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Exact Analytical Solutions for Contaminant Transport in Rivers

Exact Analytical Solutions for Contaminant Transport in Rivers Contaminant transport processes in streams, rivers, other surface water bodies can be analyzed or predicted using the advection-dispersion equation related transport models. In part 1 of this two-part series we presented a large number of one- multi-dimensional analytical solutions of the stard equilibrium advection-dispersion equation (ADE) with without terms accounting for zero-order production first-order decay. The solutions are extended in the current part 2 to advective-dispersive transport with simultaneous first-order mass exchange between the stream or river zones with dead water (transient storage models), to problems involving longitudinal advectivedispersive transport with simultaneous diffusion in fluvial sediments or near-stream subsurface regions comprising a hyporheic zone. Part 2 also provides solutions for one-dimensional advective-dispersive transport of contaminants subject to consecutive decay chain reactions. Keywords: Contaminant transport; Analytical solutions; Surface water; Transient storage models; Solute decay chains. INTRODUCTION Contaminant transport processes in streams rivers have been analyzed predicted now for several decades using a range of mathematical models (Bencala, 1983; DeSmedt et al., 2007; Fisher et al., 1979; Runkel et al., 2003; Thomann, 1973). Because many complex often nonlinear physical, chemical biological processes affect contaminant transport in streams rivers, transport model formulations now increasingly utilize partial differential equations which must be solved using numerical methods. Exact analytical solutions are generally available only for more simplified formulations based on idealized representations of the transport domain transport processes. Yet despite the apparent simplifications, analytical solute transport models remain useful for many applications such as providing initial or approximate analyses of a variety of contaminant transport scenarios, especially for longer times, facilitating transport analyses when insufficient data are available to warrant the use of a comprehensive numerical model, for serving as a benchmark to test the accuracy of numerical models. In this two-part series, we assembled a large number of analytical solutions for modeling solute transport in streams, rivers other open surface water bodies. In part 1 (van Genuchten et al., 2013) we summarized solutions for one- multidimensional equilibrium transport, with without zero-order production first-order decay. In the current part 2 we provide solutions for transport with simultaneous first-order exchange between the river relatively immobile or stagnant water zones (transient storage models) for situations exchange with the hyporheic zone is modeled as a diffusion process (further referred to here as hyporheic zone diffusion models). Also presented in this paper are several solutions for transport of solutes subject to consecutive decay chain reactions. Most of the presented solutions were derived from solutions to mathematically very similar problems in subsurface contaminant transport (Leij Toride, 1997; Leij et al., 1993; Toride et al., 1993; van Genuchten, 1985a,b; Weerts et al., 1995). Except for those pertaining to diffusion in hyporheic zones, all solutions have been incorporated in the public-domain windows-based STANMOD software package (Simnek et al., 2000). TRANSIENT STORAGE MODELS All solutions presented in Part 1 (van Genuchten et al., 2013) hold for equilibrium contaminant transport characterized by relatively symmetrical or sigmoidal concentration distributions versus time or distance, unless modified by production degradation processes, or special initial or boundary conditions. This ideal situation generally does not occur in streams rivers because of the presence of relatively immobile or stagnant zones of water connected to the mean stream channel. Such stagnant zones include pools eddies along the river banks, water isolated behind rocks, gravel or vegetation, or relatively inaccessible water along an uneven river bottom. Fluvial sediments more generally subsurface hyporheic zones may also contribute to the presence of such relatively stagnant water. By providing sinks or sources of solute during transport, stagnant water zones typically cause tailing in observed concentration distributions, which cannot be predicted with the conventional equilibrium ADE formulation. Several conceptual approaches are possible for modeling solute exchange between the river stagnant water zones. One popular approach inherent in most transient storage models is to assume first-order mass transfer between the river stagnant water (Bencala Walters, 1983; De Smedt, 2006; LeGrMarcq Laudelout, 1985; Runkel et al., 1996; Thackston Schnelle, 1970, 2000; Wlostowski et al., 2013, many others). The resulting model assumes that no advective disper- sive transport occurs in the transient storage zone, that an average concentration can be assigned to this zone at any value of time, t, longitudinal distance, x, along the stream. A more refined approach would be to assume that exchange of contaminant with stagnant zones, or the hyporheic zone in general, occurs by diffusion (De Smedt, 2007; Grant et al., 2012; Jackman et al., 1984; Wörman, 1998). Simulating this situation requires coupling the transport equation for the stream or river with some diffusion model for the near-stream subsurface environment (De Smedt, 2007; Jackman et al., 1984). In this paper we first consider the relatively traditional quasiempirical approach based on first-order mass transfer (transient storage models), while subsequently we consider more elaborate models assuming diffusion-based formulations for exchange between the river the hyporheic zone (referred to here as hyporheic diffusion models). Assuming a constant cross-sectional area A of the river a constant longitudinal dispersion coefficient Dx, the transient storage model in its most general form is given by (e.g., Bencala et al., 1983; Runkel Chapra, 1993; Wlostowski et al., 2013) tion first-order degradation are given by Toride et al. (1993). Two solutions are listed here as an example. Case E1. We give here first the solution of Eqs. (3) (4) for the initial condition C(x,0) = Cs (x,0) = Ci the third-type inlet condition (5) Dx C C - u x x= C o = + 0 0 0 < t to (6) t to a semi-infinite profile C (, t) = 0 . x (7) The solutions for the stream the dead storage zone concentrations are, respectively, C 2 C C QL = Dx -u + (C L - C) - (C - Cs ) - µC , 2 t x A x (1) Cs A = (C - Cs ) - µ s Cs , t As (2) Ci + (Co - Ci )A(x,t) 0 < t to C(x,t) = C + (C - C ) A(x,t) - C A(x,t - t ) t > t o i o o o i (8) C Cs are concentrations of the stream storage zones, respectively (ML­3), Dx is the longitudinal dispersion coefficient (L2T­1), u is the longitudinal fluid flow velocity (LT­1), x is the longitudinal coordinate (L), t is time (T), As is the cross-sectional area of the storage zone (L2), QL is the lateral volumetric inflow rate per unit length (L3T­1L­1), CL is the concentration of the lateral inflow (ML­3), is a dead-zone storage mass transfer coefficient (T­1), µ µs are firstorder decay coefficients for the stream storage zone, respectively (T­1). The terms on the right-h side denote longitudinal dispersive advective transport, lateral inflow from groundwater, solute exchange with the storage zone. Several variations of this model have been applied to river transport; they typically require a numerical solution, particularly when A, As QL vary with