Wz

hK

zy

hK

yx

hK

xzyx

)()()(

General governing equationfor steady-state, heterogeneous, anisotropic conditions

02

2

2

2

z

h

x

h2D Laplace Eqn.

--Homogeneous and isotropic aquifer without a sink/source term.--2D flow in a profile; (Unconfined aquifer with a water table boundary condition; recharge occurs as a result of the boundary condition.)

Mathematical Model of the Toth Problem

02

2

2

2

z

h

x

h0

x

h0

x

h

0z

h

h = c x + zo

Unconfined aquifer

b = 1 m

x

z

z

x b

02

2

2

2

z

h

x

h

Aquiferb

Tothproblem

x

z

02

2

2

2

y

h

x

h

2D horizontal flow in an aquifer with constant thickness, b.

Aquiferb

x

y

Figure from Hornberger et al. 1998

b unconfined aquiferb is not constant

confined aquifer

2D horizontal flow in an aquifer with constant thickness, b.

Aquiferb

x

y

with recharge

Wz

hK

zy

hK

yx

hK

xzyx

)()()(

with a source/sink term

T

R

y

h

x

h

2

2

2

2Poisson Equation

2D horizontal flow; homogeneous and isotropic aquiferwith constant aquifer thickness, b, so that T=Kb.

RbWz

hK

zy

hbK

yx

hbK

xzyx

)()()(

2D horizontal flow

Map of Long Island, N.Y.

South Fork

Charles Edward Jacob (1914-1970)Consultant to the Town of Southampton, NY

December 1968

bocean ocean

groundwater divide

C.E. Jacob’s Conceptual Model of theSouth Fork of Long Island

R

x = 0 x = Lx = - L

We can simulate this system assuming horizontal flow in a “confined” aquifer if we assume that T= Kb.

h datum

water table

1D approximation used by C.E. Jacob

T

R

dx

hd

2

2

h(L) = 0

0dx

dh at x =0

ocean ocean

R

x = 0 x = Lx = - L

Governing Eqn.Boundary Conditions

h(x) = R (L2 – x2) / 2T

Analytical solution for 1D “confined” version of the problem

C.E. Jacob’s Model

T

R

dx

hd

2

2

h(L) = 0

0dx

dh at x =0

Governing Eqn.

Boundary conditions

R = (2 T) h(x) / ( L2 – x2)

Forward solution

Inverse solution for R

Rearrange eqn to solve for T,given value for R and h(0) = 20 ft.

Inverse solution for T

L

R = (2 T) h(x) / ( L2 – x2)

Inverse solution for R

Solve for R with h(x) = h(0) = 20 ft.

Observation wellon the groundwaterdivide

x

y

ocean

ocean

T= 10,000 ft2/dayL = 12,000 ft

2L

L

T

R

y

h

x

h

2

2

2

2

Island Recharge Problem

oceanwell

• Head measured in an observation well is known as a head target.

Targets used in Model Calibration

• The simulated head at the node representing the observation well is compared with the measured head.

• During model calibration, parameter values (e.g., R and T) are adjusted until the simulated head matches the observed value.

• Model calibration solves the inverse problem.

x

y

ocean

oceanSolve the forward problem:

GivenR= 0.00305 ft/dT= 10,000 ft2/day

Solve for h at each nodal point

2L

L = 12,000 ft

T

R

y

h

x

h

2

2

2

2

Island Recharge Problem

ocean

well

T

Rahhhhh

mji

mji

mji

mjim

ji 44

211,1,

1,1,11

,

T

R

y

h

x

h

2

2

2

2

Gauss-Seidel Iteration Formula for 2D Poisson Equation with x = y = a

Write the finite difference approximation:

Island Recharge Problem4 X 7 Grid

Water Balance

IN = Out

IN = R x AREA

Out = outflow to the ocean

Top 4 rows

Red dots represent specified head cells, which are treated as inactive nodes.

Black dots are active nodes. (Note that the nodes along the groundwater divides are active nodes.)

Head at a node is theaverage head in the areasurrounding the node.

Top 4 rows

IN =R x Area = R (L-x/2) (2L - y/2) 2L

L

Also: IN = R (2.5)(5.5)(a2)

Top 4 rows

x/2 x x

x/2Top 4 rows

OUT = Qy + Qx

Qy = K (x b) (h/y)Note: x = y Qy = T h

Qx = K (y b)(h/x)

or Qx = T h

Qx

Qy

y

Qy = (Th) /2

Welly/2

Bottom 4 rows

Qx = (T h)/2

Island Recharge Problem4 X 7 Grid

Water Budget Error