w ¼
gS
þ
¹ A
¹
2 þ
gS
þ
þ A
¹
2 tanh
3 1
2 R
f
gR
f
S
þ
þ A
¹
˜ gy
: ð
30Þ Such solutions are unique up to a constant translate in y. In
the original variables S,A, we have, in view of eqn 25, the explicit expression of random traveling fronts:
A T
, t
¼ A
¹
2 1 ¹ tanh
m K
A
K
S
A
¹
þ gR
f
S
þ
2 R
f
¹ 1
T ¹ c t
, 31
and S
T ,
t ¼
S
þ
2 1 þ tanh
m K
A
K
S
A
¹
þ gR
f
S
þ
2 R
f
¹ 1
T ¹ c t
, 32
where c is given in eqn 28 and the random travel time T
in eqn 18.
2.2 The general two component model
Let us consider the more general model without restrictive assumptions about K
A
and K
S
but still under biomass equili- brium conditions:
R
f
S
t
þ S
T
¼ ¹ m
K
A
K
S
AS=F A
, S
, 33
A
t
þ A
T
¼ ¹ g
m K
A
K
S
AS=F A
, S
, 34
m ¼ bm b ¹
YAS K
A
þ A
K
S
þ S
¹ 1
, 35
where: F
A ,
S ¼
1 þ S=K
S
þ A=K
A
þ 1 ¹ b
¹ 1
Y AS=
K
A
K
S
: 36
Making the same change of variables to u,w as in eqns 24 and 25, and following the same procedure as before,
we have the same expressions as eqns 28 and 29. The only change is that eqn 30 is modified to:
dw dy
¼ ¹ ˜
gR
f
gR
f
S
þ
þ A
¹
gS
þ
þ A
¹
w ¹
S
þ
w þ
A
¹
= G
w ,
37 where G ¼ Gw is simply FA,S with A and S written as
functions of w using eqns 25 and 29. Gw is a positive quadratic function of w, and is equal to:
G w
¼ 1 þ
gK
S
R
f
¹ 1
¹ 1
c
¹ 1
¹ 1
w þ A
¹
þ R
f
¹ 1
¹ 1
K
¹ 1
A
c
¹ 1
¹ R
f
w þ 1 ¹ c
1 ¹ R
f
c A
¹
þ 1 ¹ b
¹ 1
Y c
¹ 1
¹ 1
c
¹ 1
¹ R
f
K
A
K
S
g R
f
¹ 1
2
w þ A
¹
3 w þ 1 ¹ c
1 ¹ R
f
c A
¹
which simplifies using the wave speed formula for c to:
G w
¼ 1 þ
S
þ
w þ A
¹
K
S
gS
þ
þ A
¹
¹ A
¹
w ¹ gS
þ
K
A
gS
þ
þ A
¹
¹ 1 ¹ b
¹ 1
Y S
þ
A
¹
w þ A
¹
w ¹ gS
þ
K
S
K
A
gS
þ
þ A
¹ 2
: ð
38Þ The solutions of A and S are given by eqn 25 as w
is solved from eqn 37, see Appendix B.
