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OCR for page 28
2
Modeling of Processes
INTRODUCTION
This chapter describes what models are and how they work.
It begins by explaining the processes that control ground water
flow and contaminant transport. To understand models, it is neces
sary to describe these processes by using certain mathematical equa
tions that quantitatively describe flow and transport. The mathe
matical aspects of modeling are critical. The precise language of
mathematics provides one of the best ways to integrate and express
knowledge about natural processes. By developing an awareness of
the natural processes, the mathematics should be understandable.
Also, where the process is not well understood, this awareness pro
vides an appreciation of the limits of the mathematics. Methods of
solving the mathematical expressions are presented at the end of the
chapter.
Subsurface movement whether of water, contaminants, or
heat is affected by various processes. These processes can be related
to three different modeling problems: ground water flow, multiphase
flow (e.g., soil, water, and air; water and gasoline; or water and a
dense nonaqueousphase liquid (MAPLE, and the flow of contami
nants dissolved in ground water.
28
OCR for page 28
MODELING OF PROCESSES
29
Ground Water Flow
Of these three problems, ground water flow is the simplest to
characterize and understand In most cases, models need to consider
only two ground water flow processes: flow in response to hydraulic
potential gradients, and the loss or gain of water from sinks or
sources, recharge, or pumping from wells. Hydraulic potential gradi
ents simply represent the difference in energy levels of water and are
generated because precipitation that is added to a ground water sys
tem at high elevations has more potential energy or hydraulic head
than water added at a lower elevation (Figure 2.~. The result of
these potential differences is that water moves from areas of high po
tential to areas of lower potential. As rainfall or other recharge keeps
supplying water to the flow system, ground water continues to flow.
On a cross section, it is possible to represent the spatial variability
in hydraulic potential existing along a flow system by using what are
called equipotential lines (see Figure 2.1~. The equipotential lines are
contours of hydraulic potential within some area of interest. In some
simple situations, the direction of ground water flow is perpendicular
to these equipotential lines, as shown in Figure 2.~.
The actual distribution of hydraulic head observed for an area
depends mainly on two factors, how much and where water is added
and removed, and the hydraulic conductivity distribution that exists
in the subsurface. Consider a few examples. Figure 2.2 illustrates
the hydraulic head distribution for two different water table config
urations. The water table effectively represents the top boundary
of the saturated ground water system, and its configuration reflects
different recharge conditions. In both cases, the bottom and sides
of the section are considered to be impermeable (no flow). With a
linear water table and recharge mainly at the right end of the sys
tem, a relatively smooth regional flow system develops (see Figure
2.2a). The second water table, representing significant local areas
of recharge and discharge at three locations, shows a much different
flow pattern (see Figure 2.2b). Instead of a broad regional trend,
several small, local flow systems have developed.
Ground water flow patterns also depend on the hydraulic con
ductivity distribution. Figure 2.3 compares the pattern of ground
water flow along a cross section where all properties except the hy
draulic conductivity for each layers are kept constant. Each of the
two layers shown is defined in terms of a hydraulic conductivity in
the horizontal direction (Kh) and in the vertical direction (Kit), with
the ratio Kh/KV describing the degree of directional dependence in
OCR for page 28
30
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OCR for page 28
MODELING OF PROCESSES
0.2S
co
m
> 0.2S
up
a:
31
i ' ' 1
Equipotential Lines
a
~ Ground Surtace , Water Table
O _
OK = 1i ~
1 ~'\
\~ W~/
_  ~I _ _ 1 ~1, I I K = 10 1 1 ~
, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, . . \.
S
~I ~ l ~I l I I I I I I I I ~i ~I
b
o
O S
RELATIVE BASIN LENGTH
FIGURE 2.2 Dependence of the pattern of ground water flow on the recharge
rate, as reflected by the configuration of the water table. All other parameters
are the same in the two sections (from Freeze, 1969b).
