# Relaxation Time in porous media simulations

Hi everyone!

I get stuck at understanding the conversion problem in LBM. The problem is that I get different results with the same input of physical units. I am sorry in advance for such a total explanation of what I do, but I just want to be clear.

For example, let’s use parameters below:
N=100x100x100
L = 0.01 m
press_diff = 100 Pa
kin. visc = 1e-06 m^2/s (water, 20 C)
rho = 1000 kg/m^3

Discretization
(I follow the recommendations and algorithm from this book, chapter 7):
Freely choose tau.
For example, tau = 0.54
visc_LB = c_s^2*(tau-0.5)= 0.0133 (c_s^2 = 1/3 <= delta_x_LB=1, delta_t_LB = 1)

Conversion factors:
C_visc = kin.visc/visc_LB = 7.5e-05 (C_visc = C_l^2/C_t by definition of the visc - m^2/s)
C_l = L/L_LB = L/(delta_x_LB * N)= L/N = 1e-04
C_t = С_l^2/C_visc = 0.00013
C_u = C_l/C_t = 0.74
C_rho = 1000 ( rho_LB = 1)
C_press = C_rho * C_u^2 = 562.5

LB units:

p_LB = p_real/C_press = 0.178

It’s all I need, like in the permeability tutorial.
I obtain abs. permeability(k) and average velocity(u_av_LB) in LB units:
k_LB = 0.0034
u_av_LB = 4.41e-05

After converting to real units:
k = 3.4e-11
u_av = 3.308e-05

But if I decide to use tau = 0.52
tau changes = > C_t changes = > C_u changes => C_p changes = > p_LB changes. So, new variables:
C_t = 6.66e-05
C_u = 1.5
C_p = 2250
p_LB = 0.00444

How I understand that, If I use new tau and new p_LB, I still should get the same abs. permeability and avg. velocity in real units, but:
k_LB = 9.94e-04
u_av_LB = 6.17e-06

k = k_LB * C_l^2 = 9.94e-12
u_av = u_av_LB *C_u = 9.25e-06

The results have such a big difference between them. I checked stability and accuracy relations for tau and u_max_LB, all good. No iterations limits, I use relative energy convergence to get steady-state.

What I do wrong? If I use the recommendation of delta_t=delta_x^2(C_t=C_l^2) tau becomes = 0.500003. It’s incorrect to use I think.
The problem becomes more disturbing when it comes to real pressure gradients in porous media. Huge pressure differences make me also use “close-to-limit” tau, which I think is an incorrect approach. How to do with such things? Of course, I use more resolved geometries, like 1000**3 but is it the only way out? And I still doubt about “Have I chooses the right tau?”