review of the fundamentals of groundwater flow fundamentals.pdfdarcy’s law q: volumetric flow rate...
Post on 30-Sep-2020
0 Views
Preview:
TRANSCRIPT
Darcy’s Law
Q: volumetric flow rate [L3 T-1]h1: hydraulic head upstream [L]h2: hydraulic head downstream [L]K: hydraulic conductivity [L T-1]A: cross sectional area normal to flow direction [L2]L: distance between locations where h1 and h2 are measured [L]
Q
datum
h1 h2
LLhhKAQ 21 −=
LhhKAQ 12or −
−=
A
Review of the Fundamentals of Groundwater Flow
dldhKA
LhhKAQ −=
−−= 12
Darcy’s Law:
dldhK
AQq −==
Alternatively:
q is called specific discharge, also Darcy flux or Darcy velocity. However, q is not true flow velocity! The true flow velocity, v, also referred to as seepage velocity or pore water velocity is defined as
⎟⎠⎞
⎜⎝⎛−==
dldh
nK
nAQv
Bulk area is APorosity is nArea for flowis (nA)
Confining bed
Confining bed
Confined aquifer b
Potentiometric surfaceTransmissivity (T) forconfined aquiferT = K b
Water table or “free surface”
Confining bed
hUnconfined aquifer h
Transmissivity for unconfined aquiferT = K h
h: saturated thickness of aquifer
Darcy’s Law and Hydraulic Head1. Hydraulic Head
AL
hhKQ 12 −−=
h1 and h2 are hydraulic heads associated with points 1 and 2.The hydraulic head, or total head, is a measure of the potential of the water fluid at the measurement point.
“Potential of a fluid at a specific point is the work required totransform a unit of mass of fluid from an arbitrarily chosen stateto the state under consideration.”
Q
datum
h1 h2
hp1
hp2
z1z2
Three Types of Potentials
A. Pressure potentialwork required to raise the water pressure
Reference state
Current statez = 0P = 0v = 0
z = zP = Pv = v
V: volume
: density of water assumed to be independent of pressure
w
P
w
P PdPmm
dPVm
Wρρ
=== ∫∫ 00111
ρw
B. Elevation potentialwork required to raise the elevation
C. Kinetic potentialwork required to raise the velocity
Wm
mgdz gzZ
2 0
1= =∫
Wm
madzm
mdvdt
dz vdvvZ Z v
3 0 0 0
21 12
= = = =∫ ∫ ∫
Total [hydraulic] head:
hg
Pg
zvgw
= = + +Φ
ρ
2
2
Unit [L]
(dz = vdt)
Total potential:
Φ = + +P
gzv
wρ
2
2Unit [L2T-1]
hP
gz
vgw
= + +ρ
2
2
pressurehead [L]
elevation [L]
Kinetic term
Piezometer
datum
h1 h2
z1 z2
Pg1
ρ gPρ
2
A fluid moves from where the total head is higher to where it is lower. For an ideal fluid (frictionless and incompressible), the total head would stay constant.
Total head or hydraulic head:
gvz
gPhw 2
2
++=ρ Kinetic term
negligible
h = hydraulic head [L]= pressure head [L]
z = elevation head [L]Important: h is relative to datum (reference state)
For Groundwater Flow
datum
A B
h2
piezometers
h1
flowdirection?