0.2S
~ O
id
0.2S
UJ
o
a
o
~I ~ ~ ~
Equipotential Lines
C~rolint] Pilirf~r! WAter Ts~hl
I ~11  I K = lo ~I j 11 ~71
S
1 ~:
o
RELATIVE BASIN LENGTH
S
FIGURE 2.3 Dependence of the pattern of ground water flow on the hydraulic
conductivity distribution. The only difference in the two diagrams is the pattern
of geologic layering defined in terms of the relative hydraulic conductivities
shown (from Freeze, 1969b).
OCR for page 28
32
GROUND WATER MODELS
hydraulic conductivity. Examination of the two flow patterns shows
how changes in the hydraulic conductivity distribution can change
the character of ground water flow.
Adding or removing water also can have a significant impact
on the pattern of flow. The most important sources and sinks in
a ground water flow system are pumping or injection wells (i.e.,
point sources/~nks). These are considered internal flows of water
(fluxes). Other possibilities such as recharge or evaporation are most
often considered as boundary fluxes. Pumping lowers the hydraulic
potential at the well and in its immediate vicinity, creating what is
known as a cone of depression. The result of decreasing hydraulic
potential toward the well is the flow of water to the well. Injection
does the opposite and results in flow away from a well.
So far, only steadystate flow, or flow that does not change as a
function of tune, has been discussed. Often, however, flow systems
are transient, which means that hydraulic heads change with time,
leading to variations in flow rates. For example, water leveh decline
when a pumping well is first turned on, providing an early transient
response. In many instances when sources of recharge are available,
water levels will eventually stabilize, providing a new equilibrium or
steadystate flow system. The most important feature of a transient
flow system is the ability of water to be removed from or added
to storage in individual layers. The parameter describing the water
storage capabilities of a geologic unit is called the Unspecific storage."
For transient flow problems, its value contributes to determining the
distribution of hydraulic head at a given tune. Note that the smaller
the specific storage, the faster the ground water system will seek a
new equilibrium. Readers wishing a more detailed explanation of
this parameter and aspects of ground water flow should see Freeze
and Cherry (1979~.
M~tiphase Flow
Multiphase flow occurs when fluids other than water are mov
ing ~ the subsurface. These other fluids can include gases found
in the soil zone or certain organic solvents that do not appreciably
dissolve In water (i.e., immiscible liquids). Examples of fluids that
are immiscible with water include many different manufactured or
ganic chemical such as the cleaning solvent trichioroethylene and
preservatives such as creosote. Petroleum products such as crude oil,
heating oil, gasoline, or jet fuel are also examples.
OCR for page 28
MODELING OF PROCESSES
33
The process causing all of these phases to flow ~ again movement
in response to a potential gradient. Now, however, the situation is
more complicated because the potential causing each fluid to move
is not necessarily the same as that for water. Thus each fluid can
be moving in a different direction and at a different rate. Another
complexity Is that many characteristic parameters are no longer
constant when several fluids are present together and competing for
the same pore space. For example, the relative permeability of a
geologic unit to a particular fluid like water will be small if the
proportion of water present in a given volume of porous medium is
small and will tend to increase as the amount of water increases.
As discussed previously for water, a fluid's potential also depends
on any sources or sinks that add or remove fluid. The same idea ap
plies to multifluid systems, except that now the number of processes
increases because the effects of pumping/injection and evaporation
(volatilization) affect each of the fluids present and, in addition, there
can be transfers of mass between fluids. An example of this latter
mechanism is that some portion of a gasoline spin might dissolve in
water.
To illustrate these concepts about the theory of multiphase sys
tems, consider two problems of particular interest to this reportthe
flow of water in the unsaturated zone and the migration of organic
contaminants that are either more or less dense than water. When
studying the problem of water movement in the presence of soil gas
in the unsaturated zone, it is sometimes assumed that only the water
moves. The only effect of the gas on water movement is the variabil
ity in the parameters caused by the presence of several fluids. For
example, hydraulic conductivity varies as a function of the quantity
of water in the pores.
