Environmental Engineering Reference
In-Depth Information
to calculate the changes in water volume. The water-phase
volumetric constitutive relationship takes the form:
with problems where mass is lost from the system. Conse-
quently, the overall constitutive relationship will take the form
V
w
V
t
V
t
V
t
m
w
(σ
m
w
(u
a
−
m
t
(σ
m
t
(u
a
−
=
−
u
a
)
+
u
w
)
(2.112)
m
d
M
s
(2.115)
The
m
d
term in Eq. 2.115 is a coefficient that designates
the volume change associated with a particular amount of
mass loss. The term
M
s
designates a loss of solid mass.
In other words, a change in the mass of solids will produce
a certain amount of volume change. If the term
m
d
is zero,
it means that there is no volume change even when there is
a change in the mass of solids. It is necessary to consider
the term
m
d
as an additional soil parameter that must be
estimated, measured, or determined in some way.
The loss of solid mass appears in two places in the formu-
lation: on the left-hand side of the volumetric requirement
equation and in the last term on the right of the equal sign
in Eq. 2.115.
=
−
u
a
)
+
u
w
)
+
where:
σ
=
total stress,
u
a
=
pore-air pressure,
u
w
=
pore-water pressure,
m
w
=
coefficient of water volume change associated with
the net normal stress state variable, and
m
w
=
coefficient of water volume change associated with
the matric suction.
The total stress state may also be more complex than what
has been suggested. The assumption can generally be made
that changes in the total stress do not significantly change the
volume of water in the soil for heap leach and MSW problems.
Under this assumption, Eq. 2.112 reverts to the equation for
the SWCC written in terms of volumetric water content:
V
w
V
t
2.6.5 Model to Predict Removal of Solid Mass
The prediction of solids lost from the system involves an
independent physical process. There needs to be another
partial differential equation (or some other form of equation)
that is solved independent of the other partial differential
equations. The partial differential equation must describe the
chemical reactions taking place and in the process predict the
solid mass lost from the system with time. It is the change
in the solids volume which in turn produces a change in the
overall volume in the REV:
m
w
(u
a
−
=
u
w
)
=
θ
(2.113)
2.6.4.2 Air Flow Processes
The air flow PDE can also be derived in a similar manner
to the water flow PDE; however, air is also compressible
and involves additional terms. The air phase constitutive
relationship is used in deriving the air phase PDE.
dM
s
ρ
w
G
s
dV
s
=
(2.116)
2.6.4.3 Overall Volume Change
There must be a constitutive relationship that relates overall
volume change to the stress state variables on the left-hand
side of Eq. 2.111. The constitutive relationship needs to
change when dealing with heap leach or municipal solid
waste problems. In other words, there will be an additional
volume change associated with the loss of mass (or solids)
from the system. The regular constitutive relationship for an
unsaturated soil takes the form
V
t
V
t
where:
dM
s
=
change in mass of solids that is predicted using an
independent physical and chemical model and
dV
s
=
corresponding change in volume of solids.
There is not a one-to-one correspondence between the
change in volume of solids and the change in overall vol-
ume. There must be a constitutive relationship to relate these
two volume changes:
m
t
(σ
m
t
(u
a
−
=
−
u
a
)
+
u
w
)
(2.114)
dV
ts
V
t
m
ts
dV
s
V
t
=
(2.117)
where:
where:
dV
ts
=
m
t
=
coefficient of overall volume change associated
with the net normal stress state variable and
total or overall volume change produced by a
change in volume of solids lost from the system
and
m
t
=
coefficient of overall volume change associated
with the matric suction.
m
ts
=
soil property linking a change in volume of solids
to an overall volume change.
Overall volume change can be computed using a stress
state-strain analysis (e.g., incremental elasticity model).
There is an additional term that needs to be added to the
overall volume change constitutive relationship when dealing
1
.
0, the overall volume change bears a
one-to-one correspondence to the volume of solids lost. If
If
m
ts
=
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