Theory for dynamic longitudinal dispersion in fractures and rivers with Poiseuille flow

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 39,, doi: /2011gl050831, 2012 Theory for dynaic longitudinal dispersion in fractures and rivers with Poiseuille flow Lichun Wang, 1 M. Bayani Cardenas, 1 Wen Deng, 1 and Philip C. Bennett 1 Received 31 Deceer 2011; revised 3 Feruary 2012; accepted 5 Feruary 2012; pulished 3 March [1] We present a theory for dynaic longitudinal dispersion coefficient (D) for transport y Poiseuille flow, the foundation for odels of any natural systes, such as in fractures or rivers. Our theory descries the ixing and spreading process fro olecular diffusion, through anoalous transport, and until Taylor dispersion. D is a sixth order function of fracture aperture () or river width (W). The tie (T) and length (L) scales that separate preasyptotic and asyptotic dispersive transport ehavior are T = /(4D ), where D is the olecular diffusion coefficient, and L = 4 p 48D, where p is pressure and is viscosity. In the case of soe ajor rivers, we found that L is 150W. Therefore, transport has to occur over a relatively long doain or long tie for the classical advectiondispersion equation to e valid. Citation: Wang, L., M. B. Cardenas, W. Deng, and P. C. Bennett (2012), Theory for dynaic longitudinal dispersion in fractures and rivers with Poiseuille flow, Geophys. Res. Lett., 39,, doi: /2011gl Introduction [2] Scalar ixing and spreading processes, which are typically represented y soe diffusion or dispersion coefficient in transport equations, are fundaental to any geophysical systes and engineering applications. Mass transport driven y a concentration gradient is conventionally assued to oey a diffusive process or is at least descried y a diffusion-type equation. In a stratified flow field, the velocity profile due to shear stress enhances the ixing/spreading process resulting in so-called Fickian dispersion which is encapsulated in an effective dispersion coefficient. Taylor [1953] first showed that at soe long enough tie scale the ixing/spreading process through a tue follows Fickian ehavior with a longitudinal dispersion coefficient. Later, Fischer et al. [1979] derived a corresponding longitudinal dispersion coefficient for a river also at a large tie scale. Güven et al. [1984] analyzed horizontal transport through aquifers and showed that differences in stratified groundwater flow velocity caused y vertically-varying hydraulic conductivity would also lead to a Fickian dispersive process at soe large enough scale. In such circustances, the classical advection-dispersion (or diffusion) equation (ADE) is valid. However, at preasyptotic tie scales, the classical ADE 1 Departent of Geological Sciences, University of Texas at Austin, Austin, Texas, USA. Copyright 2012 y the Aerican Geophysical Union /12/2011GL is invalid due to anoalous early arrival and persistent tails in reakthrough curves oth in fractured and porous edia [Berkowitz, 2002]. This phenoenon is referred to as non-fickian ehavior. [3] Non-Fickian transport can e atheatically represented in any ways ut one siple approach is to define a dynaic longitudinal dispersion coefficient (D) for the ADE. Several researchers have studied and quantified dynaic D using various approaches including spatial oent analysis [Dentz and Carrera, 2007], series expansion ethods [Gill and Sankarasuraanian, 1970], center anifold description [Mercer and Roerts, 1990], and Lagrangian approach [Haer and Mauri, 1988]. These studies showed that D increases onotonically fro its value at preasyptotic tie scale to the value according to Taylor s theory. Yet, the theoretical analysis y Taylor [1953] for a tue and Fischer et al. [1979] for a river at asyptotic tie scales is not sufficiently coplete. [4] There are three key assuptions