Introduction to groundwater flow modeling: finite difference

methods

Tyson Strand

1) Darcy’s law, continuity, and the groundwater flow equation

2) Fundamentals of finite difference methods

3) FD solution of Laplace’s equation

4) FD solution of Poisson’s equation

5) Transient flow

1) Darcy’s law, continuity, and the groundwater flow equation

qx

qy

qy+dy

qx+dx

0)()( =−+−++ yydyyyxxdxxx

qqdxqqdy

Everything that goes in, must come out

00==

−+

− ++

dxdydy

qq

dxqq yydyyyxxdxxx

After dividing by the volume (area), let dx and dy go to zero

0=∂

∂+

∂∂

yq

xq yx Continuity

Darcy’s law: isotropic medium

xhKqx ∂∂

−=

yhKqy ∂∂

−=

zhKqz ∂∂

−=

Scalar form

hKkzhj

yhi

xhK ∇−=

∂∂

+∂∂

+∂∂

−= )ˆˆˆ(

Vector form

Darcy’s law + continuity (in 2D)

xhKqx ∂∂

−=yhKqy ∂∂

−= 0=∂

∂+

∂∂

yq

xq yx

02

2

2

2

=

∂∂

+∂∂

−=

∂∂

−∂∂

+

∂∂

−∂∂

yh

xhK

yhK

yxhK

x

02 =∇ h Laplace’s equation

The groundwater flow equation

thSW

zhK

zyhK

yxhK

x szzyyxx ∂∂

=−

∂∂

∂∂

+

∂∂

∂∂

+

∂∂

∂∂

Sources and sinks

Transient flow term

Darcy’s law + continuity

Simplifying assumptions

• Isotropic medium: K = Kxx = Kyy = Kzz W = f(x,y,t)

• Steady-state flow (Poisson’s equation W = f(x,y))

• Steady-state flow, no sources/sinks (Laplace’s equation)

thSWhK s ∂∂

=−∇2

KWh =∇2

02 =∇ h

Fundamentals of finite difference methods

• Discretization of space

• Discretization of (continuous) quantities

• Discretization of time

• The first spatial derivative

• The second spatial derivative

• Boundary conditions and initial conditions

• Solving the problem

Disctretization of space

(i,j)

(i,j+1)

(i,j-1)

(i-1,j) (i+1,j)

(i,j)

xdx

yLx = (Nx – 1)dx

dy

Discretization of quantitiesEach spatial location (i,j) has associated with it a head value and a vector darcy flux

(i,j)

(i,j+1)

(i,j-1)

(i-1,j) (i+1,j)h(i,j)qx(i,j)qy(i,j)

Discretization of time

t = n n+1∆t n+2∆t n+3∆t n+4∆t n+5∆t

tm+2 tm+3 tm+4 tm+5tm tm+1

Increasing time

The first spatial derivative

∆−

+∆−

=

∆∆

≅∂∂ +−

xhh

xhh

xh

xh iiii

ji

11

),( 21

(i,j)

(i,j+1)

(i,j-1)

(i-1,j) (i+1,j)xhh

xh

xh ii

ji ∆−

=∆∆

≅∂∂ −1

),(

Upstream difference

Central difference

The second spatial derivative: step 1

xhh

xh ii

i ∆−

≅∂∂ −

−

1

21

xhh

xh ii

i ∆−

≅∂∂ +

+

1

21

(i,j)

(i,j+1)

(i,j-1)

(i-1,j) (i+1,j)

NOTE: This is essentially a central differenceapproximation of the first derivative evaluatedat the mid-positions

The second spatial derivative: step 2

∂∂

∂∂

=∂∂

xh

xxh2

2

( )211

11

21

21

2

2 2x

hhhx

xhh

xhh

x

xh

xh

xh iii

iiiiii

∆+−

=∆

∆−

−∆−

=∆

∂∂

−∂∂

≅∂∂ −+

−+−+

The second derivative is just the derivative of the derivative

Boundary conditionsWhat are boundary conditions and why are they necessary?

1.0=∂∂xhConsider the simple problem:

We can solve this directly by separating variables and integrating:

Cxhxh +=→∂=∂∫ ∫ 1.01.0

There are an infinitenumber of solutionsto this equation.