distance. However, as pointed out by van Genuchten et al. (1988) Huang et al. (2006), analytical solutions are readily available for mathematically very similar problems of nonequilibrium transport in porous media (Coats Smith, 1964; Leij Toride, 1997; Toride et al., 1993; van Genuchten Wagenet, 1989; van Genuchten Wierenga, 1976). If the lateral inflow degradation terms in (1) (2) are negligible, then the dead zone storage model for constant A, As, Dx reduces to Ci + (Co - Ci ) B(x,t) 0 < t to Cs (x,t) = Ci + (Co - Ci ) B(x,t) - Co B(x,t - to ) t > to , (9) u2 t (x - u )2 A(x,t)= J (a,b) exp - 4Dx 0 Dx x + u ux u2 - exp erfc Dx 4Dx Dx d , (10) u2 t (x - u )2 B(x,t)= 1- J (b,a) D exp - 4D 0 x x x + u ux u2 d , - exp erfc Dx 4Dx Dx (11) C C C = Dx -u - (C - Cs ) , t x x 2 a =t (3) b = (t ­ ) A As (12) C s = A (C - C ) s . t As These expressions contain Goldstein's J-function which is defined as (Goldstein, 1953): (4) J (a,b) = 1- exp(-b) exp(- ) I0(2 b ) d (13) Many analytical solutions for one-dimensional nonequilibrium transport with or without accounting for zero-order produc- in which I0 is the zero-order modified Bessel function. The above solutions are for a third-type inlet boundary condition. For a first-type inlet condition, (10) (11) must be replaced by, respectively, rium ADE option in CXTFIT. In that case the average porewater velocity v, the dispersion coefficient D, the dimensionless parameters in CXTFIT must be defined in terms of transient storage model parameters as follows t (x - u )2 x2 A(x,t)= J (a,b) exp - d , 3 0 4Dx 4 D t (x - u )2 x2 B(x,t)= 1- J (b,a) exp - d . 4 D 4Dx 0 x (14) v = u D = Dx A A + As L , (16) u (15) Fig. 1 shows an application of the above solution for a thirdtype boundary condition (Eq. (6)). Like for several examples presented in part 1 (van Genuchten et al., 2013), the problem draws upon calculations parameter values used by De Smedt et al. (2005). The example involves the injection of 1 kg of a solute in the main channel of a river having a crosssectional area (A) of 10 m2, a connected storage zone (As) of 2 m2, an average flow velocity of 1 m/s in the river, a dispersion coefficient of 5 m2/s. Fig. 1 shows for three values of the dead-storage zone mass transfer coefficient () calculated solute concentrations at a distance 1000 m downstream from the injection point. The curves were obtained assuming that 1 kg of solute was injected during a time period of only 10 seconds (to = 10 in Eq. (6)). De Smedt et al. (2005) used for this purpose a Dirac function, similarly as we selected in part 1 (Fig. 1). The resulting curves however are essentially identical, i.e., our Fig. 1 here Fig. 1 of De Smedt et al. (2005). The curve for = 0 assumes no exchange of solute with the storage zone, hence could be calculated also immediately with the equilibrium transport model. L is some characteristic length used to place the transport model in dimensionless form (L = 1000 m in the current example). For the calculations of Fig. 1 we hence used in CXTFIT the parameter values v = 0.83333 m/s, D = 4.16667 m/s2, = 0.83333, = 0, 0.8333 8.333 1/s). We also used the above transient storage model to analyze the experimental data (exp. I­3) of Brevis et al. (2001) De Smedt et al. (2005) that were examined earlier in part 1 using the equilibrium transport model. The data are for a tracer experiment conducted in the Chillán River in Chile in which 157.1 g of a 20% Rhodamine WT tracer was injected at location x = 0, with measurements made at L = 4604 m downstream of the injection point. Fig. 2 shows the data along with fitted curves obtained with both the transient storage model (solid line) the stard ADE equilibrium transport model discussed in part 1 (dashed line). Parameters were estimated using the nonlinear least-squares optimization features of CXTFIT, which provided estimates of the four parameters given by Eq. (16), as well as the concentration Co of the applied tracer solution for a given value of the injection or pulse time to in Eq. (8). Transient Storage Model Equilibrium Model 3 Concentration (mg/m ) Time (s) Fig. 2. Observed (solid squares) fitted (continuous dashed lines) concentrations for tracer experiment I-3 of Brevis et al. (2001) De Smedt et al. (2005). Fig. 1. Calculated concentration distributions obtained with Eq. (8) for near-instantaneous injection (to = 10 s) of 1 kg of solute (Co = 10 g/m3) in the main channel of a stream having a cross-sectional area of 10 m2, an average flow velocity of 1 m/s, a dispersion coefficient of 5 m2/d, a connected transient storage zone of 2 m2, assuming three values of the mass transfer coefficient, . Calculations for Fig. 1 were obtained with the CXTFIT code of Toride et al. (1999) as incorporated in the STANMOD software of Simnek et al. (2000). Even though CXTFIT was derived for porous media transport problems, the code is applicable immediately to most or all models listed in this paper, as well as in part 1, by assuming a volumetric water content of 1.0 in the code. However, some care is needed when the transient storage model is simulated using the deterministic nonequilib- Assuming a very short injection time of only 10 s, similarly as in part 1 for the equilibrium analysis, we obtained the following parameter values (with their 95% confidence intervals): v = 0.414 ± 0.002 m/s, D = 2.66 ± 1.40 m2/s, = 0.8647 ± 0.0304, = 2.934 ± 0.936, Co = 1808 ± 33 mg/m3. The coefficient of determination (R2) of the fit was 0.997, the root mean square error (RMSE) 0.299 mg/m3. Using Eq. (16), the CXTFIT estimates translate to the following parameters in the transient storage model: u = 0.478 m/s, Dx = 3.07 m2/s, As/A = 0.156, = 3.52 x 01­4 s­1. The total amount of solute mass (m) injected per m2 cross-sectional can now be calculated using m = u Co to, or 8.642 g/m2. Given that a total amount of 157.1 g of tracer was applied to the river, this translates to an effective cross-sectional area of 157.1 (g)/8.642 (g/m2) or 18.2 m2 for the river channel as seen by the transient storage model. Our estimates for u, A, As are exactly the same as those obtained by De Smedt et al. (2005), while estimates of the dispersion coefficient Dx, the mass transfer coefficient differed slightly. De Smedt et al. (2005) obtained values of 3.29 ± 1.05 m2/s 2.87 x 10­4 ± 0.89 x 10-4 s­1 for these two parameters, respectively. The estimates of the transport parameters above were obtained assuming a solute injection pulse of 10 s. Similarly as for the equilibrium analysis in part 1, the same results were obtained when the total mass was assumed to be applied instantaneously as modeled using the Dirac solution of Eqs. (1) (2) (not further given here) as used also by De Smedt et al. (2005), for pulse times to up to about 100 s. We also obtained essentially the same results when assuming a first-type inlet boundary condition, which again shows that differences between the first- third-type solutions generally are very small at locations having relatively large values of the dimensionless distance variable ux/Dx. Case E2. We also give the solution here of the general transient storage model given by Eqs. (1) (2) for degradation in both the stream storage zone, as well as lateral inflow (QL is assumed to be constant). Analytical solutions are given for an initially solute-free river system (Ci = 0) subject to boundary conditions (6) (7). The solutions for the stream storagezone concentrations are, respectively (Toride et al., 1993), E(x, ) = 2 2 u exp - (x - u ) Dx 4Dx 2 x + u ux - u exp erfc Dx 4Dx Dx (23) x - ut (x - ut)2 1 u 2t - F(x,t) = 1- erfc exp - 2 Dx 4Dx t 4Dx t x + ut ux 1 ux u 2t + 1+ + exp erfc 2 Dx Dx 4Dx t Dx (24) a= 2 + µs b= ( + µ s )(t - ) A . As (25) The above solution is for a third-type inlet condition. The solution for a first-type condition is exactly the same, except for Eqs. (23) (24) which must be replaced by, respectively, Co A1(x,t) + B1(x,t) 0 < t to C(x,t) = Co A1(x,t) + B1(x,t) - Co A1(t - to ) t > to E(x, ) = (17) 2 (x - u )2 x exp - 3 4Dx 4 Dx (26) F(x, ) = 1- (18) Co A2 (x,t) + B2 (x,t) 0 < t to C s(x, t) = Co A2 (x,t) + B2 (x,t) - Co A2 (t - to ) t > to , t µs Q A1(x,t)= J (a,b)exp - - ( µ + L ) E(x, ) d , A 0 + µs x - u 1 erfc 2 4Dx 1 x + u ux - exp erfc 2 4Dx Dx . (27) (19) B1(x,t)= µs QLC L t Q - ( µ + L ) F(x, ) d , J (a,b) exp - A 0 A + µs (20) A2 (x,t)= t 1- J (b,a) + µs 0 µs Q i exp - - ( µ + L ) E(x, )d , A + µs (21) B2 (x,t)= QLC L t 1- J (b,a) ( + µs ) A 0 µs Q i exp - - ( µ + L ) F(x, )d A + µs (22) in which Additional one-dimensional analytical solutions for the transient storage model are readily derived from the solutions given by Toride et al. (1993a) for transport in porous media. They are all incorporated in the CXTFIT code (Toride et al., 1999) as part of STANMOD. Multidimensional versions of the transient storage nonequilibrium model can be obtained by adding terms for the transverse dispersive flux similarly as for the equilibrium case (see Eq. (41) of van Genuchten et al., 2012). Because the nonequilibrium effects generally manifest themselves only in the longitudinal flow direction, solutions for multidimensional transient storage models can often be deduced from available onedimensional nonequilibrium solutions. We refer to Leij et al. (1993) for a complete set of analytical solutions for threedimensional nonequilibrium transport, again for porous media transport. Green's functions have proved to be particularly convenient for constructing solutions for nonequilibrium transport in multiple dimensions for different mathematical conditions (cf. Leij van Genuchten, 2000). Many or most of the multi-dimensional solutions are included in the 3DADE (Leij Bradford, 1994) N3DADE (Leij Toride, 1997) computer programs for equilibrium nonequilibrium transport, respectively. Like CXTFIT for one-dimensional transport, 3DADE (but not N3DADE) includes parameter estimation capabilities to estimate selected transport parameters from observed contaminant concentration distributions versus distance /or time. HYPORHEIC ZONE DIFFUSION MODELS The transient storage models in the previous section use a first-order mass transfer equation to account for solute exchange between the river relatively stagnant zones along the river banks or bottom. This conceptual picture can be refined by using Fick's law to describe solute diffusion from the stream into the stagnant water zones, the fluvial sediment or more generally the entire hyporheic zone (Jackman et al., 1985; Runkel et al., 2003; Wörman, 1998). Similar problems have been described solved analytically for contaminant transport in fractured or macroporous media (Sudicky Frind, 1982; Tang et al., 1981; van Genuchten, 1985a; van Genuchten et al., 1984). Typically, the porous media solutions account for advective-dispersive transport through well-defined fractures or soil macropores with simultaneous diffusion from the fractures into the surrounding soil matrix. Many of these solutions are again readily applied to river systems. Here we consider two cases, one for vertical diffusion from the river into its sediments at the bottom (Case F1), one for radial diffusion from a semi-circular stream into the surrounding subsurface (Case F2). centration in the sediment, Cs(x,z,t), is now also a function of the vertical distance z below the river bottom. Advective transport as well as longitudinal diffusive transport in the hyporheic zone are ignored. We present here solutions of the above hyporheic zone transport/diffusion model subject to the initial boundary conditions C(x,0) = Cs (x, z,0) = Ci , Dx C C - u x x= Co = + 0 0 0 < t to t to , (33) (34) C (, t) = 0 . x (35) The analytical solution for the stream concentration is (van Genuchten, 1985a) Ci + (Co - Ci ) A(x,t) 0 < t to C(x,t) = Ci + (Co - Ci ) A(x,t) - Co A(x,t - to ) t > to , (36) Fig. 3. Schematic of river system with a rectangular hyporheic zone. Case F1. Assuming a rectangular geometry of the river hyporheic zone (Fig. 3), the transport/diffusion model may be written as (see also Sudicky Frind, 1982) 1 2u + 2 2 Dx 0 u 2 + p + m 2D x u · + p sin 2t - m x 2Dx ux exp - pz 2Dx (37) C 2 C C J z = Dx -u - , 2 t x d x J z = - s Ds Cs z , z=0 (28) in which (29) d -m cos 2t - m x m = (0 < z zo ), (30) Rs Cs 2 Cs = Ds t z 2 o - 1 2 2 12 + 2 , p = o + 1 , 2 (38a,b) o = (39) Cs (x,0,t) = C(x,t), Cs (x, zo ,t) = 0, z (31) Ds 2d 2 Rs (40) (32) C Cs are solute concentrations in the stream sediment, respectively, d is the depth of the river, z is vertical distance below the river, Jz is the vertical solute diffusive flux into the hyporheic zone (ML­2T­1), zo is the effective depth of the sediment, s is the volumetric water content of the sediments, Ds is the apparent solute diffusion coefficient in the sediment (L2T­1), Rs is a solute retardation factor accounting for linear sorption/exchange in the sediment (­). Note that, as before, the stream solute concentration, C(x,t), is a function of longitudinal distance, x, time, t, but that the solute con- D u2 + s2 s H1, 4Dx d D x Ds 2 2d Rs Dx (41a) s Ds d 2 Dx H2 , (41b) (42a) H1( ) = sinh( zo / d) - sin( zo / d) , cosh( zo / d) + cos( zo / d) H 2 ( ) = sinh( zo / d) + sin( zo / d) . cosh( zo / d) + cos( zo / d) (42b) Rs Cs Ds Cs = r t r r r (a < r ro ) , (49) The above solution holds for a third-type inlet boundary condition. The solution for a first-type boundary condition is exactly the same, except that (37) must be replaced by Cs (x,a,t) = C(x,t) , Cs (x,ro ,t) =0, r (50) ux 1 2 d + exp - p z sin 2t - m x . (43) 2 0 2Dx (51) The same solutions also hold for the slightly simpler situation diffusion in the sediments occurs over a semiinfinite region, i.e., zo . Eqs. (42a,b) then reduce to unity, Eqs. (41a,b) become a ro are as shown in Fig. 2, r is the radial coordinate D u2 + s2 s , 4Dx d D x Ds 2 2d Rs Dx (44a) s Ds d 2 Dx (44b) If dispersion in the river system is neglected (Dx 0), A(x,t) for the first- third type boundary conditions both reduce to (see also Skopp Warrick, 1974) Fig. 4. Schematic of river system with cylindrical hyporheic zone. D x 1 2 + exp - s s H1 2 0 2d 2u The solutions for this case is the same as for Case F1, except for the following expressions (van Genuchten et al., 1984) D 2 ( ut - x ) H D x d 1 i sin s - s s H2 2d 2uRs 2d 2u (45) Ds a 2 Rs u2 2 4Dx (52) which holds for t > x/u. Eq. (45) similar expressions for A(x,t) pertaining to transport problems which neglect dispersion in the river system are understood to be zero for t < x/u. Eq. (45) for zo may be expressed in a much simpler alternative form (Grisak Pickens, 1981; Tang et al., 1981). 