Groundwater
Involves study of subsurface flow in saturated soil media (pressure greater
than atmospheric); Groundwater (GW) constitutes ~30% of global total
freshwater, ~99% of global liquid freshwater
Basic definitions:
 An “aquifer” is a geologic unit capable of storing/transmitting significant
amounts of water; Flow still governed by Darcy Law (P > 0)
 Unconfined aquifers:
 Pores saturate by “pooling up” on top of an impervious or low
conductivity layer
 Aquifer upper boundary is the water table (w.t.) [p=0 at w.t.]
 Aquifer supplied by recharge from above
 Elevation of w.t. changes as storage in aquifer changes (analogous
to flow in streams)
 Piezometric surface (h=P/ρg + z) corresponds to w.t.
 Confined aquifers:
 Saturated soil that is bounded above and below by low conductivity
layers
 Boundary of aquifer does not change in time (analogous to pipe
flow)
 Major recharge occurs upstream or via leaky confining layers
 Water generally under high pressure; piezometric surface is above
top of aquifer
Unconfined vs. Confined Aquifers
Figure 6.1.3 (p. 144)
Types of aquifers (from U.S. Bureau of Reclamation (1981)).
Water Storage in Aquifers
Unconfined aquifers:
 storage changes correspond directly to change in water table level
(w.t. increases = water going into storage and vice versa)
storage parameter: “specific yield” or storage coeff. (Sy) = volume of
GW released per unit decline in water table (per unit area)
Sy typically : 0.05  0.35; [Sy ]  []
Physically, 0  Sy   s
"Specific retention"= s  Sy
Confined aquifers:
 storage changes correspond mainly to compression of aquifer as
weight of overlying material is transferred from liquid to solid grains
(change in porosity) when water is removed (or vice versa)
storage parameter: “specific storage” (Ss)
 Vol. of water released from storage 
Ss = 
 h

Vol. of aquifer
 change in porosity change in piez. head
Ss typically : 5  10 5  5  10 3 m 1; [Ss ]  [L1 ]
Also often use “storativity” (S):
S  Ss  b; where b is the aquifer thickness; [S]  []
Confined aquifer
Unconfined aquifer
Figure 6.1.5 (p. 145)
Illustrative sketches for defining storage coefficient of (a)
confined and (b) unconfined aquifers (from Todd (1980)).
Derivation of 2D GW Flow Equation
From mass balance considerations the 3D GW flow equation is given by:
Ss
h   h    h    h 
  Kx    Ky    Kz 
t x  x  y  y  z  z 
Provided the necessary boundary conditions (2 per spatial dimension) and
an initial condition we could solve the above 2nd order PDE for h(x,y,z,t)
However, GW flow in aquifers largely consists of horizontal flow (in the xy plane)
Dupuit approx.: We can simplify the equation for these cases by integrating
the above equation with respect to z, assuming no vertical flow inside
aquifer (qz= 0) and that the aquifer has a horizontal lower boundary (h =
h(x,y,t))
We can integrate from z=0 to z=h’, where h’= h (water table) for
unconfined aquifers and h’=b (aquifer thickness) for confined aquifers:
h
h
   h    h    h  
h
S
dz

0 s t 0  x  K x x   y  K y y   z  K z z  dz
This equation consists of four terms (one on LHS; three on RHS). We will
integrate them one-by-one. First (term #1):
h
h
h
h
S
dz

Ss dz
0 s t
t 0
h
(for unconfined aquifers) OR
t
h h
 Ss  b =S (for confined aquifers); S  storativity
t
t
 Sy
Next (term #2):
h
h
  h  
  h 
   h  
0  x  K x x  dz  x 0  K x x  dz  x  K x x z 0 



h 
  K x h  (unconfined) OR
x 
x 
h


h    h 
K
b
 x
=
 Tx  (confined); T =K  b=transmissivity
x 
x  x  x 
Similarly (term #3),
h
h
   h  
  h  
  h 
0  y  K y y  dz  y 0  K y y  dz  y  K y y z 
0 

h


h 
K
h
(unconfined) OR
y


y 
y 


h    h 
K
b
=
Ty
(confined);
y
y 
y  y  y 
Finally (term #4),
h
h
h
h
h
   h  
K
dz

K

K

K
z
z
z
0  z  z z  
z 0
z h 
z 0
= Recharge (through top of aquifer) - 0 (if no leakage from bottom)
R
Putting the four terms back together to get 2D GW Flow Equations:
Confined aquifers:
S
h   h    h 
  Tx    Ty   R
t x  x  y  y 
which is a linear PDE; R > 0 only if leaky upper boundary
Unconfined aquifers:
Sy
h  
h   
h 
  K x h    K yh   R
t x 
x  y 
y 
which is a nonlinear PDE in h. To make the equation linear use:
h 1  2
h 1  2

