Nonlocal reactive transport with physical, chemical ...

Advancesin Wafer Resources, Vol. 20, Nos 5-6, pp. 293-308, 1997 G 1997 Published by Elsevier Science Ltd All rights reserved, Printed in Great Britain 1-6 0309-1708/97/$17,00+0.00

PII:S0309-1708(96)OO05

Nonlocal reactive transport with physical, chemical, and biological heterogeneity Bill X. Hu, John H. Cushman* & Fei-Wen Deng Math Science Department and Department of Agronomy, Purdue University, W. Lafayette, IN 47907, USA (Received 14 July 1995; accepted 30 November 1995) When a natural porous medium is viewed from an eulerian perspective, incomplete characterization of the hydraulic conductivity, chemical reactivity, and biological activity leads to nonlocal constitutive theories, irrespective of whether the medium has evolving heterogeneity with fluctuations over all scales. Within this framework a constitutive theory involving nonlocal dispersive and convective fluxes and nonlocal sources/sinks is developed for chemicals undergoing random linear nonequilibrium reactions and random equilibrium firstorder decay in a random conductivity field. The resulting transport equations are solved exactly in Fourier–Laplace space and then numerically inverted to real space. Mean concentration contours and various spatial moments are presented graphically for several covariance structures. ~ 1997 Published by Elsevier Science Ltd. All rights reserved

space and/or time) or higher-order derivatives (the degree of the highest-order derivative is a measure of the extent of nonlocality). A good example of a nonlocal constitutive theory is provided by Koch and Brady,ll Cushman,’ and Deng et al.6 These authors develop convolution-Fickian models for dispersion of a conservative solute:

INTRODUCTION At

the

outset

of any

modeling

effort

for chemical

transport in the subsurface, one must choose a frame of reference. Two such frames are generally available, Lagrangian and Eulerian. Recent theoretical results for simple reactive contaminants2>8 suggest that if local dispersion is neglected and fully nonlocal constitutive equations are used in an eulerian frame, then the eulerian and lagrangian spatial moments through the third are equivalent. However, if local scale dispersion is included in the eulerian analysis, then the lagrangian and eulerian expected spatial moments differ. This comes as no surprise since current Lagrangian techniques neglect local-scale dispersion. Thus because the Eulerian frame is currently more general, we choose it in subsequent analysis. The authors 3 have also recently shown that it is imperative to use nonlocal constitutive models when studying transport of reactive chemicals on the reservoir-scale in a physically and chemically heterogeneous environment. Use of quasi-Fickian type theories generate significant error. Before proceeding further it is advantageous to discuss the concept of nonlocality within the framework of theories of transport in porous media. A constitutive theory is said to be nonlocal if it involves integrals (over

~= H‘ R3 f)

D(x – y, t – t’) vY~(y,

t’)dydt’

(1)

where q is dispersive flux, D is a dispersion tensor, and C is the mean concentration.This constitutive relation is nonlocal in both space and time, and when localized by factoring VC out of the integral and integrating D it leads to the local quasi-Fickian constitutive relation previously mentioned (cf., Gelhar and Axness7). Such an approximation is reasonable for obtaining lower-order spatial moments of conservative solutes, but it is wholly inadequate for nonconservative species.3

ANALYSIS It is assumed on the local scale that the total concentration (solution phase plus sorbed phase), CT, satisfies the following equation: 8CT — = v“(d~”vcq

dt

* To whom correspondenceshould be addressed. 293

– V“(VTCT)

–

Y(CT)

(2)

294

B. X. Hu et al.

where dT is the total dispersion tensor for the combined sorbed and solution phases, VT is the total velocity, and Y(C~) is the total degradation rate (solution plus sorbed). If we assume that both the local convective and local dispersive fluxes for the sorbed phase are negligible, and further if the degradation processes in both solution and sorbed phases are first-order, Y(CT) = K,.C + KJ

and s =

(6h)

S+s

Substituting eqn (6) into eqn (4) we obtain 6’(C + c) + 8(s + s) = d, a2(c + c) at at ‘ ax:

+ C)- (FC+ kC)(C + C) —8( Pi+ Wj)(C’

(3)

8Xj

then (2) becomes

8C as

~

+~

= V“(dvc)

- (R.+ k,y)(~ + s) – V“(VC) – KCC – K,,S

(4)

where C is the solution concentration and S is the sorbed phase concentration (which is defined as sorbed solute mass per formation solid volume), d is the localscale dispersion tensor in solution phase, Kc and K,, are the solute degradation rates in the solution and sorbed phases, respectively, and where V is the locally homogeneous but microscopically stochastic Darcy velocity. It is further assumed that the mean flow is constant in the xl-direction so that V = (V, O,O) and that the dispersion tensor is diagonal and constant with d, the Darcy-scale longitudinal, d2 the Darcy-scale transversehorizontal and d3 the Darcy-scale transverse-vertical dispersion coefficients, respectively. It is important to note that in the Eulerian framework we do not neglect local-scale dispersion as is commonly done in the Lagrangian framework (cf., Dagan5). We further assume the concentrations C and S are related by a rate equation of the form as — = at

