Lecture 7
Solving Groundwater Equations
7.1
Boundary and Initial Conditions
Any problem of water flow through porous media occurs within some constraints in space and time. For
example, in the case of the column experiments that we discussed in previous lectures, the spatial constraints
are the physical boundaries of the columns. The temporal constraints are the beginning and end of the
experiments. For a natural groundwater system, such constraints are not easy to define. Every groundwater
system is connected to the global hydrologic cycle, so to define a spatial constraint that does not involve the
entire globe requires some careful judgement. Similarly, there are no straightforward beginnings and ends
for a ground water system.
At a basic level, the groundwater equation describes the distribution of hydraulic head h in space and
time. From this distribution, it is straightforward to know qx , qy . and qz at all locations as well. (HINT:
Darcy’s Law). Therefore, we can write h and q as h(x, y, z, t) and q(x, y, z, t), respectively.
The groundwater equations that we developed in the previous lecture describe the conditions within
the spatial boundaries, but not at the boundaries. At the boundaries, we need to specify a different set of
conditions that describe the nature of the boundary. Mathematically, the constraints that we specify at
spatial boundaries are known as boundary conditions. Simply put, boundary conditions are the restrictions
that we have to put on the groundwater equations before we even attempt to solve the problems.
In general, there are three types of boundary conditions. This classification is based on what we know on
those boundaries. To illustrate the meanings of the different boundary conditions, we will consider simple
column experiments as shown in Figure 7.1. The experiments involve a cylindrical column of sand with
water being driven to flow through it under different scenarios. The water that flows out of the column fills
the bottom bath and the excess overflows.
Dirichlet Boundary Condition: (also known as Type I Boundary Condition) is the simplest condition
and applies to part (a) of the sand column experiments. The water column at the top surface of the column
is kept at constant elevation. The value of the pressure head at the top of the sand column is known at all
times h = xb . In general, we have a Type I boundary condition when the value of h is known at all times
for the duration of interest. It can change with time but often times it is a constant value. It can be an
artificially imposed condition or a natural state of the system. Mathematically, it is described as
h(x, y, z, t)|Γ = f (x, y, z, t)
(7.1)
where f denotes some known function. The mathematical notation |Γ denotes that the expression is evaluated
at the boundaries of the domain of interest. Equation 7.1 simply states that the value of the pressure head h
at a given point along the boundary (x, y, z) is defined by a known function f that may depend on location
and time. In the above example f is a constant f (x, y, z, t) = xb .
Neumann Boundary Condition: (also known as Type II Boundary Condition) applies to boundaries where
we know the flux density going through the boundary surface. In part (b) of the illustrative example, water
13
Figure 7.1: Definition of boundary conditions
is pushed through the top part of the column using a pump that delivers water at a known flux density.
Therefore, this boundary qualifies as a Type II boundary condition. Mathematically, it is described as
qn (x, y, z) = f (x, y, z, t) (x, y, z) ∈ Γ
(7.2)
where f is a known function. Here we wrote the flux density as qn , because the flux condition is defined
normal (perpendicular) to any surface over which it applies. In all the illustrative examples shown above,
we know that water does not flow in or out through the container (outer surface) of the sand column. Here,
qn = 0 and it is considered a Type II boundary condition.
Cauchy Boundary Condition: (also known as Type II Boundary Condition) corresponds to boundaries
where we do not know the exact values of neither h nor q. But some relation between those two variables
must be known. It is considered a combination of Type I and Type II boundary conditions. We will revisit
this condition after flow in unsaturated soils is introduced.
When the solution that is being sought involves time, it is necessary to have knowledge of the conditions
at some starting point. This knowledge is known as initial condition. For groundwater problems, this
requirement refers to the hydraulic head distribution at all points in the problem domain. Mathematically,
this is written as
h(x, y, z, 0) = h0 (x, y, z) (x, y, z) ∈ Γ
(7.3)
Here h0 denotes the distribution of h at all locations when t = 0. When the groundwater problem is described
using Equation ?? time is not a relevant variable, because neither h nor q change with time. Therefore, it
is not necessary to know the initial condition to solve these type of problems. For this reason, steady state
problems are also known as Boundary Value Problems. If the problem is not steady, both boundary and
initial conditions are needed to solve the problems.
7.2
Steady Flow in Confined Aquifers
A confined aquifer is defined is a permeable porous medium sandwiched between two impervious layers.
The impervious layers typically have permeability that is several orders of magnitude lower than the aquifer
14
so that we can safely ignore flow of water into or out of them. The simplest confined aquifer has uniform
thickness b and is horizontal. A vertical cross-section of such aquifer is illustrated in Figure 7.2.