adopted y oth Taylor [1953] and Fischer et al. [1979]: (1) the Peclet nuer is sufficiently large (i.e., advection doinated transport) so as to ignore longitudinal diffusion; (2) the longitudinal advective ass flux is alanced y transverse diffusive ass flux, and the gradient of the cross-sectional averaged concentration in the longitudinal direction is at steady-state; and (3) the gradient of the cross-sectional averaged concentration in the longitudinal direction is uch greater than the gradient of concentration fluctuations. The validity of the aove assuptions has since een ignored and susequent studies have either retained these assuptions or circuvented the y following approaches that are not directly ased on the coplete transport equations. Here, we develop a ore general theory that does not require the first two assuptions, and using this theory we derive a closed-for expression for the dynaic longitudinal dispersion coefficient. Afterwards, we analyze the tie and length scales for distinguishing Fickian (asyptotic) and non-fickian (preasyptotic) transport regies. 2. Theoretical Developent 2.1. Two-Diensional Transport Model [5] Fischer et al. [1979] investigated the longitudinal dispersion coefficient for two-diensional, steady-state, fully-developed, lainar flow etween parallel plates. In this case, the advection-diffusion equation with appropriate oundary conditions is: t þ u ¼ D C 2 þ D C ð1þ C ¼ 0; 0 x < ; t ¼ 0 ð2þ 1of5

2 ¼ 0; y y ¼ =2 or =2 ð3þ C ¼ C 0 ; x ¼ 0; t > 0 ð4þ The cross-sectional averaging translates (11) into: t ¼ D C 2 ð13þ C ¼ 0; x ¼ ; t > 0 ð5þ where C is the concentration, x is the longitudinal direction, y is the transverse direction, u is the x-direction velocity that is only a function of y (unifor in x), D is olecular diffusion coefficient, t is tie, is the aperture of parallel plates (or the river width W), and C 0 is the constant inlet concentration One-Diensional Macroscopic Transport Model With Dynaic Dispersion Coefficient [6] Concentration and velocity in (1) can e decoposed into cross-sectional ean and fluctuation coponents: C ¼ C þ C and u ¼ u ð6þ where C and u are cross-sectional ean coponents, and C and u are fluctuations aout the ean. According to Taylor [1953], C can e descried y a one-diensional acroscopic transport odel written as: t þ u C ¼ D 2 C 2 To otain an explicit expression for D, we start fro the asic ADE (1). Sustituting (6) into (1) yields: t þ þ u C þ u t C C ¼ D 2 þ 2 C 2 þ 2 C Taking a coordinate transforation (9) and applying the chain rule (10) (see equations (S1) (S3) in the auxiliary aterial) 1 lead to: t þ ð7þ ð8þ x ¼ x ut and t ¼ t ð9þ ¼ ; ¼ D t ¼ u þ t C 2 þ 2 C 2 þ 2 C ð10þ ð11þ Instead of neglecting longitudinalrolecular diffusion, we retain it and apply the averaging 1 /2 /2 (equation (11)) dy, and note that: where u is the cross-sectional averaged value of u. The second ter in (13) needs to e further analyzed as it is unknown and ideally it should e expressed in ters of 2 C 2. To this end, we sutract (13) fro (11), which gives the transport equation for C : u ¼ D C 2 þ 2 C ð14þ Since the longitudinal diffusion of C is uch less than transverse diffusion, i.e., 2 C 2 C 2 y, and since u 2 and u are the products of two typically relatively sall fluctuationrelated ters therefore aking the uch saller than the other ters in (14), and that we take the difference of these sall ters, we can effectively ignore the. Then (14) siplifies to: ¼ D C ð15þ At the asyptotic tie scale, when longitudinal advection is alanced y transverse diffusion, equiliriu can e assued resulting in: u C ¼ D C ð16þ The solution to (16) with oundary conditions (siilar to (3)): is: C ¼ ¼ 0; y ¼ =2 and =2 ð17þ y Z y Z y u dydy þ C D ð Þ ð18þ By definition, and allowing for the extra ter R /2 u C ( /2)dy = 0, the unknown second ter in (13) is therefore (see equations (S4) (S6) in the auxiliary aterial): u ¼ 1 Z C =2 Z y Z y D 2 u u dydydy ð19þ Sustituting (19) into (13) and expressing the result ack in the ordinary coordinate syste results in: Z 1 =2 u dy ¼ 0; Z 1 =2 C dy ¼ 0 ð12þ t þ u C ¼ D 1 Z =2 Z y Z! y C u u dydydy D 2 ð20þ 1 Auxiliary aterials are availale in the HTML. doi: / 2011GL of5