Kr(K~C – S) = Kf C – KbS

(5)

(7)

where repeated indices imply summation. Because the mean flow is constant and in the xldirection with ~1 = V, the mean equation corresponding to eqn (7) is

–KCC– kCc– K,YS– krs

(8)

and the mean removed equation is 2 ~

●~

.

dig

– 1

ac

–‘i~

V&–

~i~

ax,

,

G — – K(C– k,~ – kCc+ kCc + dxj

– @ – k,y~ – kfs + k,ys

(9)

In deriving eqn (9) we have assumed vi is divergence-free. Substitute eqn (6) into eqn (5) to obtain a(~;

s) = (~f +kf)(C+

C) - (I?h

+kb)(~+s)

(10)

The mean equation corresponding to eqn (10) is

where Kd is the partition coefficient which controls localscale chemical exchange, Kr is a reaction rate parameter between solution and sorbed phases, Kf = K,& is the forward reaction rate and Kh = K, is the backward reaction rate. In subsequent analysis it is assumed in K (K is conductivity), Kf, Kb, Kc and K, are random, so that C, S, and V are also random variables. In the usual fashion decompose the log-hydraulic conductivity, partition coefficient, degradation rates, velocity, and concentrations into means and fluctuations about the means, in K = F + f

(6a)

Kf = Kf + kf

(6b)

Kh = ~b i- kb

(6c)

Kc = K,, + k
Kc= 001 day-’, and K, = 0.003 day-’; (a) mass in solution phase; (b) first longitudinal moment; (c) second longitudinal moment; (d) second transverse moment; (e) skewness.

Nonlocal reactive transport with physical, chemical, and biological heterogeneity Recall some basic properties of Laplace and Fourier transforms: L[df/dt] = L@ f] -f(0)

(14a)

-L[j- *, g] = L[ f]L[g]

(14b)

F[d”f/dx’] = (ik)’F[f]

(14C)

F[f *Xg] = F[f]lqg]

(14d)

F[fg] = (27r-~F[f]

*~ I’[g]

(14e)

where N is the dimensionality of the system. Here the asterisk indicates the convolution operator and the subscript indicates the variable with respect to which it operates. Apply space-Fourier and time-Laplace transforms to eqns (9) and (12) and use (14) to get

297

Let -.—1 (w+ ‘~Kf B (k, ti) = W +

+ XC + dik~ + ikl ~

(19)

W+ &

and w + KS G(k, w) = ;(k, w)—

(20)

W+ &

Then eqn (18) becomes ~ = – G(k, w)[kfc –

– kb~ + kfc – @

—-

– kbs + kbS)”]

6’C ac = ‘ ~(k, (.J) ‘im + ‘i~ – dxi [( -) —.

+ (kcc + kCc – k,c)” + (k.~ + k,s – ~)’

(21) 1 Take inverse Laplace and Fourier transforms of eqn (21) to obtain

1

tw(y, t’) C(X> t)= – ~ ~,B(x – y,t– t’)vi(y) ~yi J/

[

—. - (/kcc+ /kcc- /ccc)” –

IZ$– (k,~ + k,s

– ~-)’

+ ‘i(Y)

(15)

and

‘C(Y’ dyi “)- ~

(y, t’)+ kC(y)C(y, t’)

+ kc(y)c(y, t’) – k,c(y, t’) + k,(y)~(y,

+ (kf C – k/$ + kfc i = (w + &-l [~f:

+ k,(y)s(y,

—– kfC – k@ + kbs)”]

t’) – k,s(y,

t — ~ ~, G(x – y,t–

(16)

H

In deriving eqns (15) and (16) we assumed C(X,O) = O and S(X,O) = O, respectively. Substitute eqn (16) into (15) to obtain

t’)]dydt’

t’)[kf(y)~(y, t’)

– kb(y)~(y, t’) + kf (y)c(y, t) – kf c(y> t’)

1

– kb(y)s(y, t’) + k~S(y, t’) dydt’ w;

+

w + K,r z ~ [K~c+ (kf C - kbs + kfc —– kfC – kbs + kbs)”]

–

Kc~– (k,~

——. + k,c – kCc)A– (k,~ + k,s – k,s)” (17)

and rearrange to find

t’)

(22)

Equation (22) is implicit and obtained without approximation Two types of correlation fuctions will be obtained in subsequent analysis. One depends on space alone, and the other involves both space and time. We hencefore assume correlation functions involving space alone are stationary, but correlation functions involving both space and time are not. Under these assumptions, after multiplying eqn (22) by Wj(x)jtaking expectations, and neglecting triplet correlations we obtain t B(x – J’,t – t’) VjC(X, t)= –