Figure 2.4 shows a relationship between hydraulic conductivity
(K) and pressure head (fib). According to Freeze and Cherry (1979),
pressure head is one component of the total energy water possesses
at a point. Several features should be noted. As the pressure head
becomes smaller (more negative), the soil becomes drier and the hy
draulic conductivity decreases. Much less water wall move through
a dry soil than through a wet soil. Another feature ~ that if the
soil is drying out there is one ¢K relationship and if it is wet
ting there is another. Further, repeated wetting and drying cause
the relationship to be defined by the scanning curves that join the
wetting and drying curves at intermediate points (Figure 2.43. In
OCR for page 28
34
GROUND WATER MODELS
0.03
c)
I
> 0.02
it
8 0.01
Cot o
I 300 200 100 0 100 200 300
Unsaturated
Saturated
Drying ~;
. I I
~ K o
Scanning Curves
Wetting
PRESSURE HEAD, ~ (cm H2O)
FIGURE 2.4 Example of the relationships between pressure head and hydraulic
conductivity for an unsaturated soil (modified from Freeze, 1971a).
most multifluid systems, hydraulic conductivity and other parame
ters commonly exhibit this kind of "hysteretic" behavior, and yet for
many applications, these types of sitespecific data are not available.
The progress of a wetting front moving into a dry soil can be
described in terms of either potentials or volumetric water content,
0.14

t
_ ~
MOISTURE CONTENT
FIGURE 2.5 The distribution of water in the unsaturated zone can be de
scribed in terms of pressure head and moisture content. Results presented for a
combined saturated and unsaturated flow system illustrate how pressure head
in particular is continuous across the water table (from Freeze, 1971b).
OCR for page 28
MODELING OF PROCESSES
35
defined as the ratio of the volume of water in the voids to the total
volume of voids. Figure 2.5 illustrates how both are used to define a
wetting zone near the top of the ground. The water table is clearly
illustrated by the zero pressure contour and the total porosity contour
(complete saturation). Given that moisture contents are easier to
measure than potentials, the former are used more frequently to
describe real systems.
A more complicated case to consider is a flow involving an im
miscible fluid and water in the subsurface. Eventually, a distinction
has to be made between a fluid that is less dense than water and
one that is more dense. However, where an organic liquid is spilled
on the ground surface, both fluids will move much the same way
through the unsaturated zone (Figure 2.6a and b). The free organic
liquid in a homogeneous medium moves vertically downward, leaving
a residual trail of organic contaminants. Each pore through which
the free organic liquid moves retains some of the contaminant (resicl
ual saturation) in a relatively immobile state. Thus, if the volume
of spilled liquid ~ small and the unsaturated zone is relatively thick,
no free liquid may reach the water table. Of course, free liquid may
reach the water table over extended periods of time, and dissolved
organic liquid may be conducted by water flow.
It is when the free liquids begin to approach the top of the
capillary fringe above the water table that the differences in density
begin to affect transport. The capillary fringe is a zone above the
water table where the pores are completely saturated with water but
the pressure heads are less than atmospheric (Freeze and Cherry,
1979~. A contaminant that is lighter than water will mound and
spread, following the dip of the water table (Figure 2.6a). A fluid
that is heavier than water will spread slightly and keep moving
downward. This fluid will ultimately mound on the bottom of the
aquifer or on a lowpermeability bed within the aquifer and move in
whatever direction the unit is dipping (Figure 2.6b). Thus water and
the organic liquid need not move in the same direction.
To understand the details of multicomponent flow, it is essential
to study the concepts of wettability, imbibition and drainage, and
relative permeability. A discussion of these topics is, however, beyond
the scope of this report. Readers can refer to Bear (1972) and
Greenkorn (1983) for en overview of the basic theory. The key point
to remember is that, as in the case of water, the permeability of the
material through which these fluids are moving plays a major role in
controlling the direction and rate of flow. In the case of multiphase
OCR for page 28
36
GROUND WATER MODELS
a
Ground Surface
. _ _
Oil Phase
Oil Components
Unsaturated Zone
Capillary Fringe
Dissolved in Water~ __ Saturated
Zone
~/////'~0~:
b
Ground Surface
_v _
~ ~
, ~ ~ ~
1: .~#~. 1
t..~ :!
