Chapter 7
Horizontal plane Groundwater flow equation
Peter­Wolfgang Gräber
Systems Analysis in Water Management
7.1 Dupuit assumption and ballance equation
The description of the rotationally symmetric groundwater flow field is based on horizontal plane flow processes, in which the vertical flow vector is neglected. The transfer
of the three dimensional flow regime into a two dimensional mathematical description
takes place with consideration of the
Dupuit assumption:
• The potential lines h = const. run parallelly to the z axis. This means that the
vertical component of the groundwater flow (vz → 0) is equal to zero. This can
be realized by an infinitely large vertical flow resistance (specific permeability
coefficient in z direction (kz → ∞) or by a no gradient gauge level:
vz =
∂h
=0
∂z
• The horizontal speed is constant during the entire through flow height of the
aquifer. It means the vertical gradients of the horizontal flow components are
equal to zero.
∂vy
∂vx
=
=0
(7.1)
∂z
∂z
• The horizontal speed is proportional to the decline gradient of free the surface
according to the Darcy law:
vy = −ky ·
∂h
∂y
(7.2)
vx = −kx ·
∂h
∂x
(7.3)
The force equilibrium law is set up under the condition that only pressure force,
gravity force, capillary force and internal friction are effective. Inertia forces, adhesive
force, turbulent friction forces and others are small enough to be negligible. Since the
groundwater movement is regarded as saturated filter flow, we know following law from
Darcy:
�v = −k · grad h
(7.4)
The Darcy law is only as long valid as its by its derivation existing preconditions
fulfil. Thus it loses its validity if the above neglected forces increase. For the practical
Systems Analysis in Water Management
Peter­Wolfgang Gräber
7.1. Dupuit assumption and ballance equation
groundwater flow procedures the validity of the Darcy law can be assumed with
sufficient accuracy. Only directly in the proximity of well with large filter velocity a
breach of this law can occur.
With the Dupuit assumption the balance equation for horizontal plane groundwater
flow is built up. The specific flow rate�q , refer to flow field width of 1m, can be
calculated to:
�q =
�D
(7.5)
�v dz
z=a
D
D=
through flow thickness



 M aquifer thickness in confined


 zR
positon of the free groundwater surface in unconfined
Then the balance equation is:
�
div �q = n0 +
�D
z=a

S0 dz  ·
∂h
−w
∂t




aquifer



(7.6)
w
sources/sinks
n0
storage coefficient at the free groundwater surface due to gravimetric effects
S0
elastic storage coefficient, which works within the aquifer
The summary expression for the storage capability is designated with S as general
storage coefficient:
�

�D
S = n 0 +
S0 dz 
(7.7)
S=
z=a

�zR



S0 dz
 n0 +
z=0




�M
z=0
S0 dz



unconfined 

confined
aquifer
(7.8)




195
CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
If the gravimetric storage coefficient is substantially larger than the sum of all elastic
effects in vertical direction:
�zR
S0 dz
(7.9)
z=a
The storage coefficient S can take the following values:



 n 0 ≈ na ≈ ne
unconfined
S=
M
�



S0 dz
confined
z=0




aquifer
(7.10)



It results that the storage coefficient is only dependent on the gravimetric coefficient
in the case of a free groundwater surface and a small through flow thickness aquifer
(D << 100m).
For the effect water height h:







 h
confined
h=
aquifer




 zR
unconfined 
Thus the balance equation, also as continuity equation, is written in the form:
div �q = S ·
∂h
−w
∂t
(7.11)
With the equations 7.5 and 7.6 we get the horizontal plane groundwater flow equation
in the following form:
�

�D
∂h
div −
−w
(7.12)
k dz grad h = S ·
∂t
z=a
According to definition of h, grad h is independent on z thus it can be pulled out of
integral. For further writing simplification the integral of permeability coefficient, the
term transmissibility T , is introduced:
T =
�D
k dz
(7.13)
z=a
This integral of transmissibility will be analysed numerically poorly as the permeability
coefficient is only expressed as step function and not a continuous function.
196
Thus the horizontal plane groundwater flow equation in the representation of
the water height is:


∂h


−w
confined
div (T grad h) = S ·
∂t
aquifer
(7.14)

∂zR


− w unconfined
div (T grad zR ) = S ·
∂t
7.2 Potential illustration
An integral transform for solving partial differential equation of underground flow processes in the former chapter was used, which yields the value of transmissibility. Now
in this section another integral transform is applied, the so called Girinskij potential
Φ also a relatively simple solution, thus the horizontal plane groundwater flow equation
is illustrated in potential expression.
The Girinskij potential Φ is defined as:
Φ(x� y) =
�D
g(z) · (h(x� y� z) − z) dz
(7.15)
z=a
In this equation the function g(z) characterizes the dependence of permeability coefficient k on height z:
k(x� y� z) = k(x� y) · g(z)
(7.16)
For the following considered unstratified auqifer follows:
g(z) = 1
∂h
= 0; h �= f (z)) and the as∂z
sumption of lower bound of the aquifer a equal to zero (a = 0), the integral yields two
solutions:
Here with the validity of the Dupuit assumption (
Φ(x� y) =
�D
z2
(h − z) dz = h · z −
2
�
z=0


