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Semi-analytical Solutions of Multiprocessing Non-equilibrium Transport Equations with Linear and Exponential Distance-Dependent Dispersivity

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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.

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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.

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Correspondence to Pramod Kumar Sharma.

Appendices

Appendix 1: Analytical Solution for Linear Distance-Dependent Dispersion Coefficient

Substitute Eq. (5b) into Eq. (16),

$$ \left(Kx{V}_a+{D}_0\right)\frac{d^2Y}{d{x}^2}+{V}_a\left(k-1\right)\frac{dY}{dx}-\varPsi Y=0 $$
(A1)

Defining new variable \( \xi =\sqrt{\left(kx{V}_a+{D}_0\right)} \), Eq. (A1) can be written as:

$$ {\xi}^2\frac{d^2Y}{d{\xi}^2}+\left(1-\frac{2}{k}\right)\frac{dY}{d\xi }-{\left(\frac{2}{k{V}_a}\right)}^2\varPsi {\xi}^2Y=0 $$
(A2)

Equation (A2) has the form of the following Bessel equation by Abramowitz and Stegun (1972):

$$ {\xi}^2\frac{d^2Y}{d{\xi}^2}+\left(1-\gamma \right)\xi \frac{dY}{d\xi }-\left(-{\lambda}^2{\eta}^2{\xi}^{2\eta }+{\gamma}^2-{\gamma}^2{\eta}^2\right)Y=0 $$
(A3)

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:

$$ Y={\xi}^{\gamma}\left\{{B}_1{K}_{\gamma}\left(\lambda \xi \right)+{B}_2{I}_{\gamma}\left(\lambda \xi \right)\right\} $$
(A4)

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:

$$ \overline{C_a}={\xi}^{\gamma}\left\{{B}_1{K}_{\gamma}\left(\lambda \xi \right)+{B}_2{I}_{\gamma}\left(\lambda \xi \right)\right\}+{A}_4/{A}_3 $$
(A5)

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:

$$ \overline{C_a}={\xi}^{\gamma }{B}_1{K}_{\gamma}\left(\lambda \xi \right)+{A}_4/{A}_3 $$
(A6)

One can obtain the solution of \( \overline{C_a} \) after getting the value of B 1 with different inlet boundary conditions as mentioned below:

  1. 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)
  2. 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.

  3. 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.

  4. 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

$$ \left[{a}_1\left(1-{e}^{-{b}_1x}\right){V}_a+{D}_0\right]\frac{d^2Y}{d{x}^2}+\left({a}_1{b}_1V{e}^{-{b}_1x}-{V}_a\right)\frac{dY}{dx}-\varPsi Y=0 $$
(A14)

Defining a new parameter \( z=H{e}^{b_1x} \), where H = 1 + D 0/(a 1 V a )Equation (30) can be simplified as:

$$ z\left(1-z\right)\frac{d^2Y}{d{z}^2}+z\left(1-\frac{1}{a_1{b}_1H}\right)\frac{dY}{dz}-\frac{1}{H{a}_1{V}_a{b_1}^2}\varPsi Y=0 $$
(A15)

Equation (A15) has the form of the following Gauss hypergeometric equation (Abramowitz and Stegun 1972; Gao et al. 2010):

$$ z\left(1-z\right)\frac{d^2Y}{d{z}^2}+z\left(Q-\left(1+m+n\right)\right)\frac{dY}{dz}-mnY=0 $$
(A16)

where Q = 0 and

$$ m=\frac{1}{2{a}_1{b}_1H}\left[-1+\sqrt{\left(1+\frac{4{a}_1H}{V_a}\varPsi \right)}\right] $$
(A17)
$$ n=\frac{1}{2{a}_1{b}_1H}\left[-1-\sqrt{\left(1+\frac{4{a}_1H}{V_a}\varPsi \right)}\right] $$
(A18)

As 1 ≤ z < ∞, the solution of Eq. (A17) can be written in terms of the hypergeometric function as follows by Abramowitz and Stegun (1972):

$$ Y={B}_3{z}^{-m}F\left(m,m+1;m-n+1;{z}^{-1}\right)+{B}_4{z}^{-n}F\left(n,n+1;n-m+1;{z}^{-1}\right) $$
(A19)

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

$$ \begin{array}{l}\overline{C_a}={B}_3{z}^{-m}F\left(m,m+1;m-n+1;{z}^{-1}\right)\\ {}\kern3.72em +{B}_4{z}^{-n}F\left(n,n+1;n-m+1;{z}^{-1}\right)+{A}_4/{A}_3\end{array} $$
(A20)

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:

$$ \overline{C_a}={B}_3{z}^{-m}F\left(m,m+1;m-n+1;{z}^{-1}\right)+{A}_4/{A}_3 $$
(A21)

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:

  1. 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)
  2. 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)
  3. 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.

  4. 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):

$$ {\alpha}_e=\frac{{\displaystyle \underset{0}{\overset{x_0}{\int }}\alpha (x)dx}}{{\displaystyle \underset{0}{\overset{x_0}{\int }}dx}} $$
(A29)

For linear distance-dependent model, the effective dispersivity can be written as:

$$ {\alpha}_e=\frac{1}{2}k{x}_0 $$
(A30)

For exponential distance-dependent dispersion model, the dispersivity can be written as:

$$ {\alpha}_e=\frac{a_1\left({x}_0+{e}^{-b{x}_0}/{b}_1\right)-{a}_1/{b}_1}{x_0} $$
(A31)

Appendix 4: Optimization Approach

The parameter estimation by ordinary least square problem can be given by objective function:

$$ \underset{b}{ \min \phi }={\left({C}^{*}-\widehat{C}\right)}^T\left({C}^{*}-\widehat{C}\right) $$
(A32)

where b is the parameter vector given as b = (b 1b 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:

$$ {b}^{i+1}={b}^i+\varDelta b $$
(A33)

Levenberg-Marquardt algorithm is used as the optimization algorithm, which is written as:

$$ \left({J}^T\;J+\beta\;{D}^TD\right)\;\varDelta b=-{J}^T\;e $$
(A34)

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).

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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

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