The conceptual model for
volatilization from the vadose zone source zone is as follows. Only contaminants
with modified (dimensionless) Henry's Law constants 10-7 are
assumed to volatilize.(a) There is no contaminant
gas generation (e.g., caused by microbial metabolic processes) in the source
zone, as there sometimes is in landfills. There may initially be a layer
of uncontaminated soil above the contaminant source zone. If this clean
layer exists initially, it is assumed to be relatively porous (the same
as the rest of the soil in the source zone); i.e., it is not a compacted
cap layer that has gaseous permeability or vapor diffusion properties that
are significantly lower than those within the contaminated zone. The source
zone is assumed to be a so-called "well-mixed reactor," which means that
the vapor concentration of any contaminant in the air-filled pores is spatially
uniform throughout the source zone. Hence, no vapor diffusion takes place
within the source zone. Contaminant vapor is assumed to diffuse through
the clean layer from the uppermost edge of the source zone toward the soil
surface (and then be lost as volatilization into the atmosphere). In addition,
note that even though the contaminant also resides in the aqueous phase
in the source zone, it is assumed that upward diffusive transport in the
water phase is negligible compared to the vapor transport. Furthermore,
it is assumed that contaminant vapor diffuses through the clean layer quickly
enough that contaminant within the source zone cannot redistribute itself
fast enough to maintain the source zone at its current dimensions (with
the overall total concentration falling over time uniformly at all locations
due to the volatilization process). Rather, the conceptual model assumes
that contaminants are preferentially stripped away from the uppermost end
of the source zone to satisfy the volatilization flux. This means that
the thickness of the source zone decreases over time as the uppermost boundary
recedes to deeper depths (and the clean layer gets thicker). The vapor
diffusion process is assumed to be quasi-steady-state, which means that
the flux is always described by a steady-state diffusion expression, even
though the thickness of the diffusion zone is changing over time and the
contaminant's vapor concentration in the source zone may change over time,
as the top of the source zone recedes. This conceptual model (which predicts
recession of an infinitely steep contamination front) is an idealization
of the actual diffusive stripping process (which would predict recession
of a sigmoidally shaped front).
If a clean layer of soil
does not initially exist above the contaminated source zone, the stripping
away of contaminants from the top of the source zone (as described above)
quickly creates one. When volatilization is the only loss process occurring,
the thickness of this clean layer continues to grow over time, once it
has been formed.
The theory implemented
in the source-term release module to estimate the mass flux to volatilization
is based on the derivations of Thibodeaux and Hwang (1982). Their conceptual
model for the source zone is similar to that described above, except that
in their model a clean layer always exists initially, the source zone always
contains a NAPL phase, and the contaminant mass partitioned into the aqueous
and solid-sorbent phases is negligible. Their previous work had only considered
a pure NAPL phase (i.e., a NAPL phase composed of only one constituent).
Thibodeaux and Hwang (1982) extend their previous theory to apply to contaminants
that are present as dilute constituents of a NAPL phase mixture.
With the conceptual model
of Thibodeaux and Hwang (1982), the mass flux equation for volatilization
from a source zone through a clean layer is given by
where
Dvi is the effective vapor diffusion coefficient
of contaminant i in soil (cm2 yr-1)
z is the distance from the initial
position of the soil surface to the top of the source zone at time t (cm).
With volatilization from the
upper boundary of the source zone being the only loss route, the overall
total concentration of a contaminant in the source zone is always equal
to its initial concentration. Using this fact, the rate of reduction of
contaminant mass in the source zone due to recession of the top boundary
is given by
Based on mass balance considerations for any contaminant individually,
the volatilization mass loss rate must equal the rate of reduction of contaminant
mass in the source zone due to recession of the top boundary. Equating
the right-hand sides of Equations 5.73 and 5.74, and rearranging the resulting
equation, gives the recession rate of the top of the source to be
Assuming that the contaminant's vapor concentration in the source zone
also remains constant in time, Equation 5.75 can be rearranged and integrated
to obtain an analytical expression for the clean layer thickness, z, as
a function of time. When this expression is substituted into Equation 5.73,
the mass flux equation for volatilization becomes
There are, however, a number of reasons why the theory of Thibodeaux
and Hwang (1982) is not adequate for our purposes. Firstly, their theory
does not explicitly describe how to predict loss flux from source zones
that do not contain a NAPL phase (or that contain one initially that disappears
during the course of the simulation). Secondly, their theory is not applicable
to a contaminant when it is an intermediate or primary constituent of the
NAPL phase. In the limit of dilute NAPL-phase concentration for the contaminants
under study, Thibodeaux and Hwang (1982) appropriately used a type of Henry's
Law to represent the contaminant vapor concentration in Equations 5.73
through 5.76. But they did not specify how to do the calculations when
Henry's Law was not valid. Thirdly, because this theory is merely a simple
modification of their theory for a pure NAPL phase, it predicts the rate
of increase of clean layer thickness based on the removal of only the contaminant
being analyzed at that time. Hence, if multiple contaminants in a NAPL
phase are to be analyzed, their theory would only be internally self-consistent
if all other constituents of the NAPL phase are volatilizing at a rate
that would cause the clean layer thickness to increase at the same rate.