They are referred toas a family of curves

Cxh += 1.0h

x

We must specify avalue of h at a knownposition x in order tosolve for C.

For a second (spatial) derivative, two boundary conditionsmust be specified. There are 3 types of bc’s that we can apply

1) Head is specified at a boundary

- Called Dirichlet conditions

2) Flow (first derivative of head) is specified at a boundary

- Called Neumann conditions

3) Some combination of 1) and 2)

- Called mixed conditions

Initial conditions

Equivalent to boundary conditions, except that the boundaryis now temporal instead of spatial

Typically, the state of the system (i.e. head values) arespecified at time t = 0

thSW

zhK

zyhK

yxhK

x szzyyxx ∂∂

=−

∂∂

∂∂

+

∂∂

∂∂

+

∂∂

∂∂

Question: How many boundary and initial conditionsWould be needed to solve the above governing equation?

Solving the problem

After Wang & Anderson, 1982

Set of differentialequations

(Mathematical model)

Set of algebraicequations

(discrete model)

Finite DifferenceFinite Element

Fieldobservations

Analytical solution (Not always possible)

Calculustechniques

Iterative ordirect methods

Approximatesolution

CompareIf possible

Recall how we arrived at Laplace’s equation

xhKqx ∂∂

−=yhKqy ∂∂

−= 0=∂

∂+

∂∂

yq

xq yx

02

2

2

2

=

∂∂

+∂∂

−=

∂∂

−∂∂

+

∂∂

−∂∂

yh

xhK

yhK

yxhK

x

02 =∇ h Laplace’s equation

We are focusing on iterative methods

Solving Laplace’s equation (2D)

02

2

2

2

=∂∂

+∂∂

yh

xh02 =∇ h

( )2),1(),(),1(

2

2 2x

hhhxh jijiji

∆

+−≅

∂∂ −+

( )2)1,(),()1,(

2

2 2y

hhhyh jijiji

∆

+−≅

∂∂ −+

Let ∆x = ∆y

( )2),1(),(),1(

2

2 2x

hhhxh jijiji

∆

+−≅

∂∂ −+

( )2)1,(),()1,(

2

2 2y

hhhyh jijiji

∆

+−≅

∂∂ −+

+

( )[ ]),()1,()1,(),1(),1(2 41

jijijijiji hhhhhx

−+++∆ +−+−

( )[ ]),()1,()1,(),1(),1(2 41

jijijijiji hhhhhx

−+++∆ +−+−

Set equal to zero

04 ),()1,()1,(),1(),1( =−+++ +−+− jijijijiji hhhhh

The above equation is the basic finite difference solution to Laplace’s equation

Now we rearrange the previousequation so that we can implement

it into our regular grid

Solve for h(i,j)

4)1,()1,(),1(),1(

),(+−+− +++

= jijijijiji

hhhhh

Now what do we do?

(i,j)

Specify boundary conditions

Guess at initial values for headat all (i,j) locations

Begin iterations

What does it mean to “iterate”?

Given: We know the boundary conditions and haveinitial guess values for head(position)

Do: Update head values at every (i,j) locationbased on previous equation (including boundaries if applicable)

While: Convergence criteria are not met

Convergence

1) Compare obtained head values at a giveniteration to those at the previous iteration

2) Continue to iterate as long as head valuescontinue to change within some preset limit

3) Test the solution (to be described)

Testing a computational model

Comparison to other analytical results/field data (validation?)

Changing the grid spacing

Changing the convergence criterion

Mass balance check

Validation/verification

1) Analytical solution is known (uncommon)

2) Comparison to field data

3) Test on simple situations for which 1 or 2are known

4) Check model predictions

Possible model problems

Stability

Robustness

Sample problem: Laplace’s equation

Lecture 2• Poisson’s equation

– Digression: Inflow, outflow, and sign conventions• Finite difference form for Poisson’s equation• Example programs solving Poisson’s equation• Transient flow

– Digression: Storage parameters• Finite difference form for transient gw flow

equation (explicit methods & stability)• Example transient flow program • Implicit iterative methods • Example transient flow program, fully implicit

Solving Poisson’s equation

−==∇

TyxR

KWh ),(2 What is W?

