Subsurface Hydrology Unsaturated Zone Hydrology Groundwater Hydrology (Hydrogeology )

Water Table R = P - ET - RO P E ET RO R Groundwater waste

Processes we need to model Groundwater flow calculate both heads and flows (q) Solute transport – requires information on flow (velocities) calculate concentrations v = q /  = K I /  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) a spreadsheet (e.g., EXCEL)

Components of a Mathematical Model Governing Equation Boundary Conditions Initial conditions (for transient problems) The governing equation for solute transport problems is the advection-dispersion equation. In full solute transport problems, we have two mathematical models: one for flow and one for transport.

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. Links to MODFLOW

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 Impermeable Rock Groundwater divide Groundwater divide Homogeneous, isotropic aquifer 2D, steady state Head specified along the water table

Homogeneous, isotropic aquifer Toth Problem with contaminant source Impermeable Rock Groundwater divide Groundwater divide 2D, steady state Contaminant source

Processes to model 1. Groundwater flow 2. Transport (a) Particle tracking: requires velocities and a particle tracking code. calculate path lines (b) Full solute transport: 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 Impermeable Rock Groundwater divide Groundwater divide advection-dispersion eqn 2D, steady state Contaminant source

Processes we need to model Groundwater flow calculate both heads and flows (q) Solute transport – requires information on flow (velocities) calculate concentrations v = q/n = K I / n 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. True flow paths Linear flow paths assumed in Darcy’s law Figures from Hornberger et al. (1998) The deviation of flow paths from the linear Darcy paths is known as dispersion.

In addition to advection, we need to consider two other processes in transport problems. Dispersion Chemical reactions Advection-dispersion equation with chemical reaction terms.

Dispersion Advection Chemical Reactions Source/sink term Change in concentration with time  is porosity; D is dispersion coefficient; v is velocity. Allows for multiple chemical species

advection-dispersion equation groundwater flow equation

advection-dispersion equation groundwater flow equation

1D, transient flow; homogeneous, isotropic, confined aquifer; no sink/source term Flow Equation: Transport Equation: Uniform 1D flow; longitudinal dispersion; No sink/source term; retardation

1D, transient flow; homogeneous, isotropic, confined aquifer; no sink/source term Flow Equation: Transport Equation: Uniform 1D flow; longitudinal dispersion; No sink/source term; retardation

Assumption of the Equivalent Porous Medium (epm) Representative Elementary Volume REV

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 Specific Storage S s =  V / (  x  y  z  h) K x, K y, K z are components of the hydraulic conductivity tensor.

Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow div q = - S s (  h  t) +R* (Law of Mass Balance) q = - K grad h (Darcy’s Law) div (K grad h) = S s (  h  t) – R *

Figure taken from Hornberger et al. (1998) Darcy column  h/L = grad h q = Q/A Q is proportional to grad h

q = - K grad h K is a tensor with 9 components

K = K xx K xy K xz K yx K yy K yz K zx K zy K zz Principal components of K

q = - K grad h Darcy’s law grad h qequipotential line grad hq Isotropic Kx = Ky = Kz = K Anisotropic Kx, Ky, Kz

q = - K grad h

x z x’ z’ globallocal  K xx K xy K xz K yx K yy K yz K zx K zy K zz K’ x K’ y K’ z [K] = [R] -1 [K’] [R]

Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow div q = - S s (  h  t) +R* (Law of Mass Balance) q = - K grad h (Darcy’s Law) div (K grad h) = S s (  h  t) – R *

 x  y  z = change in storage OUT – IN = = -  V/  t S s =  V / (  x  y  z  h)  V = S s  h (  x  y  z) tt tt

OUT – IN = = -  V tt

Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow div q = - S s (  h  t) +W* (Law of Mass Balance) q = - K grad h (Darcy’s Law) div (K grad h) = S s (  h  t) – W *

2D confined: 2D unconfined: Storage coefficient (S) is either storativity or specific yield. S = S s b & T = K b