A simulation algorithm is elaborated for liquid moving into porous media along the predefined directions of velocities.
Aims
- To develop liquid moving model for porous environment.
- To make time-step calculations.
Solution
- To use liquid predefined moving direction and the weight of the each direction.
- To make liquid velocity based on the liquid predefined moving direction.
- To use liquid relaxation time in order to define time the liquid moving part is done in.
- To interpolate the liquid count after moving for the regular grid.
Solution actions
- Calculate the distribution function for the liquid in each predefined direction value for all the grid points.
- Sum up the values of distribution functions at each grid point in each direction in order to get the density of the liquid.
- Compute the velocity of the liquid at each point of the grid.
- Calculate local equilibrium function for each direction at each point of the grid. By using time step and calculated velocity we get how far the liquid can move in the considered direction.
Liquid predefined moving directions and velocities
The velocity of the liquid in the certain grid point is calculated using:
where p is pressure.
To nd out local distribution of the liquid the equilibrium function should be used in the way of:
where p is the pressure, w is the weighting function, fi is the liquid currently analyzed potential
moving direction and u is the vector of the liquid velocity in a [V x; V y]T form.
The algorithm of the application is:
- Calculate the liquid each potential directions values for all of the grid points.
- For each grid point sum the each direction values in order to get the density of the liquid in
each point of the grid. - Compute the velocity of the liquid in each point of the grid.
- Calculate local equilibrium function for each direction of the each point of the grid. By this
step we get how far the liquid can move in the potential direction. - Compute the time the liquid can move from its position to the new one in the potential
direction. - Calculate the liquid new position after advactive moving step made from the regular grid point
position for every grid point. - By using interpolation nd out the liquid position in the regular grid by using the liquid new
position after advactive moving step made from the regular grid point position for every grid
point.
Interpolation
When the liquid moved from the regular grid point to the advective one the value on the liquid in the regular grid point the liquid moved from can be found within the area of 9 point of the analyzed grid.
The central point value is unknown because the liquid moved from it, but the 8 points around the advective point are regular with the previous time step or after interpolation got values.
To solve Ac=B equation, where c – unknown vector the relationship coefficients (size=9).
A – matrix. Each row contains the same set of freely chosen X and/or Y elements of the Taylor series, where x and y component values are the differences between the central advective point and the current point coordinates each row uses coordinates of the only one point of the analyzed grid.
B – vector. Each element is a value at the point in 3×3 matrix used for interpolation.
To calculate the liquid amount in the centre of the regular grid interpolating area it is needed to sum of multiplications of the conforming by index c vector elements and the matrix A Taylor series elements, where x and y are the differences between the central advective point and the coordinates of the point the values should be found for.
Calculation results
