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Contaminated water flow modelling through the porous media by using fractional advection-dispersion equation (FADE) Contaminated water flow modelling through the porous media by using fractional...
Vaganov, Yurii V.; Kadyrov, Marsel A.; Drugov, Denis A.; Tugushev, Oskar A.
2023-06-01 00:00:00
GEOLOGY, ECOLOGY, AND LANDSCAPES INWASCON https://doi.org/10.1080/24749508.2022.2154924 RESEARCH ARTICLE Contaminated water flow modelling through the porous media by using fractional advection-dispersion equation (FADE) Yurii V. Vaganov, Marsel A. Kadyrov, Denis A. Drugov and Oskar A. Tugushev Department of Oil and Gas Deposits Geology, Tyumen Industrial University, Tyumen, Russia ABSTRACT ARTICLE HISTORY Received 4 November 2022 This study aims to evaluate the validity of Fractional Advection-Dispersion equation (FADE) in Accepted 30 November 2022 breakthrough curves in saturated homogenous soil media, i.e. clay, and to examine the three primary restraints, pore water velocity, dispersion coefficient, and order of fractional differen - KEYWORDS tiation which are impacting solute transport behavior (Fickian or Non-Fickian). In this study, the Fractional advection FADE framework used to characterize the transport process at depth 50 cm soil (clay). FADE dispersion equation; Main based on FORTRAN, to estimate the parameters of FADE including fractional differentia - simulation; fluid flow; porous tion (λ), the dispersive coefficient (D), and the average pore-water velocity (v). If the value of is media equal to or greater than 2, the transport is said to be Fickian otherwise is non-Fickian. In this study, the fractional differentiation of non-Fickian behavior was found 1.85 which is less than 2. On the other hand, early long-time tailing to the soil media showed an increase in the dispersion coefficient (D) for FADE and higher values in differentiation coefficient (λ = 1.85– 1.99). The results of breakthrough curves (BTCs) of relative concentration (C/C0) show that it is best fitted by using FADE. For the assessment of fitting, Root Mean Square Error (RMSE) and determination Coefficient was used to find the quality of fit. ðv; D; λÞ . 1. Introduction belongings, affecting the collection of oil, steel shops, and controlling topographical procedures, for exam- Groundwater quality issues are partly due to the ple, auxiliary improvement of the outdoor layer. increasing water demand with the increasing popula- Contaminant relocation in clay wealthy soils may tion all over the world, and partly because a great also anyhow be vital due to the fact in some zones portion of water resources, particularly the surface those soils are set over aquifers and alongside the water conduits, has been contaminated. People are streams. An important parameter for breaking down- contaminating water sources by their unawareness, flow in sedimentary conditions is the penetrability of for example, misuse of pesticides, fertilizer, industrial argillaceous units, with massive importance becoming waste material, seepage, and underground gas tanks. a member of the relationship amongst porousness and Many agricultural chemicals, such as herbicides, pes- porosity (Neuzil, 1986). Clay a permeable media ticides and nitrate have been discovered in the shallow picked up the keenness of researchers due to the fact groundwater system. Particularly, those chemicals that its little measured elements and traditionally treasured threaten drinking water supplies and in agriculture stone debris which offers earthen substances singular fields are contaminated or toxic at higher concentra- houses, such as cation alternate capacities, synergist tions and with time their concentration tends to capacities, plastic behavior whilst wet, swelling beha- decrease. Although their concentration is low, how- vior, and coffee porousness (Bergaya & Lagaly, 2013; ever, it is necessary to take steps to stop further con- Zaheer et al., 2021). tamination in contaminated regions. Therefore, when For most parts of the water remediation and treat- these contaminants pass into the freshwater systems, ment, clay is utilized. It is a profoundly compelling they will pose a great risk to the health of the public. substance in the granular frame, utilized for the treat- The movement through the porous medium causes ment and refinement of wastewater and muck dewa- the contaminant to disperse into the other fluid. tering. Fundamentally clay is made of such minerals Contaminated water is unacceptable for ingesting and other protected mixes liking to