Abstract
In this paper, a semi-analytical solution of multiprocess non-equilibrium (MPNE) transport equations with linear and exponential distance-dependent dispersivity is developed. The model has been used to simulate the laboratory experimental data of chloride and fluoride solutes through a 15 m long heterogeneous soil column. It is observed that a better fit to the observed breakthrough curves is obtained, when the mass transfer between advective and non-advection region is considered. It is also observed that both linear and exponential distance-dependent dispersion models give a good fit to the observed breakthrough curves, however, the exponential distance-dependent dispersion model gives a much better fit. The estimated transport parameters can be used for simulation of observed data of reactive plume through the porous media at field scale and also for the remediation of contaminated subsurface soil.



Similar content being viewed by others
Explore related subjects
Discover the latest articles and news from researchers in related subjects, suggested using machine learning.References
Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Government Printing Off, Washington DC
Bear J (1972) Dynamics of fluids in porous media. Elsevier, New Yorker 764pp
Brusseau ML, Jessup RE, Rao PSC (1989) Modeling the transport of solutes influenced by multiprocess nonequilibrium. Water Resour Res 25(9):1971–1988
Brusseau ML, Jessup RE, Rao PSC (1992) Modelling solute transport influenced by multi- process nonequilibrium and transformation reactions. Water Resour Res 28(1):175–182
Chen JS, Ni CF, Liang CP (2008) Analytical power series solutions to the two-dimensional advection-dispersion equation with distance-dependent dispersivities. Hydrol Process 22:4670–4678
de Hoog, FR, Knight JH, Stokes AN (1982) An improved method for numerical inversion of Laplace transforms. SIAM J Sci Stat Comput
Dentz M, Kinzelbach H, Attinger S, Kinzelbach W (2000) Tempral behavior of solute cloud in a heterogeneous porous medium 1. Point-like injection. Water Resour Res 36(12):3591–3614
Dentz M, Gouze P, Carrera J (2011) Effective non-local reaction kinetics for transport in physically and chemically heterogeneous media. J Contam Hydrol 222–236
Gao G, Feng S, Zhan H, Huang G, Ma X (2009) Evaluation of anomalous solute transport in a large heterogeneous soil column with mobile-immobile model. J Hydrol Eng 14(9):966–974
Gao G, Zhan H, Feng S, Fu B, Ma Y, Huang G (2010) A new mobile‐immobile model for reactive solute transport with scale‐dependent dispersion. Water Resour Res 46, W08533. doi:10.1029/2009WR008707
Gelhar LW (1993) Stochastic subsurface hydrology. Prentice-Hall, Englewood Cliffs
Huang K, Toride N, van Genuchten MT (1995) Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns. Transp Porous Media 18(3):283–302
Hunt B (1998) Contaminant source solutions with scale-dependent dispersivities. J Hydrol Eng 3(4):268–275
Hunt B (2002) Scale-dependent dispersion from pit. J Hydrol Eng 7(2):168–174
Joshi N, Ojha CSP, Sharma PK (2012) A non-equilibrium model for reactive contaminant transport through fractured porous media: model development and semi analytical solution. Water Resour Res 48, W10511. doi:10.1029/2011WR011621
Kool JB, Parker JC (1988) Analysis of the inverse problem for transient unsaturated flow. Water Resour Res 24(6):817–830
Leij FJ, Skaggs TH, Van Genuchten MT (1991) Analytical solutions for solute transport in three‐dimensional semi‐infinite porous media. Water Resour Res 27(10):2719–2733
Logan JD (1996) Solute transport in porous media with scale-dependent dispersion and periodic boundary conditions. J Hydrol 184(3–4):261–276
Mishra S, Parker JC (1990) Analysis of solute transport with a hyperbolic scale‐dependent dispersion model. Hydrol Process 4(1):45–57
Neville J, Ibaraki M, Sudicky EA (2000) Solute transport with multiprocess nonequilibrium: a semi-analytical solution approach. J Contam Hydrol 44:141–159
