Geology 727

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Geology 727. Transport Modeling in Groundwater. Subsurface Hydrology. Unsaturated Zone Hydrology. Groundwater Hydrology (Hydrogeology ). R = P - ET - RO. ET. ET. P. E. RO. R. waste. Water Table. Groundwater. v = q /  = K I / . Processes we need to model

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Geology 727

Transport Modeling in Groundwater

Subsurface Hydrology

Unsaturated Zone Hydrology

Groundwater Hydrology

(Hydrogeology)

R = P - ET - RO

ET

ET

P

E

RO

R

waste

Water Table

Groundwater

v = q /  = K I /

• Processes we need to model
• Groundwater flow
• calculate both heads and flows (q)
• Solute transport – requires information on flow (velocities)
• calculate concentrations

Darcy’s law

Types of Models

• Physical (e.g., sand tank)
• Analog (electric analog, Hele-Shaw)
• Mathematical

Types of Solutions of Mathematical Models

• Analytical Solutions: h= f(x,y,z,t)
• (example: Theis eqn.)
• Numerical Solutions
• Finite difference methods
• Finite element methods
• Analytic Element Methods (AEM)

Finite difference models

• may be solved using:
• a computer programs (e.g., a FORTRAN program)

Components of a Mathematical Model

• Governing Equation
• Boundary Conditions
• Initial conditions (for transient problems)

In full solute transport problems, we have two

mathematical models: one for flow and one for transport.

The governing equation for solute transport problems is the advection-dispersion equation.

Flow Code: MODFLOW

•  USGS code
•  finite difference code to solve the groundwater flow equation
• MODFLOW 88
• MODFLOW 96
• MODFLOW 2000

Transport Code: MT3DMS

•  Univ. of Alabama
•  finite difference code to solve the advection-dispersion eqn.

The pre- and post-processor

Groundwater Vistas

links and runs MODFLOW and MT3DMS.

Introduction to solute transport modeling

and

Review of the governing equation

for groundwater flow

Conceptual Model

A descriptive representation

of a groundwater system that incorporates an interpretation of the geological,hydrological, and geochemical conditions, including information about the boundaries of the problem domain.

Toth Problem

Head specified along the water table

Groundwater

divide

Groundwater

divide

Homogeneous, isotropic aquifer

Impermeable Rock

Toth Problem with contaminant source

Contaminant source

Homogeneous, isotropic aquifer

Groundwater

divide

Groundwater

divide

Impermeable Rock

Processes to model

• Groundwater flow
• Transport
• Particle tracking: requires velocities and a particle tracking code. calculate path lines
• (b)Full solutetransport: requires velocites and a solute transport model. calculate concentrations

Topo-Drive

Finite element model of a version of the Toth Problem for regional flow in cross section. Includes a groundwater flow model with particle tracking.

Toth Problem with contaminant source

Contaminant source

Groundwater

divide

Groundwater

divide

Impermeable Rock

v = q/n = K I / n

• Processes we need to model
• Groundwater flow
• calculate both heads and flows (q)
• Solute transport – requires information on flow (velocities)
• calculate concentrations

Requires a flow model and a solute transport model.

Groundwater flow is described by Darcy’s law.

This type of flow is known as advection.

Linear flow paths

assumed in Darcy’s law

True flow paths

The deviation of flow paths from

the linear Darcy paths is known

as dispersion.

Figures from Hornberger et al. (1998)

with chemical reaction terms.

In addition to advection, we need to consider two other processes in transport problems.

• Dispersion
• Chemical reactions

Allows for multiple

chemical species

Dispersion

Chemical

Reactions

Source/sink term

Change in concentration

with time

• is porosity;

D is dispersion coefficient;

v is velocity.

groundwater flow equation

groundwater flow equation

Flow Equation:

1D, transient flow; homogeneous, isotropic,

confined aquifer; no sink/source term

Transport Equation:

Uniform 1D flow; longitudinal dispersion;

No sink/source term; retardation

Flow Equation:

1D, transient flow; homogeneous, isotropic,

confined aquifer; no sink/source term

Transport Equation:

Uniform 1D flow; longitudinal dispersion;

No sink/source term; retardation

Assumption of the

Equivalent Porous Medium

(epm)

REV

Representative Elementary Volume

Dual Porosity Medium

Figure from Freeze & Cherry (1979)

Review of the derivation of the

governing equation for

groundwater flow

General governing equation

for groundwater flow

Kx, Ky, Kz are components

of the hydraulic conductivity

tensor.

Specific Storage

Ss = V / (x y z h)

Law of Mass Balance + Darcy’s Law =

Governing Equation for Groundwater Flow

---------------------------------------------------------------

div q = - Ss (h t) +R* (Law of Mass Balance)

q = - Kgrad h (Darcy’s Law)

div (K grad h) = Ss (h t)–R*

Darcy column

Q is proportional

q = Q/A

Figure taken from Hornberger et al. (1998)

K is a tensor with 9 components

Principal components of K

Kxx Kxy Kxz

KyxKyy Kyz

Kzx KzyKzz

K =

Darcy’s law

q

equipotential line

q

Isotropic

Kx = Ky = Kz = K

Anisotropic

Kx, Ky, Kz

global

local

z

z’

x’

x

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

K’x 0 0

0 K’y 0

0 0 K’z

[K] = [R]-1 [K’] [R]

Law of Mass Balance + Darcy’s Law =

Governing Equation for Groundwater Flow

---------------------------------------------------------------

div q = - Ss (h t) +R* (Law of Mass Balance)

q = - Kgrad h (Darcy’s Law)

div (K grad h) = Ss (h t)–R*

V = Ssh (x y z)

t

t

OUT – IN =

x y z

= change in storage

= -V/ t

Ss = V / (x y z h)

OUT – IN =

= - V

t

Law of Mass Balance + Darcy’s Law =

Governing Equation for Groundwater Flow

---------------------------------------------------------------

div q = - Ss (h t) +W* (Law of Mass Balance)

q = - Kgrad h (Darcy’s Law)

div (K grad h) = Ss (h t)–W*

2D confined:

2D unconfined:

Storage coefficient (S) is either storativity or specific yield.

S = Ss b & T = K b