Derivation of the Advection-Dispersion Equation (ADE) Assumptions 1. Equivalent porous medium (epm) (i.e., a medium with connected pore space or a densely fractured medium with a single network of connected fractures) 2. Miscible flow (i.e., solutes dissolve in water; DNAPLs and LNAPLs require a different governing equation. See p. 472, note 15.5, in Zheng and Bennett.) 3. No density effects

Density-dependent flow requires a different governing equation. See Zheng and Bennett, Chapter 15. Figures from Freeze & Cherry (1979) Derivation of the Advection-Dispersion Equation (ADE) Darcys law: h1 h 2 h1 Q KA s

h2 q = Q/A s advective flux fA = q c f = F/A How do we quantify the dispersive flux? h1 h2

fA = advective flux = qc f = fA + fD s How about Ficks law of diffusion? FDiff c 2 c1 DdA s

where Dd is the effective diffusion coefficient. Dual Porosity Domain Figure from Freeze & Cherry (1979) Ficks law describes diffusion of ions on a molecular scale as ions diffuse from areas of higher to lower concentrations. We need to introduce a law to describe dispersion, to account for the deviation of velocities from the average linear velocity

calculated by Darcys law. Average linear velocity True velocities We will assume that dispersion follows Ficks law, or in other words, that dispersion is Fickian. This is an important assumption; it turns out that the Fickian assumption is not strictly valid near the source of the contaminant. porosity c 2 c1

fD D s where D is the dispersion coefficient. Porosity () Mathematically, porosity functions as a kind of units conversion factor. for example: qc=vc Later we will define the dispersion coefficient in terms of v and therefore we insert now: c 2 c1 fD D

s Case 1 Assume 1D flow qx s and a line source Case 1 porosity

Advective flux Assume 1D flow h 2 h1 fA qxc [ K ]c vxc x qx Dispersive flux s c 2 c1 fD Dx

x D is the dispersion coefficient. It includes the effects of dispersion and diffusion. Dx is sometimes written DL and called the longitudinal dispersion coefficient. Assume 1D flow Case 2 qx s and a point source

Advective flux fA = q x c c 2 c1 fDx Dx ( ) x Dispersive fluxes c 2 c1 fDy Dy ( ) y c 2 c1

fDz Dz ( ) z Dx represents longitudinal dispersion (& diffusion); Dy represents horizontal transverse dispersion (& diffusion); Dz represents vertical transverse dispersion (& diffusion). Continuous point source Average linear velocity Instantaneous point source center of mass

Figure from Freeze & Cherry (1979) Instantaneous Point Source Gaussian longitudinal dispersion transverse dispersion Figure from Wang and Anderson (1982) Derivation of the ADE for 1D uniform flow and 3D dispersion

(e.g., a point source in a uniform flow field) vx = a constant vy = vz = 0 f = fA + fD c 2 c1 fDx Dx( ) x c 2 c1 fDy Dy( ) y

c 2 c1 fDz Dz ( ) z Mass Balance: Flux out Flux in = change in mass Porosity () There are two types of porosity in transport problems: total porosity and effective porosity. Total porosity includes immobile pore water, which contains solute and therefore it should be accounted for when determining the total mass in the system. Effective porosity accounts for water in interconnected pore

space, which is flowing/mobile. In practice, we assume that total porosity equals effective porosity for purposes of deriving the advection-dispersion eqn. See Zheng and Bennett, pp. 56-57. Definition of the Dispersion Coefficient in a 1D uniform flow field vx = a constant vy = vz = 0 Dx = xvx + Dd Dy = yvx + Dd Dz = zvx + Dd

where x y z are known as dispersivities. Dispersivity is essentially a fudge factor to account for the deviations of the true velocities from the average linear velocities calculated from Darcys law. Rule of thumb: y = 0.1x ; z = 0.1y ADE for 1D uniform flow and 3D dispersion 2c 2c 2c c

c Dx 2 Dy 2 Dz 2 v x t x y z No sink/source term; no chemical reactions Question: If there is no source term, how does the contaminant enter the system? Simpler form of the ADE

2c c c D 2 v x t x Uniform 1D flow; longitudinal dispersion; No sink/source term; no chemical reactions Question: Is this equation valid for both point and line sources? There is a famous analytical solution to this form of the

ADE with a continuous line source boundary condition. The solution is called the Ogata & Banks solution. Effects of dispersion on the concentration profile no dispersion dispersion t1 t2 t3 t4 (Freeze & Cherry, 1979, Fig. 9.1) (Zheng & Bennett, Fig. 3.11)

Effects of dispersion on the breakthrough curve Instantaneous Point Source Gaussian Figure from Wang and Anderson (1982) Breakthrough curve Concentration profile

long tail Microscopic or local scale dispersion Figure from Freeze & Cherry (1979) Macroscopic Dispersion (caused by the presence of heterogeneities) Homogeneous aquifer Heterogeneous aquifers

Figure from Freeze & Cherry (1979) Dispersivity () is a measure of the heterogeneity present in the aquifer. A very heterogeneous porous medium has a higher dispersivity than a slightly heterogeneous porous medium. Dispersion in a 3D flow field z global

local z x Kxx Kxy Kxz K= x Kx 0 0

Kyx Kyy Kyz 0 Ky 0 Kzx Kzy Kzz 0 0 Kz

[K] = [R]-1 [K] [R] h h h qx Kxx Kxy Kxz x y z h h h

qy Kyx Kyy Kyz x y z h h h qz Kzx Kzy Kzz x y z

Dispersion Coefficient (D) D = D + Dd D represents dispersion Dd represents molecular diffusion Dxx Dxy Dxz D = Dyx Dyy Dyz Dzx Dzy Dzz In general: D >> Dd c c c

fDx Dxx Dxy Dxz x y z c c c fDy Dyx Dyy Dyz x y z

c c c fDz Dzx Dzy Dzz x y z In a 3D flow field it is not possible to simplify the dispersion tensor to three principal components. In a 3D flow field, we must consider all 9 components of the dispersion tensor. The definition of the dispersion coefficient is more complicated for 2D or 3D flow. See Zheng and Bennett,

eqns. 3.37-3.42. Recall, that for 1D uniform flow: Dx = xvx + Dd Dy = yvx + Dd Dz = zvx + Dd General form of the ADE: Expands to 9 terms Expands to 3 terms

(See eqn. 3.48 in Z&B)