CE 574
Reactive transport in porous media

6.5 Advection Dispersion Equation (ADE)

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By combining the transport processes outlined above, we can derive an expression for the mass conservation of a non-reactive solute as follows:

∂ ( ϕ C ) ∂ t = − ∇ ⋅ ( J a d v + J d i s p )
(1)

Substitution of Eqn. (1) and (13) into Eqn. (16) yields

∂ ( ϕ C ) ∂ t = − ∇ ⋅ ( ϕ v C ) + ∇ ⋅ ( ϕ D h ∇ C )
(2)

Equation (17) is the classical Advection-Dispersion equation (ADE). For one-dimensional systems, Equation (17) is simplified into

∂ ( ϕ C ) ∂ t = − ∂ ( ϕ v C ) ∂ x + ∂ 2 ( ϕ D h C ) ∂ x 2
(3)

Analytical solution of ADE is available for homogeneous porous media [Zheng and Bennett, 2002]:

C = C 0 2 [ erfc ( L − v t 2 D h t ) + exp ⁡ ( v L D h ) erfc ( L + v t 2 D h t ) ]
(4)

with the initial and boundary conditions:

C ( x , 0 ) = 0 x ≥ 0 C ( 0 , t ) = C 0 t ≥ 0 C ( ∞ , t ) = 0 t ≥ 0
(5)

where C0 is the injecting concentration of the tracer, and erfc(B) is complementary error function:

erfc ( B ) = 1 − erf ( B ) = 1 − 2 π ∫ 0 B e − t 2 d t ∼ 1 − 1 − exp ⁡ ( − 4 B 2 π )
(6)
Here, erf(B) is error function, a special function of sigmoid shape that ocuurs in probability, statistics, and partial differential equations describing diffusion.