2 s Ds a 2 Dx + H1 , (53a) 2Ds 2 ro2 Rs Dx 2 s Ds a 2 Dx H2 , (53b) x DR s s erfc s 2d u 2t - ux (x < ut) . (46) Alternative expressions for the solution of the above transport/diffusion problem for a first-type inlet boundary condition were given by Sudicky Frind (1982) assuming a finite zo, Tang et al. (1981) for the case zo . These two studies also presented solutions for the sediment concentration, Cs. Case F2. This example is the same as F1, except that the stream has a semi-circular cross-sections as shown in Fig. 4. Diffusion now takes place in a radial direction. For this case, Eqs. (28) through (32) must be replaced by H N 1 ( M1 - M 2 ) + N 2 ( M1 + M 2 ) 2 2 N1 + N 2 (54a) H N 1 ( M1 + M 2 ) - N 2 ( M1 - M 2 ) 2 2 N1 + N 2 (54b) M1( ) = Ber1(o )Ker1( ) - Bei1( o )Kei1( ) - Ker1( o )Ber1( ) + Kei1( o )Bei1( ), M 2 ( ) = Ber1(o )Kei1( ) + Bei1( o )Ker1( ) (55a) J C 2 C C = Dx -u - r , 2 t x a 2 x J r = -2 a a Da Cs r r=a (47) - Ker1( o )Bei1( ) - Kei1( o )Ber1( ), N1( ) = Bei1(o )Ker( ) + Ber1( o )Kei( ) - Kei1( o )Ber( ) + Ker1( o )Bei( ), (55b) (48) (56a) N 2 ( ) = Bei1(o )Kei( ) - Ber1( o )Ker( ) - Kei1( o )Bei( ) - Ker1( o )Ber( ) (56b) E1 E2 E3 E4 . The transport equations for this problem are (61) in which o = ro/a, Ber, Bei, Ker Kei represent Kelvin functions (Olver, 1970). For the case of a semi-infinite radial diffusion system (ro ), H1 H2 reduce to C1 2 C1 C = Dx - u 1 - µ1C1 , t x x 2 Ci 2 Ci C = Dx - u i + µ i-1Ci-1 - µ iCi 2 t x x (62a) H1( ) = - 2 Ker( ) Ker '( ) + Kei( ) Kei'( ) Ker 2 ( ) + Kei 2 ( ) (57a) (i = 2,3,4) , (62b) H 2 ( ) = 2 Kei( ) Ker '( ) - Ker( ) Kei'( ) Ker 2 ( ) + Kei 2 ( ) (57b) Ci represents the concentration of the i-th species, µi is the i-th first-order degradation coefficient. Eqs. (62a,b) are solved for the initial boundary conditions The solutions again simplify when longitudinal dispersion in the river is assumed to be negligible (Dx = 0). A(x,t) in Eq. (37) becomes then Ci (x, 0) = 0 (i = 1,4) , g i(t) 0 0 < t to t to , (63) 2 s Ds x 1 2 + exp - H1 2 0 a 2u Dx Ci = Ci - u x x=0+ (58) (64) D 2 ( ut - x ) d 2 s Ds x i sin s 2 - H2 2 a uRs a u Ci (, t) = 0 x (65) which also hold for ro , provided H1 H2 are given by (57a,b). A good approximation of (58) for ro is (van Genuchten et al., 1984) in which gi(t) are the prescribed input concentration functions. We give here the solution for the very general situation the input concentrations are given by x D R D x s s exp - s 2 s erfc s a u 2t - ux a u g1(t) = B1 e (66a) (x < ut) , (59) g2 (t) = B2 e g3 (t) = B4 e (60) + B3 e + B5 e (66b) - 3t which was found to give accurate results for the condition + B6 e (66c) - 4t x ut < x + ua Rs Ds g4 (t) = B7 e + B8 e + B9 e - 3t + B10 e (66d) The above general solutions for rectangular cylindrical streambeds involve integrals of expressions that are the product of decaying exponential functions rapidly oscillating sinusoidal functions. Because of the oscillatory properties of the integrs as a function of the integration parameter , direct numerical integration of the integrals using Gaussian quadrature or related techniques often leads to very poor results, even with an excessive number of quadrature terms. We refer to Rasmuson Neretnieks (1981) van Genuchten et al. (1984) for efficient procedures to evaluate the integrals. CONSECUTIVE DECAY CHAINS The final set of analytical solutions in this paper concerns the movement of solutes involved in sequential first-order decay reactions. Typical examples are the transport of interacting nitrogen species, pesticides their degradation products, chlorinated hydrocarbons, organic phosphates, pharmaceuticals, radionuclides, interacting toxic trace elements (e.g., from acid mine drainage or other anthropogenic or natural sources). We consider here the transport of four species (Ei, i = 1,..,4) involved in the consecutive decay chain Bi i are constants. The multiple terms in Eqs. (66) may describe possible decay processes in the contaminant source, /or account for finite release rates from the source into the river system. For one particular release mechanism, the constants Bi are related to each other through the Bateman equations (Bateman, 1910). A detailed description of this situation is given by van Genuchten (1985b). The analytical solution of the above transport problem is (van Genuchten, 1985b) C * (x,t) 0 < t to i Ci (x,t) = C * (x,t) - e- ito C * (x,t - t ) t > t , i o o i * C B1F110 , (67) (68) * C B2 F210 + B3 F220 + µ1B1 (S - F210 + F110 ) , µ2 - µ1 12 (69) * C3 = B4 F310 + B5 F320 + B6 F330 + + + µ2 B2 (S - F + F210 ) µ3 - µ2 23 310 µ2 B3 (S - F + F220 ), µ3 - µ2 23 320 (70) Here we give one illustrative hypothetical application of the consecutive decay chain solution given by Eqs. (67)­(76). Fig. 5 shows calculated distributions versus distance for a threemember decay chain (E1E2E3) assuming a finite pulse type injection (to = 500 s) of solute E1 with concentration Co=1 g/m3, using mostly the same parameters as before for Fig. 1 (i.e., u = 1 m/s Dx = 5 m2/s), with values of 0.004, 0.001 0.002 s­1 for the degradation coefficients of solutes E1, E2 E3, respectively. * C4 = B7 F410 + B8 F420 + B9 F430 + B10 F440 + + + µ3 B4 (S - F410 + F310 ) + µ4 - µ3 34 µ3 B5 (S - F420 + F320 ) + µ4 - µ3 34 µ3 B6 (S - F430 + F330 ), µ4 - µ3 34 (72) Fig. 5. Calculated distributions versus distance for a three-member decay chain (E1E2E3) upon injection of a finite pulse (to = 500 s) of solute E1 having a concentration Co of 1 g/m3 (u = 1 m/s, Dx = 5 m2/s, µ 0.004 s­1, µ 0.001 s­1, µ3 = 0.002 s­1). (71) Sij = F jji - Fiji , u x - t (u - )x Fijk = exp(-aijk t) exp erfc 4Dx t 2Dx u + x + t (u + )x u + exp (73) erfc u- 4Dx t 2Dx x + ut ux 2u 2 + 2 exp - µit erfc ( µi aijk ), 2 4Dx t Dx -u 1 x - ut (x - ut)2 u 2t + Fijk = exp(-aijk t) erfc exp - Dx 4Dx t 4Dx t 2 x + ut ux 1 ux u 2t ( µi = aijk ), - 1 + + exp erfc 2 Dx Dx 4Dx t Dx (74) A useful computer program for evaluating the above consecutive chain transport solutions is the CHAIN code of van Genuchten (1985b), also incorporated into STANMOD. We note that the decay chain solutions in this section assume no sorption of all species involved. The CHAIN code is for the more realistic case sorption can occur, the retardation factors (Ri) of the individual species can be different. Finally, we note similar decay chain solutions for multidimensional transport with unequal retardation factors, but for first-type boundary conditions without the Bateman source equations, are discussed in a recent paper by Quezada et al. (2004). CONCLUDING REMARKS In this two-part paper we collected a large number of analytical solutions for contaminant transport in rivers surface water bodies. The solutions in part 1 pertain to one-dimensional longitudinal transport in streams rivers, longitudinal transport lateral dispersion in rivers larger surface water bodies. The current part 2 focused on nonequilibrium transport caused by the presence of stagnant water zones (transient zone models), simultaneous longitudinal advectivedispersive transport in a river diffusion into out of the hyporheic zone. We also provided several solutions for the transport of solutes involved in consecutive decays chains. Most of the solutions were derived from solutions to mathematically very similar problems in subsurface contaminant transport. Except for solutions pertaining to diffusion in hyporheic zones, all solutions have been incorporated in the public-domain windows-based STANMOD software package (Simnek et al., 2000). This software package also includes parameter estimation capabilities, hence may be a convenient tool for analyzing observed contaminant concentration j aijk = µ - µj i k=0 k >0 (75) = u 2 + 4Dx ( µi - aijk ) . (76) The solution for a first-type inlet boundary condition is the same as the above third-type inlet solution, except for the term Fijk (Eqs. (73) (74)), which should be replaced by 1 x - t (u - )x Fijk = exp(-aijk t) exp erfc 4Dx t 2Dx 2 x + t (u + )x 1 . + exp erfc 2 2Dx 4Dx t (77) distributions versus distance /or time. While inherently less flexible than more comprehensive numerical models for contaminant transport in streams rivers, we believe that the analytical solution can be very useful for simplified analyses of alternative contaminant transport scenarios, as well as for testing of numerical models. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Hydrology and Hydromechanics de Gruyter

Exact Analytical Solutions for Contaminant Transport in Rivers

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de Gruyter