gP wρ
Hydraulic gradient
Q Kh h
LA
Kh h
LA
KiA Kdhdl
A
=−
= −−
= − = −
1 2
2 1
datum
h1 h2
L
Gradient of h with respect to xat point A is:
dhdx
hx
h hx x
Ax
=
≅−−
→∆
∆∆0
2 1
2 1
lim
h2
h1
x1 x2
A
h
x
x
y
θ
xh∂∂
yh∂∂
dldh
h=4
32
1
Magnitude and direction of gradient vector in 2D
22
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
=yh
xh
dldh
xhyh∂∂∂∂
= arctanθ
Extension of Darcy’s LawDarcy’s Law in 1D:
dldhKq −=
( )
kzhKj
yhKi
xhKkqjqiqq
kzhj
yhi
xhhhgrad
dldh
zyx ∂∂
−∂∂
−∂∂
−=++=
∂∂
+∂∂
+∂∂
=∇==
zhKq
yhKq
xhKq zyx ∂
∂−=
∂∂
−=∂∂
−=∴ ;;
i, j, k: unit vectors
NOTE: above equations are valid only if K is scalar, i.e., aquifer is isotropic. In other words, K does not vary with direction at any location.
Darcy’s Law in 3D:
In an anisotropic aquifer, K is a second-order tensor.It has 9 components in the Cartesian coordinate system:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
zzzyzx
yzyyyx
xzxyxx
KKKKKKKKK
K
Darcy’s Law becomes
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
∂∂
∂∂
∂∂
zhyhxh
zzzyzx
yzyyyx
xzxyxx
z
y
x
KKKKKKKKK
qqq
formmatrix inor
( )hgradKq ⋅−=
Darcy’s Law with expanded terms:
zhK
yhK
xhKq
zhK
yhK
xhKq
zhK
yhK
xhKq
zzzyzxz
yzyyyxy
xzxyxxx
∂∂
−∂∂
−∂∂
−=
∂∂
−∂∂
−∂∂
−=
∂∂
−∂∂
−∂∂
−=
( )zyxjixhKq
jiji
,,3,2,1,
:notationtensor In
=
∂∂
−=
NOTE Kxx, Kyy, Kzz are principal components of the K tensor; parallel or normal to the direction of the maximum or minimum K.Kij (where i ≠ j) are cross terms of the K tensor, representing contributionsto the flow rate in direction i from the gradient in direction j.
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
zzzyzx
yzyyyx
xzxyxx
KKKKKKKKK
K
x
y
zKyy
Kzz
Kxx
xy
z
Kyy
Kzz
Kxx
If the Cartesian coordinateaxes are aligned with the principal directions of the K tensor,the cross terms are all equal to zero:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
zz
yy
xx
KK
KK
000000
General K tensor: Aquifer formation
Use only one index for the K tensor:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
z
y
x
KK
KK
000000
zhKq
yhKq
xhKq zzyyxx ∂
∂−=
∂∂
−=∂∂
−= ;;
We have components of specific discharge:
or in tensor notation:
( )zyxixhKq
iii
,,3,2,1=∂∂
−=
q
grad(h)
In isotropic media, K is scalar, specific discharge isparallel to hydraulic gradient
q
grad(h)
In anisotropic media, K is tensor or vector, specific discharge is NOT parallel to hydraulic gradient
GROUNDWATER STORAGEa) Storage Concept
V AhV A hw
w
==∆ ∆
if: Q1 = Q2
∆∆
Vh
w ==
00
Steady State
Storage Coefficient: Volume of water that an aquifer releases from or takes into storage per unit surface area of the aquifer per unit change in head.
