GEOS 4430/5310 Lecture Notes: Regional Flow
and Flow Nets
Dr. T. Brikowski
Fall 2013
0
file:regional flow.tex,v (1.26), printed October 15, 2013
Vertical Averaging
I
from a regional point of view, typical aquifers are wide and
very thin (aspect ratio 0.1)
I
simplest way to view them is constant in the z dimension, i.e.
take vertical average of aquifer properties and groundwater
flow
I
Ignores vertical flow, usually an OK assumption
I
the most important areal properties of aquifers are
transmissivity T , and storativity S
Transmissivity
I
amount of water that can be transmitted by an aquifer
(horizontally) through a unit width given a unit head gradient
I
T = Kb, units are
I
values range widely, unconsolidated sand aquifers have
2
conductivity values range from 5.0x10−6 − 5.0x10−2 m
sec
I
Measuring transmissivity: derived from observations of well
tests or found by laboratory measurements of K and field
measurements of b
I
see typical values (Fig. 2)
L2
t
Storativity
I
I
S, storage coefficient or storativity: The amount of water
stored or released per unit area of aquifer given unit head
change
Ss , specific storage
I
I
The amount of water stored or released per unit volume of
aquifer given unit head change
S = Ss b, where b is the aquifer thickness
I
storage change is accomplished via compression of the aquifer
1
,
matrix and the fluid. Fluid compressibility β = 4.4x10−10 Pa
1
−8
while typical (sand) aquifer compressibility α = 1x10 Pa
I
Ss = ρf g (α + φβ), where ρf is the fluid density, g the
gravitational constant, and φ the porosity. Obtained using the
control volume approach.
I
typical values of S (dimensionless) are 5.0x10−3 –5.0x10−5
I
Measuring storativity: derived from observations of multi-well
tests
K Ranges
Figure 1: Relative ranges of hydraulic conductivity (after BLM Hydrology
Manual, 1987?).
T Ranges
Figure 2: Relative ranges of transmissivity and well yield (after BLM
Hydrology Manual, 1987?). The irrigation-domestic boundary lies at
2
∼ 0.214 m
sec .
Effect of Scale on Measured K
Figure 3: Effect of tested volume (i.e. heterogeneity) on measured K
(Bradbury and Muldoon, 1990).
Flow Equation: Confined Aquifers
I
for steady-state system (no change in storage), areal flow
(vertical averaging). Flow equation is (see Darcy’s Law
Notes):
∂2h
∂2h
+
= 0
(1)
∂x 2
∂y 2
I
with storage:
∂2h
∂2h
S ∂h
·
+
=
∂x 2
∂y 2
T ∂t
(2)
Unconfined Flow
I
Dupuit Assumptions: all flow horizontal
I
I
I
hydraulic gradient (∇h) is equal to slope of the water table
equipotential lines are vertical (no vertical flow, or qz = 0)
Error in this assumption (Bear, 1972):
0 < error <
1
∂2h
∂x 2
∂h 2
∂z
2
+ ∂h
∂z
≈0
I
implies smooth free-surface:
I
this view is inaccurate for “thick” aquifers (Fig. 5)
Unconfined Flow Representative Control Volume
Free Surface
h 1
Q 1
Q 2
h 2
∆y
∆x
Figure 4: Mass fluxes for control volume in uniform unconfined flow field.
Note free surface (water table) is top surface of volume.
Derivation of Unconfined Flow Equation
I
the total discharge or flux Q1 and Q2 can be specified as:
dh ∆yh1
Q1 = q1 · ∆yh1 = −K dx
x1
∆y dh2 2
= −K
Given du
= 2u du
dx
dx
dx
2
x1
dh Q2 = q2 · ∆yh2 = −K dx x ∆yh2
2
∆y dh2 = −K
2 dx x2
Derivation of Unconfined Flow Equation (cont.)
I
Steady state mass balance (really a volume balance),
including recharge onto the free surface:
rate of
rate of
−
= {Source/Sink}
mass in
mass out
Q1 − Q2 = R∆x∆y

 2
dh dh2 −
dx x 
K ∆y  dx x1
2
−

 =R
2
∆x
|
{z
}
Central difference for
−
K
2
K
2
d
dx
d2 h 2
dx 2
=R
d2 h 2
dx 2
= −R
Derivation of Unconfined Flow Equation (cont.)
