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The role of molecular diffusion in dispersion theory
Journal of Hydrology, 19 (1973) 361-365 © North-Holland Publishing Company, Amsterdam - Printed in The Netherlands
Short Communication THE ROLE OF MOLECULAR DIFFUSION IN DISPERSION THEORY Z. WEINBERGER and S. MANDEL
Hebrew University, Jerusalem (IsraeO (Accepted for publication January 4, 1973)
ABSTRACT Weinberger, Z. and Mandel, S., 1973. The role of molecular diffusion in dispersion theory. J. Hydrol., 19: 361-365. The analytic derivations of the equations governing dispersive flow assume that different solutions entering an interstitial flow channel are completely mixed at the exit node of the flow channel. The physical mechanisms which can effect this mixing are turbulence and molecular diffusion between the solutions in the channel. We have examined these mechanisms and find for the interstitial velocities ordinarily encountered in groundwater flow that turbulence is not effective as a mixing mechanism whereas molecular diffusion is. For molecular diffusion to be efficient as a mixing mechanism the interstitial velocities should be less than: u < 2.25 ks h2 where k is the diffusion coefficient of the salt solution; s the mean length of the interstitial channels and h the channel diameter. This velocity places an upper limit to the range of validity of the equations of dispersion theory as presently developed.
DISCUSSION AND CONCLUSIONS
The theoretical derivations of the equations of dispersive flow are based on analyses of network models in which the interstices of the porous medium are replaced by straight elementary channels. Lagrangian analyses of such models have been made by De Josselin de Jong (1958) and by Saffman (1959). More recently the present authors have reformulated the foundations of dispersion theory using an Eulerian description (1972). The fundamental assumption of all these studies is that the different solutions entering an elementary channel at an entrance node are completely mixed at the exit node. The theory of dispersive flow for situations where the mixing hypothesis is not valid has yet to be developed. Qualitatively we can, however,
362
" Z. W E I N B E R G E R
A N D S. M A N D E L
say that parameters, which in ordinary dispersive theory are independent of flow velocities, become dependent upon interstitial flow velocities in a complex manner when the mixing hypothesis is not satisfied. In the Lagrangian formulation the complete mixing hypothesis is introduced by stating that the path of elementary tagged particles through the channels is a Markovian process and in the Eulerian description it is the solutions themselves which are mixed in the channels. The two formulations of the mixing hypothesis are of course equivalent. The nature of the mixing process itself is not considered further in the above studies. Two mechanisms suggest themselves as being responsible for mixing: turbulence and molecular diffusion. Let us examine these effects in turn. Turbulent flow will develop in a channel when the value of the Reynolds number R = u6/v becomes greater than a constant whose value is dependent on the channel geometry; R = Reynolds number; u = mean fluid velocity; 6 = geometric parameter channel, diameter for cylinders; and v = kinematic viscosity ~ 0.1 cm 2/sec. For cylindrical channels the Reynolds number for the onset of turbulence is greater than 2,300 even for highly irregular inflow (Prandtl, 1952). Assuming a channel diameter of 0.1 ram, we find for the onset of turbulent flow that: u > 2 m/sec
(1)
Such a velocity is orders of magnitude greater than natural flow velocities in porous media. Although natural flow channels are tortuous even for the short distances between nodes, the intuitive extrapolation of the result of eq. 1 is that the flow in the elementary channels is highly laminar. Another, and possibly stronger, argument for laminar flow in the elementary channels is provided by Darcy's law which is valid for the bulk flow of liquid in a porous medium over a wide range of flow rates. If the flow patterns in the elementary channels were to change from laminar to turbulent with increasing bulk flows, then Darcy's simple linear relations should cease to describe the bulk flow. We may therefore conclude that turbulence is not efficient as a mixing mechanism. Let us now examine the molecular diffusion between the two solutions flowing in a laminar manner in the elementary channels. For simplicity we assume that the two solutions at the entrance node of the elementary channel have equal flow rates. Let the height of the two-dimensional channel be h, and S the length of the channel (see Fig. 1). We will assume that the velocity along the channel height is uniform (not Poiseuille) and equal to its mean velocity u. The time spent by the solutions in the channel is then given by S/u. From elementary considerations it follows that the concentration distribu-
THE ROLE OF MOLECULAR DIFFUSION IN DISPERSION THEORY
/),-
363
C
,.