Ogata A (1970) Theory of dispersion in a granular medium. US Geol Surv Prof Paper 411-I
Pang L, Hunt B (2001) Solutions and verification of scale-dependent dispersion model. J Contam Hydrol 53(1):21–39
Papapetridis K, Paleologos EK (2012) Sampling frequency of groundwater monitoring and remediation delay at contaminated sites. Water Resour Manag 26(9):2673–2688
Pickens JF, Grisak GE (1981) Modeling of scale dependent dispersion in hydrogeologic systems. Water Resour Res 17(6):1701–1711
Sharma PK, Srivastava R (2012) Concentration profiles and spatial moments for reactive transport through porous media. J Hazard Toxic Radio Waste 16(2):125–133
Swami D, Sharma PK, Ojha CSP (2014) Simulation of experimental breakthrough curves using multiprocess non-equilibrium model for reactive solute transport in stratified porous media. Sadhana Indian Acad Sci 39(Part 6):1425–1446
van Genuchten MT (1981) Analytical solutions for chemical transport with simultaneous, zero-order production and first-order decay. J Hydrol 49(3):213–233
van Genuchten MT, Wierenga PJ (1976) Mass transfer studies in sorbing porous media: I. Analytical solutions. Soil Sci Am J 40(4):473–480
Yates SR (1990) An analytical solution for one-dimensional transport in heterogeneous porous media. Water Resour Res 26(10):2331–2338
Yates SR (1992) An analytical solution for one-dimensional transport in heterogeneous porous media with an exponential dispersion function. Water Resour Res 28(8):2149–2154
Acknowledgments
Authors would like to acknowledge the financial support by DST New Delhi through a sponsored research project no. DST-456-CED. Authors are very thankful to reviewers for good suggestions for improving the manuscript.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: Analytical Solution for Linear Distance-Dependent Dispersion Coefficient
Substitute Eq. (5b) into Eq. (16),
Defining new variable \( \xi =\sqrt{\left(kx{V}_a+{D}_0\right)} \), Eq. (A1) can be written as:
Equation (A2) has the form of the following Bessel equation by Abramowitz and Stegun (1972):
where, γ = 2/k, \( \lambda =\frac{2}{k{V}_a}\sqrt{\varPsi } \), η = 1
Therefore, from Abramowitz and Stegun (1972), a general solution of Eq. (A3) can be written as:
where B 1 and B 2 are two constants, which are used to satisfy the boundary conditions.
The solute concentration in the advective region in Laplace domain can be written as:
In terms of the lower boundary condition given by Eq. (6c), \( d\overline{C_a}/d\xi \) must remain finite when ξ → ∞, thus B 2 must remain zero based on the properties of modified Bessel functions of Abramowitz and Stegun (1972). Then Eq. (A5) becomes:
One can obtain the solution of \( \overline{C_a} \) after getting the value of B 1 with different inlet boundary conditions as mentioned below:
-
Case 1A:
Constant concentration boundary condition
When the value of molecular diffusion coefficient D 0 is not equal to zero, then the following expression has been obtained:
$$ {B}_1=\frac{C_0/P-{A}_4/{A}_3}{\xi_0^{\gamma }{K}_{\gamma}\left(\lambda {\xi}_0\right)} $$(A7)When the value of molecular diffusion coefficient D 0 is equal to zero, then the following expression is been obtained:
$$ {B}_1=\left(\frac{C_0}{P}-\frac{A_4}{A_3}\right)\frac{\lambda^{\gamma }}{2^{\gamma -1}\varGamma \left(\gamma \right)} $$(A8) -
Case 1B:
Pulse type boundary condition
For the case of pulse type boundary condition the Eqs. (A7) and (A8) can be written as:
$$ {B}_1=\frac{\left({C}_0/P\right)\left(1-{e}^{\left(-P{t}_0\right)}\right)-{A}_4/{A}_3}{\xi_0^{\gamma }{K}_{\gamma}\left(\lambda {\xi}_0\right)} $$(A9)$$ {B}_1=\left(\frac{C_0}{P}\left(1-{e}^{\left(-P{t}_0\right)}\right)-\frac{A_4}{A_3}\right)\frac{\lambda^{\gamma }}{2^{\gamma -1}\varGamma \left(\gamma \right)} $$(A10)Equation (A9) needs to be used when D0 is zero, otherwise Eq. (A10) has to be used.