3 where p is the pressure gradient. (25) is widely-known as Cuic Law for flow through fractures [Ge, 1997]. Fro (25), the cross-sectional ean and fluctuation velocity coponents are calculated as: u ¼ 2 p and u ¼ 2 p 1 y ð26þ Therefore, in a paraolic (Poiseuille) velocity field (25), (26), the asyptotic (21) and dynaic (24) D can e respectively expressed as: Figure 1. D/D Taylor as a function of diensionless tie t = 4tD / following (28) and Dentz and Carrera [2007]. Therefore, the D in (7) at asyptotic tie scale (D Taylor ) is: D ¼ D 1 Z =2 Z y Z y u u dydydy D ð21þ However, (21) is only valid assuing longitudinal advection is alanced y transverse diffusion. Siplification of (15) into (16) is not valid if the goal is to descrie the transient dispersive processes. Therefore, in accordance with initial condition (2) and the no-flux oundary condition (17), we solved (15) directly through a unique Green function [Polyanin, 2002]: Gy; ð h; tþ ¼ 1 þ 2 X np cos ð y þ =2 Þ nphþ cos ð =2 Þ n¼1 exp D n 2 p 2 ðt tþ ð22þ The unknown second ter in (13) is now expressed as: u Z ( =2 ¼ 1 u Z t Z ) =2 u C Gy; h; t 0 ð Þdhdt dy ð23þ Assuing that the ter C is constant with tie, we are ale to extract it fro the integral operation and then do the anipulation as we did to otain (21). Finally, we get the expression for the dynaic D in the ordinary coordinate syste: D ¼ D þ 1 Z =2 u Z t Z =2 0 u Gðy; h; tþdhdtdy ð24þ 2.3. Paraolic Flow Model [7] The closed for expressions of (21) and (24) require the solution for the flow field which is wellknown for Poiseuille flow. The velocity field for fullydeveloped pressure-gradient driven flow in etween two parallel no-slip walls, e.g., fracture surfaces or river anks, is descried y: u ¼ 2 p 8 1 4y2 ð25þ 72 u D ¼ D þ ð p 6 D Þ2 X 1 n n¼1 6 D ¼ D þ ðuþ2 210D ð27þ ½ cos ð np Þþ1Š2 1 exp D n 2 p 2 t 3. Results and Discussion of Threshold Scales ð28þ [8] Our exact expression for dynaic D is valid across all transport regies (Figure 1) diffusive, anoalous, and Taylor dispersion and our theory agrees conceptually and qualitatively with results of scaling relationships for dynaic D derived via spatial oent analysis of nuerical siulation results [Latini and Bernoff, 2001]. Dentz and Carrera [2007] presented a theory, also derived through spatial oent analysis, for apparent dynaic D. Our results are equivalent to Dentz and Carrera [2007] despite the very different approaches (Figure 1). Beyond the diffusion regie, the dynaic D would increase asyptotically towards the value predicted y (27), which has een shown through less direct ethods [Gill and Sankarasuraanian, 1970; Güven et al., 1984; Haer and Mauri, 1988; Mercer and Roerts, 1990; Dentz and Carrera, 2007]. In contrast to previous studies, we developed an exact and coplete expression for dynaic D y direct solution of the general transport equations. [9] Within the anoalous preasyptotic regie, the dynaic D is found to increase rapidly at sall tie scale t < 0.1 (t =4tD / ), and it then varies slowly at a relatively large tie scale t > 0.5 and gets close to its axiu when t = 1 which corresponds to the tie it takes a particle to travel fro the center to the side oundary. By t = 1, the initial concentration would e copletely seared out, and the dynaics of the cross-sectional averaged concentration follows a acroscopic transport odel with constant D (27). [10] The tie scale for Taylor dispersion is therefore t = 1 or: T ¼ = ð4d Þ ð29þ whereas the counterpart length scale is: L ¼ u = ð4d Þ ð30þ Below T and L, the classical ADE with constant D is invalid and transport is anoalous. While this ehavior is well studied theoretically and experientally for other systes such as in heterogeneous porous edia [Güven 3of5

