Application of temporal moments to interpret solute transport with time-dependent dispersion


Temporal moments of solute transport through porous media are calculated to analyze the time average spatial distribution of solute plume. Simulation of spatially and temporally distributed breakthrough curves (BTCs) is computationally rigorous and lacking the explanation about overall plume evolution within porous media. However, temporal moment provides an attractive and simple solution to study the plume behavior. In this study, temporal moments are presented to interpret solute plume behavior in heterogeneous porous media such as hydraulically coupled stratified porous media with different time-dependent dispersion models. Governing equations of solute transport have been solved numerically using Crank-Nicolson scheme, and further solute concentration data has been utilized to calculate moments of solute concentration using numerical integration. The effect of various parameters such as mass-transfer coefficient, pore-water velocity, time-dependent dispersion coefficients, and porosity of mobile region on the transport of solute has been studied through sensitivity analyses. Temporal moment analysis revealed that the mass recovery, mean residence time, and variance are sensitive to the estimated parameters. Numerical results suggested that the asymptotic time-dependent dispersion function with mobile–immobile model represents the plume spreading through heterogeneous porous media in a more realistic manner.

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\( C_{0} \) :

Injected concentration of solute source, (M/L3)

\( C_{m} \) :

Solute concentration in the mobile region at any time t, (M/L3)

\( C_{im} \) :

Solute concentration in the immobile region at any time t, (M/L3)

\( D_{\left( t \right)} \) :

Time-dependent hydrodynamic dispersion coefficient along the flow velocity, (L2/T)

\( D_{0} \) :

Maximum or uniform dispersion coefficient, (L2/T)

\( D_{m} \) :

Effective diffusion coefficient, (L2/T)

\( D_{macro} \left( x \right) \) :

Macro-dispersion coefficient calculated using second temporal moment

\( f \) :

Fraction of sorption sites that equilibrate instantly with the mobile regions

\( K_{A} \) :

Asymptotic time-dependent dispersion coefficient, (T)

\( K_{dm} \) :

Distribution coefficient of the linear sorption process in the mobile region, (L3/M)

\( K_{dim} \) :

Distribution coefficient of the linear sorption process in the immobile region, (L3/M)

\( K_{L} \) :

Linear time-dependent dispersion coefficient, (T)

\( M_{0} \) :

Zeroth absolute temporal moment

\( M_{1} \) :

First absolute temporal moment

\( M_{2} \) :

Second absolute temporal moment

\( M_{n} \) :

nth absolute temporal moment

\( \mu_{lm} \) :

First-order transformation coefficient for solution phase in the mobile region, (T−1)

\( \mu_{lim} \) :

First-order transformation coefficient for solution phase in the immobile region, (T−1)

\( \mu_{sm} \) :

First-order transformation coefficient for sorbed phase in the mobile region, (T−1)

\( \mu_{sim} \) :

First-order transformation coefficient for sorbed phase in the immobile region, (T−1)

\( \mu_{1} \) :

First normalized temporal moment

\( \mu_{2} \) :

Second normalized temporal moment

\( \mu_{n} \) :

nth order normalized temporal moment

\( \omega \) :

First order mass transfer coefficient (T−1)

\( q \) :

Flow rate, (L/T)

\( \rho_{b} \) :

Bulk density of the porous medium, (M/L3)

\( t \) :

Total simulation time, (T)

\( t_{p} \) :

Pulse duration, (T)

\( T_{1} \left( x \right) \) :

First temporal moment

\( T_{2} \left( x \right) \) :

Second central temporal moment

\( \theta_{m} \) :

Volumetric water content of the mobile region

\( \theta_{im} \) :

Volumetric water content of the immobile region

\( \theta \) :

Total volumetric water content of the porous media

\( v_{m} \) :

Mobile pore water velocity, (L/T)

\( V\left( x \right) \) :

Solute velocity calculated using first temporal moment

\( x \) :

Spatial coordinate taken in the direction of the fluid flow, (L)


Advection dispersion equation


Advection-dispersion model with asymptotic time-dependent dispersion


Advection-dispersion model with linear time-dependent dispersion


Mobile–immobile model


Mobile–immobile model with asymptotic time-dependent dispersion function


Mobile–immobile model with constant dispersion function


Mobile–immobile model with linear time-dependent dispersion function


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All the work has been carried out at Indian Institute of Technology, Mandi during Master’s study of first author. First author would like to thank Indian Institute of Technology, Mandi for supporting this study. The authors would like to thank the reviewer and editor for their constructive and valuable comments.

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Correspondence to Abhay Guleria.

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Guleria, A., Swami, D., Joshi, N. et al. Application of temporal moments to interpret solute transport with time-dependent dispersion. Sādhanā 45, 159 (2020).

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  • Moment analysis of solute concentration
  • heterogeneous porous media
  • mobile–immobile model
  • mass-transfer coefficient
  • time-dependent dispersion