SOIL SCIENCE SOCIETY OF AMERICAJOURNAL

VOL. 53

SEPTEMBER-OCTOBER 1989

No. 5

DIVISION S-l-SOIL PHYSICS
Two-Site/Two-Region Models for Pesticide Transport and Degradation:
Theoretical Development and Analytical Solutions
M. Th. van Genuchten* and R. J. Wagenet
persion equation (CDE), or of alternative equations,have been widely used as models of chemical transport and transformation in soil-water systems. Although such models are limited to conditions ofsteady-state water flow in homogeneous soils, andhence are primarily useful only for interpreting laboratory experiments, valuable information has been
gained through their use. The application of analyticalmodels to the study of pesticide transport in soils has
actually been a standard practice for many years

ABSTRACT
Analytical solutions are presented for two convection-dispersion
type transport models useful for studying simultaneous pesticide
sorption and degradation. One solution is for the familiar two-site
sorption model in which adsorption-desorption proceeds kineticallyon one fraction of the sorption sites, and at equilibrium on the remaining sites. Another solution holds for two-region (or mobile-immobile liquid phase) transport appropriate for aggregated or fractured media. The transport models account for degradation in boththe solution and sorbed phases. The dimensionless analytical solutions for the two-site and two-region models are shown to be identical; they contain up to six independent dimensionless parameters:a column Peclet number, a retardation factor, a coefficient partitioning the soil/chemical system in equilibrium and nonequilibrium
parts, a rate coefficient, and two dimensionless degradation coefficients. One of the two independent degradation coefficients may beeliminated when the solution and sorbed phase degradation rate coefficients are assumed to be identical, or when, with additional but
reasonable assumptions, adsorbed phase degradation is assumed tobe negligible.

(Lindstrom et al., 1967; Kay and Elrick, 1967; Dav-idson et al., 1968; Davidson and McDougal, 1973;
O'Connor et al., 1976; Rao et al., 1979; Enfield andCarsel, 1981; Jury et al., 1983, 1984; Zhong et al.,1986). Models of this type have been reviewed a number of times (van Genuchten and Cleary, 1979; Nielsen et al., 1986), and have recently been placed within
the perspective of other approaches to describe solutetransport (Addiscott and Wagenet, 1985; Jury, 1984).
A variety of analytical solutions are currently available (e.g., van Genuchten and Alves, 1982; Javandel
et al., 1984), some of which have now been included
into optimization packages usable on a routine basis
for interpretation of laboratory or field tracer experiments (Parker and van Genuchten, 1984).
Pesticide transport is influenced by a large numberof simultaneous processes and properties. Among thenonequilibrium processes influencing transport is thepossible division of sorption sites into two types. The
two-site sorption concept presumes that sorption or
exchange sites in soils can be classified into two fractions: one fraction (Type-1) on which sorption is assumed to be instantaneous, and another fraction(Type-2) on which sorption is considered to be timedependent. The resulting two-site model has been successfully used to describe the transport of a numberof solutes (Selim et al., 1977; Cameron and Klute,
1977; Rao et al., 1979; DeCamargo et al., 1979; Hoffman and Rolston, 1980; Parker and Jardine, 1986)
that interact with a solid phase composed of such different constituents as soil minerals, organic matterand various oxides. Studies of the sorption of 2,4-D(2,4-dichlorophenoxy acetic acid) and atrazine (2chloro-4-ethylamine-6-isopropylamino-l,3,5 triazine)

ONCERN about the fate of chemicals introducedC
into soil-water systems has recently intensified.
It is often necessary to estimate the behavior of achemical in the field without substantial knowledge ofthe interaction of the chemical with the solid phaseor its degradation rate. As new chemicals are proposed
for future use, or as closer scrutiny is given to thosealready in use, it is necessary to utilize scientificallysound, comprehensive tools to evaluate the potential
behavior of these chemicals in the environment. Wellconstructed tools in the form of models describing
transport in soil-water systems also serve the complementary purpose of increasing our understanding ofbasic processes affecting chemical fate.
Analytical solutions of the classical convection-dis-M.Th. van Genuchten, USDA-ARS, U.S. Salinity Lab., 4500 Glenwood Dr., Riverside, CA 92501; and R.J. Wagenet, Dep. of Agronomy, Cornell Univ., Ithaca, NY 14853. Joint contribution from the
U.S. Salinity Lab. and the Dep. of Agronomy, Cornell Univ. Received 29 June 1988. 'Corresponding author.
Published in Soil Sci. Soc. Am. J. 53:1303-1310 (1989).