M2

 M ·h−
2
=
2

z

R

2
Peter­Wolfgang Gräber
confined
�D
(7.17)
0




aquifer
(7.18)


unconfined 
Systems Analysis in Water Management
CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
With consideration of the Darcy law the specific volume flow is:
�D
�v dz
(7.19)
(−k · grad h)dz
(7.20)
�q =
z=0
and with �v = −k grad h follows:
�q =
�D
z=0
Since k and h by definition are not functions of z, we can write k outside the integral
and exchange the order of the deviation(the gradient) and the integration.
We get:
�q = −k · grad
�D
(7.21)
h dz
z=0
�q = −k · grad Φ
(7.22)
The horizontal plane groundwater flow equation in potential form:
∂h
−w
div (k grad Φ) = S
∂t
∂zR
−w
div(k grad Φ) = S
∂t
confined




aquifer
(7.23)


unconfined 
This PDE can be transferred into a uniform potential expression, if the definition
for the Girinskij potential is used separately, according to confined and unconfined
conditions:
S
∂h ∂Φ
∂h
=S·
·
∂t
∂Φ
∂t

1 ∂Φ
S ∂Φ


 S·
·
=
M ∂t
M ∂t
=

S ∂Φ
1
∂Φ

 S·√
=
·
zR ∂t
2Φ ∂t
198
(7.24)
M2
with Φ = M h −
2
zR2
with Φ =
2
confined






ununconfined 
aquifer
(7.25)
7.2. Potential illustration
We get:
Φ
M
+
M
2
∂h
1
=
∂Φ
M
h=
(7.26)
(7.27)
or:
zR =
√
2Φ
(7.28)
1
∂zR
=√
∂Φ
2Φ
(7.29)
Assuming a homogeneous, isotropic aquifer, i.e. k = const., then k can be calculated from the divergence and the division of the right side. With introduction of the
transmissibility and the geohydraulic time constant we get:
div (grad Φ) = a
∂Φ w
−
∂t
k
with:
a=
T =
S
T
�D
z=0
(7.30)
k dz =



 k·M


 k · zR
confined




GWL
(7.31)


unconfined 
Now we have found a universally valid PDE, which is linear and analytically solvable.
However it must be noted that the linearity is not exact under free groundwater surface
conditions, i.e. with unconfined aquifer, since the geohydraulic time constant a is a
function of zR . In this case a temporal average value will be taken for T and also for
a. The following approximation has been well proved for a:
ã ≈
at=0 − 2at
3
(7.32)
For extreme drawdown ratios over 10% of groundwater level this equation is only
approximately valid. Often the water level change of a standpipe , i.e. the drawdown,
is of interest instead of the Girinskij potential.
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CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
Therefore the potential difference between the output potential Φ0 and the current
potential Φ is used. In unstratified aquifer, i.e. with k(z) = const. and thus g(z) = 0
follows:
Z = Φt=0 − Φ =
=
�D
(ht=0 − z) dz −
z=0



