A descriptive representation of a groundwater system that incorporates an interpretation of the geological & hydrological conditions. Generally includes information about the water budget.

Conceptual Model

a set of equations that describesthe physical and/or chemical processesoccurring in a system.

Mathematical Model

• Governing Equation

• Boundary Conditions• Specified head (1st type or Neumann) constant head

• Specified flux (2nd type or Dirichlet) no flux

Components of a Mathematical Model

• Initial Conditions (for transient conditions)

Mathematical Model of the Toth Problem

Laplace Equation

2D, steady state

02

2

2

2

zh

xh0

xh 0

xh

0zh

h = c x + zo

Types of Solutions of Mathematical Models

• Analytical Solutions: h= f(x,y,z,t) (example: Theis eqn., Toth 1962)

• Numerical SolutionsFinite difference methodsFinite element methods

• Analytic Element Methods (AEM)

Toth Problem

z

xAnalytical Solution Numerical Solution

02

2

2

2

zh

xh

continuous solution discrete solution

0xh0

xh

0zh

h = c x + zo

Mathematicalmodel

Toth Problem

z

xAnalytical Solution Numerical Solution

02

2

2

2

zh

xh

h(x,z) = zo + cs/2 – 4cs/2 …

z

xcontinuous solution discrete solution

(eqn. 2.1 in W&A)

0xh0

xh

0zh

h = c x + zo

Mathematicalmodel

Toth Problem

z

xAnalytical Solution Numerical Solution

02

2

2

2

zh

xh

h(x,z) = zo + cs/2 – 4cs/2 …

hi,j = (hi+1,j + hi-1,j + hi,j+1 + hi,j-1)/4

z

xcontinuous solution discrete solution

(eqn. 2.1 in W&A)

0xh0

xh

0zh

h = c x + zo

Mathematicalmodel

433243 ECDDD

Example of spreadsheet formula

Hinge line

Add a water balance& compute waterbalance error

INOUT

OUT – IN = 0

Hinge line

Q= KIA

Hinge line

Add a water balance& compute waterbalance error

1 m

x

Q = KIA=K(h/z)(x)(1)

A

x=z Q = K h

z

z

x

(x/2) x x (x/2)

Mesh centered grid: area needed in water balance

No FlowBoundary

watertablenodes

x=z Q = K h

x x

Block centered grid: area needed in water balance

No flow boundary

watertablenodes

K as a Tensor

div q = 0

q = - K grad h

Steady state mass balance eqn.

Darcy’s law

grad h

q equipotential line

grad hq

Isotropic AnisotropicKx = Kz Kx Kz

z

x

div q = 0q = - K grad h

steady state mass balance eqn.

Darcy’s law

Scalar1 component

Magnitude Head (h)

Vector3 components

Magnitude and direction

q & grad

Tensor9 components

Magnitude, direction and magnitude changing with direction

Hydraulic conductivity (K)

02

2

2

2

2

2

zh

yh

xh

div q = 0

q = - K grad h

steady state mass balance eqn.

Darcy’s law

Assume K = a constant(homogeneous and isotropic conditions)

Laplace Equation

0)()()(

zhK

zyhK

yxhK

xzyx

Governing Eqn. for TopoDrive

2D, steady-state, heterogeneous, anisotropic

x

z

x’

z’

global local

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

K’x 0 0

0 K’y 0

0 0 K’z

bedding planes

Kxx 0 0

0 Kyy 0

0 0 Kzz

qx

qy

qz

= -

zhyhxh

q = - K grad h

xhKq xxx

yhKq yyy

zhKq zzz

K =

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

Kxx ,Kyy, Kzz are the principal components of K

K is a tensor with 9 components

q = - K grad h

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

qx

qy

qz

= -

zhyhxh

q = - K grad h

zhK

yhK

xhKq xzxyxxx

zhK

yhK

xhKq

zhK

yhK

xhKq

zhK

yhK

xhKq

zzzyzxz

yzyyyxy

xzxyxxx

WthS

zhK

yhK

xhK

z

zhK

yhK

xhK

yzhK

yhK

xhK

x

szzzyzx

yzyyyxxzxyxx

)(

)()(

WthS

zhK

zyhK

yxhK

xszyx

)()()(

storagein change)(

W

zq

yq

xq zyx

thSs

This is the form of the governing equation used in MODFLOW.

x

z

x’

z’

global local

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

K’x 0 0

0 K’y 0

0 0 K’z

bedding planes

x

z

x’

z’global local

0xh

zhKq xzx

grad h q’

zhKq zzz

)'' ('xhKq xx

0)'(''

zhKq zz

Kz’=0

Assume that there is no flow across impermeable bedding planes

q

zhK

yhK

xhKq

zhK

yhK

xhKq

zhK

yhK

xhKq

zzzyzxz

yzyyyxy

xzxyxxx

x

z

x’

z’

global local

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]

bedding planes

q’q