, 44:81 | Cite as

Non-reactive solute transport modelling with time-dependent dispersion through stratified porous media

  • ABHAY GULERIAEmail author


We present a numerical solution of the mobile–immobile model (MIM) with time-dependent dispersion coefficient to simulate solute transport through heterogeneous porous media. Observed experimental data of non-reactive solute transport through hydraulically coupled stratified porous media have been simulated using asymptotic and linear time-dependent dispersion functions. Non-Gaussian breakthrough curves comprising long tails are simulated well with the MIM incorporating asymptotic time-dependent dispersion model. The system is under the strong influence of physical nonequilibrium, which is evident by variable mass transfer coefficient estimated at different down-gradient distances. Asymptotic time-dependent functions are capable of capturing the rising limb of the solution phase breakthrough curves with improved accuracy, whereas tailing part simulation capabilities are similar for both asymptotic and linear time-dependent dispersion functions. Further, the temporal moment analysis demonstrated increased spreading, variance for linear dispersion model as compared with asymptotic dispersion model. It is also observed that the first-order mass transfer coefficient varies inversely with travel distance from the input source. It can be concluded from the study that MIM with time-dependent dispersion function is simpler yet sensitive to account for medium’s heterogeneity in a better manner even for small observation distances from the source.


Nonreactive solute mobile–immobile model time-dependent dispersion temporal moment of solute concentration stratified porous media 

List of symbols


injected concentration of solute source (M/L3)


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


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

D( t)

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


maximum dispersion coefficient (L2/T)

\( D_{m} \)

effective diffusion coefficient (L2/T)


asymptotic time-dependent dispersion coefficient (T)


linear time-dependent dispersion coefficient (T)


zeroth absolute temporal moment of solute concentration


first absolute temporal moment of solute concentration


second absolute temporal moment of solute concentration


first normalized temporal moment of solute concentration


second central temporal moment of solute concentration

\( \theta_{m} \)

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)

\( q \)

flow rate (L/T)

\( \omega \)

first-order mass transfer coefficient (T–1)

\( f \)

fraction of adsorption sites that equilibrate instantly with the mobile regions

\( \mu_{lm} \)

first-order decay coefficient for degradation of solute in the mobile solution phase (T–1)

\( \mu_{lim} \)

first-order decay coefficient for degradation of solute in the immobile solution phase (T–1)

\( \mu_{sm} \)

first-order decay coefficient for degradation of solute in the mobile region adsorbed solid phase (T–1)

\( \mu_{sim} \)

first-order decay coefficient for degradation osolute in the immobile region adsorbed solid phase (T–1)

\( \mu_{1} \)

first normalized temporal moment of solute concentration

\( \mu_{2 } \)

second normalized temporal moment of solute concentration

\( K_{d} \)

distribution coefficient of the linear sorption proc(L3/M)

\( \rho_{b} \)

bulk density of the porous medium (M/L3)

\( x \)

spatial coordinate taken in the direction of the fluid flow (L)


advection–dispersion equation


asymptotic time-dependent dispersion function


linear time-dependent dispersion function


mobile–immobile model


mobile–immobile model with asymptotic time-dependent dispersion function


mobile–immobile model with linear time-dependent dispersion function


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Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.School of EngineeringIndian Institute of Technology MandiKamandIndia

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