Environmental Engineering Reference
In-Depth Information
16.6.8 Water Phase Continuity
The continuity equation for the water phase can be obtained
by equating the time derivative of the water phase constitu-
tive equation to the divergence of Darcy's flow law:
incorporated into the formulation since vapor flow can be
relevant. Surface boundary conditions for the air, liquid water,
water vapor, and heat flow equations are also summarized.
The soil-atmospheric boundary conditions incorporate the
microclimatic conditions at a site.
k w
ρ w g
∂ε v
∂t
∂(u a
u w )
2 u w
β w 1
+
β w 2
=−
16.7.1 Air Phase Partial Differential Equation
The differential equation for one-dimensional, K 0 ,a r
flow during consolidation requires that one term be added
when dealing with nonisothermal conditions. The term
takes into consideration the effect of temperature changes
on air flow. Pore-air pressures are consequently affected.
One-dimensional air flow in the y -direction can be expressed
as follows (Dakshanamurthy and Fredlund, 1981):
∂t
∂k w
∂x
1
ρ w g
∂u w
∂x
∂k w
∂y
∂u w
∂y
∂k w
∂z
∂u w
∂z
∂k w
∂y
+
+
+
+
(16.67)
16.6.9 Air Phase Continuity
The continuity equation for the air phase can be obtained
from the time derivative of the air phase constitutive
equation and the net mass rate of air flow as described by
Fick's law:
2 u a
∂y 2
∂u a
∂t =−
∂u w
∂t +
∂T
∂t
c v
C a
+
C at
(16.69)
u a
u w
D a
ρ a ¯
∂ε v
∂t
1
ρ a ¯
where:
2 u a +
β a 1
+
β a 2
=−
u a
∂t
u a
∂D a
∂x
∂D a
∂y
∂D a
∂z
C a =
interaction coefficient associated with the air
phase PDE,
∂u a
∂x +
∂u a
∂y +
∂u a
∂z
( 1
S)n
u a
∂t
¯
×
+
is, (m 2 /m 1 k )/ [1
m 2 /m 1 k
that
u a
¯
S) n/ ¯
u a m 1 k ]
( 1
(16.68)
m 1 k =
coefficient of air volume change with respect to a
where:
change in net normal stress, d σ y
u a ,for K 0
loading,
D a
=
coefficient of transmission for the air phase,
c v
=
coefficient of consolidation with respect to the air
phase, that is,
D a
ω a /RT K
ρ a
=
density of air,
u a
¯
=
absolute pore-air pressure (i.e.,
u a =
¯
u a
u atm ),
1
u a
=
gauge pore-air pressure,
u a m 1 k 1
m 2 /m 1 k
,
u atm =
¯
atmospheric pressure (i.e., 101 kPa),
¯
( 1
S) n
S
=
degree of saturation, and
C at =
interaction temperature coefficient associated with
the air phase PDE, that is
1
T
n
=
porosity.
,
( 1
S)n
u a
¯
The solution of a coupled analysis for two-phase flow
through an unsaturated soil involves the simultaneous solu-
tion of Eqs. 16.64, 16.65, 16.66, 16.67, and 16.68.
m 2 /m 1 k )
u a m 1 k +
( 1
¯
( 1
S)n
T K =
absolute temperature (i.e., T K =
T
+
73 . 16 ), K,
temperature, C, and
T
=
t
=
time.
16.7 WATER, AIR FLOW, AND NONISOTHERMAL
SYSTEMS
The above equation does not take into account any non-
linear variation which might be present in the coefficient of
transmission, D a . Variations in the coefficient of transmis-
sion with respect to space (i.e., ∂D a /∂y ) are assumed to be
negligible.
Air flow and water flow may not only occur as a result of a
load being applied to the soil. It is also possible that moisture
infiltration and temperature fluctuations may produce ther-
mal gradients within a soil mass that result in fluid flow.
Nonisothermal conditions are closely related to microcli-
matic changes in the field. Seasonal temperature changes can
cause air and water flow and consequently volume changes
within a soil mass. A nonisothermal condition often occurs
under highway and airfield pavements or under the shallow
foundations of lightweight structures.
Following is a summary of the formulation of one-
dimensional flow under nonisothermal conditions in an
unsaturated soil. Temperature gradients are incorporated
into the formulation of flow. Water vapor flow is also
16.7.2 Fluid and Vapor Flow Equation
for Water Phase
The differential equation for fluid and vapor flow under non-
isothermal conditions is similar to the water flow equation
for one-dimensional consolidation with the addition of one
term. The additional term accounts for the water vapor flow
due to diffusion and advection processes (Wilson, 1990).
Water vapor flow can be relevant under nonisothermal con-
ditions. The one-dimensional fluid and vapor flow equation
 
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