Transcript Subsurface Hydrology - UW

 2c
c
c
D 2 v
 R
x
t
x
Subsurface Hydrology
Unsaturated Zone Hydrology
Groundwater Hydrology
(Hydrogeology)
R = P - ET - RO
ET
ET
P
E
RO
R
Water Table
Groundwater
waste
Processes we need to model
• Groundwater flow
calculate both heads and flows (q)
v = q /  = K I /
• Solute transport – requires
information on flow (velocities)
calculate concentrations
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)
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.
• 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
Head specified along the water table
Groundwater
Groundwater
Homogeneous, isotropic aquifer
divide
divide
Impermeable Rock
2D, steady state
Toth Problem with contaminant source
Contaminant source
Groundwater
Groundwater
Homogeneous, isotropic aquifer
divide
divide
Impermeable Rock
2D, steady state
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
Contaminant source
Groundwater
divide
Groundwater
advection-dispersion eqn
divide
Impermeable Rock
2D, steady state
Processes we need to model
• Groundwater flow
calculate both heads and flows (q)
v = q/n = K I / n
• 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)
In addition to advection, we need to consider
two other processes in transport problems.
• Dispersion
• Chemical reactions
Advection-dispersion equation
with chemical reaction terms.
Allows for multiple
chemical species
Dispersion
Advection
Chemical
Reactions
Change in concentration
with time
Source/sink term
 is porosity;
D is dispersion coefficient;
v is velocity.
advection-dispersion equation
groundwater flow equation
h

h

h

h
Ss

( Kx ) 
( Ky ) 
( Kz )  W *
t
x
x
y
y
z
z
advection-dispersion equation
groundwater flow equation
h

h

h

h
Ss

( Kx ) 
( Ky ) 
( Kz )  W *
t
x
x
y
y
z
z
Flow Equation:
 2h
h
T 2  S
t
x
1D, transient flow; homogeneous, isotropic,
confined aquifer; no sink/source term
Transport Equation:
 2c
c
c
D 2 v
 R
x
t
x
Uniform 1D flow; longitudinal dispersion;
No sink/source term; retardation
Flow Equation:
 2h
h
T 2  S
t
x
1D, transient flow; homogeneous, isotropic,
confined aquifer; no sink/source term
Transport Equation:
 2c
c
c
D 2 v
 R
x
t
x
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

h

h

h
h
( Kx ) 
( Ky ) 
( K z )  Ss
W *
x
x
y
y
z
z
t
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 = - K grad h
(Darcy’s Law)
div (K grad h) = Ss (h t) –R*
Darcy column
h/L = grad h
Q is proportional
to grad h
q = Q/A
Figure taken from Hornberger et al. (1998)
q = - K grad h
K is a tensor with 9 components
Kxx Kxy Kxz
K=
Kyx Kyy Kyz
Kzx Kzy Kzz
Principal components of K
Darcy’s law
q = - K grad h
q
equipotential line
grad h
Isotropic
Kx = Ky = Kz = K
q
grad h
Anisotropic
Kx, Ky, Kz
q = - K grad h
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
z
global
local
z’
x’

x
K’x 0
0
Kyx Kyy Kyz
0
K’y
0
Kzx Kzy Kzz
0
0
K’z
Kxx Kxy Kxz
[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
h
qx   K x
x
h
qy   K y
y
h
qz   K z
z
Law of Mass Balance + Darcy’s Law =
Governing Equation for Groundwater Flow
--------------------------------------------------------------div
q = - Ss (h t) +R* (Law of Mass Balance)
q = - K grad h
(Darcy’s Law)
div (K grad h) = Ss (h t) –R*
OUT – IN =
qx
qy qz
(


 W *) x y z
x
y
z
= change in storage
= - V/ t
Ss = V / (x y z h)
V = Ss h (x y z)
t
t
OUT – IN =
qx
qy qz
(


 W *)
x
y
z
h
qx   K x
x
h
qy   K y
y
h
qz   K z
z
= - V
t
h
 Ss
t

h

h

h
h
( Kx ) 
( Ky ) 
( K z )  Ss
W *
x
x
y
y
z
z
t
Law of Mass Balance + Darcy’s Law =
Governing Equation for Groundwater Flow
--------------------------------------------------------------div
q = - Ss (h t) +W* (Law of Mass Balance)
q = - K grad h
(Darcy’s Law)
div (K grad h) = Ss (h t) –W*

h

h

h
h
( Kx ) 
( Ky ) 
( K z )  Ss
W *
x
x
y
y
z
z
t
2D confined:
2D unconfined:

h

h
h
(Tx ) 
(Ty )  S
W
x
x
y
y
t

h

h
h
( hKx ) 
( hKy )  S
W
x
x
y
y
t
Storage coefficient (S) is either storativity or specific yield.
S = Ss b & T = K b