This clearly would not be true in general. Fourthly, because their equations
for a given contaminant assume that the contaminant concentration in the
NAPL phase is constant in time, their theory would only be internally self-consistent
if other constituents of the NAPL phase are being removed at rates that
keep the relative composition of the remaining NAPL constant in time. This
clearly also would not be true in general.
Therefore, when volatilization is the only mass loss process,
a more appropriate way for the source-term release module to calculate
the mass flux is as follows. In general, the mass flux equation for volatilization
from a source zone through a clean layer is still given by Equation 5.73.
However, when mass balance arguments are made as in Thibodeaux and Hwang
(1982) (i.e., for each contaminant individually), a separate source zone
boundary recession rate is calculated for each contaminant. If the module
actually used all of these individual recession rates, what this would
mean in terms of predicted behavior in the overall problem scenario is
that a vertical spatial gradient in NAPL-phase composition would immediately
develop as more highly volatile compounds are preferentially removed from
the upper portions of the source zone. While this may be physically reasonable,
it is inconsistent with our assumption that the source zone is a "well-mixed
reactor." Because the NAPL phase partitioning theory assumes that all NAPL
components are in a zone of the same size, as soon as a spatial gradient
in NAPL-phase composition developed (because of the recession rate theory),
the module would be calculating NAPL-phase mole fractions that are inconsistent
with this conceptualization (and errors caused by the inconsistency would
be compounded over time).
If a contaminant is not a component of the NAPL phase, its phase
partitioning behavior does not depend on any components of the NAPL phase,
or any other contaminants in general. Furthermore, all contaminants that
cannot be part of a NAPL phase (i.e., contaminants with KHi<
10-7) are assumed to be nonvolatile (i.e., are assigned values
of Dvi that are identically equal to zero); which means that
there is no change in position of the top boundary of the source zone due
to volatilization when calculated individually for these contaminants.
Many of these contaminants can decay/degrade into other contaminants, but
none of these other contaminants are contaminants that may be a component
of the NAPL phase. In addition, no contaminant that may be a component
of the NAPL phase can decay/degrade into any of the non-NAPL phase contaminants.
All of this means that the overall source zone can be conceptualized as
two separate problems: one that consists of all of the non-NAPL contaminants
and one that consists of all the NAPL components. The problem that consists
of all non-NAPL contaminants will have a single, "well-mixed reactor" source
zone over time by default. For the problem that consists of all NAPL-phase
components, the source-term release module uses the individual recession
rates calculated for each component of the NAPL phase to obtain a single
overall rate of recession of the upper source zone boundary for these contaminants.
To do this, first consider that the volume of the
source zone at any time is given by Equation 5.69, and the thickness of
the source zone at any time is given by
Using Equations 5.69 and 5.77, the rate of reduction of contaminant
mass in the source zone due to recession of the top boundary is given by
(Note that Equation 5.78 is analogous to Equation 5.74.) Again, equating
the right-hand sides of Equations 5.73 and 5.78, and rearranging the resulting
equation, gives the recession rate of the top of the source to be
Because the vapor concentration in Equation 5.79 is not a simple function
of the contaminant mass when a NAPL phase is present, and because the contaminant
mass is not constant in time, Equation 5.79 cannot be rearranged and integrated
to obtain a general analytical expression for the clean layer thickness,
z, as a function of time. Hence, even when volatilization is the only loss
process occurring, the source-term release module must solve a mathematical
problem (for each contaminant) that consists of simultaneously solving
Equations 5.73, and 5.79.
For contaminants that are components of the NAPL phase, the module
then calculates a single updated position of the top boundary of the source
zone by taking an average of the individual updated positions predicted
by the solution of Equation 5.79, weighted by the mole fraction of the
contaminant:
(In Equation 5.80, the subscript i on
the variable z denotes that this is the updated position of z as calculated
for an individual contaminant.) With this approach, contaminants that are
present in the NAPL phase in higher proportions will have a greater influence
on how much the top boundary of the source zone recedes.
When a NAPL phase exists in the source zone, the vapor concentration
of the contaminant is controlled by the composition of the NAPL phase.
In this case, the source-term release module calculates the value of Cvi
in Equations 5.73 and 5.79 by the phase partitioning theory described in
Section 2.2.4 (which uses different methods, depending on the mole fraction
[or concentration] of the contaminant in the NAPL phase).
When no NAPL phase exists in the source zone, the vapor concentration
can be calculated by a simple partitioning relation (i.e., Equation 2.8
in Section 2.2.3). If the volume of the source zone is given by Equation
5.69, and the thickness, h, of the source zone is given by Equation 5.77,
these equations can be substituted into Equation 2.8, and the resulting
expression can be substituted into Equation 5.73. Therefore, in this case
(when volatilization is the only loss process occurring and no NAPL phase
exists), the mass flux equation for contaminant loss from the source zone
by volatilization is given by
Similarly, the equation giving the rate of recession of the top boundary
of the source zone (when no NAPL phase exists) is given by
The above derivations relate to the case where a clean layer always
exists above the contaminated source. When the soil is initially contaminated
all the way up to the soil surface (i.e, when zt
= 0) Equation 5.73 or 5.81 would predict an infinite volatilization mass
flux initially because the thickness of the diffusion zone, z, would be
zero. This is, of course, not what would happen in the real-world situation.