Fig. 1. Model elementary flow channel.
tion at the exit n o d e will be given b y (Carslaw and Jaeger, 1959):
C(x) = ~1( C
+ C ~ ) - ~ ( C2~ - C
a)
~ = 1 e x p (- Kn27r h 2 :S)cosmrhsin u
2
(2)
w h e r e C a = u n i f o r m c o n c e n t r a t i o n o f i n f l o w f r o m x = 0 to x -- h/2; C~ = unif o r m c o n c e n t r a t i o n o f i n f l o w f r o m x = hi2 to x = h ; K = d i f f u s i o n c o e f f i c i e n t o f salt s o l u t i o n ~ 0.94 c m : / d a y ; and x = vertical c o o r d i n a t e in c h a n n e l measu r e d f r o m the c h a n n e l base. Choosings/h ~ 10 and h ~ 0.01 cm, we find:
z? KS
- h2
~ 9,400 cm/day
(3)
F o r u less t h a n the a b o v e value, the first t e r m o f the e x p a n s i o n o f eq. 2 will be s u f f i c i e n t l y a c c u r a t e f o r o u r e s t i m a t i o n s . F r o m eq. 2 we have t h e n as an a p p r o x i m a t i o n for the c o n c e n t r a t i o n at x = 0:
1 c(o) = ~(c. + c ~ ) - 2(C ~ - c ) . ~
(4a)
where: 7 = e x p lI
K~2~
uh 2 1
= exp
9,400
u
and at x = h:
1 +c~)+ 2(C~ - C ). C(h) = -~(C
(4b)
L e t us a r b i t r a r i l y c h o o s e as o u r m e a s u r e o f h o m o g e n e i t y t h a t the d i f f e r e n c e o f c o n c e n t r a t i o n at the e x t r e m e s cff the exit n o d e s o f the c h a n n e l is less t h a n
1% o f
t h e m e a n c o n c e n t r a t i o n , i.e., that:
IC(O) - C(h)l _ 4 I C o - C¢~I ½(C~ + C ) ~r %(C,~ + Co) " ")' < 0"01
(5)
364
Z. WEINBERGER AND S. MANDEL
For a worst case analysis, we assume that C a >> C a then the ratio of eq. 5 becomes: 8
- ~ / < 0.01 71"
or:
7r2 K S t , t < - h2
(6)
. 800
which, for the numerical parameters chosen above, gives: u < 17 m / d a y i.e., for velocities normally encountered in groundwater flow the mechanism of molecular diffusion ensures homogeneity of the solutions at the exit nodes of elementary channels. A less conservative estimate for the restrictions on u may be obtained by adopting for the measure of homogeneity the relative difference in the mean concentrations of the top and b o t t o m halves of the channel instead of the concentration at the extremes. The mean concentrations are obtained by integration of eq. 2 for the two halves of the channel. Following the reasoning which leads to eq. 6 we obtain: zr2K S
KS
h 2 lnSOO
h2
u<---2.25-rr2
or:u=<
20 m/day
We remark that, for real flow patterns, the time taken for the fluid to reach the end of a channel is greater for the boundary region (x = 0 and x = h) than the center, a consequence of which is that the exiting fluid is even more homogeneous than that predicted by the uniform flow approximations of this analysis. The above analysis does, however, imply that the equations of dispersion, which have been derived with the help of the complete mixing hypothesis from the analysis of models in which the velocities of flow in the elementary channels are randomly distributed, must be applied with caution when short term phenomena are measured. In particular the first tracer particles measured in field experiment may not have diffused sufficiently in the elementary channels since the condition of eq. 6 may be violated for rapidly transversed paths. It will then be expected that gravitational forces will control the spatial distribution of tracer particles at the start of a temporal measurement of injected tracer particles. Eq. 6 complements the lower b o u n d for the velocity of interstitial flow for which dispersion theory remains valid discussed by Saffman (1959, 1960). For
THE ROLE OF MOLECULAR DIFFUSION IN DISPERSION THEORY
365
the equations of dispersive flow to be valid the flow velocities must be sufficiently large so that molecular diffusion can be neglected when compared to the dispersion due to the random hydraulic flow through the elementary channels. This lower bound is given by (Saffman, 1960): K u>>-~which for representative values of parameters chosen above yields: u >~ 10 c m / d a y .
REFERENCES Carslaw, H.S. and Jaeger, J.C., 1959. Conduction of Heat in Solids. Oxford University Press, London, 2nd ed., De Josselin de Jong, G., 1958. Longitudinal and transverse diffusion in granular deposits. Trans. Am. Geophys. Union, 39: 6 7 - 7 4 . Mandel, S. and Weinberger, Z., 1972. Analysis of a network model for dispersive flow. J. Hydrol., 16: 147-157. Prandtl, L., 1952. Essentials of Fluid Dynamics. Blackie, Glasgow. Saffman, P.G., 1959. A theory of dispersion in a porous medium. J. Fluid Mech., 6: 321-349. Saffman, P.G., 1960. Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries. J. Fluid Mech., 7: 194-208.