(h ); h 
(h )
x 2 x
y 2 y
h
h h Sy  2
Sy
; Sy

h
[linearized about h0 ]
t
h0 t 2h0 t
h
 
Substituting:
Sy h2
  1 h2    1 h2 
  Kx

Ky
R
2h0 t
x  2 x  y  2 y 
Which is linear in the variable h2.
By expressing both equations in linear form, we can write a unified
governing linear 2D GW flow equation for a homogeneous/isotropic
aquifer:
 2 2 R
C1
 2 2
t x
y
C2
where
Unconfined

h2 / 2
Sy / (Kh0 )
C1
K
C2
Confined
h
S /T
T
For pumping problems (flow toward a single pumping well) it is useful to
write the governing equation in cylindrical coordinates:
C1
 1     R

 r  
t r r r
C2
NOTE: We will often use this equation as the starting point for solving
steady-state and/or 1D problems. In the case of steady-1D problems, the
above PDE reduces to an Ordinary Differential Equation (ODE) which we
can generally easily integrate.
The linearity of the governing equation also allows for the application
of the principle of superposition.
Figure 6.4.4 (p. 159)
Well hydraulics for a confined aquifer.
Figure 6.4.6 (p. 161)
Well hydraulics for an unconfined aquifer.
Drawdown for Transient Pumping from a Confined
Aquifer
s(r,t)  H 0  h(r,t) 
Q
W (u)
4 T
Sr 2
u
4Tt
Tabulated Well Function
Table 6.5.1 (p. 163)
Values of W(u) for Various Values of u.
Steady-state pumping (single well in
infinite confined aquifer)
h(r) 
 r
Q
ln    h1
2 T  r1 
or using the undisturbed head (H 0 ) at the radius of influence (R ):
h(r) 
Q
 r
ln    H 0
2 T  R 
which can also be expressed in terms of drawdown:
s(r)  H 0  h(r)  
Q
 r
ln  
2 T  R 
Steady-state pumping (single well in
infinite unconfined aquifer)
h 2 (r) 
Q  r
ln    h12
 K  r1 
or using the undisturbed head (H 0 ) at the radius of influence (R ):
h 2 (r) 
Q  r
ln    H 02
 K  R 
which can also be expressed in terms of drawdown:
Q  r 