H

O R3

x

— = -~

(kf C - k# + kfc - ~

—- kbs + kbs)”

W ‘c ‘-(kCC+ ( ‘i% + ““i% – f3Xj ) —- (k$~ + k,s - k,s)”

—

—kCc- k,c)”

W(X

m(y, t’) — ~yi + Vjk.(x – Y)C(Y, t’)

[

+Vjk.(x –y)~(y, t’) 1 dydt’

—

t

HO (18)

– y)

G(x – y, t – t’)[~jkf(x – Y)C(Y, t’) R3

– ?Jjkb(X – y) A$(y7t’)]dydt’

(23)

B. X. Hu et al.

5

A

Kc= 0.03

-

-5

0

5

5

10

15

20

25

30

35

40

45

I

I

I

I

I

I

I

I

I

50

Kc= 0.01

B ~

0.1

0.05 0.03 0.02 0.01

15

20

25

0.005

-5

0

5

10

30

35

40

45

50

X(m)

5

c

Kc= 0.003

E. s’ -5

o

I

1

I

I

5

10

15

20

I

25

1

30

I

35

I

I

40

45

X(m) Fig. 3. Concentration contours for various K, (K.,= K,/B) at t = 40 daY‘or ‘f = ‘h = ‘aY-’”

1 50

299

Nonlocal reactive transport with physical. chemical, and biological heterogeneity In a similar fashion (see Appendix) one finds kfc, kbc, k,c, and k$c. Apply Laplace and Fourier transforms to eqn (23) to obtain x ~=.—

1

{[(

(27r)3

B*k @$)iki +B*kV,kc +G*k vjkf C ‘1 (24)

+ [(i*k~~k,)- (G *~~j~~)]~} Again

in a similar

k:s = ;3 (k, @

f e-&(t-t’) [~fc(x, t’) + q(x)c(x, 10

;Jk, u) = (w+ ~b)-’

(25)

Multiply eqn (25) by kb(x), take ensemble averages, and neglect triplet correlations to yield J

‘ -~~(r-~’)[~f~(x, ~e

KfA

{[ (27r)3

B*,

k,k$ (34)

Apply space-Fourier and time-Laplace transforms to eqns (8) and (11) to obtain x—

x

z—

x—

– (26)

—

–G*kk‘1 ,:k~ + k$kfi(0) ‘}

t’) + kbkf(0)C(x, t’)

– kbkb(())~(x, t’)] dt’

–djkf C – ikl VC – ikj$

+ WS – s. =

KCC –k:c – K,,S – k:s

(35)

and

Take Laplace and Fourier transforms of eqn (26) to yield & = (u+ x~)-’ [I&c

+ ~(o)c

- ZX(o)$]

(36)

(27) Insert eqns (A5) and (28) into eqn (36), and rearrange the terms to obtain

Insert eqn (A6) into eqn (27) to obtain k:s = ;I (k, w): + ;2(k, u)~

(33)

and

WC – t.

kbs(x, t) =

{-&[(B*@k

+B*kk:kC + 8 *, ‘1 k.:kf + k,,kf(0) ‘1

– kb(X)~(X, t’) + kf (x)c(x, t’) – kfc(x, t’) – kb(X)S(X, t’) + kbs(x, t’)]dt’

(32)

+ ;4(k, w)~

;4(k, w) = – (W+ &-l

0

transforms to

where

fashion one obtains k~c, k~c, k~c, and

k~c (see Appendix). Take Laplace and Fourier inverse transforms of eqn (16) to obtain s=

Apply time-Laplace and space-Fourier eqn (31), then use eqn (A8) to yield

(28)

s=&[&

+;,(k, w):]/;,

(k, W)

(37)

with where j (k, u) = (w+ &-l

and

{-&’[(B*k@ik

+ B*, k;k, + 6 *, kl;kf + kfikf(o) ‘1 ‘}

(29)

kjz, – G*, kfkb

1

{*[(’*,=)

- ( C.k=)]+m(”)}

and (30)

f -R,((-t’) [~f~(x, t’) + k,~f(0)C(x, “) ~e

– k,,kb(o)~(x, t’)] dt’

~z(k, W) = Kf – &[(’*kfi)ik’+’*k=

Following the same procedure, one can also multiply eqn (25) by k,, take expectations and neglect triplet correlations to obtain

/

‘)

(38)

+ ~Jk, w)

~Jk, w) = – (LJ+ &-l

k,,s(x, t) =

1

1

j*, fl(k, w) = 1 +— — (27r)3 w + Kfi [(–

(31)

+6*,

Ik;~f]

–j(k,

w)

(39)

Insert eqns (24), (A7), (32) and (37) into eqn (35), and rearrange terms to obtain C(k, w) = ~-’ (k, u) ICo(k) + ~(k, w)~o(k)]

(40)

B. X. Hu et al.

300

0.6

—

Kc=O.003 --Kc=O.01 –––- Kc=O.03

a \ \

b

50 -

0.5i ‘!.. 40 -

x- 30 -

20 — 10 -

I

t 0.0t 0

1

I

I

I

I

I

20

40

60

80

100

Kc=O.003

--------- Kc=O.01 –––- Kc=O.03

o 20

0

40

60

80

100

t

c

d

120 100 -

...... ......