Short Communication THE ROLE OF MOLECULAR DIFFUSION IN DISPERSION THEORY Z. WEINBERGER and S. MANDEL
Hebrew University, Jerusalem (IsraeO (Accepted for publication January 4, 1973)
ABSTRACT Weinberger, Z. and Mandel, S., 1973. The role of molecular diffusion in dispersion theory. J. Hydrol., 19: 361-365. The analytic derivations of the equations governing dispersive flow assume that different solutions entering an interstitial flow channel are completely mixed at the exit node of the flow channel. The physical mechanisms which can effect this mixing are turbulence and molecular diffusion between the solutions in the channel. We have examined these mechanisms and find for the interstitial velocities ordinarily encountered in groundwater flow that turbulence is not effective as a mixing mechanism whereas molecular diffusion is. For molecular diffusion to be efficient as a mixing mechanism the interstitial velocities should be less than: u < 2.25 ks h2 where k is the diffusion coefficient of the salt solution; s the mean length of the interstitial channels and h the channel diameter. This velocity places an upper limit to the range of validity of the equations of dispersion theory as presently developed.
DISCUSSION AND CONCLUSIONS
The theoretical derivations of the equations of dispersive flow are based on analyses of network models in which the interstices of the porous medium are replaced by straight elementary channels. Lagrangian analyses of such models have been made by De Josselin de Jong (1958) and by Saffman (1959). More recently the present authors have reformulated the foundations of dispersion theory using an Eulerian description (1972). The fundamental assumption of all these studies is that the different solutions entering an elementary channel at an entrance node are completely mixed at the exit node. The theory of dispersive flow for situations where the mixing hypothesis is not valid has yet to be developed. Qualitatively we can, however,
362
" Z. W E I N B E R G E R
A N D S. M A N D E L
say that parameters, which in ordinary dispersive theory are independent of flow velocities, become dependent upon interstitial flow velocities in a complex manner when the mixing hypothesis is not satisfied. In the Lagrangian formulation the complete mixing hypothesis is introduced by stating that the path of elementary tagged particles through the channels is a Markovian process and in the Eulerian description it is the solutions themselves which are mixed in the channels. The two formulations of the mixing hypothesis are of course equivalent. The nature of the mixing process itself is not considered further in the above studies. Two mechanisms suggest themselves as being responsible for mixing: turbulence and molecular diffusion. Let us examine these effects in turn. Turbulent flow will develop in a channel when the value of the Reynolds number R = u6/v becomes greater than a constant whose value is dependent on the channel geometry; R = Reynolds number; u = mean fluid velocity; 6 = geometric parameter channel, diameter for cylinders; and v = kinematic viscosity ~ 0.1 cm 2/sec. For cylindrical channels the Reynolds number for the onset of turbulence is greater than 2,300 even for highly irregular inflow (Prandtl, 1952). Assuming a channel diameter of 0.1 ram, we find for the onset of turbulent flow that: u > 2 m/sec
(1)
Such a velocity is orders of magnitude greater than natural flow velocities in porous media. Although natural flow channels are tortuous even for the short distances between nodes, the intuitive extrapolation of the result of eq. 1 is that the flow in the elementary channels is highly laminar. Another, and possibly stronger, argument for laminar flow in the elementary channels is provided by Darcy's law which is valid for the bulk flow of liquid in a porous medium over a wide range of flow rates. If the flow patterns in the elementary channels were to change from laminar to turbulent with increasing bulk flows, then Darcy's simple linear relations should cease to describe the bulk flow. We may therefore conclude that turbulence is not efficient as a mixing mechanism. Let us now examine the molecular diffusion between the two solutions flowing in a laminar manner in the elementary channels. For simplicity we assume that the two solutions at the entrance node of the elementary channel have equal flow rates. Let the height of the two-dimensional channel be h, and S the length of the channel (see Fig. 1). We will assume that the velocity along the channel height is uniform (not Poiseuille) and equal to its mean velocity u. The time spent by the solutions in the channel is then given by S/u. From elementary considerations it follows that the concentration distribu-
THE ROLE OF MOLECULAR DIFFUSION IN DISPERSION THEORY
/),-
363
C
,.