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SOIL SCI. SOC. AM. J., VOL. 53, SEPTEMBER-OCTOBER 1989

(Rao et al., 1979) indicate that the two-site model may
well be suitable for these pesticides.
Another nonequilibrium model that has been founduseful for modeling solute transport is the "two-region' or "bi-continuum" transport model (Coats andSmith, 1964; Gaudet et al., 1977; van Genuchten and
Cleary, 1979; Nkedi-Kizza et al., 1984) in which thesorption rate is thought to be limited by the rate at
which solutes are transported by diffusion to the exchange sites. This conceptualization has led to physical nonequilibrium models that partition soil water
into mobile (flowing) and stagnant (immobile or nonflowing) regions. The approach in effect assumes thatthe pore-water velocity distribution is bimodal: convective-dispersive transport is confined to only a fraction of the liquid-filled pores, while the remainder ofthe pores have stagnant water (e.g., dead-end porewater, intra-aggregate water, thin liquid films aroundparticles). In models of this type, solute exchange between the two liquid phases is frequently considered
to be a first-order rate process. We note that similar"mobile-immobile" type two-region transport models
have also been used for modeling solute transport inriver systems containing dead zones (Bencala andWalters, 1983; LeGrand-Marcq and Laudelout, 1985).
Microbial degradation is another process of importance to pesticide transport. Biodegradation of organic
chemicals has classically been presumed to occur primarily in the liquid phase of the soil. Whereas degradation in the sorbed phase has at times been measured (Marshman and Marshall, 1981; Moyer et al.,
1972; Scott et al., 1983), other studies with such herbicides as 2,4-D (Ogram et al., 1985), diquat (1,1'ethylene-2,2'-bypyridylium ion) (Weber and Cole,1968) and triazine (Moyer et al., 1972) indicate thatthe sorbed phase may not always be available for deg-

radation. Thus, the separation of pesticide degradation into relative contributions of the sorbed and solution phases is a continuing concern for soil scientists
and soil microbiologists dealing with the environmental fate of pesticides.
Analytical solutions of two-site and two-region typemodels with solution and sorbed phase degradation
are needed to fully interpret pesticide transport experiments involving laboratory soil columns. Such experiments in turn are designed to estimate future pesticide behavior in field soils (e.g., Zhong et al., 1986).

d(8c)

[1]

dx

dt

where c is thevolume-averaged solution concentration (withunits M Lr3), 0 is the volumetric water content (L3 Lr3 orL°), ft/ is a first-order decay coefficient for degradation fromthe liquid phase (T-1), x is distance (L), t is time (T), Ja isa transfer rate fromthe solution to the sorbed phase due tosorption (M Lr3 T"1), and Js is the solute flux density (M Lr2
T-1),

[2]
in which D is the dispersion coefficient (L2 T-') and q thevolumetric water flux density (L T-')- Combining Eq. [1]and [2] gives

Equation [1] is based on a mass balance of input andoutputs for the solution phase. A similar mass balance forthe sorbed phase leads to

ds

[4]

P — = Ja at

where s is the 3sorbed concentration (A/°), p is the soil bulk
density (M Lr ), and n1s is the first-order sorbed phase degradation coefficient (T- ). Equation [4] states that the change
in sorbed concentration equals the transfer rate (Ja) from thesolution to the sorbed phase minus decay from the sorbed
phase. We will further use the term "sorption rate" whenreferring to Ja. Equation [3] and [4] hold for the solution and sorbed
phases, respectively. Adding the two equations eliminates Ja
and leads to a transport equation for the soil system as a
whole

d(8c)
dt

P

dt

.