3 STATISTICAL ANALYSIS OF RANDOM FRONTS 3.1 General formalism
Because T is a random variable, so are nutrient concentra- tion Ax,t, substrate concentration Sx,t and biomass con-
centration mx,t. They may be estimated with their ensemble means expected values:
hA x
, t
i ¼ Z
`
A T
, t
f T
; x ,
x dT
, 39
hS x
, t
i ¼ Z
`
S T
, t
f T
; x ,
x dT
40 hm
x ,
t i ¼
Z
`
m T
, t
f T
; x ,
x dT
41 where AT,t, ST,t and mðT; tÞ are given earlier, and
f ðT ; x; x
Þ is the probability density function pdf of the
travel time Tx,x of a particle from x
to x. Note that after averaging, the dependence on x,t in eqns 39, 40 and 31
is not of the form x ¹ ct. Hence strictly speaking, hAi, hSi, and hmi are not traveling waves. However, they do behave like
traveling waves in the sense that they move at constant speeds to leading order in time. The evaluation of the expected
concentrations reduces to that of travel time pdf, which only depends on the water velocity and does not vary with the
nutrient or substrate wave front speed. The uncertainty asso- ciated with the estimation can be evaluated with the variances:
j
2 A
x ,
t ¼ h
A
2
x ,
t i ¹ hA
x ,
t i
2
¼ Z
`
A
2
T ,
t f
T ; x
, x
3 dT ¹ hA x
, t
i
2
, ð
42Þ j
2 S
x ,
t ¼
Z
`
S
2
T ,
t f
T ; x
, x
dT ¹ hS x
, t
i
2
, 43
j
2 m
x ,
t ¼
Z
`
m
2
T ,
t f
T ; x
, x
dT ¹ hm x
, t
i
2
: 44
As mentioned earlier, the randomness in vx may stem from q andor nx. In the case of one-dimensional,
steady-state flow, q can be either a deterministic or random constant. The latter represents the ensemble of 1-D
columns of different q or a single column with random boundary conditions. In this case the travel time moments
can, however, only be given via approximation. In this study, we consider the case that q ; q
is a specified 106
J. Xin, D. Zhang
constant such that the spatial variability in porosity is the only source of randomness in the velocity vx ¼ q
nx. The porosity is assumed to be stationary such that its mean
hni is constant and its covariance C
n
x,x
1
only depends on the relative distance r ¼ x ¹ x9. The covariance function C
n
is taken to be exponential: C
n
r ¼
j
2 n
exp ¹ lrl
l
n
45 where j
2 n
and l
n
are the variance and correlation scale of porosity, respectively.
In this case, the travel time can be rewritten as: T
x ,
x ¼
Z
x x
n x
9 q
dx9: 46
Hence, its moments are given exactly as: hT
x ,
x i ¼
hni q
x ¹ x ,
47 j
2 T
x ¼
1 q
2
Z
x x
Z
x x
C
n
x 9 ¹ x0
dx9 dx0 48
¼ 2
q
2
Z
x x
x ¹ x 9
C
n
x 9 ¹ x
dx9 ¼
2j
2 n
q
2
l
n
x ¹ x ¹
l
2 n
1 ¹ exp ¹ x ¹ x
l
n
: It is seen that the travel time variance increases sublinearly
with x ¹ x .
If the travel time obeys a lognormal distribution, its pdf can be described with the first two moments:
f T
; x ,
x ¼
1 2p
1=2
Tj
ln T
exp ¹
{ln T
¹ h ln
T i}
2
2j
2 ln
T
, 49
where hln
T i ¼ ¹ 0:5 ln
[ hTi
2
þ j
2 T
] þ
2 ln hTi
and j
2 ln
T
¼ ln
[ hTi
2
þ j
2 T
] ¹
2 ln hTi
. Although other forms of travel time pdfs are possible, it is found by Zhang and Tche-
lepi
19
for a similar problem of random Buckley-Leverett displacement that the impact of distributional forms may be
neglected. With eqn 49 the means and variances of concen- trations in eqns 39–44 can be evaluated by numerical
integrations, say via Simpson’s rule.
Another quantity of interest is the substrate removal in the domain segment from x
to x
1
: R
t ; x
1
, x
¼ Z
x
1
x
R
f
n x
[ S
þ
¹ S
x ,
t ]
dx ,
50 which can be estimated with its expected value:
hR t
; x
1
, x
i ¼ Z
x
1
x
R
f
[ hniS
þ
¹ h n
x S
x ,
t i
] dx
¼ Z
x
1
x
R
f
hniS
þ
¹ Z
1
Z
`
n x
S T
, t
f [
n x
; T
x ,
x ]
dn dT dx:
ð 51Þ
Here f[nx;Tx,x ] is the joint pdf of the porosity n
at x and the travel time from x to x. It can easily be shown
that the cross covariance between nx and Tx,x is given
as: hn9
x T
9 x
, x
i ¼ l
n
j
2 n
q 1 ¹ exp ¹
x ¹ x l
n
, 52
where the superscript 9 indicates deviation from the mean. With this and eqn 48, we obtain immediately that the
correlation function
between n
x and
T x,x
, r
nT
x ¹ x ¼ h
n 9
x T
9 x
, x
i= [
j
n
j
T
x ]
→ 0 as x ¹ x → `.
That is to say, nx and Tx,x become uncorrelated when
the distance x ¹ x is large relative to l
n
. Physically speaking, this is so because the influence of nx on
T x,x
becomes smaller and smaller as the distance x ¹ x
increases. Below we evaluate the statistical moments for two spe-
cific models.
3.2 The nutrient-deficient model