Fig. 1. Model elementary flow channel.
tion at the exit n o d e will be given b y (Carslaw and Jaeger, 1959):
C(x) = ~1( C
+ C ~ ) - ~ ( C2~ - C
a)
~ = 1 e x p (- Kn27r h 2 :S)cosmrhsin u
2
(2)
w h e r e C a = u n i f o r m c o n c e n t r a t i o n o f i n f l o w f r o m x = 0 to x -- h/2; C~ = unif o r m c o n c e n t r a t i o n o f i n f l o w f r o m x = hi2 to x = h ; K = d i f f u s i o n c o e f f i c i e n t o f salt s o l u t i o n ~ 0.94 c m : / d a y ; and x = vertical c o o r d i n a t e in c h a n n e l measu r e d f r o m the c h a n n e l base. Choosings/h ~ 10 and h ~ 0.01 cm, we find:
z? KS
- h2
~ 9,400 cm/day
(3)
F o r u less t h a n the a b o v e value, the first t e r m o f the e x p a n s i o n o f eq. 2 will be s u f f i c i e n t l y a c c u r a t e f o r o u r e s t i m a t i o n s . F r o m eq. 2 we have t h e n as an a p p r o x i m a t i o n for the c o n c e n t r a t i o n at x = 0:
1 c(o) = ~(c. + c ~ ) - 2(C ~ - c ) . ~
(4a)
where: 7 = e x p lI
K~2~
uh 2 1
= exp
9,400
u
and at x = h:
1 +c~)+ 2(C~ - C ). C(h) = -~(C
(4b)
L e t us a r b i t r a r i l y c h o o s e as o u r m e a s u r e o f h o m o g e n e i t y t h a t the d i f f e r e n c e o f c o n c e n t r a t i o n at the e x t r e m e s cff the exit n o d e s o f the c h a n n e l is less t h a n
1% o f
t h e m e a n c o n c e n t r a t i o n , i.e., that:
IC(O) - C(h)l _ 4 I C o - C¢~I ½(C~ + C ) ~r %(C,~ + Co) " ")' < 0"01
(5)
364
Z. WEINBERGER AND S. MANDEL
For a worst case analysis, we assume that C a >> C a then the ratio of eq. 5 becomes: 8
- ~ / < 0.01 71"
or:
7r2 K S t , t < - h2
(6)
. 800
which, for the numerical parameters chosen above, gives: u < 17 m / d a y i.e., for velocities normally encountered in groundwater flow the mechanism of molecular diffusion ensures homogeneity of the solutions at the exit nodes of elementary channels. A less conservative estimate for the restrictions on u may be obtained by adopting for the measure of homogeneity the relative difference in the mean concentrations of the top and b o t t o m halves of the channel instead of the concentration at the extremes. The mean concentrations are obtained by integration of eq. 2 for the two halves of the channel. Following the reasoning which leads to eq. 6 we obtain: zr2K S
KS
h 2 lnSOO
h2
u<---2.25-rr2
or:u=<
20 m/day
We remark that, for real flow patterns, the time taken for the fluid to reach the end of a channel is greater for the boundary region (x = 0 and x = h) than the center, a consequence of which is that the exiting fluid is even more homogeneous than that predicted by the uniform flow approximations of this analysis. The above analysis does, however, imply that the equations of dispersion, which have been derived with the help of the complete mixing hypothesis from the analysis of models in which the velocities of flow in the elementary channels are randomly distributed, must be applied with caution when short term phenomena are measured. In particular the first tracer particles measured in field experiment may not have diffused sufficiently in the elementary channels since the condition of eq. 6 may be violated for rapidly transversed paths. It will then be expected that gravitational forces will control the spatial distribution of tracer particles at the start of a temporal measurement of injected tracer particles. Eq. 6 complements the lower b o u n d for the velocity of interstitial flow for which dispersion theory remains valid discussed by Saffman (1959, 1960). For
THE ROLE OF MOLECULAR DIFFUSION IN DISPERSION THEORY
365
the equations of dispersive flow to be valid the flow velocities must be sufficiently large so that molecular diffusion can be neglected when compared to the dispersion due to the random hydraulic flow through the elementary channels. This lower bound is given by (Saffman, 1960): K u>>-~which for representative values of parameters chosen above yields: u >~ 10 c m / d a y .
REFERENCES Carslaw, H.S. and Jaeger, J.C., 1959. Conduction of Heat in Solids. Oxford University Press, London, 2nd ed., De Josselin de Jong, G., 1958. Longitudinal and transverse diffusion in granular deposits. Trans. Am. Geophys. Union, 39: 6 7 - 7 4 . Mandel, S. and Weinberger, Z., 1972. Analysis of a network model for dispersive flow. J. Hydrol., 16: 147-157. Prandtl, L., 1952. Essentials of Fluid Dynamics. Blackie, Glasgow. Saffman, P.G., 1959. A theory of dispersion in a porous medium. J. Fluid Mech., 6: 321-349. Saffman, P.G., 1960. Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries. J. Fluid Mech., 7: 194-208.