assimilate a wide but moreover for agribusiness purposes. Regular sedi- scope of contaminants and will embody suspended mentary media, for example shale and clay, accom- solids, numerous natural mixes, and toxicants. Clay modate low-permeability porous media. It has can be scattered into the treatment tank physically assumed a vital activity in securing groundwater with an estimating scoop or ceaselessly with a dry CONTACT Marsel A. Kadyrov kadyrovma@tyuiu.ru Department of Oil and Gas Deposits Geology, Tyumen Industrial University, ul. Volodarskogo 38, Tyumen 625000, Russia © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the International Water, Air & Soil Conservation Society(INWASCON). This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2 Y. V. VAGANOV ET AL. feed framework. An exact depiction of the transport of consider this atypical transport is to stretch out con- contaminants in permeable media is imperative to ventional Brownian movement to partial Brownian assess and remediate tainting in soils and aquifers. movement. Numerous specialists (Hewett, 1986; Hydrodynamic scattering includes the procedures of Neuman, 1990) have built up a few distinctive fractal mechanical scattering and sub-atomic dissemination. models dependent on partial Brownian movement. The mechanical scattering is brought about by variety Clay may be scattered into the remedy tank bodily in liquid speed in the pore space of permeable media, with an estimating scoop or frequently with a dry feed and dispersion is brought about by atoms’ arbitrary framework. Traditional Brownian motion is regularly movement. Dispersity is normally used to portray the used to painting solute shipping in permeable media. scattering conduct of solutes in permeable media. The attribute of Brownian motion lies in that it pre- The development of step-by-step genuine and dicts an immediate increment of fluctuation for numerically strong association techniques, the ever- motion eliminates with time. collective depth of PCs has stimulated the analysts to Theoretically, the scale effect of the dispersion coef- extra substantial usage of numerical codes recently. ficient will not exist, if soils in the column are perfectly The exhaustive usage of numerical fashions is more- “homogeneous.” However, it is impossible in practice over altogether improved with the aid of using their to make such a perfect “homogeneous” column. accessibility in each of the overall populace and com- Therefore, the heterogeneity of the soil particles and mercial enterprise areas (Šimůnek, 2006). Fractional non-uniformity in packing the soil columns will result advection-dispersion equation (FADE) is conse- in an enrichment of the dispersion with the travel quently used to explain the solute shipping via time. In FADE, the dispersion flux is proportional to a saturated porous media with the aid of using a step the fractional derivative of solute concentration rather forward curve. The created Fractional Advection- than the integer derivative assumed in ADE. The Dispersion Equation (FADE) relies upon the idea of FADE model has been used to simulate solute trans- the Lévy movement and is instrumental for mimicking port through overland flow (Deng et al., 2006) satu- the transportation of contaminants. The appropria- rated soils (Huang et al., 2006), and streams and rivers. tion of the contaminant attention, as opposed to FADE was used to simulate the non-Fiction transport time, is not in Gaussian shipping behavior. The crux process for the conservative solute. Anyway, the of FADE primarily based totally approach is the gra- unpredictability of multi-directional partial subordi- dient of fractional derivatives of contaminant atten- nates extraordinarily confines the utilization of FADE tion and dispersion flux being proportionate to every in multi-dimensional issues. The spatial FADE depicts different, moreover the exponent of the fractional the spread of solute mass over substantial separations derivatives well-known shows the final results of het- through a convolutional fractional derivative. The spa- erogeneous composition of porous media on transpor- tial FADE is increasingly valuable for depicting the tation of the contaminant (Benson, 1998). There are super-diffusive quick transport including long late numerous techniques for figuring out FADE. time following quicker than Fiction development Considering the Markov technique circumstance rates (Pachepsky et al., 2000). As reported by Cortis Zaslavsky (1994) constructed up a Fokker Planck- and Berkowitz, it is important to precisely