-
Case 1C:
Constant flux type boundary condition
When the value of molecular diffusion coefficient D 0 is not equal to zero, then following expression has been obtained:
$$ {B}_1=2\gamma \frac{C_0/P-{A}_4/{A}_3}{\lambda {\xi}_0^{\gamma +1}{K}_{\gamma +1}\left(\lambda {\xi}_0\right)} $$(A11)When the value of molecular diffusion coefficient, D 0 = 0, the value of dispersion coefficient, D(x) = 0 at x =0, and constant flux type boundary condition becomes equal to constant concentration type boundary condition.
-
Case 1D:
Instantaneous solute input boundary condition
When the value of molecular diffusion coefficient D 0 is not equal to zero, one obtains
$$ {B}_1=2\gamma \frac{M/{V}_a-{A}_4/{A}_3}{\lambda {\xi}_0^{\gamma +1}{K}_{\gamma +1}\left(\lambda {\xi}_0\right)} $$(A12)When the value of molecular diffusion coefficient D 0 is equal to zero, the following expression results:
$$ {B}_1=\left(\frac{M}{V_a}-\frac{A_4}{A_3}\right)\frac{\lambda^{\gamma }}{2^{\gamma -1}\varGamma \left(\gamma \right)} $$(A13)
Appendix 2: Analytical Solution for Exponential Dispersion Coefficient
Substituting the value of exponential dispersion coefficient from Eq. (5c) into Eq. (16), we get
Defining a new parameter \( z=H{e}^{b_1x} \), where H = 1 + D 0/(a 1 V a )Equation (30) can be simplified as:
Equation (A15) has the form of the following Gauss hypergeometric equation (Abramowitz and Stegun 1972; Gao et al. 2010):
where Q = 0 and
As 1 ≤ z < ∞, the solution of Eq. (A17) can be written in terms of the hypergeometric function as follows by Abramowitz and Stegun (1972):
where F(m, m + 1; m − n + 1; z − 1) and F(n, n + 1; n − m + 1; z − 1) are the Gauss hyper geometric functions, B 3 and B 4 are integration constants whose values depend on the boundary conditions.
Therefore, the solute concentration in the advective region in the Laplace domain is
In terms of the outlet boundary condition, for \( d\overline{C_a}/dz \) to remain finite when z → ∞; B 4 must remain zero as n < 0 by Abramowitz and Stegun (1972). Equation A20 becomes:
The solution of Eq. (A21) is absolutely convergent for D 0 > 0 since the radius of convergence for the hypergeometric function is given by 1/z. When D 0 > 0, H > 0, then 1/z is always less than unity by Abramowitz and Stegun (1972).