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0042-790X
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10.2478/johh-2013-0032
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Abstract

Contaminant transport processes in streams, rivers, other surface water bodies can be analyzed or predicted using the advection-dispersion equation related transport models. In part 1 of this two-part series we presented a large number of one- multi-dimensional analytical solutions of the stard equilibrium advection-dispersion equation (ADE) with without terms accounting for zero-order production first-order decay. The solutions are extended in the current part 2 to advective-dispersive transport with simultaneous first-order mass exchange between the stream or river zones with dead water (transient storage models), to problems involving longitudinal advectivedispersive transport with simultaneous diffusion in fluvial sediments or near-stream subsurface regions comprising a hyporheic zone. Part 2 also provides solutions for one-dimensional advective-dispersive transport of contaminants subject to consecutive decay chain reactions. Keywords: Contaminant transport; Analytical solutions; Surface water; Transient storage models; Solute decay chains. INTRODUCTION Contaminant transport processes in streams rivers have been analyzed predicted now for several decades using a range of mathematical models (Bencala, 1983; DeSmedt et al., 2007; Fisher et al., 1979; Runkel et al., 2003; Thomann, 1973). Because many complex often nonlinear physical, chemical biological processes affect contaminant transport in streams rivers, transport model formulations now increasingly utilize partial differential equations which must be solved using numerical methods. Exact analytical solutions are generally available only for more simplified formulations based on idealized representations of the transport domain transport processes. Yet despite the apparent simplifications, analytical solute transport models remain useful for many applications such as providing initial or approximate analyses of a variety of contaminant transport scenarios, especially for longer times, facilitating transport analyses when insufficient data are available to warrant the use of a comprehensive numerical model, for serving as a benchmark to test the accuracy of numerical models. In this two-part series, we assembled a large number of analytical solutions for modeling solute transport in streams, rivers other open surface water bodies. In part 1 (van Genuchten et al., 2013) we summarized solutions for one- multidimensional equilibrium transport, with without zero-order production first-order decay. In the current part 2 we provide solutions for transport with simultaneous first-order exchange between the river relatively immobile or stagnant water zones (transient storage models) for situations exchange with the hyporheic zone is modeled as a diffusion process (further referred to here as hyporheic zone diffusion models). Also presented in this paper are several solutions for transport of solutes subject to consecutive decay chain reactions. Most of the presented solutions were derived from solutions to mathematically very similar problems in subsurface contaminant transport (Leij Toride, 1997; Leij et al., 1993; Toride et al., 1993; van Genuchten, 1985a,b; Weerts et al., 1995). Except for those pertaining to diffusion in hyporheic zones, all solutions have been incorporated in the public-domain windows-based STANMOD software package (Simnek et al., 2000). TRANSIENT STORAGE MODELS All solutions presented in Part 1 (van Genuchten et al., 2013) hold for equilibrium contaminant transport characterized by relatively symmetrical or sigmoidal concentration distributions versus time or distance, unless modified by production degradation processes, or special initial or boundary conditions. This ideal situation generally does not occur in streams rivers because of the presence of relatively immobile or stagnant zones of water connected to the mean stream channel. Such stagnant zones include pools eddies along the river banks, water isolated behind rocks, gravel or vegetation, or relatively inaccessible water along an uneven river bottom. Fluvial sediments more generally subsurface hyporheic zones may also contribute to the presence of such relatively stagnant water. By providing sinks or sources of solute during transport, stagnant water zones typically cause tailing in observed concentration distributions, which cannot be predicted with the conventional equilibrium ADE formulation. Several conceptual approaches are possible for modeling solute exchange between the river stagnant water zones. One popular approach inherent in most transient storage models is to assume first-order mass transfer between the river stagnant water (Bencala Walters, 1983; De Smedt, 2006; LeGrMarcq Laudelout, 1985; Runkel et al., 1996; Thackston Schnelle, 1970, 2000; Wlostowski et al., 2013, many others). The resulting model assumes that no advective disper- sive transport occurs in the transient storage zone, that an average concentration can be assigned to this zone at any value of time, t, longitudinal distance, x, along the stream. A more refined approach would be to assume that exchange of contaminant with stagnant zones, or the hyporheic zone in general, occurs by diffusion (De Smedt, 2007; Grant et al., 2012; Jackman et al., 1984; Wörman, 1998). Simulating this situation requires coupling the transport equation for the stream or river with some diffusion model for the near-stream subsurface environment (De Smedt, 2007; Jackman et al., 1984). In this paper we first consider the relatively traditional quasiempirical approach based on first-order mass transfer (transient storage models), while subsequently we consider more elaborate models assuming diffusion-based formulations for exchange between the river the hyporheic zone (referred to here as hyporheic diffusion models). Assuming a constant cross-sectional area A of the river a constant longitudinal dispersion coefficient Dx, the transient storage model in its most general form is given by (e.g., Bencala et al., 1983; Runkel Chapra, 1993; Wlostowski et al., 2013) tion first-order degradation are given by Toride et al. (1993). Two solutions are listed here as an example. Case E1. We give here first the solution of Eqs. (3) (4) for the initial condition C(x,0) = Cs (x,0) = Ci the third-type inlet condition (5) Dx C C - u x x= C o = + 0 0 0 < t to (6) t to a semi-infinite profile C (, t) = 0 . x (7) The solutions for the stream the dead storage zone concentrations are, respectively, C 2 C C QL = Dx -u + (C L - C) - (C - Cs ) - µC , 2 t x A x (1) Cs A = (C - Cs ) - µ s Cs , t As (2) Ci + (Co - Ci )A(x,t) 0 < t to C(x,t) = C + (C - C ) A(x,t) - C A(x,t - t ) t > t o i o o o i (8) C Cs are concentrations of the stream storage zones, respectively (ML­3), Dx is the longitudinal dispersion coefficient (L2T­1), u is the longitudinal fluid flow velocity (LT­1), x is the longitudinal coordinate (L), t is time (T), As is the cross-sectional area of the storage zone (L2), QL is the lateral volumetric inflow rate per unit length (L3T­1L­1), CL is the concentration of the lateral inflow (ML­3), is a dead-zone storage mass transfer coefficient (T­1), µ µs are firstorder decay coefficients for the stream storage zone, respectively (T­1). The terms on the right-h side denote longitudinal dispersive advective transport, lateral inflow from groundwater, solute exchange with the storage zone. Several variations of this model have been applied to river transport; they typically require a numerical