Figure 7.2: Confined Aquifer
We specify a coordinate system as shown in the figure. We insert piezometers at different depths at two
locations: A (x = 0) and B (x = L). We find out that the hydraulic head at all levels in A are identical
(h(0, y) = h1 ) and, likewise, the hydraulic head at all levels in B are identical (h(0, y) = h2 ). (NOTE:
the top of the lower impervious layer is considered the reference plane) We also notice that h1 > h2 . We
notice that the water levels do not fluctuate with time. Therefore, the groundwater is a steady state. From
laboratory measurement of core samples we also find out that the hydraulic conductivity is uniform and
isotropic. Therefore, the equation that describes this particular groundwater is Equation ??. Because we are
concerned here only on two-dimensional cross section, we rewrite the groundwater equation in two dimensions
as,
∂2h ∂2h
+ 2 =0
(7.4)
∂x2
∂z
In order to solve this problem, we need to specify the problem we need to specify the boundary conditions
at the four boundaries. At the left and right boundaries, we know the hydraulic heads (Type I boundary
conditions)
h(0, y) → h1
h(L, y) → h2
At the top and bottom boundaries we know that the vertical flux density is zero (Type II boundary conditions)
qz (x, 0) → 0
qz (x, b) → 0
15
With these constraints we know the full description of the problem. These constraints can be inputed into
a groundwater computer model and get the relevant solution for the problem.
It turns out that this problem is rather simple and can be solved without requiring a computer model.
Notice that h is uniform at all depths in the two locations and therefore everywhere, so dh/dz = 0. Thus,
Equation 7.4 can be rewritten as
d2 h
∂2h
= 2 =0
(7.5)
2
∂x
dx
If we integrate this once we get
dh
=C
dx
where C constant of integration. Recall that Darcy’s law is written as
qx = K
dh
dx
Therefore, we also notice that C = qx /K. Because this is constant, we know that h must change linearly
with x (Hint: derivative of a straight line is constant). So we have
qx = K
h1 − h2
L
Now if we consider a portion of the unconfined aquifer that has a width of w (measured normal to cross-section
shown in the figure), the discharge (volume per unit time) between the locations A and B is Q = qx × w × b
Q = wbK
h1 − h2
L
(7.6)
The problem gets complicated if the thickness b is not constant and/or the aquifer is not horizontal.
7.3
Steady Flow in Unconfined Aquifers
An unconfined aquifer is defined as a groundwater system that is bounded by a free water table at the
top. Water table is the location below ground surface where the pores are filled with water. More strictly,
it is defined as the location where the pressure head is zero. Practically it is defined as the level of water in
a well that is not being pumped.
Consider an unconfined aquifer made of a thick porous medium resting on top of a flat impervious layer.
We have two wells at two locations: A (x = 0) and B (x = L) with water levels h1 and h2 , respectively. We
do not know the location of the water table in between A and B. A vertical cross-section of such aquifer is
illustrated in Figure 7.3.
As in the case of the confined aquifer, this unconfined aquifer is also described by Equation 7.4. The
boundary conditions are also identical. The only difference between the two problems is in the geometry of
the system. In the case of the unconfined aquifer, the location of the top boundary is unknown. Therefore,
we can define the top boundary condition at this stage only conceptually.
For now, instead of attempting to solve this complex problem we will introduce a simplification introduced
by a French engineer Dupuit1 . The Dupuit approximation states the horizontal flow at any location given
by
∂h
(7.7)
qx = −K
∂x
is uniform vertically. Therefore, the horizontal discharge (equivalent to EQuation 7.6) must be
Q = −qx × w × h(x)
(7.8)
1 Trivia: Dupuit is responsible for numerous concepts in microeconomics including cost-benefit analysis and setting of toll
fees for bridges
16
Figure 7.3: Unconfined Aquifer
The only difference is that in the case of the confined aquifer the thickness is constant (b), whereas thickness
of the unconfined aquifer it is not constant and unknown (h(x)). Substituting Equation 7.7 in 7.8 we get,
Q
∂h
= −Kh(x)
w
∂x
(7.9)
Integration of Equation 7.9 leads to
2Qx
wK
2
We can find the value of c = h1 by noting that at point A (x = 0) we have h = h1 . Therefore,
h(x)2 = c −
h(x)2 = h21 −
2Qx
wK
(7.10)
(7.11)
By substituting the value of h at point B (x = L), which is h2 , in Equation 7.11 we get
Q=Kw
h21 − h22
2L
(7.12)
If we want to calculate the location of the water table, we would first calculate the discharge Q and substitute
this in 7.11. Note that Equation 7.11 is quadratic. Therefore, the water table between A and B has a
parabolic shape. Strictly speaking, this shape is not very accurate, especially close to the two wells. But
it is possible to show mathematically that the discharge (Equation 7.12) is accurate. Using the algebraic
relation a2 − b2 = (a − b)(a + b) we can expand Equation 7.12 as,
Q=Kw
(h1 + h2 ) (h1 − h2 )
2
L
(7.13)
This equation is similar to Equation 7.6 where the thickness of the confined aquifer b is equivalent to the
average thickness of the unconfined aquifer b ≈ (h1 + h2 )/2. This implies that we can accurately estimate
the discharge of a steady unconfined aquifer by simply assuming the hydraulic head changes linearly from
point A to point B.