~ CHC Phase
Unsaturated Zone
Capillary Fringe
K2 K1
CHC Dissolved
in Water
Saturated
_ _ _ _ _ _ _ _
' Zone
l
_ I_
~////////////////////////////////////////i/////~,
FIGURE 2.6 The flow of a nonaqueousphase liquid that is (a) less dense
than water (oil) and (b) more dense than water (chlorohydrocarbon, CHC) in
the unsaturated and saturated zones. In both cases the contaminants are also
transported as dissolved compounds in the ground water (from Schwille, 1984~.
systems, a relative permeability is defined for each fluid with values
ranging between zero and one as the relative abundance of each fluid
(i.e., saturation) changes. The key point here is that as more fluids
are introduced to the pore space, more of the pore space is devoted
to the relatively immobile state of each fluid and therefore less pore
space is devoted to liquid flow.
OCR for page 28
MODELING OF PROCESSES
37
The distribution of an NAP L in the subsurface is described quan
titatively in terms of the relative saturation of the NAPL, which is
given by the ratio of the volume of the NAPL to the total pore vol
ume. In other words, it describes what proportion of the pore volume
is filled with the NAPL. A relative saturation can be defined for each
one of the organic liquids and water. This kind of description is
generally not used in field settings because of the detailed study that
is necessary. Instead, presence/absence indications are used, as illus
trated in Figure 2.6. The results of computer simulations normally
characterize relative fluid saturations. Expressing the distribution of
fluids in terms of relative saturation is analogous to expressing the
moisture content in terms of unsaturated flow.
Dissolved Contanunant Passport
One of the reasons why problems involving dissolved contam
inants are so difficult to model is the number and complexity of
controlling processes. The processes can be divided into two groups:
(1) those responsible for material fluxes and (2) sources or sinks for
the material. For the problem of contaminant migration these are the
mass transport and mass transfer processes, respectively (Table 2.1~.
A brief discussion of each of the processes listed in Table 2.1 follows,
with a general assessment of its impact on contaminant transport.
Advection
Advection is the primary process responsible for contaminant
migration in the subsurface. Mass is transported simply because the
ground water in which it is dissolved is moving in a flow system.
In most cases, it can be assumed that dissolved mass is transported
in the same direction and with the same velocity as the ground
water itself. For example, given the conditions of flow described by
the equipotential lines and flowlines of Figure 2.7a, it is a simple
matter to define the plume of dissolved contaminants in terms of the
streamtubes that pass through the source. A streamtube is defined an
the area between two adjacent Bowlines. When Bowlines are equally
spaced, the discharge of water through each is the same (Freeze and
Cherry, 1979~. This simple approach assumes that the density of the
contaminated fluid is about the same as that of the ground water.
The mean velocity of contaminant migration can also be assumed to
be the same as the mean ground water velocity (or seepage velocity).
OCR for page 28
38
tTROUND WATER MODELS
TABLE 2.1 A Summary of the Processes Important in Dissolved Contaminant
Transport and Their Impact on Contaminant Spreading
Process Definition Impact on Transport
Mass transport
1. Advection Movement of mass as a Most important way of
consequence of ground transporting mass away
water flow. from source.
2. Diffusion Mass spreading due to An attenuation mechanism
molecular diffusion in of second order in most
response to concentration flow systems where
gradients. advection and dispersion
dominate.
3. Dispersion Fluid mixing due to effects An attenuation mechanism
of unresolved hetero that reduces contaminant
geneities in the per concentration in the
meability distribution. plume. However, it
spreads to a greater extent
than predicted by
advection alone.
