Transcript Document

Part 2:slides 41-80
SHORT COURSE ON
MODELING
FLOW AND SOLUTE TRANSPORT
IN THE SUBSURFACE
by
JACOB BEAR
Professor Emeritus, Technion—Israel Institute of Technology,
Haifa, Israel
Lectures presented at the Instituto de Geologia, UNAM,
Mexico City, Mexico, December 6--8, 2003
Copyright © 2002 by Jacob Bear, Haifa Israel. All Rights Reserved.
To use, copy, modify, and distribute these documents for any purpose
is prohibited, except by written permission from Jacob Bear.
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PIEZOMETRIC HEAD
z = Elevation of point

= Specific weight of water
p = Pressure in the water
hz
p

Piezometric head is measured
with respect to a
DATUM LEVEL.
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CLASSIFICATION OF AQUIFERS ACCORDING
TO THE PIEZOMETRIC HEAD SYSTEM
CONFINED AQUIFER: bounded from above and below
by impervious formations. Piezometric surface above
ceiling of aquifer.
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ARTESIAN AQUIFER
Portion of a confined aquifer in which the
piezometric head is above ground surface.
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PHREATIC AQUIFER: bounded from above
by a phreatic surface.
PERCHED AQUIFER: local phreatic aquifer above
the phreatic surface of a major one.
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LEAKY (PHREATIC OR CONFINED)
AQUIFER: bounded from above and/or below
by an aquitard.
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MULTIPLE AQUIFERS
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Distorted scale!!
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AQUIFERS ARE VERY THIN DOMAINS, RELATIVE
TO HORIZONTAL DISTANCES OF INTEREST
CONFINED AQUIFER
Very small vertical flow component.
CONFINED AQUIFER WITH PARTIALLY PENETRATING
WELLS
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Phreatic aquifer
River
Note the
streamlines
Leaky-confined aquifer
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Recall: distorted scale!
LOCAL vs. REGIONAL PROBLEMS.
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Flow in a vertical cross-section, with horizontal
water table
Flow near inflow
And outflow boundaries.
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MOTION EQUATIONS FO SINGLE PHASE
FLOW: DARCY's LAW
Two ways for presenting Darcy’s law
The MOTION EQUATION is an APPROXIMATE
FORM of the AVERAGED MOMENTUM BALANCE
EQUATION FOR A FLUID
---the NAVIER STOKES EQUATIONS
(neglecting inertial effects and terms that express energy
dissipation as a result of momentum transfer within the fluid).
Or….the French engineer in charge of the water system in
the city of Dijon, Henry Darcy, 1856, ..…conducted
experiments on sand packed filter columns,
reaching EMPIRICAL CONCLUSION
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h1  h2
Q  KA
L
p
hz
rg
K = coefficient of proportionality called hydraulic conductivity.
Q = volume of fluid per unit time passing through a column of
constant cross-sectional area, A and length L.
h1, h2 = elevations of inflow and exit reservoirs of the column.
z = elevation of the point at which the piezometric head is
measured, above a datum level.
p, r = fluid's pressure and mass density.
z = elevation of the point at which the piezometric head is measured.
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p, r = fluid's pressure and mass density
52
In a compressible fluid, r = r(p), we define
HUBBERT's POTENTIAL (Hubbert, 1940):
dp
h  h (x, t )  z  
po g r ( p)
*
*
p
In Darcy’s law,
h1 - h2 = Dh = Energy loss across the column due to
friction at the microscopic solid-fluid interface.
HYDRAULIC GRADIENT:
h1  h2
h( x)  h( x  Dx)
J =


L
L
x
=
x+Dx
Flow
x
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SPECIFIC DISCHARGE:
Q
q   discharge per unit time through a unit
A
area normal to the flow.
(…NOT “apparent velocity”, “Darcy velocity”, etc).
q  KJ
So far... flow through a finite length, L.
What happens AT A POINT?
Ds
s
s+ 1
Ds
2
s
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s- 1
Ds
2
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Ds
s
1
s- Ds
2
q
s
1
s+ Ds
2
Consider flow in a column aligned in the direction of the
unit vector s. Along the column, h = h(s). Use Darcy's law
for the segment:
h s  1 Ds  h s  1 Ds
2
2
qs ( s)  K
Ds
In the limit, as Ds
0, we obtain:
lim
Ds 0
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h s  1 Ds  h s 1 Ds
2
2
Ds
dh
 .
ds
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Thus, AT A POINT:
dh
dh
qs   K
 KJ ss ; h  h(s, t ); J ss   ;
ds
ds
with qs considered positive when it coincides with the
positive direction of the s-axis.
FLOW TAKES PLACE FROM HIGH PIEZOMETRIC
HEAD (ENERGY) TO LOW PIEZOMETRIC HEAD.
NOT NECESSARILY FROM HIGH TO LOW PRESSURE.
Do not use the piezometric head when the density
varies (by temperature and/or concentration changes).
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VELOCITY, V, is the distance traveled per unit
f
time.
VV
(Omit averaging symbol)
Q
q
V