KW

yh

xh

=∂∂

+∂∂

2

2

2

2

byxRW ),(

−=

Where b is the thickness of the aquifer (z dimension)

Inflow, outflow, and sign conventions

Think about the magnitudes of qout and qinWhat does it mean in terms of sources/sinks?

x

02

2

=∂∂xh

02

2

>∂∂xh

Sign conventions

02

2

<∂∂xh

02

2

<∂∂xh

There must be recharge (or someother source) at the REV (steady-state assumption)

02

2

>∂∂xh There must be discharge (or some

other sink) at the REV(steady-state assumption)

−==∇

TyxR

KWh ),(2

• You need to be careful as to how things are defined

• W MUST be defined discharge (sink) positive• It is more intuitive to define a recharge (source)

function R(x,y) that is positive for recharge (sources) and negative for discharge (sinks)

• This will also help give us an intuitive understanding of the transient gw flow equation

Back to Poisson’s equation

TyxRh ),(2 −=∇ Proceeding as we did for Laplace’s equation

( )[ ]

TR

hhhhhx

jijijijijiji

),(),()1,()1,(),1(),1(2 41

−=−+++∆ +−+−

Solving for h(i,j)

( )

∆++++= +−+− T

Rxhhhhh ji

jijijijiji),(

2

)1,()1,(),1(),1(),( 41

Solving Poisson’s equation numerically

• Basically, we can proceed exactly as we did for Laplace’s equation, using the previous finite difference approximation for h(i,j)

• Define boundary conditions• Set initial guess values• Iterate• Check results

Sample problem: Poisson’s equation with uniform recharge/discharge

Sample problem: Poisson’s equation with point source

recharge/discharge

Solving transient flow problems

btyxR

thShK s

),,(2 −∂∂

=∇

We’ve learned how to take care of everything except the time derivative

thh

th

th m

jimji

∆

−=

∆∆

≈∂∂ +

),(1),( Forward difference approximation

thh

th

th m

jimji

∆

−=

∆∆

≈∂∂ −1

),(),( Backward difference approximation

Central difference approximation

thh

th

th m

jimji

∆

−=

∆∆

≈∂∂ −+

2

1),(

1),(

= BAD NEWS

• The central difference approximation is unconditionally unstable (for time derivative)

• This method should NOT be used to estimate the first time derivative

Storage parameters• S = storage coefficient, or storativity

– Dimensionless– Measures the volume of water expelled (absorbed) per

unit surface area per unit head change

• Ss = specific storage = S/b– Units [1/L] also called elastic storage coeff.– Measures the water volume per unit aquifer volume that

is expelled (stored) due to compressibility (matrix & water) per unit head change

– Used for confined units or the saturated parts of an unconfined unit

– b = aquifer thickness for confined units, use bs = thickness of the saturated region for unconfined systems

Specific yield• Sy = specific yield

– Dimensionless– Used to describe unconfined systems– Ratio of water volume that drains from a saturated

region of aquifer due to gravity forces per unit aquifer volume

– Generally several orders of magnitude larger than bsSsexcept in very fine grained systems

• Relationship between Sy, Ss, and S

ssy SbSS +=Fetter, 1994

Storage parameters (concl.)

• In unconfined systems, the specific storage can generally be neglected– Except in very fine grained media

• In confined systems, only the specific storage is pertinent (so long as head values remain above the upper confining unit)

• In general, for computational models, it is best to use the storage coefficient (storativity)

• Definitions of storage parameters are not always consistent in literature, be careful

Transient flow• So, let’s consider the transient flow equation in

the form

TtyxR

th

TS

yh

xh ),,(

2

2

2

2

−∂∂

=∂∂

+∂∂

Assume the left hand size is negative (out > in):then, by the convention we’ve set , either rechargeis positive or the time rate of change of storage is negative (right?)