if: Q1 Q2≠
∆∆
Vh
w ≠≠
00
Non-Steady State
SV
A hw=
∆∆
LL L
3
2
⎡⎣⎢
⎤⎦⎥
Q2
Q1
∆h
h
Storage tank
A
b) Unconfined Aquifer (Specific Yield)
n: Porosity; assume 100% drainable
nhA
VS wy =∆
∆=
hnAVnAhV
w
w
∆=∆=
Sy : specific yield
Typically, not 100% drainable
hfnAVfnAhV
w
w
∆=∆=
fnhA
VS wy =∆
∆=
(f < 1)
Sr specific retention
Storage from unconfinedaquifer involves physicaldewatering
A: area
h
aquifer
h∆
ry SSn +=
c) Confined Aquifer
∆∆
VA
hw ∝
∆∆
VA
S hw =
S Sbs = [L-1]
SV
A hw=
∆∆
water inputb
piezometer
pumpwaterlevel ∆h
∆Vw
No physical dewatering involved.Storage coefficient for confined aquifer
63 1010 −− −≈S usually << porosity
S : storage coefficient, dimensionless
Ss : specific storage
A
I. Derivation of the Groundwater Flow Equation
A. Continuity Principle (conservation of mass)inflow - outflow = storage
B. Darcy’s LawSpecify how inflow and outflow are calculated
∆
“Control Volume”
∆
∆
y
∆xz
Qxleft side
Qz top
Qz bottom
Qx right side
Qyfront
Qy back
(x,y,z)
yxzzyxqQ
yxzzyxqQ
zxzyyxqQ
zxzyyxqQ
zyzyxxqQ
zyzyxxqQ
yxthzS
-
ZZ
ZZ
YY
YY
XX
XX
S
∆∆⎟⎠⎞
⎜⎝⎛ ∆
+
∆∆⎟⎠⎞
⎜⎝⎛ ∆
−
∆∆⎟⎠⎞
⎜⎝⎛ ∆
+
∆∆⎟⎠⎞
⎜⎝⎛ ∆
−
∆∆⎟⎠⎞
⎜⎝⎛ ∆
+
∆∆⎟⎠⎞
⎜⎝⎛ ∆
−
∆∆∆∆
∆∆
∆
2,, = top
2,, =bottom
,2
, =front
,2
, =back
,,2
=right
,,2
=left
)( =storage
sink Q + topQ+front Q +right Q =outflowsource Q + bottom Q+back Q +left Q =inflow
storage=outflow inflow
ZYX
ZYX
+
q x x y z y z q x x y z y z
q x y y z x z q x y y z x z
q x y z z x y q x y z z x y
Q S ht
x y z
X X
Y Y
Z Z
SS S
−⎛⎝⎜
⎞⎠⎟
− +⎛⎝⎜
⎞⎠⎟
+ −⎛⎝⎜
⎞⎠⎟
− +⎛⎝⎜
⎞⎠⎟
−⎛⎝⎜
⎞⎠⎟
− +⎛⎝⎜
⎞⎠⎟
+ =
∆∆ ∆
∆∆ ∆
∆∆ ∆
∆∆ ∆
∆∆ ∆
∆∆ ∆
∆∆
∆ ∆ ∆
2 2
2 2
2 2
, , , ,
, , , ,
, , , ,
Divide both sides by ∆ ∆ ∆x y z
zyxQq
thSq
zq
yq
xq
thS
zyxQ
z
zzyxqzzyxq
y
zyyxqzyyxq
x
zyxxqzyxxq
SSSSSSS
ZYX
SSS
ZZ
YYXX
∆∆∆=
∂∂
=+∂∂
−∂∂
−∂∂
−
∆∆
=∆∆∆
+∆
⎟⎠⎞
⎜⎝⎛ ∆−−⎟
⎠⎞
⎜⎝⎛ ∆+
−
∆
⎟⎠⎞
⎜⎝⎛ ∆−−⎟
⎠⎞
⎜⎝⎛ ∆+
−∆
⎟⎠⎞
⎜⎝⎛ ∆−−⎟
⎠⎞
⎜⎝⎛ ∆+
−
where
2,,
2,,
,2
,,2
,,,2
,,2
zhKq
yhKq
xhKq
zz
yy
xx
∂∂
−=
∂∂
−=
∂∂
−=
thSq
zhK
zyhK
yxhK
x SSSzyx ∂∂
=+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
Heterogeneous/anisotropic with sink/sourcetransient 3D 0≠
∂∂
th
General Flow EquationThree dimensional, transient, heterogeneous, anisotropic
Laplace’s EquationSteady-state condition in homogeneous, isotropic medium without sink/source
02
2
2
2
2
2
=∂∂
+∂∂
+∂∂
zh
yh
xh
( ) ( ) ( ) ( )22
2
2
2
2
2
∇=∂∂
+∂∂
+∂∂
zyx
Laplacian Operator
∇ =2 0h
Two-dimensional
thbSbq
yhbK
yxhbK
x SSSyx ∂∂
=+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
thSq
yhT
yxhT
x ssyx ∂∂
=+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂ *
Homogeneous & Anisotropic (in terms of transmissivity)
thSq
yhT
xhT ssyx ∂
∂∂∂
∂∂
=++ *2
2
2
2
Homogeneous & Isotropic
th
TS
Tq
yh
xh ss
∂∂
∂∂
∂∂
=++*
2
2
2
2
b: saturated thickness
II. Mathematical model of groundwater flow
A. Governing Equation
02 =∇ h (assumptions?)