solution, particularly when A, As QL vary with distance. However, as pointed out by van Genuchten et al. (1988) Huang et al. (2006), analytical solutions are readily available for mathematically very similar problems of nonequilibrium transport in porous media (Coats Smith, 1964; Leij Toride, 1997; Toride et al., 1993; van Genuchten Wagenet, 1989; van Genuchten Wierenga, 1976). If the lateral inflow degradation terms in (1) (2) are negligible, then the dead zone storage model for constant A, As, Dx reduces to Ci + (Co - Ci ) B(x,t) 0 < t to Cs (x,t) = Ci + (Co - Ci ) B(x,t) - Co B(x,t - to ) t > to , (9) u2 t (x - u )2 A(x,t)= J (a,b) exp - 4Dx 0 Dx x + u ux u2 - exp erfc Dx 4Dx Dx d , (10) u2 t (x - u )2 B(x,t)= 1- J (b,a) D exp - 4D 0 x x x + u ux u2 d , - exp erfc Dx 4Dx Dx (11) C C C = Dx -u - (C - Cs ) , t x x 2 a =t (3) b = (t ­ ) A As (12) C s = A (C - C ) s . t As These expressions contain Goldstein's J-function which is defined as (Goldstein, 1953): (4) J (a,b) = 1- exp(-b) exp(- ) I0(2 b ) d (13) Many analytical solutions for one-dimensional nonequilibrium transport with or without accounting for zero-order produc- in which I0 is the zero-order modified Bessel function. The above solutions are for a third-type inlet boundary condition. For a first-type inlet condition, (10) (11) must be replaced by, respectively, rium ADE option in CXTFIT. In that case the average porewater velocity v, the dispersion coefficient D, the dimensionless parameters in CXTFIT must be defined in terms of transient storage model parameters as follows t (x - u )2 x2 A(x,t)= J (a,b) exp - d , 3 0 4Dx 4 D t (x - u )2 x2 B(x,t)= 1- J (b,a) exp - d . 4 D 4Dx 0 x (14) v = u D = Dx A A + As L , (16) u (15) Fig. 1 shows an application of the above solution for a thirdtype boundary condition (Eq. (6)). Like for several examples presented in part 1 (van Genuchten et al., 2013), the problem draws upon calculations parameter values used by De Smedt et al. (2005). The example involves the injection of 1 kg of a solute in the main channel of a river having a crosssectional area (A) of 10 m2, a connected storage zone (As) of 2 m2, an average flow velocity of 1 m/s in the river, a dispersion coefficient of 5 m2/s. Fig. 1 shows for three values of the dead-storage zone mass transfer coefficient () calculated solute concentrations at a distance 1000 m downstream from the injection point. The curves were obtained assuming that 1 kg of solute was injected during a time period of only 10 seconds (to = 10 in Eq. (6)). De Smedt et al. (2005) used for this purpose a Dirac function, similarly as we selected in part 1 (Fig. 1). The resulting curves however are essentially identical, i.e., our Fig. 1 here Fig. 1 of De Smedt et al. (2005). The curve for = 0 assumes no exchange of solute with the storage zone, hence could be calculated also immediately with the equilibrium transport model. L is some characteristic length used to place the transport model in dimensionless form (L = 1000 m in the current example). For the calculations of Fig. 1 we hence used in CXTFIT the parameter values v = 0.83333 m/s, D = 4.16667 m/s2, = 0.83333, = 0, 0.8333 8.333 1/s). We also used the above transient storage model to analyze the experimental data (exp. I­3) of Brevis et al. (2001) De Smedt et al. (2005) that were examined earlier in part 1 using the equilibrium transport model. The data are for a tracer experiment conducted in the Chillán River in Chile in which 157.1 g of a 20% Rhodamine WT tracer was injected at location x = 0, with measurements made at L = 4604 m downstream of the injection point. Fig. 2 shows the data along with fitted curves obtained with both the transient storage model (solid line) the stard ADE equilibrium transport model discussed in part 1 (dashed line). Parameters were estimated using the nonlinear least-squares optimization features of CXTFIT, which provided estimates of the four parameters given by Eq. (16), as well as the concentration Co of the applied tracer solution for a given value of the injection or pulse time to in Eq. (8). Transient Storage Model Equilibrium Model 3 Concentration (mg/m ) Time (s) Fig. 2. Observed (solid squares) fitted (continuous dashed lines) concentrations for tracer experiment I-3 of Brevis et al. (2001) De Smedt et al. (2005). Fig. 1. Calculated concentration distributions obtained with Eq. (8) for near-instantaneous injection (to = 10 s) of 1 kg of solute (Co = 10 g/m3) in the main channel of a stream having a cross-sectional area of 10 m2, an average flow velocity of 1 m/s, a dispersion coefficient of 5 m2/d, a connected transient storage zone of 2 m2, assuming three values of the mass transfer coefficient, . Calculations for Fig. 1 were obtained with the CXTFIT code of Toride et al. (1999) as incorporated in the STANMOD software of Simnek et al. (2000). Even though CXTFIT was derived for porous media transport problems, the code is applicable immediately to most or all models listed in this paper, as well as in part 1, by assuming a volumetric water content of 1.0 in the code. However, some care is needed when the transient storage model is simulated using the deterministic nonequilib- Assuming a very short injection time of only 10 s, similarly as in part 1 for the equilibrium analysis, we obtained the following parameter values (with their 95% confidence intervals): v = 0.414 ± 0.002 m/s, D = 2.66 ± 1.40 m2/s, = 0.8647 ± 0.0304, = 2.934 ± 0.936, Co = 1808 ± 33 mg/m3. The coefficient of determination (R2) of the fit was 0.997, the root mean square error (RMSE) 0.299 mg/m3. Using Eq. (16), the CXTFIT estimates translate to the following parameters in the transient storage model: u = 0.478 m/s, Dx = 3.07 m2/s, As/A = 0.156, = 3.52 x 01­4 s­1. The total amount of solute mass (m) injected per m2 cross-sectional can now be calculated using m = u Co to, or 8.642 g/m2. Given that a total amount of 157.1 g of tracer was applied to the river, this translates to an effective cross-sectional area of 157.1 (g)/8.642 (g/m2) or 18.2 m2 for the river channel as seen by the transient storage model. Our estimates for u, A, As are exactly the same as those obtained by De Smedt et al. (2005), while estimates of the dispersion coefficient Dx, the mass transfer coefficient differed slightly. De Smedt et al. (2005) obtained values of 3.29 ± 1.05 m2/s 2.87 x 10­4 ± 0.89 x 10-4 s­1 for these two parameters, respectively. The estimates of the transport parameters above were obtained assuming a solute injection pulse of 10 s. Similarly as for the equilibrium analysis in part 1, the same results were obtained when the total mass was assumed to be applied instantaneously as modeled using the Dirac solution of Eqs. (1) (2) (not further given here) as used also by De Smedt et al. (2005), for pulse times to up to about 100 s. We also obtained essentially the same results when assuming a first-type inlet boundary condition, which again shows that differences between the first- third-type solutions generally are very small at locations having relatively large values of the dimensionless distance variable ux/Dx. Case E2. We also give the solution here of the general transient storage model given by Eqs. (1) (2) for degradation in both the stream storage zone, as well as lateral inflow (QL is assumed to be constant). Analytical solutions are given for an initially solute-free river system (Ci = 0) subject to boundary conditions (6) (7). The solutions for the stream storagezone concentrations are, respectively (Toride et al., 1993), E(x, ) = 2 2 u exp - (x - u ) Dx 4Dx 2 x + u ux - u exp erfc Dx 4Dx Dx (23) x - ut (x - ut)2 1 u 2t - F(x,t) = 1- erfc exp - 2 Dx 4Dx t 4Dx t x + ut ux 1 ux u 2t + 1+ + exp erfc 2 Dx Dx 4Dx