Chemical mass transfer
4. Radioactive decay
5. Sorption
6. Dissolution/
. . .
precipitation
Irreversible decline in the
activity of a radionuclide
through a nuclear
reaction.
Partitioning of a contaminant
between the ground water
and mineral or organic
solids in the aquifer.
The process of adding
contaminants to, or
removing them from,
solution by reactions
dissolving or creating
various solids.
7. Acid/base Reactions involving a
reactions transfer of protons (H+).
An important mechanism for
contaminant attenuation
when the halflife for
decay is comparable to or
less than the residence
time of the flow system.
Also adds complexity in
production of daughter
products.
An important mechanism
that reduces the rate at
which the contaminants
are apparently moving.
Makes it more difficult to
remove contamination at a
site.
Contaminant precipitation is
an important attenuation
mechanism that can
control the concentration
of contaminant in
solution. Solution
concentration is mainly
controlled either at the
source or at a reaction
front.
Mainly an indirect control
on contaminant transport
by controlling the pH of
ground water.
OCR for page 28
68
G
IIJ
m
z
~ 10
o
a:
1d
GROUND WATER MODELS
, Constant Head Nodes
it.
.
. ~Kh= ~=00
Kh = v = 105 cm/s
Kh = 107 cm/s; Kv = 10~ cm/s
Kh = It = 10 3 cm/s
COLUMN NUMBERS
FIGURE 2.13 Model system from the previous figure subdivided by a rect
angular grid system. Nodes are defined in the center of each grid cell. By
assigning Kh = Kv = 0.0 in the area at the top left, it is effectively excluded
from the calculation.
2.7. Included are a brief description of the methods and a few key
references that can be used to obtain more detailed information.
The last topic that needs to be addressed in this section involves
the mathematical techniques for solving matrix equations. In most
models, the major computational effort comes in solving the system
of mode! equations. In general, there are two basic methods. In one
approach the entire system of equations is solved simultaneously with
direct methods, providing a solution that is exact, except for machine
roundoff error. In the second approach, iterative methods obtain a
solution by a process of successive approximation, which involves
making an initial guess at the matrix solution and then improving
this guess by some iterative process until an error criterion is satisfied.
Direct methods have two main disadvantages. The first is that
a computer may not be able to store the large matrices or solve
the system in a reasonable time when the number of nodes is large.
Sometimes this problem can be dealt with to some extent by us
ing sparse matrix solvers and various nodenumbering schemes. The
second problem with direct methods is the roundoff errors. Be
cause many arithmetic operations are performed, roundoff errors
can accumulate and significantly influence results for certain types
of matrices.
Iterative schemes avoid the need for storing large matrices. This
OCR for page 28
69
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OCR for page 28
70
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71
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OCR for page 28
72
GROUND WATER MODELS
feature makes them attractive for solving problems with many un
knowns. Numerous schemes have been developed; a few of the
more commonly used ones include successive overrelaxation methods
(Varga, 1962), the alternatingdirection implicit procedure (Douglas
and Rachford, 1956), the iterative alternatingdirection implicit pro
cedure (Wachpress and Habetler, 1960), and the strongly implicit
procedure (Stone, 1968~. Because operations are performed many
times, iterative methods also suffer from potential roundoff errors.
The efficiency of iterative methods depends on an initial estimate
of the solution. This makes the iterative approach less desirable
for solving steadystate problems (Narasimhan et al., 1978~. To
speed up the iterative process, relaxation and acceleration factors are
used. Unfortunately, the definition of best values for these factors
commonly is problem dependent. In addition, iterative approaches
require that an error tolerance or convergence criterion be specified
to stop the iterative process. This, too, may be problem dependent.
All of these parameters must be specified by the mode} user.
According to Narasimhan et al. (1977) and Neuman and Nara
simhan (1977), perhaps the greatest limitation of the iterative
schemes is the requirement that the matrix be well conditioned.