[5]

This equation is valid irrespective of whether sorption is anequilibrium or kinetic process. Equilibrium sorption presumes that solute exchange between the liquid and solid
phases occurs at such a rate that at all times the sorbed
concentration, s, is uniquely related algebraically to the so-

In this paper we shall develop the governing equations
and relevant analytical solutions for these two-site andtwo-region transport models with degradation. For
completeness, we shall also formulate the classicalequilibrium and one-site kinetic transport models
with degradation. In a later study (Gamerdinger et al.,1989) we will use the analytical models to analyze a
set of column displacement experiments with atrazine
and 2,4,5-T (2,4,5-trichlorophenoxy acetic acid).
LINEAR EQUILIBRIUM SORPTION
We first derive the governing equation for classical "Fickian"-based, deterministic solute transport with degradation.
Consider a soil system (Fig. 1) composed of a liquid phase
involving convective and diffusive/dispersive transport, anda solid phase subject to chemical sorption or exchange. The
continuity equation for the liquid phase is

Equilibrium:-^—————

rr = k rr
dt
dt

Kinetic:= ap (kc-s)

JFig. 1. Schematic of the one-site equilibrium and kinetic sorption
transport models with degradation.

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VAN GENUCHTEN & WAGENET: MODELS FOR PESTICIDE TRANSPORT AND DEGRADATION
Liquid Phaselution concentration, c. If the sorption process is linear, thisleads tos = kc[6]where k is an empirical distribution coefficient (M~' L3).
Substituting Eq. [6] in [5] gives
dt

dx

dX

[(l-f)kc-s ][7]where R and fi are the retardation factor and total (or effective) degradation constant, respectively

R = 1+ pk/B

[8]
[9]

Note that when the two degradation constants n, and ns are
identical (e.g., for radioactive decay), Eq. [9] reduces to n =H,R. For chemical or microbial degradation, however, thetwo rate coefficients likely have different values.
Equation [7] holds for transient fluid flow. For steadystate flow in a uniform medium (constant 6 and p), the transport equation becomes

where v = q/6 is the average pore water velocity. Analyticalsolutions of this model for various initial and boundary conditions are given elsewhere (van Genuchten and Alves,
1982).
ONE-SITE KINETIC ADSORPTION
Assuming linear kinetic sorption, the sorption rate, Ja,
from the solution to the sorbed phase is given by
Ja = ap(kc - s),[11]
where a is a first-order kinetic rate coefficient. Equations [3]and [4] become now respectively,- ~ qc)- ap(kc - s) -

at

c)£
dt

= a(kc - s) - ns

[12a]
[12b]

which is the one-site kinetic model for transient flow. For
steady-state water flow in a uniform medium, Eq. [12a] reduces to

[,„

dx2
It is important to recognize that Eq. [13] holds only for theliquid phase, and as such only contains a decay term for theliquid phase (last term of Eq. 1 3). By making use of Eq.
[12b], the sorption rate term may be eliminated from Eq.
[13] to give
dc
pds
d2c
dc
dt

Fig. 2. Schematic of the two-site partial equilibrium, partial kinetic
sorption transport model with degradation.
step-by-step development that Lindstrom's 1976 formulation is incomplete.
The one-site model is a special case of the two-site transport model to be discussed next. Thus, analytical solutions
for the two-site model are directly applicable also to theabove one-site kinetic model.
TWO-SITE EQUILIBRIUM/KINETIC
SORPTION
Governing Equations
The derivation for this case proceeds in the same fashion
as for the one-site model. Since the two-site model itself has
been explained in detail elsewhere, only a capsulized derivation is presented here to indicate the treatment of thedegradation terms. Following the notation of van Genuchten (1981) and Parker and van Genuchten (1984), we haveat equilibrium for the Type-1 (equilibrium) and Type-2 (kinetic) sites, respectively

sl=fkc= (\-f)kc

[15a]
[15b]

where / is the fraction of exchange sites assumed to be atequilibrium, and where the subscripts 1 and 2 refer to the
two Type-1 and Type-2 sorption sites, respectively. Total
adsorption, s, is given bys = + s2[16]
which at equilibrium becomes equivalent to Eq. [6].
Figure 2 shows schematically a soil made up of the liquid
phase, a Type-1 solid phase, and a Type-2 solid phase. BothJai and Ja2 are the sorption rates from the liquid into theType-1 and Type-2 regions, respectively. Mass transport in
the liquid phase is still described by Eq. [3], given that Ja
now becomes the sum of/„, and Ja2. The mass balances for
the Type-1 and Type-2 are similar to Eq. [4]

[17a]

P— =
at

[17b]

dt

This equation holds for the soil system as a whole and is
identical to Eq. [5] for steady-state water flow. Hence, theone-site kinetic model is also given by Eq. [12b] and [14].
Note that the sorbed phase decay term appears now in Eq.
[12b] as well as [14]. This decay term was ignored by Lindstrom (1976) in his classical derivation of the analytical solution for first-order kinetic solute transport with decay (seeEq. [1] of Lindstrom, 1976). We conclude from the above