4 Figure 2. D/D Taylor as a function of diensionless length x/ (following equation (28)) showing the different length scales L for fractures with different apertures (following equation (31)). et al., 1984; Koch and Brady, 1987; Gelhar, 1993], to our knowledge, our work is the first direct solution of the prole for classic Poiseuille flow with few assuptions in the general transport theory and without resorting to spatial oent analysis. [11] Non-Fickian transport through fractures has een extensively studied at preasyptotic tie scales, ut few have highlighted the significance of length scales. To this end, we investigate the effect of aperture () and pressure gradient on L in straight fractures. Sustitution of u in (26) into (30) gives: L ¼ 4 p 48D ð31þ which shows strong sensitivity (fourth order) to. To further ephasize the iportance of, we calculated L for different using typical values of susurface pressure gradients and D = /s (typical for salt) (Figure 2). Additionally and perhaps ore iportantly, since u is a second order function of, the dynaic D depends on via a sixth order function. Moreover, L is only linearly related to the pressure gradient. Both oservations highlight as a critical paraeter. [12] Our theory for predicting dynaic D is also applicale to rivers. Since transverse velocity variation is typically 100 or ore ties ore effective copared to vertical velocity variation in causing longitudinal dispersion [Fischer et al., 1979], it is reasonale to siplify the strea into a planfor 2D transport prole, not different fro a fracture. For a strea with unifor water depth, Fischer Figure 3. D/D Taylor as a function of diensionless length x/ W (following equation (28)) showing the different length scales L for rivers with different widths W (following equation (31)). et al. [1979] showed that (21) is still valid ut with the transverse ixing coefficient ɛ t in the place of D. The transverse ixing coefficient for natural rivers is given y [Deng et al., 2001]: " ɛ t ¼ 0:145 þ u # W 1:38 u H ð32þ 3520u H where u * is the shear velocity descriing shear stress-related otion, W is the width, and H is the depth. In the sae sense, (24) can e odified y using ɛ t in the place of D for rivers. [13] Our theory works well for predicting D of natural straight rivers (Tale 1) y using reported u *, W, and H to calculate ɛ t. ɛ t is further applied y replacing D in (31) to estiate L. But unlike the significant effect of on L (which is analogous to W for rivers), L for the rivers we analyzed only varies over 1 2 orders of agnitude (Figure 3 and Tale 1) since the pressure gradient in wide streas is generally less than that in narrow streas, and the pressure gradient also corresponds to a relatively sall range (1 2 orders of agnitude) in ean velocity (Tale 1). Nonetheless, our calculations show that L varies around ties the strea width. This is uch larger than the rule of thu of 25 strea widths suggested as a ixing length when conducting strea tracer studies [Day, 1977]. Alternatively, Rutherford [1994] proposed a sei-epirical ethod for L: u L ¼ 0:23Hu ð33þ where is a coefficient which varies fro 1 10 for rough channels and where the denoinator is an approxiation for Tale 1. Coparison of Measured and Theoretical Dispersion Coefficients (D) Calculated Using Equation (28) at Asyptotic Tie Scale and Fro Deng et al. [2001], and Coparison of Theoretical Length Scales (L) Calculated With Equation (31) a River W () H () u (/s) u * (/s) D ( 2 /s) Measured Value Equation (28) Deng et al. T (hr) L/W ( ) L (Equation (31)) (k) L (Equation (33)) (k) Antieta Creek, MD Tangipahoa River, LA Bighorn River, WY Minnesota River Coite River Missouri River a The noralized L/W is calculated using L fro equation (31). 4of5