t Dx (24) a= 2 + µs b= ( + µ s )(t - ) A . As (25) The above solution is for a third-type inlet condition. The solution for a first-type condition is exactly the same, except for Eqs. (23) (24) which must be replaced by, respectively, Co A1(x,t) + B1(x,t) 0 < t to C(x,t) = Co A1(x,t) + B1(x,t) - Co A1(t - to ) t > to E(x, ) = (17) 2 (x - u )2 x exp - 3 4Dx 4 Dx (26) F(x, ) = 1- (18) Co A2 (x,t) + B2 (x,t) 0 < t to C s(x, t) = Co A2 (x,t) + B2 (x,t) - Co A2 (t - to ) t > to , t µs Q A1(x,t)= J (a,b)exp - - ( µ + L ) E(x, ) d , A 0 + µs x - u 1 erfc 2 4Dx 1 x + u ux - exp erfc 2 4Dx Dx . (27) (19) B1(x,t)= µs QLC L t Q - ( µ + L ) F(x, ) d , J (a,b) exp - A 0 A + µs (20) A2 (x,t)= t 1- J (b,a) + µs 0 µs Q i exp - - ( µ + L ) E(x, )d , A + µs (21) B2 (x,t)= QLC L t 1- J (b,a) ( + µs ) A 0 µs Q i exp - - ( µ + L ) F(x, )d A + µs (22) in which Additional one-dimensional analytical solutions for the transient storage model are readily derived from the solutions given by Toride et al. (1993a) for transport in porous media. They are all incorporated in the CXTFIT code (Toride et al., 1999) as part of STANMOD. Multidimensional versions of the transient storage nonequilibrium model can be obtained by adding terms for the transverse dispersive flux similarly as for the equilibrium case (see Eq. (41) of van Genuchten et al., 2012). Because the nonequilibrium effects generally manifest themselves only in the longitudinal flow direction, solutions for multidimensional transient storage models can often be deduced from available onedimensional nonequilibrium solutions. We refer to Leij et al. (1993) for a complete set of analytical solutions for threedimensional nonequilibrium transport, again for porous media transport. Green's functions have proved to be particularly convenient for constructing solutions for nonequilibrium transport in multiple dimensions for different mathematical conditions (cf. Leij van Genuchten, 2000). Many or most of the multi-dimensional solutions are included in the 3DADE (Leij Bradford, 1994) N3DADE (Leij Toride, 1997) computer programs for equilibrium nonequilibrium transport, respectively. Like CXTFIT for one-dimensional transport, 3DADE (but not N3DADE) includes parameter estimation capabilities to estimate selected transport parameters from observed contaminant concentration distributions versus distance /or time. HYPORHEIC ZONE DIFFUSION MODELS The transient storage models in the previous section use a first-order mass transfer equation to account for solute exchange between the river relatively stagnant zones along the river banks or bottom. This conceptual picture can be refined by using Fick's law to describe solute diffusion from the stream into the stagnant water zones, the fluvial sediment or more generally the entire hyporheic zone (Jackman et al., 1985; Runkel et al., 2003; Wörman, 1998). Similar problems have been described solved analytically for contaminant transport in fractured or macroporous media (Sudicky Frind, 1982; Tang et al., 1981; van Genuchten, 1985a; van Genuchten et al., 1984). Typically, the porous media solutions account for advective-dispersive transport through well-defined fractures or soil macropores with simultaneous diffusion from the fractures into the surrounding soil matrix. Many of these solutions are again readily applied to river systems. Here we consider two cases, one for vertical diffusion from the river into its sediments at the bottom (Case F1), one for radial diffusion from a semi-circular stream into the surrounding subsurface (Case F2). centration in the sediment, Cs(x,z,t), is now also a function of the vertical distance z below the river bottom. Advective transport as well as longitudinal diffusive transport in the hyporheic zone are ignored. We present here solutions of the above hyporheic zone transport/diffusion model subject to the initial boundary conditions C(x,0) = Cs (x, z,0) = Ci , Dx C C - u x x= Co = + 0 0 0 < t to t to , (33) (34) C (, t) = 0 . x (35) The analytical solution for the stream concentration is (van Genuchten, 1985a) Ci + (Co - Ci ) A(x,t) 0 < t to C(x,t) = Ci + (Co - Ci ) A(x,t) - Co A(x,t - to ) t > to , (36) Fig. 3. Schematic of river system with a rectangular hyporheic zone. Case F1. Assuming a rectangular geometry of the river hyporheic zone (Fig. 3), the transport/diffusion model may be written as (see also Sudicky Frind, 1982) 1 2u + 2 2 Dx 0 u 2 + p + m 2D x u · + p sin 2t - m x 2Dx ux exp - pz 2Dx (37) C 2 C C J z = Dx -u - , 2 t x d x J z = - s Ds Cs z , z=0 (28) in which (29) d -m cos 2t - m x m = (0 < z zo ), (30) Rs Cs 2 Cs = Ds t z 2 o - 1 2 2 12 + 2 , p = o + 1 , 2 (38a,b) o = (39) Cs (x,0,t) = C(x,t), Cs (x, zo ,t) = 0, z (31) Ds 2d 2 Rs (40) (32) C Cs are solute concentrations in the stream sediment, respectively, d is the depth of the river, z is vertical distance below the river, Jz is the vertical solute diffusive flux into the hyporheic zone (ML­2T­1), zo is the effective depth of the sediment, s is the volumetric water content of the sediments, Ds is the apparent solute diffusion coefficient in the sediment (L2T­1), Rs is a solute retardation factor accounting for linear sorption/exchange in the sediment (­). Note that, as before, the stream solute concentration, C(x,t), is a function of longitudinal distance, x, time, t, but that the solute con- D u2 + s2 s H1, 4Dx d D x Ds 2 2d Rs Dx (41a) s Ds d 2 Dx H2 , (41b) (42a) H1( ) = sinh( zo / d) - sin( zo / d) , cosh( zo / d) + cos( zo / d) H 2 ( ) = sinh( zo / d) + sin( zo / d) . cosh( zo / d) + cos( zo / d) (42b) Rs Cs Ds Cs = r t r r r (a < r ro ) , (49) The above solution holds for a third-type inlet boundary condition. The solution for a first-type boundary condition is exactly the same, except that (37) must be replaced by Cs (x,a,t) = C(x,t) , Cs (x,ro ,t) =0, r (50) ux 1 2 d + exp - p z sin 2t - m x . (43) 2 0 2Dx (51) The same solutions also hold for the slightly simpler situation diffusion in the sediments occurs over a semiinfinite region, i.e., zo . Eqs. (42a,b) then reduce to unity, Eqs. (41a,b) become a ro are as shown in Fig. 2, r is the radial coordinate D u2 + s2 s , 4Dx d D x Ds 2 2d Rs Dx (44a) s Ds d 2 Dx (44b) If dispersion in the river system is neglected (Dx 0), A(x,t) for the first- third type boundary conditions both reduce to (see also Skopp Warrick, 1974) Fig. 4. Schematic of river system with cylindrical hyporheic zone. D x 1 2 + exp - s s H1 2 0 2d 2u The solutions for this case is the same as for Case F1, except for the following expressions (van Genuchten et al., 1984) D 2 ( ut - x ) H D x d 1 i sin s - s s H2 2d 2uRs 2d 2u (45) Ds a 2 Rs u2 2 4Dx (52) which holds for t > x/u. Eq. (45) similar expressions for A(x,t) pertaining to transport problems which neglect dispersion in the river system are understood to be zero for t < x/u. Eq. (45) for zo may be expressed in a much simpler alternative form (Grisak Pickens, 1981; Tang et al., 1981). 