The mass transport equation for the system as a whole follows by adding the contributions of Eq. [3] and [17a,b], andnoting that Ja = Jal + Ja2
_

=

dt

dt

dx

dx

[18]

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SOIL SCI. SOC. AM. J., VOL. 53, SEPTEMBER-OCTOBER 1989

Because Type-1 sites are always at equilibrium, sorption
onto these sites is also given by the time-derivative of Eq.
[15a]

et

at

By using a first-order kinetic sorption rate law analogous toEq. [11], and by making use of Eq. [15b], the mass balance
equation for the Type-2 sites becomes= a[(l - f)hc - s2] -[20]

Substituting Eq. [19] and [20] into Eq. [18] and using Eq.
[1 5a] to eliminate s, from the equilibrium sorbed phase degradation term leads to

fpk)c
dt

dX

ap[(\ - f)kc - s2] - 0M/c - pptl c.

[21]

Hence, the complete two-site model is given by Eq. [20] and[21].

(i -0;

[19]

[31b]

)R ~^ ~ u(Cl ~ C2) ~ ^2

where

3R — \)L
H,L
M,,G
+
v
v

*
and

(Oni + fpkfisl}
a

[32a]

Analytical solutions of Eq. [31a,b] can be derived using
Laplace transforms as shown in detail, among others, byLindstrom and colleagues (Lindstrom and Narasimhan,
1973; Lindstrom and Stone, 1974; Lindstrom, 1976). We
omit here the relatively straightforward but extremely
lengthy derivation. The general analytical solution for our
problem is

•
c(zT
Cl(Z
J)) =

'

ca(z,T) - ca(z,T - T0) T>T0

Analytical Solution
For steady-state flow in a uniform system, Eq. [21] reduces

(z,T)

m,
[33b]

0 < T < T0

- c6(z,r - r0) T > i

to

where
dx

,0= !T g(z,r) J(a,b) dr

[34a]

Jo

[22]
C2(z,0 = -^-w + 77 J0

Equations [20] and [22] are solved for an initially solutefree, semi-infinite soil profile subject to a pulse-type inputc(x,0) = 5t(jc,0) = s2(x,0) = 0[23]

rfr [34b]

ro = vt0/L[35]/(a,/?) = 1 - e-» f" fr*I0[2jb\] dX[36a]boundary condition. Thus, the auxiliary conditions are of

the form£(Z,T) [1 - JM

Jo

[24]

fc =

a =

(co + ^(r - r)[36b>(;]+

andwhere C0 is the input concentration, and t0 is the time duration of the applied solute pulse. To obtain a convenient
dimensionless system of equations consistent with those of
van Genuchten (1981, p. 27), the following variables are
introduced[26a,b]
T = vt/L
z = x/LR

1 + pk/0 Rm= 1 + fpk/d[28a,b]« = a(l - &)RL/v,[29]
where L is an arbitrary distance from the input boundary(e.g., column length). We also define the dimensionless concentrations c, and c2«(^T) = ^,T)v

in which G(z,t) represents the analytical solution of
dG

1 d2G

dG

G

[38]

for the same initial and boundary conditions as before. For
volume-averaged resident concentration (Kreft and Zuber,
1978; Parker and van Genuchten, 1984), g(z,r) is given by
WJ?

PR(u + r,)0R

1/2
7T/3/?

Equations [22] and [20] become then respectively,

[37]

0r "

exp -

P(pRz --

[39]

VAN GENUCHTEN & WAGENET: MODELS FOR PESTICIDE TRANSPORT AND DEGRADATION

1307

where n,m is the mobile liquid phase degradation coefficient,Ja[ is the transfer rate from the liquid to the solid phase ofthe "mobile" region, while Ja2 is due to diffusion-like exchange between the mobile and stagnant liquid zones. Analogous to Eq. [4], the mass balance for the "mobile" sorbed
concentration (sm) isfm
JP —— =
01

[42]

where nsm is the sorbed phase degradation coefficient for themobile region. Adding Eq. [41] and [42] gives the transport
equation for the mobile region as a wholedsdt—— =dc

a2 ~Jo3 " "i m*t\ mc i m

sm •

[43]

A similar mass balance without the convection-dispersion
terms holds for the stagnant region as a whole (subscript im)
Fig. 3. Schematic of the two-region (mobile-immobile) transportmodel with degradation.