An illconditioned matrix can drastically affect the rate of conver
gence or even prevent convergence. An example of an illconditioned
matrix is one in which the main diagonal terms are much smaller
than other terms in the matrix.
More recently, a semiiterative method has gained popularity
(Gresho, 1986~. This method, or class of methods known as conju
gate gradient methods, was first described by Hestenes and Stiefel
(1952~. It is widely used to solve linear algebraic equations where
the coefficient matrix is sparse and square (Concus et al., 1976~.
One advantage of the conjugate gradient method is that it does not
require the use or specification of iteration parameters, thereby elim
inating this partly subjective procedure (Manteuffe} et al., 1983~.
Kuiper (1987) compared the efficiency of 17 different iterative meth
ods for the solution of the nonlinear threedimensional ground water
flow equation. He concluded that, in general, the conjugate gradient
methods did the best.
Numerical methods by their very nature, yield approximate so
lutions to the governing partial differential equations. The accuracy
of the solution can be significantly affected by the choice of numerical
parameters, such as the size of the spatial discretization grid and the
length of time steps. Those using ground water models and those
OCR for page 28
MODELING OF PROCESSES
73
making management decisions based on mode! results should always
be aware that tradeoffs between accuracy and cost will always have
to be made. If the grid size or time steps are too coarse for a given
problem, it is possible to generate a numerical solution that converges
on an answer that has an excellent mass balance but is still inaccu
rate. furthermore, if iteration parameters are not properly specified,
the solution may not converge. It is hoped that this will show up as a
mass balance error, which will be noted by the user. This indicates,
however, the importance of a mass balance in numerical models.
REFERENCES
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eling of porous media contamination by organic compounds, 1. Equation
development. Water Resources Research 21~1), 1118.
Abriola, L. M., and G. F. Pinder. 1985b. A multiphase approach to the modeling
of porous media contamination by organic compounds, 2. Numerical
simulation. Water Resources Research 21~1), 1926.
Ahlstrom, S. W., H. P. Foote, R. C. Arnett, C. R. Cole, and R. J. Serne.
1977. Multicomponent Mass Transport Model: Theory and Numerical Im
plementat ion (discreteparcelrandomwalk version). BNWL2 127. Battelle
Northwest Laboratories, Richland, Wash.
Alexander, M. 1985. Biodegradation of organic chemicals.
Environmental
Science and Technology 19, 106.
Anderson, M. P. 1979. Using models to simulate the movement of contaminants
through ground water flow systems. Critical Reviews in Environmental
Control 9~2), 97156.
Anderson, M. P. 1984. Movement of contaminants in groundwater: Groundwa
ter transportadvection and dispersion. Pp. 3745 in Groundwater Con
tamination. Studies in Geophysics. National Academy Press, Washington,
D.C.
Atlas, R. M. 1981. Microbial degradation of petroleum hydrocarbons: An
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Baehr, A. L., and M. Y. Corapcioglu. 1987. A compositional multiphase
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solution. Water Resources Research 23~1), 201214.
Bates, J. K., and W. B. Seefeldt. 1987. Scientific Basis for Nuclear Waste
Management X. Materials Research Society, Symposium Proceedings, Vol.
84, 829 pp.
Bear, J. 1972. Dynamics of Fluids in Porous Media. Elsevier, New York, 764
PPe
Bear, J. 1979. Hydraulics of Groundwater. McGrawHill, New York, 569 pp.
Bredehoeft, J. D., and G. F. Pinder. 1973. Mass transport in Bowing ground
water. Water Resources Research 9, 194210.
Cherry, J. A., R. W. Gillham, and J. F. Pickens. 1975. Contaminant hydroge
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Cleary, R. W., and M. J. Ungs. 1978. Groundwater Pollution and Hydrology,
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OCR for page 28
74
GROUND WATER MODELS
Concus, P., G. Golub, and D. O'Leary. 1976. Sparse Matrix Computations.
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Corapcioglu, M. Y., and A. L. Baehr. 1987. A compositional multiphase model
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