When D 0 = 0 and H = 1, then z is always less than unity as x > 0. When D 0 = 0 and H = 1. The expressions of B 3 in Eq. (A21) can be easily derived and the solutions of solute concentration in the advective region with exponential dispersion coefficient can be obtained with different inlet boundary conditions, which have been given below:
-
Case 2a:
For constant concentration boundary condition
When the value of molecular diffusion coefficient D 0 is not equal to zero, then the following expression has been obtained:
$$ {B}_3=\frac{C_0/P-{A}_4/{A}_3}{H^{-m}F\left(m,m+1;m-n+1;{H}^{-1}\right)} $$(A22)When the value of molecular diffusion coefficient D 0 is equal to zero, then following expression has been obtained:
$$ {B}_3=\frac{\left[{C}_0/P-{A}_{11}/{A}_{10}\right]\varGamma \left(-n+1\right)\varGamma \left(-n\right)}{\varGamma \left(m-n+1\right)\varGamma \left(-m-n\right)} $$(A23) -
Case 2b:
Pulse type boundary condition
For pulse type boundary condition the Eqs. (A22) and (A23) can be written as:
$$ {B}_3=\frac{\left({C}_0/P\right)\;\left(1-{e}^{\left(-P{t}_0\right)}\right)-{A}_4/{A}_3}{H^{-m}F\left(m,m+1;m-n+1;{H}^{-1}\right)} $$(A24)$$ {B}_3=\frac{\left[\left({C}_0/P\right)\left(1-{e}^{\left(-P{t}_0\right)}\right)-{A}_4/{A}_3\right]\;\varGamma \left(-n+1\right)\varGamma \left(-n\right)}{\varGamma \left(m-n+1\right)\varGamma \left(-m-n\right)} $$(A25) -
Case 2c:
For constant flux type boundary condition
When the value of molecular diffusion coefficient D 0 is not equal to zero, then following expression has been obtained:
$$ {B}_3=\frac{V_a\left({C}_0/P-{A}_{11}/{A}_{10}\right)}{H^{-m}\left\{{D}_0{b}_1mF\left(m+1,m+1;m-n+1;{H}^{-1}\right)+{V}_aF\left(m,m+1;m-n+1;{H}^{-1}\right)\right\}} $$(A26)When the value of D 0 is equal to zero, the value of dispersion coefficient, D(x) equal to zero at x = 0, and constant flux type boundary condition becomes equal to constant concentration type boundary condition.
-
Case 2d:
For instantaneous solute input boundary condition
When the value of molecular diffusion coefficient D 0 is not equal to zero, then following expression has been obtained:
$$ {B}_3=\frac{\left(M-{V}_a\;{A}_4/{A}_3\right)}{H^{-m}\left\{{D}_0{b}_1mF\left(m+1,m+1;m-n+1;{H}^{-1}\right)+{V}_aF\left(m,m+1;m-n+1;{H}^{-1}\right)\right\}} $$(A27)When the value of molecular diffusion coefficient D 0 is equal to zero, then following expression has been obtained:
$$ {B}_3=\frac{\left(\left(M/{V}_a\right)-\left({A}_4/{A}_3\right)\right)\varGamma \left(-n+1\right)\varGamma \left(-n\right)}{\varGamma \left(m-n+1\right)\varGamma \left(-m-n\right)} $$(A28)
Appendix 3: Effective Dispersivity for Distance Dependent Dispersion Models
Effective dispersivity is obtained by averaging the local dispersivity over the entire travel domain. It can be written as (Mishra and Parker 1990):
For linear distance-dependent model, the effective dispersivity can be written as:
For exponential distance-dependent dispersion model, the dispersivity can be written as:
Appendix 4: Optimization Approach
The parameter estimation by ordinary least square problem can be given by objective function:
where b is the parameter vector given as b = (b 1, b 2, … b p )T; C*(x, t) is the vector of observed concentration and Ĉ(x, t) is vector of model predicted concentrations obtained by solving the direct problem for a given parameter vector b. At every iteration, the parameter correction Δb is determined as:
Levenberg-Marquardt algorithm is used as the optimization algorithm, which is written as:
where J is the sensitivity matrix; e is the vector of residuals of C* − Ĉ. The columns of the Jacobian matrix J contain the partial derivatives of the residuals e with respect to the elements of parameter vector b. The Jacobian has dimension n × p where n is the number of observations and p is the number of unknown parameters to be estimated. β is a positive scalar and D = diag(d 1, d 2, …, d p ) is a scaling matrix that takes into account differences in the magnitude of the sensitivities of the different parameters, the elements of D are updated as given by Kool and Parker (1988).
Rights and permissions
About this article
Cite this article
Sharma, P.K., Ojha, C.S.P., Swami, D. et al. Semi-analytical Solutions of Multiprocessing Non-equilibrium Transport Equations with Linear and Exponential Distance-Dependent Dispersivity. Water Resour Manage 29, 5255–5273 (2015). https://doi.org/10.1007/s11269-015-1116-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11269-015-1116-6
Keywords
Profiles
- Sanjay Kumar Shukla View author profile