For flux-averaged or flowing concentrations, which are characteristic of column effluent curves (van Genuchten and Parker, 1984), g(z,r) becomes
W7J

dt

i - (1 - /)p/Wi« .
[44]
Solute exchange between the mobile and immobile liquid
regions is approximated by the expression
Ja2 = <*(Cm ~ Cim)

(a

z PpR

PtfRz - r)~

[40]

TWO-REGION TRANSPORT MODEL WITH
DEGRADATION
Governing Equations
The derivation of the two-region model with first-orderdegradation also parallels the no-decay case (van Genuchten, 1981; p. 12-16). The inclusion of degradation into
the two-region model appears somewhat more complicatedthan for the two-site model. This is because local degradation rates inside aggregates can be radically different fromthose occurring at aggregate edges due to varying O2 and/or
microbial activities. To keep the formulation general, we
need provisions for different degradation coefficients in themobile and immobile liquid phases, as well as for different
degradation coefficients for the mobile and immobile sorbed
phases of the soil. Thus, a total of four degradation rate
coefficients are possible in the two-region model, as compared to three for the two-site model.
Figure 3 shows again schematically the most important
fluxes in a "two-region" soil system with decay. Convectivedispersive transport is confined to a mobile liquid region(subscript m) while the presence of solute in an immobile
region (subscript im) depends on liquid diffusion from themobile to the immobile liquid. The solid phase is also partitioned, in this case into a fraction / that equilibrates instantaneously with the mobile fluid, and another fraction(1 — /) that equilibrates with the immobile liquid. Using
the same notation as before, the transport equation for themobile liquid phase (subscript m) is
dc
— Ja\

—

[41]

[45]

in which the model-specific a is a first-order mass transfer
coefficient describing the rate of transfer between the mobile
and immobile liquid phases. Sorption in the inter- and intraaggregate regions of the soil is described with linear isotherms of the form
sm = kcm
sim = kcim.
[46a,b]
Substituting Eq. [45] and [46a,b] into [43] and [44] gives
finally the transient-flow two-region transport model withdegradationfpk)cndt

dx\'"'-m ax ~ qc"a(cm - cim) ~ f)Pk]ci

M,M + fpknsm)cm[47a]= a(cm - cim)h (1 - f)pknam}Clm .

[47b]

Analytical Solution
Assuming steady-state flow in uniform soil, Eq. [47a,b]
reduce to- a(cm - cim) -fpknsm)cm

[48a]

[48b]

We use the same initial and boundary conditions as for thetwo-site model (replace c by cm in Eq. [23] through [25]).
Introduce further the model-specific dimensionless parameters (T and z are as before)

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SOIL SCI. SOC. AM. J., VOL. 53, SEPTEMBER-OCTOBER 1989

Table 1. Expressions for the dimensionless parameters | and »j in Eq. [31a,b] for the one-site, two-site and two-region transport models.
One-site modelTwo-site model

Solution and sorbed
phase degradation
rates are independent(0« - 1) *„Degradation
everywhere
the same7) =Degradation
only in theliquid phaser, = 0Degradation
only in thesorbed phaseTwo-region modelf£:„ = (1 - 0)* ^Ml

1) = 0f = 0t =1 -(!-«**!

(SR — 0m) ^im, = [(1 - 0)* - <t>lm] ^«„/«,*«.

P = VmL/Dn

e + Pk[49]

co = aL/^f[50a,b]= c/m/C0.
[51a,b]The dimensionless transport equations become then identical to Eq. [31a,b], provided % and r\ are now denned as
L

'"

' "

^

[52a]