I
Similarly for two dimensions
∂ 2 h2
∂ 2 h2
2R
+
= −
2
2
∂x
∂y
K
Flow Equation: Unconfined Aquifers
Same development as for confined flow, but now height of control
volume is h. For a homogeneous unconfined aquifer we obtain the
Boussinesq Equation:
∂ 2 h2
∂ 2 h2
+
= 0
2
∂x
∂y 2
(3)
Sy ∂h
∂ 2 h2
∂ 2 h2
+
=
·
2
2
∂x
∂y
T ∂t
(4)
with storage:
where Sy is the specific yield, or the amount of water that will
gravity drain from a unit volume of aquifer.
Unconfined Flow Approximation
Figure 5:
Approximations and true geometry of free surface Freeze and Cherry (after Fig. 5.14, 1979)).
Dashed lines are equipotentials, vectors show fluid flow. Top image has a seepage face at E-D, unsaturated flow
crosses the water table. For the middle image K ≡ 0 in the unsaturated zone, the water table is a flowline, bottom
figure shows Dupuit-Forchheimer flow (horizontal only).
Definitions: Rock Property Fields
I
Homogeneity: refers to uniform aquifer parameters (i.e.
inhomogeneity is the norm in earth materials)
I
Isotropy: refers to directional dependence of material
properties. Usually only hydraulic conductivity is treated as
anisotropic (directionally dependent). Anisotropy is described
by giving values along the principal axes of an ellipsoid (e.g.
the hydraulic conductivity ellipsoid, Fig. 6)
Hydraulic Conductivity Ellipsoid
z
sqrt(K z )
sqrt(K x )
x
Figure 6: Illustration of the hydraulic conductivity ellipsoid for an
anisotropic medium Freeze and Cherry (after Fig. 2.10, 1979)).
Multi-dimensional Darcy’s Law
I
need to describe flow in 2- and 3-D situations, must express
Darcy’s Law in multi-dimensional form
I
~ as a vector (Fig. 7) the components of the specific
consider q
discharge are given by (allowing for anisotropy):
qx = −Kx
∂h
∂x
qy = −Ky
∂h
∂y
The components are related to one another by
q
|~
q | = qx2 + qy2
qy
tan θ =
qx
(5)
Vector Velocity
y, j
q
qy
θ
x, i
qx
~ in two dimensions
Figure 7: Vector representation of q
3-D Darcy’s Law
I
can simply add a Z -component to (5) provided the principal
components of the hydraulic conductivity are aligned with the
coordinate axes. In this case (5) can be written in vector form




qx
Kx
 qy  = −  Ky  · ∂h , ∂h , ∂h
(6)
∂x ∂y ∂z
qz
Kz
~ = −K · ∇h
q
(7)
I
the most general 3D form of (7) (when principal components
of conductivity are arbitrarily oriented) expresses conductivity
as a second-rank tensor (matrix) (Bear, 1972),


Kxx Kxy Kxz
K =  Kyx Kyy Kyz 
(8)
Kzx Kzy Kzz
3-D Darcy’s Law (cont.)
I
I
I
this tensor is symmetric, i.e. Kxy = Kyx
the term Kij refers to the i component of conductivity acting
in the j direction (e.g. U. Cambridge notes)
compare to the stress tensor from structural geology
Layered Heterogeneity
I
I
Geologic layering often introduces macroscopic anisotropy
(Fig. 8)
An equivalent homogeneous anisotropic conductivity can be
derived (eqns. 3.41-3.42, Fetter, 2001)
I
~ must be constant across each layer,
since specific discharge q
for vertical flow
qz =
I
K2 ∆h2
Kz ∆h
K1 ∆h1
=
= ... =
d1
d2
d
where Kz is the equivalent homogeneous vertical hydraulic
conductivity
rearranging:
Kz =
qz d
=
∆h
qz d
qz d1
K1
+
d
Kz = Pn
qz d2
K2
di
i=1 Ki
+ ...
(9)
Layered Heterogeneity (cont.)
I
similarly the equivalent horizontal conductivity is found by
noting that the total horizontal discharge Qx is just the sum of
the individual layer discharges:
qh =
n
X
∆h
Ki di ∆h
= Kx
d l
l
i=1
or rearranging
Kx =
n
X
Ki di
i=1
d
(10)
Layered Heterogeneity (cont.)
I
e.g. for two layers of 1 m thickness, one with K=10 m
d , the
m
other with 100 d , Kh is dominated by the most transmissive
layer, Kz is dominated by least transmissive
2
1
+ 100
2
==
0.11
10 100
Kh =
+
2
2
= 110
Kz =
I
1
10
the moral here is that complicated problems can often be
simplified by combining a bit of geology and math
Effect of Layering on Anisotropy
Figure 8: Anisotropy in hydraulic conductivity caused by layering. Freeze
and Cherry (1979, , Fig. 2.9).