2 s Ds a 2 Dx + H1 , (53a) 2Ds 2 ro2 Rs Dx 2 s Ds a 2 Dx H2 , (53b) x DR s s erfc s 2d u 2t - ux (x < ut) . (46) Alternative expressions for the solution of the above transport/diffusion problem for a first-type inlet boundary condition were given by Sudicky Frind (1982) assuming a finite zo, Tang et al. (1981) for the case zo . These two studies also presented solutions for the sediment concentration, Cs. Case F2. This example is the same as F1, except that the stream has a semi-circular cross-sections as shown in Fig. 4. Diffusion now takes place in a radial direction. For this case, Eqs. (28) through (32) must be replaced by H N 1 ( M1 - M 2 ) + N 2 ( M1 + M 2 ) 2 2 N1 + N 2 (54a) H N 1 ( M1 + M 2 ) - N 2 ( M1 - M 2 ) 2 2 N1 + N 2 (54b) M1( ) = Ber1(o )Ker1( ) - Bei1( o )Kei1( ) - Ker1( o )Ber1( ) + Kei1( o )Bei1( ), M 2 ( ) = Ber1(o )Kei1( ) + Bei1( o )Ker1( ) (55a) J C 2 C C = Dx -u - r , 2 t x a 2 x J r = -2 a a Da Cs r r=a (47) - Ker1( o )Bei1( ) - Kei1( o )Ber1( ), N1( ) = Bei1(o )Ker( ) + Ber1( o )Kei( ) - Kei1( o )Ber( ) + Ker1( o )Bei( ), (55b) (48) (56a) N 2 ( ) = Bei1(o )Kei( ) - Ber1( o )Ker( ) - Kei1( o )Bei( ) - Ker1( o )Ber( ) (56b) E1 E2 E3 E4 . The transport equations for this problem are (61) in which o = ro/a, Ber, Bei, Ker Kei represent Kelvin functions (Olver, 1970). For the case of a semi-infinite radial diffusion system (ro ), H1 H2 reduce to C1 2 C1 C = Dx - u 1 - µ1C1 , t x x 2 Ci 2 Ci C = Dx - u i + µ i-1Ci-1 - µ iCi 2 t x x (62a) H1( ) = - 2 Ker( ) Ker '( ) + Kei( ) Kei'( ) Ker 2 ( ) + Kei 2 ( ) (57a) (i = 2,3,4) , (62b) H 2 ( ) = 2 Kei( ) Ker '( ) - Ker( ) Kei'( ) Ker 2 ( ) + Kei 2 ( ) (57b) Ci represents the concentration of the i-th species, µi is the i-th first-order degradation coefficient. Eqs. (62a,b) are solved for the initial boundary conditions The solutions again simplify when longitudinal dispersion in the river is assumed to be negligible (Dx = 0). A(x,t) in Eq. (37) becomes then Ci (x, 0) = 0 (i = 1,4) , g i(t) 0 0 < t to t to , (63) 2 s Ds x 1 2 + exp - H1 2 0 a 2u Dx Ci = Ci - u x x=0+ (58) (64) D 2 ( ut - x ) d 2 s Ds x i sin s 2 - H2 2 a uRs a u Ci (, t) = 0 x (65) which also hold for ro , provided H1 H2 are given by (57a,b). A good approximation of (58) for ro is (van Genuchten et al., 1984) in which gi(t) are the prescribed input concentration functions. We give here the solution for the very general situation the input concentrations are given by x D R D x s s exp - s 2 s erfc s a u 2t - ux a u g1(t) = B1 e (66a) (x < ut) , (59) g2 (t) = B2 e g3 (t) = B4 e (60) + B3 e + B5 e (66b) - 3t which was found to give accurate results for the condition + B6 e (66c) - 4t x ut < x + ua Rs Ds g4 (t) = B7 e + B8 e + B9 e - 3t + B10 e (66d) The above general solutions for rectangular cylindrical streambeds involve integrals of expressions that are the product of decaying exponential functions rapidly oscillating sinusoidal functions. Because of the oscillatory properties of the integrs as a function of the integration parameter , direct numerical integration of the integrals using Gaussian quadrature or related techniques often leads to very poor results, even with an excessive number of quadrature terms. We refer to Rasmuson Neretnieks (1981) van Genuchten et al. (1984) for efficient procedures to evaluate the integrals. CONSECUTIVE DECAY CHAINS The final set of analytical solutions in this paper concerns the movement of solutes involved in sequential first-order decay reactions. Typical examples are the transport of interacting nitrogen species, pesticides their degradation products, chlorinated hydrocarbons, organic phosphates, pharmaceuticals, radionuclides, interacting toxic trace elements (e.g., from acid mine drainage or other anthropogenic or natural sources). We consider here the transport of four species (Ei, i = 1,..,4) involved in the consecutive decay chain Bi i are constants. The multiple terms in Eqs. (66) may describe possible decay processes in the contaminant source, /or account for finite release rates from the source into the river system. For one particular release mechanism, the constants Bi are related to each other through the Bateman equations (Bateman, 1910). A detailed description of this situation is given by van Genuchten (1985b). The analytical solution of the above transport problem is (van Genuchten, 1985b) C * (x,t) 0 < t to i Ci (x,t) = C * (x,t) - e- ito C * (x,t - t ) t > t , i o o i * C B1F110 , (67) (68) * C B2 F210 + B3 F220 + µ1B1 (S - F210 + F110 ) , µ2 - µ1 12 (69) * C3 = B4 F310 + B5 F320 + B6 F330 + + + µ2 B2 (S - F + F210 ) µ3 - µ2 23 310 µ2 B3 (S - F + F220 ), µ3 - µ2 23 320 (70) Here we give one illustrative hypothetical application of the consecutive decay chain solution given by Eqs. (67)­(76). Fig. 5 shows calculated distributions versus distance for a threemember decay chain (E1E2E3) assuming a finite pulse type injection (to = 500 s) of solute E1 with concentration Co=1 g/m3, using mostly the same parameters as before for Fig. 1 (i.e., u = 1 m/s Dx = 5 m2/s), with values of 0.004, 0.001 0.002 s­1 for the degradation coefficients of solutes E1, E2 E3, respectively. * C4 = B7 F410 + B8 F420 + B9 F430 + B10 F440 + + + µ3 B4 (S - F410 + F310 ) + µ4 - µ3 34 µ3 B5 (S - F420 + F320 ) + µ4 - µ3 34 µ3 B6 (S - F430 + F330 ), µ4 - µ3 34 (72) Fig. 5. Calculated distributions versus distance for a three-member decay chain (E1E2E3) upon injection of a finite pulse (to = 500 s) of solute E1 having a concentration Co of 1 g/m3 (u = 1 m/s, Dx = 5 m2/s, µ 0.004 s­1, µ 0.001 s­1, µ3 = 0.002 s­1). (71) Sij = F jji - Fiji , u x - t (u - )x Fijk = exp(-aijk t) exp erfc 4Dx t 2Dx u + x + t (u + )x u + exp (73) erfc u- 4Dx t 2Dx x + ut ux 2u 2 + 2 exp - µit erfc ( µi aijk ), 2 4Dx t Dx -u 1 x - ut (x - ut)2 u 2t + Fijk = exp(-aijk t) erfc exp - Dx 4Dx t 4Dx t 2 x + ut ux 1 ux u 2t ( µi = aijk ), - 1 + + exp erfc 2 Dx Dx 4Dx t Dx (74) A useful computer program for evaluating the above consecutive chain transport solutions is the CHAIN code of van Genuchten (1985b), also incorporated into STANMOD. We note that the decay chain solutions in this section assume no sorption of all species involved. The CHAIN code is for the more realistic case sorption can occur, the retardation factors (Ri) of the individual species can be different. Finally, we note similar decay chain solutions for multidimensional transport with unequal retardation factors, but for first-type boundary conditions without the Bateman source equations, are discussed in a recent paper by Quezada et al. (2004). CONCLUDING REMARKS In this two-part paper we collected a large number of analytical solutions for contaminant transport in rivers surface water bodies. The solutions in part 1 pertain to one-dimensional longitudinal transport in streams rivers, longitudinal transport lateral dispersion in rivers larger surface water bodies. The current part 2 focused on nonequilibrium transport caused by the presence of stagnant water zones (transient zone models), simultaneous longitudinal advectivedispersive transport in a river diffusion into out of the hyporheic zone. We also provided several solutions for the transport of solutes involved in consecutive decays chains. Most of the solutions were derived from solutions to mathematically very similar problems in subsurface contaminant transport. Except for solutions pertaining to diffusion in hyporheic zones, all solutions have been incorporated in the public-domain windows-based STANMOD software package (Simnek et al., 2000). This software package also includes parameter estimation capabilities, hence may be a convenient tool for analyzing observed contaminant concentration j aijk = µ - µj i k=0 k >0 (75) = u 2 + 4Dx ( µi - aijk ) . (76) The solution for a first-type inlet boundary condition is the same as the above third-type inlet solution, except for the term Fijk (Eqs. (73) (74)), which should be replaced by 1 x - t (u - )x Fijk = exp(-aijk t) exp erfc 4Dx t 2Dx 2 x + t (u + )x 1 . + exp erfc 2 2Dx 4Dx t (77) distributions versus distance /or time. While inherently less flexible than more comprehensive numerical models for contaminant transport in streams rivers, we believe that the analytical solution can be very useful for simplified analyses of alternative contaminant transport scenarios, as well as for testing of numerical models.

Journal

Journal of Hydrology and Hydromechanics – de Gruyter

Published: Sep 1, 2013

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