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WOLFRAM NOTEBOOK
Transport and Deposition of Colloid in Rock Fractures
Transport and Deposition of Colloid in Rock Fractures
This Demonstration describes the dynamics of transport and deposition of colloidal particles in the walls of a rock fracture. The partial differential equation governing colloid transport in a one-dimensional fracture idealized as two parallel plates can be written as [1]:
∂c(t,x)
∂t
2
b
∂(t,x)
*
c
∂t
D
c
∂c(t,x)
∂
2
x
∂c(t,x)
∂x
where is the liquid-phase colloid concentration, is the coordinate along the fracture, is time, 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 side of this equation represents the mass flux of colloid onto the surfaces of the fracture and can be expressed as:
c
x
t
D
c
U
b
*
c
∂(t,x)
*
c
∂t
κU
b
where is the fracture surface deposition coefficient. For a semi-infinite fracture, the initial and Cauchy boundary conditions are:
κ
c(0,x)=0
-+Uc(t,0)=U
D
c
∂c(t,0)
∂x
c
0
and
∂(t,∞)
∂x
where =1.0 is the source colloid concentration.
c
0
An analytical solution for this equation can be derived using Laplace transforms and is given by [1]:
c(t,x)=exp(1-ξ)erfc+exp(1+ξ)erfc+exp-erfc
c
0
1
1+ξ
Ux
2
D
c
x-Utξ
2
t
D
c
1
1-ξ
Ux
2
D
c
x+Utξ
2
t
D
c
U
2
b
4κ
D
c
Ux
D
c
2Uκt
2
b
x+Ut
2
t
D
c
where .
ξ=
1+
8κ
D
c
U
2
b
The breakthrough curve (red curve) does not reach the maximum value of 1 because colloid is deposited in the fracture surfaces. 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; moreover, 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.