Introduction to Flow Nets
I
Introduction
I
I
I
I
I
a graphical solution method for potential flow problems (i.e.
solving 2-D Laplace Eqn.)
always handy as an initial solution, can be refined later
through numerical modeling
there are entire books on this method
Assumptions: homogeneous, saturated, isotropic, steady-state,
no storage (incompressible water & matrix), Darcy’s Law valid
First Step: Define Boundary Conditions
I
I
I
constant-head boundary
no-flow boundary (also constant-flow boundary)
water-table boundary (line of known head at water-table, i.e.
only found in unconfined aquifers, cross-sectional problems)
Construction of Flownets
See Fig. 9
I
sketch model area, specify boundary conditions
I
draw trial flow lines, extending between constant-head
boundaries (i.e. perpendicular to them). Parallel to no-flow
boundaries near those boundaries.
I
draw trial equipotentials perpendicular to trial flow lines,
always perpendicular to them and spaced to form “squares”
with the flow lines. Parallel to constant-head boundaries,
perpendicular to no-flow boundaries
I
adjust repeatedly until an orthogonal set of equipotential and
flow lines is generated
Flownet Construction Methodology
Figure 9: General steps in developing a flownet. After (Fig. 4.11, Fetter,
2001).
Quantitative Flow Nets
I
if care is taken to make areas bounded by equipotentials and
streamlines “square”, discharge through the region of flow can
be calculated
I
in most systems, these will be “curvilinear squares”. The best
way to correctly make such a square is to ensure that the
sides enclose and are tangent to a circle
I
Fetter (eqn 4-55, 2001) gives a formula for this discharge, see
also Freeze and Cherry (eqn. 5.7, 1979).
Q = K ∆h
m
n
(11)
where m is the number of streamtubes (number of streamlines
minus one), and n is the number of head divisions (number of
equipotentials minus one)
Heterogeneity & Flow-Line Refraction
I
just as in refraction of light, when water passes from one
medium to another with a different conductivity (i.e. different
velocity)
I
the angle of refraction is Fetter (Fig. 4.13, 2001)
tan σ1
K1
=
K2
tan σ2
(12)
where σ1 is the angle between the incoming streamline and
the normal to the interface, and σ2 is the angle with the
outgoing streamline (Fig. 10A)
I
i.e. for a streamline entering a region of lower conductivity,
σ2 < σ1 , and the outgoing streamline will bend toward the
normal to the boundary
I
this is exactly analogous to Snell’s Law for light rays
Streamline Refraction
Figure 10: Streamline refraction. B) K2 > K1 , C) K2 < K1 . As in
Snell’s Law, the streamline is closer to the normal in the region of lowest
velocity (high index of refraction or low hydraulic conductivity). After
(Fig. 4.14, Fetter, 2001).
Flow direction for anisotropic aquifers
I
flow direction is not perpendicular to equipotential lines in
anisotropic media (i.e. those having Kx 6= Ky , Fig. 6,
resulting flownet in Fig. 11)
I
flow direction can be estimated graphically when constructing
flow nets for anisotropic media (Fig. 12)
Anisotropic Flow Net
Figure 11: Flownet for anisotropic conditions (Ky > Kx ). After (Fig.
4.10, Fetter, 2001).
Determining Anisotropic Flow Direction
Figure 12: After (Fig. 4.9, Fetter, 2001).
Example Flownets
Figure 13: Flownet for seepage from a channel. a) heterogeneous
1
x
, b) heterogeneous anisotropic KKUL = 50, K
isotropic KKUL = 50
Kz = 10.
After (Fig. 3.6.3, Todd and Mays, 2005).
Interpreting Potentiometric Surface
Figure 14: Example contour map of potentiometric surface with flow
lines. Section 2 has better prospect for a productive well. After (Fig.
3.6.5, Todd and Mays, 2005).
Streamfunction
I
I
Flownets rely on an important property of the flow equation
(Laplace’s Equation): isopotentials and the resulting flow lines
are orthogonal
this can be described mathematically in two useful ways:
I
I
I
define a streamfunction (Ψ), contours of which represent
flow/stream lines. This formulation is popular in fluid
dynamics and aeronautics.
define a complex potential, the real part of which represents
head, and the complex part is the streamfunction.
Analytic Element Modeling (AEM)
I
I
I
This formulation is the basis for a branch of groundwater
modeling known as “AEM” or Analytic Element Modeling (see
Haitjema, 1995; Strack, 1989)
see 2011 GSA Annual Meeting Special Session for examples
common “superimposed analytic solutions” applications are
the dipole, representing a fixed-potential boundary (e.g. river,
see Fig. 15)
Image Wells
Figure 15: Image well method (dipole) (Fig. 2.4, Strack, 1989).
