Averaging the before and after distribution functions

Hi. I’m experimenting with bulk porous media in my simulations (Dardis - McCloskey type stuff). I’ve been reading a paper by Walsh called “A new partial-bounceback lattice-Boltzmann method for fluid flow through heterogeneous media”. In it, Walsh calculates the macroscopic velocity as

rho * u = sum( 0.5 * c_a * ( f_a(in) + f_a(out) ) )

I’ve seen this kind of averaging in other places too, and in Walsh’s paper, it’s vital for getting the correct porosity when the fluid driver is a body force. The problem is, I don’t understand why you’d do that kind of averaging. Could somebody tell my why or provide me with the reference(s) where this topic is dealt with? Thanks!