s(r)  H 0  h(r)  H 0   H 02 
ln   
 R  

K

0.5
Superposition of GW solutions
The linearity of the governing equations in GW flow allow
for the application of the principle of superposition of
solutions. This means we can “build-up” the solution to a
more complicated problem by summing up solutions to
simpler problems (that when added together represent the
actual GW problem). In the context of pumping wells, we
have solutions for a single well in an infinite aquifer; we
would like to extend these solutions to more realistic
problems.
Cases where this idea is useful in pumping problems include:
1. An infinite aquifer with multiple wells
2. Non-infinite aquifers with a no-flux BC
3. Non-infinite aquifers with a fixed-head BC
4. Combinations of the above cases
Superposition of Multiple Wells
Single well solutions are only strictly applicable for the idealized conditions
they were derived under.
However linearity of the governing equation allows for superposition of
single well solutions.
For the case of multiple wells in a confined aquifer, the actual drawdown
resulting from all wells can be obtained simply by summing up the
drawdown from each single well. Note: Summation of drawdowns not
strictly applicable for unconfined aquifers since equations not linear in h
(and therefore not linear in drawdown s).
sactual (x, y,t)  s1 (x, y,t)  s2 (x, y,t)  ...  sN (x, y,t)
Method of Images – No-flux Boundaries
s(x,y) = ?
In realistic problems we are often dealing with a non-infinite aquifer
(i.e. there is some sort of boundary within the radius of influence of
the pumping well). In the case above there is an impermeable
boundary meaning there is no flux at that boundary.
We are interested in solving for the head (or drawdown) profile under
these conditions. The differences between the actual head
(drawdown) and that predicted by a single pumping well (in an infinite
aquifer) solution are:
i) The gradient in head must be zero at the boundary, and
ii) the presence of the boundary will reduce the flow to the
well from that portion of the domain, thereby reducing the
head (increasing the drawdown) everywhere as the flow must
be replaced from other portions of the domain.
We could attempt to solve this more complicated problem numerically,
or we can use superposition to our advantage to determine the actual
solution.
Method of Images – No-flux Boundaries
To model the realistic problem, we can use the concept of an “image
well” to model the effects of the impermeable boundary. Namely, if
we place an image well pumping at the same rate as the real well at a
symmetric distance on the other side of the boundary (i.e. as a mirror
image) and add up the drawdowns (which we can do due to
superposition) we will ensure that:
i) The gradient in head is zero at the boundary, and
it
ii) all flow that would have been captured by the real well if
were in an infinite aquifer will instead be captured by the
image well, thereby increasing the drawdown everywhere in
the real domain.
Hence the drawdown anywhere in the domain for the real problem is:
sactual (r,t)  sreal (r,t)  simage (r,t)
Where the drawdown solutions on the right-hand-side are those for an single
well in an infinite aquifer. Note: The image well must use a different
coordinate since it is as a different location.
Method of Images – No-flux Boundaries
We can illustrate this with an example. As we’ve seen, for an infinite
aquifer with a single well, the drawdown field would look like the
following:
Symmetric
cone of
depression
Method of Images – No-flux Boundaries
If we have a non-infinite aquifer, i.e. with a no-flux boundary, the
single well solution obviously no longer holds:
Head
gradient not
equal to zero
at boundary
However, we can conceptualize the boundary condition as a pumping
image well (pumping at the same rate):
Image well
drawdown
(assuming an
infinite aquifer)
Single well
drawdown
(assuming an
infinite aquifer)
Method of Images – No-flux Boundaries
If the drawdown from the two single well solutions are summed up,
we get the actual resultant drawdown:
Note:
-- Head gradient
is zero at
boundary
-- Drawdown is
increased
compared to
single well
solution.
Method of Images – Fixed-head Boundaries
The other type of non-infinite aquifer condition we often must deal
with is that of a stream boundary. For simplicity, the stream is often
modeled as a fixed-head boundary condition.
We are interested in solving for the head (or drawdown) profile under
these conditions. The differences between the actual head
(drawdown) and that predicted by a single pumping well (in an infinite
aquifer) solution are:
i) The head must equal the fixed-head at the boundary, and
ii) the presence of the boundary will increase the flow to the
well from that portion of the domain, thereby increasing the
head (decreasing the drawdown) everywhere as the flow is
augmented by the fixed-head boundary.
We could attempt to solve this more complicated problem numerically,
or we can again use superposition to our advantage to determine the
actual solution.
Method of Images – Fixed-head Boundaries
To model the realistic problem, we can again use the concept of an
“image well” to model the effects of the fixed head boundary.
Namely, if we place an image well recharging (negative pumping rate)
at the same rate as the real well at a symmetric distance on the other
side of the boundary and add up the drawdowns (which we can do due
to superposition) we will ensure that:
i) The head is equal to the fixed value at the boundary, and
ii) all flow that would have been unavailable to the real well
if it were in an infinite aquifer (due to drawdown) will
instead be supplied by the image well, thereby decreasing the
drawdown everywhere in the real domain.
Hence the drawdown anyhwere in the domain for the real problem is:
sactual (r,t)  sreal (r,t)  simage (r,t)
Where the drawdown solutions on the right-hand-side are those for an single
well in an infinite aquifer. Note: The image well must use a different
coordinate since it is as a different location.
Method of Images – Fixed-head Boundaries
Using the same example as before , but now with a recharge image
well, if the drawdown from the two single well solutions are summed
up, we get the actual resultant drawdown:
Note:
-- Drawdown is
zero at the
boundary
-- Drawdown is
decreased
compared to single
well solution.
Numerical Example
 The governing equation for 2D steady-state GW flow in a
homogeneous/isotropic confined aquifer (with recharge/
pumping) is:
2 h 2 h
1
 R(x, y)  Q(x, y) 



2
2
T
x
y
h  piezometric surface
T  aquifer transmissivity
R(x, y)  aquifer recharge
Q(x, y)  pumping from aquifer
Assume island is square, with a head h=0 meters along the
boundary:
h=0
h(x,y)=?
y
x
Spatially varying recharge and non-infinite aquifer make analytical
solutions (even with superposition) difficult if not impossible
 Need to solve numerically
Case 1:
 Spatially variable recharge [more recharge away from (x,y) = (0,0)]:
R(x, y)  R x 2  y 2
Case 2:
 Spatially variable recharge with pumping (at 5 wells):
R x 2  y 2  Q(xi , yi ) i [1...5]
In general, 2D/3D GW problems must be solved
numerically, i.e. using MODFLOW
Figure 6.9.2 (p. 184)
Cell map used for the digital computer model of the Edwards
(Balcones fault zone) aquifer (after Klemt et al. (1979)).
Figure 6.9.4b (cont.)
Water levels for Barton Springs—Edwards Aquifer. (b)
Perspective block diagram of 1981 water levels viewed from the
east side of the aquifer (after Wanakule (1985)).