th
KSh S
∂∂
=∇2 (assumptions?)
Simpler Examples:
thSq
zhK
zyhK
yxhK
x SSSzyx ∂∂
=+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
Assumptions: 3-D, transient, heterogeneous, anisotropic,with sinks/sources(principal K components aligned with coordinate axes)
B. Initial ConditionsFor any transient problem, the initial head distributionmust be known in order to solve for head changes with time, i.e.,
Q
t = 0
t = t h0
( ) ),,(0,,, zyxfzyxh =
( ) 00,,, hzyxh =For this example,
where f(x,y,z) is known
C. Boundary ConditionsHeads, fluxes or some combination of the twomust be known at boundaries in order to solve for head changes with space in the interior of the flow field.
(Wang and Anderson, 1982)
h1 h2
1. Specified head (Dirichlet condition)
( ) 10 ),,,(,,, Γ∈= tzyxhtzyxhwhere h0(x,y,z,t) is known
For example,h=h1 on the left boundaryh=h2 on the right boundary
2. Specified flux (Neumann condition)
2),,,( Γ∈=∂η∂
− tzyxqhK o
where qo(x,y,z,t) is a known flux across the boundary.
A special case is zero flux boundary condition, i.e.,
20 Γ∈=∂η∂
−h
Γ
η
+++ + ++ +
+ +++
QUESTION: A river could be treatedas either a specified-head or specified flux boundary. What’s the difference between the two treatments?
3. Combination of 1 & 2 (Cauchy condition)also referred to as head-dependent boundary condition
b/ K/
h=hb
q/h
the flux into the aquiferthrough the aquitardis dependent on the headin the aquifer, i.e.,
3// Γ∈−
−=∂η∂
−b
hhKhK b
QUESTION: Could a river also be treated as a head-dependent boundary condition? If so, what’s the difference with the previous two treatments?
III. Solutions of Mathematical Model
Solutions to a mathematical model of groundwater flow can be obtained analytically or numerically.
Analytical Solution: Head h is expressed explicitly as mathematical formula (function) of x, y, z, t.
For example, steady-state flow in a 1-D confined aquifer:
Q
L
h1 h2
( )
slope!constant
check
12
121
⇒
−=
−+=
Lhh
dxdh
xL
hhhxh
Example of Analytical Solutions
0=∂∂
=⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
thS
xhT
x
1. Governing equation: 2. Boundary conditions:
( )h x hX = =0 1
( )h x hX L= = 2
h1 h2b
Ldatum
x
( )
( )
Th
xd hdx
ddx
dhdx
dx dx
dhdx
addx
h dx adx
h x ax b
∂∂
2
2
2
20 0 0= → = ⎛⎝⎜
⎞⎠⎟ =
= =
= +
∫ ∫
∫ ∫
with boundary condition (1): x = 0, h(x) = h1we found: b = h1with boundary condition (2): x = L, h(x) = h2we found: a h h
L=
−2 1
∴ =−
+h x h hL
x h( ) 2 11
Numerical Solution:Head h is solved approximately at predefined “nodal points” as illustrated below. Numerical solution is typically obtained through a computer code.
(Wang and Anderson, 1982)
top related