Solving the Flow Equation
Many of the problems in hydrogeology involve solving the flow
equation (2) or (4). This will generally involve several of the
following approaches:
I
analytic solution: exact formula for simple situations (usually
homogeneous, isotropic, simplified geometry), generally
derived by direct integration of the flow equation
I
graphical solution: usually flownets
I
numerical model: required for complex situations, 3-D flow,
etc. Currently Modflow is the computer program of choice for
groundwater models
Steady Confined Flow Analytic Solution
I
assuming horizontal confined flow in an isotropic
homogeneous aquifer (Fig. 16)
I
the 2-D flow equation (1) becomes:
d2 h
dx 2
=0
I
seeking a solution for h(x) we integrate the equation twice to
obtain the general solution:
Z
Z
dh
d2 h
dx =
0 dx → dx
= C1 →
dx 2
Z
Z
dh
C1 dx → h(x) = C1 x + C2
dx dx =
I
then we use boundary conditions to determine the particular
solution for the case shown in Fig. 16:
I
at x = 0,
h = h1 , which implies C2 = h1
Steady Confined Flow Analytic Solution (cont.)
I
I
at x = L, h = h2 . Substituting these values h2 = C1 L + h1
1
and C1 = h2 −h
L
then the particular solution is:
h =
h2 − h1
· x + h1
L
(13)
which is the equation for a straight line, as indicated in Fig. 16
Confined Flow Example
Figure 16: Confined flow system example for computation of analytic
solution. Note homogeneity assures that h(x) is a straight line. After
(Fig. 4.16, Fetter, 2001).
Steady Unconfined Flow Analytic Solution
I
assuming horizontal unconfined flow in an isotropic
homogeneous aquifer (Fig. 17)
I
the 2-D flow equation (3) becomes:
d2 h2
dx 2
=0
I
seeking a solution for h(x) we integrate the equation twice to
obtain the general solution:
Z
Z
2
d2 h 2
dx =
0 dx → dh
dx = C1 →
dx 2
Z
Z
dh2
C1 dx → h2 (x) = C1 x + C2
dx dx =
I
then we use boundary conditions to determine the particular
solution for the case shown in Fig. 17:
I
at x = 0,
h = h1 , which implies C2 = h12
Steady Unconfined Flow Analytic Solution (cont.)
I
at x = L, h = h2 . Substituting these values h22 = C1 L + h12
h2 −h2
and C1 = 2 L 1
I
then the particular solution is (eqn. 4.71, Fetter, 2001):
r
h22 − h12
h =
· x + h12
(14)
L
I
allowing for uniform recharge w along the water table, the
final form is (eqn. 4.70, Fetter, 2001):
r
h22 − h12
w
+ h12 +
(L − x)x
(15)
h =
L
K
I
several other relationships can be derived from (15), e.g.
discharge per unit width q 0 :
I
dh
using Darcy’s Law q 0 = −Kh dx
and noting that
2
−K
dh
then q 0 = − 2 dx
du 2
dx
= 2u du
dx
Steady Unconfined Flow Analytic Solution (cont.)
I
differentiating the square of (15)
−K ( h1 − h2
w
q (x) = −
+ (L − 2x)
2
L
K
K (h1 − h2 )
L
=
− w
−x
2L
2
0
Unconfined Flow Example
Figure 17: Unconfined flow with recharge example for computation of
analytic solution. Note free surface assures that h(x) is a parabolic line.
After (Fig. 4.19, Fetter, 2001).
References
Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York, NY
(1972)
Bradbury, K.R., Muldoon, M.A.: Hydraulic conductivity determinations
in unlithified glacial and fluvial materials. Special technical pub.,
ASTM (1990)
Fetter, C.W.: Applied Hydrogeology. Prentice Hall, Upper Saddle River,
NJ, 4th edn. (2001), http://vig.prenhall.com/catalog/
academic/product/0,1144,0130882399,00.html
Freeze, R.A., Cherry, J.A.: Groundwater. Prentice-Hall, Englewood Cliffs,
NJ (1979)
Haitjema, H.M.: Analytic Element Modeling of Groundwater Flow.
Academic Press, San Diego, CA (1995), iSBN 0-12-316550-4
Strack, O.D.L.: Groundwater Mechanics. Prentice Hall, Englewood Cliffs,
NJ (1989)
Todd, D.K., Mays, L.W.: Groundwater Hydrology. John Wiley & Sons,
Hoboken, NJ, 3rd edn. (2005), http://www.wiley.com/WileyCDA/
WileyTitle/productCd-EHEP000351.html