A 
EFFECTIVE POROSITY. Part of the void space is
not available to fluid flow, due to dead-end pores
(immobile Fluid), zone with fine grained material.
V 
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q
eff
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HYDRAULIC CONDUCTIVITY, K (dims. L/T).
Can be defined as: Specific discharge per unit gradient
(in 1-d flow in an isotropic porous medium). The hydraulic
conductivity depends of fluid properties and void space
configuration (width of passages and tortuosity).
rg
g
K k
k


 = dynamic viscosiy
n = kinematic viscosity
PERMEABILITY, k (dims. L2), depends only on void space
Configuration.
UNITS for HYDRAULIC CONDUCTIVITY:
m/d, cm/s, ft/d, gal/d-ft2 , (SI: m/s)
UNITS for PERMEABILITY:
m2, cm2, ft2 , (SI: m2)
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In petroleum Engineering:
1cm3 /s/cm 2  1centipoise
1darcy 
1atmosphere
k r g Dh
Q
 L
FORMULAE FOR PERMEABILITY:
Empirical and semi-empirical formulae:
k  Cd
2
(and many other formulae)
C = dimensionless coefficient.
d = effective grain diameter, say d10.
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Changes of permeability with time, due to
Compaction by external load.
Dissolution of solid.
Precipitation.
Clogging by fines,
Biological activities...
Shrinkage of clay soil.
RANGE OF VALIDITY OF DARCY’s LAW
Experiments:
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In pressurized conduits, and channels, the
(dimensionless) REYNOLDS NUMBER, Re, expresses
the ratio of inertial to viscous forces acting on a fluid.
Helps to distinguish between LAMINAR FLOW, at low Re
and turbulent flow at higher Re.
By analogy, for flow through porous media:
Re 
qd

,
d = some representative (microscopic) length characterizing
void space, e.g. d10. n = kinematic viscosity of fluid (e.g.,
Darcy's law is valid as long as the Re, that indicates the
magnitude of the inertial forces relative to the viscous drag
ones, does not exceed a value of about 1 (but sometimes
as high as 10):
Re 1.
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EXTENSIONS OF DARCY'S LAW:
To three dimensions.
To compressible fluids.
To inhomogeneous porous medium.
To inhomogeneous porous medium.
To anisotropic porous medium.
Remember: Darcy's law is a simplified form of the averaged
momentum balance equation.
THREE-DIMENSIONAL FLOW
For a homogeneous isotropic porous medium,
K = a constant SCALAR
q  K grad h ( K h)
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Since the specific discharge is a VECTOR:
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q  K grad h ( K h)
J