1.0 80 : x

x“ 60 -

0.540 — Kc=O.003 --Kc=O.01 –––- Kc=O.03

20 -

o 20

0

40

60

— Kc=O.003 -Kc=O.01 –––- Kc=O.03

1/,,,,,,,,,,,,,,,,,,,,,,,,,

100

80

t

0.0 0

20

40

60

80

100

t

0.8,

e

0.7

//

/

/’

0.6 0.5 ~ 0.4 5 ; 0.3 0.2 — -----

0.1 / 0.0 -0.1 , o

Kc=0.003 Kc=O.01 Kc=O.03

I

I

I

I

I

20

40

60

80

100

t

Fig.4. Spatial moments for a reactivechemicalat various K. (K, = RJ3) as a function of time (~)for ~f = & = 1day-’; (a) mass in solution phase; (b) first longitudinal moment; (c) second longitudinal moment; (d) second transverse moment; (e) skewness.

301

Nonlocal reactive transport with physical, chemical, and biological heterogeneity eqn (8) to obtain

where

#c t?c as ~ + ~ – dJ ~

~(k, w) = w + ajk~ + ikl V + K,

+

=

/

. ,. + B*k vjk,. + G *k vjkf

dc(y, “) dydtl

dyi

{

H ~

R3B(X

e

HO

B(x – y, t – t’)

0

-K,(l-/’)B(x – y, e

t– H –y,

(k, w)

t’ – t“)

/R3

X [k,,k,,(x – y)~(y, t“) + k,,k,,(x – y)~(y, t“)]dydt” t — G(x t’)[Vjkf(x– Y)c(y,t’) O R3 t – Vjkb(X – y)~(y, t’)]dydt’ – () G(x – y, t – t’)

A 1- *[P*’=) (27r) -‘*kvJkb

/(W + &)~l

t’)+ k,.k,,(x – Y) L$(Y>t’)ldyd+

– y)~(y,

HO Vjk,r

-Kh(~-/I)B(x– y, t’ – t“)k,,vi(x – Y)

dyi

/ t’

and

mY, t’) dydtl ~yi

/ R3

0

+Kf

+~@,w)1

– Y)~(Y, t’)] dydt’

t’)k ~ikf, ~ kfk or uncorrelated with f, Hu et al. ’” studied the effect @f and fib have on reactive chemical transport under nonequilibrium sorption without degradation. In this article we focus on the effect of degradation on chemical transport. For illustrative purposes we assume a linear negative correlation between kf and f and we assume kb and f are linearly positively

Nonlocal reactive transport with physical, chemical, and biological heterogeneity

303

60 L

0.60

a

b

““””””””””” Id=o.l

1-

50

— l&l.o ‘––” 1~=2.0 ‘“–”” ld=4.0

0.55 0,50

40

0.45 z

x- 30 0,40 20

0.35 0.30

10

0.25

‘

Icf=o.l

— –––

Icf=l.0 lcf=2.0

–-

ld=4.0

k

(3,2(3~

0

o

20

40

60

100

80

20

o

40

80

60

100

t 1.0

c

d

100

80

x:

x- 60

0.5

40

20

“-’

Icf=o.l

— –––

Icf.l.o ld=2.0

‘“

— Icf.l .0 –––- lcf=2.0 –’–” lcf=4.0

–“–’” lcf=4.0

0.0

n 20

-o

60

40

100

80

.

o

I

1

I

I

20

40

60

80

I

100

t

0,7

./-

e 0.6

Icf=o,l

,/>-.-——:—-”——-= =-=----................. -”’”””””””””””””

0.5 0.4

0.2 “--’

Icf=o.l

0.1

— Id=l.o –––- lcf=2.0

0,0

–-’

.0,

lcf=4.0

~

o

20

40

60

80

100

t Fig. 5. spatial moments for a reactive chemical at various Lf as a function of time (0 for & = & = 1 day-’, K, = Ool day-’ t K, = 0.003 day “, 1= 1m, and 1,.= 0.5 m; (a) mass in solution phase; (b) first longitudinal moment; (c) second longitudinal moment; (d) second transverse moment; (e) skewness.

304

B. l’. flu et al.

correlated /kf(x) = –