TtyxR

th

TS

yh

xh ),,(

2

2

2

2

−∂∂

=∂∂

+∂∂

Let’s write the finite difference approximation for this equation

TR

thh

TS

yhhh

xhhh

mji

mji

mji

mji

mji

mji

mji

mji

mji

),(),(1),(

2)1,(),()1,(

2),1(),(),1(

)(2

)(2

−∆

−=

∆

+−+

∆

+−

+

−+−+

This is called the explicit representation, since all spatial termsare evaluated at the “old” time = m

Let ∆x = ∆y, and solve for the head at (i,j) at time index m+1

[ ]mji

mji

mji

mji

mjim

jimji hhhh

xStT

SRth

xStTh )1,()1,(),1(),1(2

),(),(2

1),(

41 −+−++ +++

∆∆

+∆+

∆∆

−=

The explicit equation is easy to solve:

• Given a set of initial conditions (& bc’s)• The head at all (i,j) locations at time index

m+1 can be calculated directly from the head values at the previous time index m

• Notice, there is no iteration, once you have an initial condition it is a simple matter to step forward in time by an amount ∆t

Stability of the explicit method• One must be careful to keep the time steps small

enough such that the explicit method remains stable

• If the time step is too large, fluctuations will develop in the head as a function of time, which will be amplified as the model continues to step through time

• The stability criteria are:

25.02 <∆∆xStT 5.02 <∆

∆xStT

2-D 1-D

Example transient problem:Explicit solution, 1D

Initial

Final

We are interested in what happensin between the initial and final states

The head at the right boundary dropsfrom blue to red at time t = 0+

The red potentiometric surface is thefinal state

Implicit methods

• Notice in the explicit formulation that we have evaluated all spatial derivatives at the old time (tm)

• Is that the best?• What if we evaluated spatial derivatives at

the new time (tm+1)?• What if we evaluated spatial derivatives

halfway between?

2),1(),(),1(

2

1),1(

1),(

1),1(

2

2

)(2

)1()(

2x

hhhx

hhhxh m

jimji

mji

mji

mji

mji

∆

+−−+

∆

+−≈

∂∂ −+

+−

+++ αα

Where α can vary between 0 (fully explicit) and 1 (fully implicit)

A similar expression can be written for the second derivativewith respect to y

We simplify the algebra by defining the following and letting ∆x=∆y

4~ )1,()1,(),1(),1(

),(+−+− +++

= jijijijiji

hhhhh

[ ] [ ]mji

mji

mji

mji hh

xhh

xyh

xh

),(),(21),(

1),(22

2

2

2 ~)1(4~4−

∆−

+−∆

≈∂∂

+∂∂ ++ αα

Put this approximation into the gw flow equation

[ ] [ ]TR

thh

TShh

xhh

x

mji

mji

mjim

jimji

mji

mji

),(),(1),(

),(),(21),(

1),(2

~)1(4~4−

∆

−=−

∆−

+−∆

+++ αα

Rearrange

( )[ ]TRx

hhhtTSxhh

tTSx m

jimji

mji

mji

mji

mji 4

~14

~4

),(2

),(),(),(

21),(

1),(

2 ∆+−−+

∆∆

=−

∆

∆+ ++ ααα

Solving the previous equation for the new head value at (i,j)

( )[ ]

∆+−−+

∆∆

+

∆∆

+= ++

TRx

hhhtTSxh

tTSx

hmjim

jimji

mji

mji

mji 4

~14

~

4

1 ),(2

),(),(),(

21),(2

1),( αα

α

This is the fundamental implicit finite difference approximationα = 0 Fully explicitα = ½ Crank-Nicolsonα = 1 Fully implicit

The above expression can be solved by iterative or direct methods

We will continue to focus on iterative methods

The advantage of implicit methods is STABILITY

To solve iteratively:- Sweep through the lattice, updating the (i,j) head

value at the new time by the above formula- Continue to sweep through the lattice, updating,

as long as the new head values keep changing- Stop when new head values do not change

within preset limits- (update as you go along:

called Gauss-Seidel iteration)- Repeat for the next time step

Fully implicit, 1D sample problem

∆

∆+

∆∆

+= ++ m

imi

mi h

tTSxh

tTSx

h )(

21

)(21)( 2

~

21

1

1)(

~ +mihWhere is now defined by

+=

++

+−+

2~ 1

)1(1)1(1

)(

mi

mim

i

hhh

References

• H. F. Wang and M. P. Anderson, 1982. Introduction to Groundwater Modeling, Academic Press, Inc.

• C. W. Fetter, 1994. Applied Hydrogeology, Macmillan College Publishing Company.

• F. W. Schwartz and H. Zhang, 2003. Fundamentals of Ground Water, Wiley publishing.