J  h J

J
x
y
z
qx  K J x , q y  K J y , qz  K J z ,
x, y , z
J
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x
 axes of Cartesian coordinate system
h
h
h
 , J y  , J z  ,
x
y
z
qx , qy , qz  and J x ,J y ,J z ,
Are components of the VECTORS q,
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Recall: The hydraulic gradient is a vector equal to
the negative of the gradient vector.
J  h
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COMPRESSIBLE FLUID:
Here, r = r (p), and we use the motion equation
(Darcy's law) written in terms of h* (= Hubbert’s
potential):
q   K h
*
INHOMOGENEOUS POROUS MEDIUM:
We use K = K(x,y,z) in Darcy’s Law.
ANISOTROPIC POROUS MEDIUM
If the permeability at a given point depends on direction,
the porous medium at that point is said to be ANISOTROPIC.
Reasons for anisotropy: layering, shape of grains, vertical
stress.
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Darcy's law for an anisotropic porous medium:
qx  K xxJ
x
 K xyJ
y
 K xzJ
z
q y  K yxJ
x
 K yyJ
y
 K yzJ
z
qz  K zxJ
x
 K zyJ
y
 K zzJ
z
This is a linear relationship between the components
q , q , q  of q and the components J
x
of
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J
y
z
x
,J y ,J
z
,
.
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The permeability is represented by NINE
COEFFICIENTS: Kxx, Kxy, Kxz,…., etc.
Kij may be interpreted as the contribution to qi by a unit
of J
.
x
Components of the SECOND RANK TENSOR OF.
HYDRAULIC CONDUCTIVITY TENSOR
Symmetric tensor, i.e.,
Kij = Kji
Six distinct components.
The coefficients Kij are non-negative.
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In matrix forms, in 3-d domain:
K xx
K xy
K xz
K = K yx
K yy
K yz
K zx
In 2-d domain:
K zy
K zz
K=
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K xx
K xy
K yx
K yy
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Other (compact) forms of Darcy"s law:
Vector form:
Indicial form:
q  K  J  K h
qi  K ijJ
subscripts i,j indicate xi, xj.
j
h
 -K ij
x j
Einstein's summation convention:
subscript (or superscript) repeated twice and only twice
in any product or quotient of terms is summed over the
entire range of values of that subscript (or superscript).
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PRINCIPAL DIRECTIONS OF A
SECOND RANK TENSOR
Kij  0 for all i  j and Kij  0 for i  j.
When principal directions are used as a coordinate system:
K xx 0
K= 0
0
0
K yy 0
0
K zz
K=
K xx
0
0
K yy
IN AN ANISOTROPIC POROUS MEDIUM,
FLOW AND GRADIENT ARE NOT CO-LINEAR!
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GENERALIZED DARCY LAW
By averaging the momentum balance equation for a
Newtonian fluid, and introducing the simplifying
assumptions:
Inertial effects are negligible relative to viscous ones.
Shear stress WITHIN the fluid is negligible in
comparison with the drag at the fluid-solid interface.
we obtain for saturated flow:
V - Vs  -
k

  p  r gz 
V, p, r,  = (average) velocity, pressure, density, and
viscosity of the fluid, respectively.
Vs = (average) velocity of the solid.
z = elevation.
k(x,y,z) = permeability tensor.
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q r   (V -Vs )   Vr
qr = -
k

  p  r gz 
For Stationary, non-deformable porous
medium, Vs = 0:
k
q = -   p  r gz 

When x,y,z are PRINCIPLE DIRECTIONS:
k xx p
qx =  x
k yy p
qy =  y
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k zz  p

qz =  rg 

  z

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MODELING FLOW IN AN AQUIFER---ESSENTIALLY HORIZONTAL FLOW
What about 2-d flow in an aquifer?
USE DARCY'S LAW
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h
Qx  - KB
 KBJ x ,
x
h
Qy  - KB
 KBJ y .
y
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….and in vector form when K and T are tensors: :
Q '  -  K B   grad h
 - T  grad h  - T  grad h  T  h  T  J .
Q’= Discharge per unit width through entire aquifer thickness.
h=h(x,y,t) = Average head along aquifer's thickness, B.
The product KB is called TRANSMISSIVITY.
When K=K(x,y,z),
T=T(x,y) =

K(x,y,z) dz,
B ( x, y )
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FLOW IN A CONFINED AQUIFER.
Recall: on phreatic surface, p = 0, h = z.
h = h(x,y,z,t) = Piezometric head.
H=H(x,y,t) = Elevation of water table above datum level.
h
H
qs  - K
- K
 - K sin 
s
s
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Introduce DUPUIT (1863) ASSUMPTION(s)
ESSENTIALLY HORIZONTAL FLOW IN THE AQUIFER .
Equivalently:
Equipotentials are vertical,
Pressure distribution is hydrostatic along theoretical,
Velocities are uniform along the vertical.
In vector notation:
Q '  - K h grad h  - Kh h
When bottom is at elevation h = h (x,y):
Q'= - K (h-h) grad h  -K (h-h) h.
K(h - h) plays the role of transmissivity of
the phreatic aquifer. It is a tensor in an
anisotropic medium.
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Use Dupuit assumption:
dH
Q '   KH ( x)
 const.
dx
Q ' dx   KH ( x)dH
Boundary conditions. What about the seepage face?
When employing the Dupuit assumption, we neglect the
seepage face.
By integration, we get the
DUPUIT-FORCHHEIMER DISCHARGE FORMULA:
2
o
2
L
h -h
Q'  K
.
2L
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HOWEVER….non-horizontal flow:
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End of part 2
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Copyright © 2000 by Jacob Bear, Haifa Israel.
All Rights Reserved.
MODELING
FLOW AND SOLUTE TRANSPORT
IN THE SUBSURFACE
by
JACOB BEAR
WORKSHOP I
The Second International conference on
Salt Water Intrusion and Coastal Aquifers
--Monitoring, Modeling and Management
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Part 2:slides 41-80
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