5 the transverse ixing coefficient. Calculations with (33) resulted in scales that are larger, and ay in fact e several orders of agnitude larger, than those calculated using (31) (Tale 1). Therefore, anoalous transport with a dynaic D in rivers can e relevant at scales uch larger than what (31) predicts. [14] We present a closed-for expression for dynaic dispersion coefficient y direct solution of the general transport equations. Using this, we analyzed the tie and length scales for typical fractures and soe large rivers when transport has copletely transitioned fro non-fickian to Fickian. Our theoretical approach can also e applied to transport of other scalars such as heat, or other related proles such as cases with ass transfer across fracture walls. Future work should focus on non-trivial oundary conditions to provide further insight on dynaic dispersive transport. [15] Acknowledgents. This aterial is ased upon work supported as part of the Center for Frontiers of Susurface Energy Security (CFSES) at the University of Texas at Austin, an Energy Frontier Research Center funded y the U.S. Departent of Energy, Office of Science, Office of Basic Energy Sciences under Award DE-SC Additional support was provided y the Geology Foundation of the University of Texas. [16] The Editor thanks the two anonyous reviewers for their assistance in evaluating this paper. References Berkowitz, B. (2002), Characterizing flow and transport in fractured geological edia: A review, Adv. Water Resour., 25, , doi: / S (02) Day, T. J. (1977), Field procedures and evaluation of a slug dilution gauging ethod in ountain streas, N. Z. J. Hydrol., 16, Deng, Z. Q., V. P. Singh, and L. Bengtsson (2001), Longitudinal dispersion coefficient in straight rivers, J. Hydraul. Eng., 127(11), , doi: /(asce) (2001)127:11(919). Dentz, M., and J. Carrera (2007), Mixing and spreading in stratified flow, Phys. Fluids, 19(1), , doi: / Fischer, H. B., E. J. List, R. C. Y. Koh, J. Ierger, and N. H. Brooks (1979), Mixing in Inland and Coastal Waters, Acadeic, San Diego, Calif. Ge, S. (1997), A governing equation for fluid flow in rough fractures, Water Resour. Res., 33(1), 53 61, doi: /96wr Gelhar, L. W. (1993), Stochastic Susurface Hydrology, Prentice-Hall, Englewood Cliffs, N. J. Gill, W. N., and R. Sankarasuraanian (1970), Exact analysis of unsteady convective diffusion, Proc. R. Soc. A., 316(1526), , doi: / rspa Güven, O., F. J. Molz, and J. G. Melville (1984), An analysis of dispersion in a stratified aquifer, Water Resour. Res., 20(10), , doi: /wr020i010p Haer, S., and R. Mauri (1988), Lagrangian approach to tie-dependent lainar dispersion in rectangular conduits. Part 1. Two-diensional flows, J. Fluid Mech., 190, , doi: /s Koch, D. L., and J. F. Brady (1987), A non-local description of advectiondiffusion with application to dispersion in porous edia, J. Fluid Mech., 180, , doi: /s Latini, M., and A. J. Bernoff (2001), Transient anoalous diffusion in Poiseuille flow, J. Fluid Mech., 441, , doi: / S Mercer, G. N., and A. J. Roerts (1990), A centre anifold description of containant dispersion in channels with varying flow properties, SIAM J. Appl. Math., 50(6), , doi: / Polyanin, A. D. (2002), Handook of Linear Partial Differential Equations for Engineers and Scientists, Chapan and Hall, Washington, D. C. Rutherford, J. C. (1994), River Mixing, John Wiley, New York. Taylor, G. (1953), Dispersion of solule atter in solvent flowing slowly through a tue, Proc. R. Soc. A., 219(1137), , doi: / rspa P. C. Bennett, M. B. Cardenas, W. Deng, and L. Wang, Departent of Geological Sciences, University of Texas at Austin, 1 University Station C9000, Austin, TX 78712, USA. (wlc309@ail.utexas.edu) 5of5