In reality, a concentration gradient immediately develops at the soil surface,
and volatilization loss is governed by a transient-state diffusion process.
Contaminant mass is stripped away from the surface and a "relatively clean"
layer begins to develop at the surface (i.e., there is a region near the
surface, above the sigmoidally-shaped contaminant front, where the contaminant
concentration is very close to zero). Eventually, when this region becomes
thick enough, the volatilization behavior becomes more like that predicted
by the idealized conceptual model used in the module. So, when no clean
layer exists initially, the module uses an alternate expression for the
volatilization flux (instead of Equation 5.73 or 5.81) to calculate a bounding
value for this flux at the initial time step. Furthermore, even when a
clean layer exists (either initially, or when one forms at subsequent time
steps), the module uses this same alternate expression to calculate a bounding
value for the flux at time steps when Equation 5.73 or 5.81 would predict
finite, but unreasonably high, values of flux (because z is nonzero but
still very small). This alternate expression is derived from the transient-state
diffusion model described above. This model, called the "Contaminated Soil
Gradient Model," was one of the volatilization scenarios that users could
select for the vadose zone source zone in previous versions of MEPAS (see
Droppo and Buck 1996). The mass flux equation for contaminant loss by volatilization
for this model is given by
where Ctio is the value of overall total concentration of
contaminant i used at the beginning of a time step in the bounding
calculation for volatization (g cm-3 or Ci cm-3).
Equation 5.83 gives the
instantaneous volatilization loss at time t since the initial incorporation
of contaminant into the soil at a spatially uniform overall total concentration
of CTio. Note that because of the infinite concentration gradient
at the soil surface at t = 0, Equation 5.83 also predicts an infinite flux
at t = 0. However, this infinite flux would only last for an infinitesimally
small time. Therefore, the "Contaminated Soil Gradient Model" actually
calculates an average value of the flux for a given time period, which
we will take to be the length of our time step. The mass flux equation
for average contaminant loss (over a time period t) by volatilization for
this model is given by
The value calculated by Equation 5.84 is considered an upper bounding
value because we realize that the volatilization rate should not be greater
than what would be predicted by the transient-state diffusion problem at
initial times, and because of the way we calculate CTio,
and the fact that we use t. Equations 5.83 and 5.84 were developed under
the assumption that no NAPL phase exists in the source zone. Implicit in
these equations is the idea that the contaminant's vapor concentration
can be calculated by simple partitioning theory (i.e., Equation 2.8). When
Equation 5.84 is used to calculate the values for mass loss at a particular
time step, CTio is calculated by
This means that when a NAPL phase is present, the implicit vapor concentration
in Equation 5.84 is greater than what it would really be because all of
the contaminant mass is considered as part of the aqueous, sorbed, and
vapor phases only. So the flux value calculated by Equation 5.84 is biased
to be an upper bound. Furthermore, by using Equation 5.84 for any time
step, the module is in effect calculating a value that is commensurate
with the idea that the contaminant's mass at the beginning of that time
step has been redistributed all the way up to the soil surface again. Again,
this produces a value biased to be an upper bound compared to using the
actual value of t.
Substituting Equation 5.85 into Equation 5.84, the volatilization
loss of a contaminant, as calculated by the bounding value, is given by
Equating the right-hand sides of Equations 5.78 and 5.86 according
to mass balance arguments, the rate of recession of the top boundary of
the source zone based on an individual contaminant (when a bounding value
for volatilization is used) is given by
Equation 5.87 is solved to get an updated value of z for any contaminant
where a bounding value is used for volatilization loss. These values of
zi for NAPL-phase components are then used by the module in conjunction with updated values
of z for other NAPL-phase components (obtained by solving either Equation
5.79 or 5.82) to calculate the single rate of recession of the top boundary
of the source zone for NAPL components by using Equation 5.80.
To summarize, for each contaminant, the mass flux equation for
contaminant loss from the source zone by volatilization is given by Equation
5.73 when a NAPL phase exists in the source zone, and Equation 5.81 when
no NAPL phase exists in the source zone, unless this value is greater than
the bounding value calculated by Equation 5.86. In this event, the mass
flux equation is given by Equation 5.86. In addition, when bounding values
are not used for volatilization fluxes, the rate of change in position
of the upper boundary of the source zone (which will recede due to volatilization
losses), based on an individual contaminant, is given by Equation 5.79
when a NAPL phase exists in the source zone, and Equation 5.82 when no
NAPL phase exists in the source zone. If a bounding value is used for the
volatilization flux, the rate of change in position of the upper boundary
of the source zone, based on an individual contaminant, is given by Equation
5.87. For contaminants that are components of a NAPL phase, Equation 5.80
is then used to calculate a single updated position of the top boundary
of the source zone for all of the NAPL-phase components.
(a)
This screening is accomplished by assigning
a value of zero to the effective vapor diffusion coefficients of all contaminants
with KHi < 10-7