Another significant simplification is possible whendegradation is assumed to be limited to the solution
phase, a situation that seems to hold for at least somepesticide-soil combinations (Weber and Cole, 1968;
Moyer et al, 1972; Ogram et al, 1985). The solid
phase degradation coefficients are then zero (M,I = H&
= 0) and £ and 77 reduce to* = */
u= 0[55a,b]where, similar to Eq. [54] and other cases shown in
Table 1, t/v = ^L/v.
L
Table 1 summarizes the expressions for £ and 77 for[52b]
the most general case of unequal degradation rates in4
the solution and sorbed phases, as well as for several
The dimensionless initial and boundary conditions for thelimiting cases involving the one-site, two-site and twotwo-site and two-region models are easily shown to be idenregion transport models. Note that for the general twotical also. Thus, the analytical solution for the two-sitesite and two-region formulations, £ includes all or partmodel also applies to the above two-region mobile-immoof the effects of both solution and sorbed phase degbile transport model with degradation.
radation. The same is true for 77 in the general tworegion formulation. The parameters £ and i\ in bothDISCUSSION
models are in that case independent, leading to a total
The above formulations for the two-site and twoof six independent dimensionless parameters: the colregion models were obtained without making any asumn Peclet number P, the retardation R, a dimensumptions about possible values of the degradation
sionless partitioning coefficient /?, a dimensionless rate
coefficients M/, M*i, Mm/> etc. While at least in theory it
coefficient for kinetic sorption or mobile-immobile
is possible that the degradation coefficients in any partype exchange w, and the two degradation rate coefticular model will attain different values, in practice
ficients £ and 77. For equal degradation in the solution
it may be nearly impossible to distinguish between
and sorbed phases, £ and i\ become dependent, andthem, with concomitant problems of parameter idenonly five independent variables can be identified (seetification. This is especially true for the two-region
also Table 1): P, R, 0, u and ^.
model which contains four potentially different degFive independent parameters also result when degradation coefficients. Thus, the number of degradation
radationin the one-site and two-site models is limited
terms needs to be reduced where possible. Fortuto
thesolution
phase. Note that £ = ^/ (= M/£/V) in
nately, it is possible to make several simplifications of
that
case,
while
i\ becomes zero (Table 1). However,
the most general case of unequal solution and sorbed
£
and
77
for
the
two-region
model remain theoreticallyphase degradation rates.
independent
becauseoftheinclusion of the mobile
One simplification =results when all rate coefficients(</>m) and immobile water (</>im) fractions in f and 77.
are the same, e.g, M/ M*I = V-& = M for the two-site
These fractions are not independently included in any
model, or M/« - M«« = MJm = fem = M for the two- other dimensionless variable (including 0 as shown by
region model. The otherwise independent parameters
van Genuchten and Cleary, 1979), and hence must be£ and t\ for the two-site (Eq. [32a,b]) and the twoconsidered separately. This could pose considerable
region (Eq. [52a,b]) models reduce then to (see also
problems, especially when some of the parameters are
Table 1)estimated from measured breakthrough data. Curve| = 0R * n = (1 - PW
[53a,b]fitting is often necessary since the partitioning of theliquid phase into mobile and immobile parts is genwhere the dimensionless parameter ^ is defined as
erally not known a priori. We can avoid this problem
by assuming that the sorption sites and the mobile.
[54]

VAN GENUCHTEN & WAGENET: MODELS FOR PESTICIDE TRANSPORT AND DEGRADATION

2

3

4

5

6

7

8

PORE VOLUME,!

Fig. 4. Effect of the dimensionless degradation coefficient, <!*, on the
shape of effluent curves calculated with the two-region model.

water fractions are distributed in the same manner
throughout the soil system, that is (Nkedi-Kizza et al,
1984)
[56]
*m = 0 » / 0 = / = / J
so that the £ and rj parameters become$ = 0*,
i, = (l-18)^.
[57a,b]Note that £ and 77 are now the same as Eq. [53a,b] for
equal degradation in the solution and sorbed phases,
except that R now has been dropped.
Figure 4 illustrates the effect of degradation on calculated effluent curves. All parameter values are thesame as used previously by van Genuchten and Cleary(1979) to demonstrate the effects of the dimensionlessvariables P, R, 0 and w on calculated effluent curves.
The one additional degradation parameter ^ in thisexample assumes that degradation is everywhere thesame (Table 1). Given the assumption inherent in Eq.
[56], the curves also apply to the special case of solution phase degradation only, provided the ^'s in Fig.
4 are multiplied by the value ofR (=3.0). Numerous
effluent curves were also calculated with the general
two-site/two-region model assuming unequal solution
and sorbed phase degradation coefficients. The curves
so obtained (not further shown here) behaved qualitatively very similar to those plotted in Fig. 4.

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SOIL SCI. SOC. AM. J., VOL. 53, SEPTEMBER-OCTOBER 1989