT data.
In this study different properties of the porous media such as porosity, permeability, specific surface area and surface to volume ratio were calculated on different domain sizes starting from 64-cube up to 520-cube of the porous media obtained from the micro-CT images. The micro-CT images were obtained from the CT scan of a core size sample which was 1.5 inches in diameter and almost three inches in length. Only a portion of the rock sample was scanned to produce images with resolution of 2.32 microns in x, y and z directions. Variation of different properties was plotted against their domain size to see its effect on the properties. The porosity and permeability were measured experimentally on the same core plug with established laboratory core analysis methods. The permeability values for the smaller domain sizes fluctuated due to the microscopic domain effect and stabilized to match the experimentally measured permeability for the domain size beyond the 500-cube. The computer code written for this simulation was first verified by reproducing the analytical results from known flow models such as, Poiseuille flow through narrow slits, flow through circular conduit, etc. Porosity and surface area were calculated by counting the appropriate pixels of the images. The ratio of porosity and specific surface area was plotted against the domain size and found to be asymptotic at the domain size larger than the 500-cube. So, the optimum domain size to simulate fluid flow through porous media for calculating permeability should be free from microscopic domain effects should be selected where the ratio of porosity and specific surface area values become asymptotic. One of the practical applications of the lattice Boltzmann model and the methodology developed is the ability to predict the flow and storage capacities of hydrocarbon reservoirs using small cuttings generated through the drilling of wells without resorting to the expensive and risky process of coring.