is the liquid-phase colloid concentration;
is the coordinate along the fracture;
is the colloid dispersion coefficient;
is the average interstitial velocity in the fracture;
is the fracture width; and
is the concentration of colloid deposited in the fracture wall expressed as mass of colloid per unit area of fracture surface. The second term on the left hand sign of this equation represents the mass flux of colloid onto the surfaces of the fracture and can be expressed as
is the fracture surface deposition coefficient. For a semi-infinite fracture the initial and boundary conditions are:
is the source colloid concentration.
An analytical solution for this equation can be derived using Laplace transforms and is given by :
Increasing the deposition coefficient results in a reduction in the liquid phase colloid concentration since the deposition coefficient determines the amount of colloid deposited on the fracture walls; the liquid phase colloid concentration decreases with decreasing fracture width because the smaller the fracture width the easier the access of the colloidal particles to the walls of the fracture. You can vary these and other parameters to follow the dynamics of the system.