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Soil Flux and Volume Change in Unsaturated-Flow Equations
Appendix 3 SOI L F L U X A N D V O L U M E C H A N G E I N U N S A T U R A T E D - F L O W EQUATION S
An alternative approach to the examination of the problem of a changing porosity (or void ratio) during flow is to assume that in addition to considerations of motion of water relative to the soil particles (i.e. a modified Darcy relationship as in eq.5.31), the flux of soil particles must satisfy the three physical conditions of: (a) continuity, (b) Newton's law of motion, and (c) rheologjcal equation of state. Since the motion of soil particles in the volume changing soil during fluid flow is slow, the acceleration considerations implicit in Newton's law of motion can be safely ignored. From conservation of mass, the equation of continuity in a fixed coordinate system can be written in general terms as:
div(p ) + j £
= 0
9s
(1)
where: q = soil flux (vector); ñ = bulk density = G p / ( 1 + e)\ p water; e = void ratio; G = specific gravity of solid particle; t = time From eq.2 and 1: s
s
s
div 9s =
+
TT7 Yt T+7 '
g r a d e
w
w
= density of (2)
( 3 )
Eq.3 which includes the second order term <7*/(l + e) · grad e has been used by Wong (1969). If the second term on the right hand side of eq.3 is ignored, the equation is similar to that used by Zaslavsky (1964). Defining the soil flux q as: s
s
q
= k
s
grad Ö
(4)
where k represents the coefficient of particle conductivity, and Ö denotes the pressure potential in the permeating pore fluid which results in particle movement it follows that since Ö depends on the void ratio e: s
430
APPENDI X 3
Q s
s
" * ~be - D
g r a d
e
5
()
grad e
$
é where D = particle diffusivity. Thus eq.3 becomes: $
r a (
1 be A S ^ div (Z) grad
e
· grad e
(6)
5
writing: ?w
= ?w s
+
7
( )
where <7 = flux of water relative to moving soil particles. Taking the divergence of both sides, we obtain: WS
div ?
w
= div <7
WS
bB = - —
+ div dq
s
(8)
By expanding and substituting, the expanded form of the continuity equation can be obtained as:
-
T
= div«7
ws
+ â ^ —
-
+ —
· grade) +
· grad.
(9)
The above form can be simplified if the second order term is ignored. Thus:
-ft'*"™*
TTe
{ft
( 1 0 )
Eq.10 represents the continuity condition which can now be combined with the modified form of the Darcy relationship as in eq.5.31 to yield the diffusion equation. If <7 is considered small in eq.10 it can be further reduced to (Wong, 1973): bB . dv = d* l vv < 7 + + - ^ (11) S
A
' Yt
÷
w s
a7
Eq.l 1 can now be used with the continuity condition shown in Chapter 5.
An alternative approach to the examination of the problem of a changing porosity (or void ratio) during flow is to assume that in addition to considerations of motion of water relative to the soil particles (i.e. a modified Darcy relationship as in eq.5.31), the flux of soil particles must satisfy the three physical conditions of: (a) continuity, (b) Newton's law of motion, and (c) rheologjcal equation of state. Since the motion of soil particles in the volume changing soil during fluid flow is slow, the acceleration considerations implicit in Newton's law of motion can be safely ignored. From conservation of mass, the equation of continuity in a fixed coordinate system can be written in general terms as:
div(p ) + j £
= 0
9s
(1)
where: q = soil flux (vector); ñ = bulk density = G p / ( 1 + e)\ p water; e = void ratio; G = specific gravity of solid particle; t = time From eq.2 and 1: s
s
s
div 9s =
+
TT7 Yt T+7 '
g r a d e
w
w
= density of (2)
( 3 )
Eq.3 which includes the second order term <7*/(l + e) · grad e has been used by Wong (1969). If the second term on the right hand side of eq.3 is ignored, the equation is similar to that used by Zaslavsky (1964). Defining the soil flux q as: s
s
q
= k
s
grad Ö
(4)
where k represents the coefficient of particle conductivity, and Ö denotes the pressure potential in the permeating pore fluid which results in particle movement it follows that since Ö depends on the void ratio e: s
430
APPENDI X 3
Q s
s
" * ~be - D
g r a d
e
5
()
grad e
$
é where D = particle diffusivity. Thus eq.3 becomes: $
r a (
1 be A S ^ div (Z) grad
e
· grad e
(6)
5
writing: ?w
= ?w s
+
7
( )
where <7 = flux of water relative to moving soil particles. Taking the divergence of both sides, we obtain: WS
div ?
w
= div <7
WS
bB = - —
+ div dq
s
(8)
By expanding and substituting, the expanded form of the continuity equation can be obtained as:
-
T
= div«7
ws
+ â ^ —
-
+ —
· grade) +
· grad.
(9)
The above form can be simplified if the second order term is ignored. Thus:
-ft'*"™*
TTe
{ft
( 1 0 )
Eq.10 represents the continuity condition which can now be combined with the modified form of the Darcy relationship as in eq.5.31 to yield the diffusion equation. If <7 is considered small in eq.10 it can be further reduced to (Wong, 1973): bB . dv = d* l vv < 7 + + - ^ (11) S
A
' Yt
÷
w s
a7
Eq.l 1 can now be used with the continuity condition shown in Chapter 5.