 M (ht=0 − h)
2
2


 (zRt=0 − zR )
2
�D
(ht − z) dz
(7.33)
z=0
confined




aquifer


unconfined 
The subscript 0 stands for conditions at time point t = 0, i.e. Φ0 = Φt=0 , h0 = ht=0 ,
zR0 = zRt=0 . In some ciations the subscript n (Φn , hn , zRn ) is used for it.
Inserting this into the PDE we get:
div (grad Z) = a
∂Z w�
+
∂t
k
(7.34)
By definition w� is the supply quantity caused by the change of potential Z, is by
definition, while w represents the supply quantity, which affects from the outside of
aquifer, e.g. the natural groundwater replenishment:
� �
w
w
w�
= −
(7.35)
k
k
k t=0
Practically the following two cases are interested:
• w� = 0, i.e. supply conditions won’t change when the regarded groundwater level
varies, and
Z
w�
= 2 , i.e. the difference of potential Z causes an additional proportional
•
k
B
supply (see section 8.1.3, page 231 supply from neighbouring layers)
If we add a supply factor B in all cases, which approaches to infinite for the first
case (B ⇒ ∞), we get the general form, the standard form of the horizontal planes
groundwater flow equation:
div (grad Z) = a
200
∂Z
Z
+ 2
∂t
B
(7.36)
To solve this PDE it is necessary to transfer the general vectorial differential expression
into a coordinate related expression (see section 2.2 arithmetic rules of the vector
algebra, page 57).
By using cartesian coordinates we get a PDE to analyse the groundwater flow processes
in connection with the ditch flow (see to section 6.1 one dimensional flow equation,
page 188). Cylindrical coordinates lead to a representation, which is very useful for
rotationally symmetrical problems (see section 8.1 Theis well equation).
7.3 Boundary conditions
Each flow process takes place in a locally and temporally defined area, i.e. it represents
a closed system, which is connected with its environment under certain conditions.
Information, energy and material can be exchanged through such couple conditions.
They are called boundary conditions. While the system is described by the PDE
and generally valid for all conditions, a unique solution will be achieved by boundary
conditions. The effect of the boundary conditions is identical to determination of
the integration constant by solving a differential equation. Boundary conditions are
impressed to the regarded system from the outside and influence independently of state
variables.
It is differentiated between boundary conditions (conditions at certain local points)
and initial conditions (conditions of reference to a time point). Besides force equilibrium and mass conservation, the initial- and boundary conditions serve explicit mathematical descriptions of the original process, the flow process. They are regarded as a
part of the mathematical model.
7.3.1 Initial conditions
In dynamic systems relative time will be discussed. The absolute time point, from which
the system behaviour is changing from static into movement, is regarded as starting
point with relative time t = 0. The initial conditions serve to define the dynamic
system state at this point. Since the state variable of horizontal planes groundwater
flow is the piezometric head h or the situation of free surface, the initial condition is a
matter of potentials within the systems , i.e. the groundwater height or the pertinent
transform potential. In the model this will be a function of h, zR or Φ dependent on
Peter­Wolfgang Gräber
Systems Analysis in Water Management
CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
location. Other conditions are also necessary for the maintenance of steady state at
the border. These are from the same type like the boundary conditions. The difference
is they are valid for t < 0.
7.3.2 Randbedingungen
Three different kinds of boundary conditions differ in physical action modes in the
groundwater flow (see figure 7.1):
1. 1st type (Dirichlet condition)
2. 2nd type (Neumann condition)
3. 3rd type (Cauchy condition)
Boundary conditions are general functions of place and time. We differentiate boundary
conditions between influence inside of flow field (e.g. well, lakes, rivers, precipitation,
evaporation), and outside effect at the edge (e.g. delimitation of flow field by rivers or
barriers). It is characteristic for boundary conditions that its effect is independent on
the flow conditions (e.g. groundwater level) of the investigation area. Generally it is
nearly impossible to find a complete analytical expression for geohydraulic boundary
conditions.
• 1st type boundary condition �Dirichlet condition) work,
if the hydraulic potential (e.g. h, zR , Z, Φ) on the boundary is known as a
function of the time t and independent on the potential, i.e. the system variables
of the investigation area. This appears e.g. in rivers, lakes or drainage:
ϕ = ϕ(x� y� t)
(7.37)
• 2nd type boundary conditions �Neumann condition) work,
if the source intensity distribution and thus the hydraulic potential gradient on
the bound are known as a function of time t. This may be aroused for example
by wells with constant flow rate, supply due to groundwater regeneration, sealing
of sheet pile wall or underground structures:
grad ϕ = grad ϕ(x� y� t)
(7.38)
• 3rd type boundary condition �Cauchy condition) work,
if in general a temporally constant flow resistance exists between a surface with
known potential distribution and the boundary of flow field. Such boundary
202
7.3. Boundary conditions
conditions work in rivers with colmation bottom layer as well as flow resistance
of lift wells:
ϕ + A grad ϕ = B(A and B are definite constants)
(7.39)
In the figure 7.1 the effect of boundary conditions on an aquifer is illustrated. We
recognize that the flow rates of 1st and 3rd type boundary conditions dependent on the
difference between the effect potential (water level h) in the aquifer and the boundary
conditions. Therefore the flow rate can vary in amount and direction.
In 2nd type boundary condition the potential of the boundary condition (possibly regarded as negative or positive pressure) can vary accordingly with the potential of the
aquifer.
With the numerical models (see to section 9.1.1 numerical methods, e.g. finite differences methods, page 246) additional boundary conditions arise in the course of the
definition of the model borders. In contrast to the original procedure, which possesses
an infinite spatial expansion, the numerical models are spatially limited due to the finite computing capacity (memory space, computing speed). Thus a considerable error
arises, which must be reduced or eliminated by suitable measures (see section 9.1.1
finite differences method, page 246).
Another problem related to boundary conditions arises in the interaction investigation
of surface and aquifer systems . A volume flow appears between the surface and the
aquifer or the unsaturated soil zone. Depending upon potential conditions an ex- or
infiltration of surface water can come out or into the aquifer. If this filtration stream
is substantially smaller than the volume flow within water, or the filtration amount
is substantially smaller than the entire storage volume of surface water, the surface
water has an effect of boundary condition on the groundwater. In the other case, if the
potential of surface water varies due to the filtration phenomenon, the surface water
may be not considered as boundary condition, but components of the system and are
coupled to the aquifer model according to suitable mathematical relations.
203
CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
Figure 7.1: effect of boundary conditions on an aquifer
204