Lattice gas simulation of Darcy flow with memory-free collisions

A lattice gas algorithm is proposed for the simulation of water flow in the unsaturated zone. Microscopic dynamics of a two-dimensional model system are defined. Up to four fluid particles occupy the sites of a square lattice. At each time step, the particles are sent to neighbouring sites according to probabilistic rules which depend on the permeability and the potential but not on the input velocities of the particles. On the macroscopic scale, the flow is described by a diffusion term and a Darcy term. Several extensions including higher dimension are discussed.List of Symbols c ⁽ⁿ⁾ constant in the definition of the rejection probabilityP forn = 1,2,3 particles at a site 0 c ⁽ⁿ⁾ 1 - D diffusion constant - D vertical extent of the system, measured in cells - E _i vector connecting a site to its neighbour in directioni - i direction of a nearest neighbour site,i = 1,..., 4 - j direction of a nearest neighbour site,j = 1,..., 4 - j mass transport (fluid flow),j = v - j ^x x-component of the flowj - k(x) spatial dependence of the permeability, user defined under the constraint 0 k 1 - k () the part of the permeability which depends on the degree of saturation (seek) - k ⁽ⁿ⁾ (x) effective permeability at a sitex that holdsn particles - L horizontal extent of the system, measured in cells - l _mac macroscopic length scale, e.g. one meter - l _mic microscopic length scale (one lattice constant) - m integer number of time steps - n (x) number of particles at the lattice sitex - N _A total number of particles on all A-sites - P probability for rejection of a randomly selected direction or set of directions - p arithmetic mean of the probability for a site to receive a particle from a particular neighbour (the average is taken over the four neighbours) - p _i ⁽ⁿ⁾ probability that one out ofn particles at a site is sent in directioni - p _ij ⁽²⁾ probability that the two particles at a site are sent in directionsi andj - t time - t _mac macroscopic time scale, e.g. one day - t _mic microscopic time scale (one time step) - v fluid velocity - x space vector, mostly two-dimensional:x = (x, y) - x horizontal component ofx - y vertical component ofx - quotient of microscopic and macroscopic time scales,t _mic /t _mac - quotient of microscopic and macroscopic length scales,l _mic /l _mac - _i p + _i is the probability that a particle is received from the neighbour atx +E _i - K(X, ) effective permeability,k =k(x)k () - correlation length - degree of saturation, used synonymously with density (homogeneous porosity) - ₀ value of a homogeneous particle density - ø(x) external potential (user defined), ø = ^gr + ^mat - ø(x) arithmetic mean of the external potential at the four sites surroundingx - ø _i external potential at the sitex +E _i - total potential, = ø + ^den - ^gr(x) gravitational potential - ^mat(x) matrix potential - ^den() density-dependent potential - _n potential depending on the occupation number - ⁽ⁿ⁾ (x) probability that sitex is occupied byn particles - ₀ ^{(n) (n)} in a system with homogeneous particle density - mac macroscopic - mic microscopic