17
7.3.1
Unconfined aquifer with horizontal stratification
Consider the unconfined aquifer is located within a porous medium made of two layers of rock with hydraulic
conductivities of K1 and K2 . The bottom layer has a thickness of a. As before, we have piezometer readings
of h1 and h2 at locations A and B that are separated by a horizontal distance of L. We can recognize two
cases of this problem: Case 1 is where h2 is higher than the boundary between the two pervious layers (i.e.,
h2 > a). Now, we can easily see that the discharge Q has two components: one that occurs in the bottom
layer and another in the top. But we do not know the exact partitioning of the total discharge. For the
bottom we write
dh
Q1 = −w a K1
dx
and for the top (the thickness of the aquifer in the top layer is h − b) we write
Q2 = −w (h(x) − a) K1
dh
dx
We can now write the total discharge as Q = Q1 + Q2
Q
dh
= (−a K1 − (h − a)K2 )
w
dx
(7.14)
Separating the the x and h terms gives
Q
dx = −a K1 dh − hK2 dh + aK2 dh
w
(7.15)
After integration we get,
Q
h2
x = −a K1 h − K2
+ aK2 h + C
w
2
The constant of integration C is obtained by substituting x = 0 and h = h1 ,
C = a K1 h1 + K2
h21
− aK2 h1
2
(7.16)
(7.17)
Substituting Equation 7.17 in 7.16 gives,
K2 2
Q
x = a K1 (h1 − h) +
(h − h2 ) − aK2 (h1 − h)
w
2 1
(7.18)
After collecting terms we get
Q
x = a K1 (h1 − h) + K2
w
(h21 − h2 )
− a(h1 − h)
2
(7.19)
We can now calculate the discharge by substituting x = L and h = h2 in Equation 7.19
Q
a K1 (h1 − h2 )
(h2 − h22 )
a(h1 − h2 )
=
+ K2 1
− K2
w
L
2L
L
(7.20)
Note that that Equation 7.20 has two components. The first component is identical to discharge of a confined
aquifer given by Equation 7.6. The second term is similar to the discharge of an unconfined simple aquifer
given by Equation 7.8 with a correction (aK2 (h1 −h2 )/L) to account for the depth that is already represented
by the bottom layer.
To draw the location of the water table, we use first determine the dicharge using Equation 7.20 and
replace this value in Equation 7.19 to determine h as a function x. But notice that h cannot be explicitly
expressed as function of x in Equation 7.19. Therefore, we would have to calculate x as a function of h
instead. for h2 < a, the error in the Dupuit approximation is big and Equation 7.19 may not be used to
draw the water table. But the discharge calculated using Equation 7.20 is still valid.
18
7.3.2
Unconfined aquifer with vertical stratification
Consider the unconfined aquifer is located within a porous medium made of two vertical blocks of rock with
hydraulic conductivities of K1 and K2 resting on a flat impervious layer. As before, we have piezometer
readings of h1 and h2 at locations A and B that are separated by a horizontal distance of L. The distance
from point A to the contact between the blocks is L1 and the distance from the contact to point B is L2 , so
that L = L1 + L2 . The piezometer reading at x = L1 is h∗ , but this value is unknown. See Figure 7.4
Figure 7.4: Vertical Stratification in Unconfined Aquifer
Now, we can write the Dupuit approximations for the two rock blocks separately,
Q1 = −waK1
dh
dx
Q2 = −waK2
dh
dx
and
Each of these equations are identical to Equation 7.9. Therefore, the solution 7.12 applies to both of them
as well and with the appropriate substitutions for h1 and h2 :
Q = K1 w
h21 − h2∗
2L1
(7.21)
Q = K2 w
h2∗ − h22
2L2
(7.22)
and
Note that we used Q = Q1 = Q2 because we also recognize that the total horizontal discharge Q does not
change with x. From Equation 7.22 we get
h2∗ = h22 +
19
2QL2
K2 w
Substituting this in Equation 7.21 we get
h21
−
h22
2Q
=
w
L1
L2
+
K1
K2
which, after rearranging, gives a formula for discharge:
Q=w
h21 − h22
2(L1 /K1 + L2 /K2 )
(7.23)
Now if we want to calculate the location of the water table, we use the discharge calculated by Equation
7.23 in either Equation 7.21 or Equation 7.22. For x ≤ L1 ,
r
2Qx
h(x) = h21 −
(7.24)
K1 w
and for x ≥ L1
r
h(x) =
h22 +
2Qx
K2 w
(7.25)
Examples that illustrate the above groundwater solutions are uploaded on the course webpage.
7.4
Exercise
Calculate the discharge and groundwater profile for an unconfined aquifer made of two vertical strata. The
distance between the two wells (locations) is 1000 m. The contact between the two aquifers is 500 m away
from each of the wells. The two strata have hydraulic conductivities of K1 = 10−4 m/s and K2 = 10−5 m/s.
The hydraulic heads at points A and B are 100 m and 80 m measured above the bottom of the aquifer (Figure
7.4).
20