determine Kolmogorov equation (FPKE) wherein the FPKE both the early arrival and late-time tailing behavior for changed into summed as much as partial request and subsurface contaminants and groundwater remedia- bought the fractional Fokker Planck-Kolmogorov cir- tion problems. Benson (1998) and Pachepsky et al. cumstance (FFPKE). A fractional dispersion circum- (2000 a, b), presented the Lévy motion-based theory, stance to painting Lévy flights making use of a change namely Fractional Advection-Dispersion Equation technique changed into inferred with the aid of using (FADE) to define and model these anomalous trans- Benson (1998), wherein partial subsidiary immedi- ports, and spatial and temporal spreading of solute ately administrator changed into characterized pri- concentration. FADE is successful in modeling satu- marily based totally on eigenvector circumstance. He rated and unsaturated transport in porous media at that factor summed up Fick’s regulation to the form (Pachepsky et al., 2000; Chakraborty, Meerschaert communicated concerning left and proper partial sub- et al., 2009) and is very useful for solute transport in sidiaries. As of late, Schumer et al. (2001) proposed soils for the separation of scale effects from the values every other method to accumulate the FADE. of dispersion coefficient. Traditional Brownian movement is frequently used FADE applicability in the early and late parts of to portray solute transport in permeable media. The BTCs shows its capability to describe the heterogene- attributes of the Brownian movement lie in that pre- ities and irregular transport behavior within the soil dicts a straight increment of fluctuation for movement columns. As talked about over, the Lévy flight por- removes with time. In any case, nonlinear increment trayal of FADE may restrain its pertinence to depict of difference of movement remove has been seen in solute transport inland developments. Second, the vast field transport tests. One conceivable plan to example of the fractional derivative λ is a key GEOLOGY, ECOLOGY, AND LANDSCAPES 3 parameter in FADE. Benson et al. gave an approach to with the intention that a time- or distance- decide λ from earlier by examining the measurements established dispersity may be avoided. How should of the hydraulic conductivity field. In any case, this the statistics concerning the transportation conduct technique requires adequate and site-explicit land esti- of solutes at 50 cm intensity scales be deducted mations to define measurable structure. In addition, closer to large dimensions and longer intervals of next to no is thought about the connection between λ time? The whole soil profile was divided into five and properties of permeable media. At long last, the alternating layers. The characteristic length multifaceted nature of multi-directional fractional (z = 10 cm) was chosen to investigate the transport derivative extraordinarily restrains the utilization of behavior at some particular depths. For maximum FADE in multi-dimensional issues. Be that as it may, components for the water remediation and remedy, the spatial FADE is a conceivably valuable model of clay is applied. It is a profoundly compelling sub- managing spatially non-local transport since it depicts stance in the granular frame, applied for the remedy the spread of solute mass over huge separations utiliz- and refinement of wastewater and muck dewatering. ing a convolutional partial derivative (Zhang et al., Fundamentally clay is a product of such minerals 2007a; Zhang and Benson, 2008). The spatial FADE and different covered mixes liking to assimilate an expands the pertinence of ADE by portraying the extensive scope of contaminants and could encom- super-diffusive quick transport including substantial pass suspended solids, several herbal mixes, and driving plume edges and quicker than-Fickian devel- toxicants. The primary aim of this research is the opment rates (Pachepsky et al., 2000). As of late, many utilization of the FADE model for the description of broadened types of FADE were created to portray the fluid flow and evaluates the solute transport solute transport in non-stationary (Zhang et al., behavior. 2007) or anisotropic (Chakraborty, Meerschaert et al., 2009) permeable media. FADE is used which possesses the remarkable capa- 2. Research methodology city for characterizing the non-Gaussian transporta- The clay soil was dried to remove the impurity con- tion conduct within the permeable medium. To all the tained in the sample, such as gravel particles, residual more characterize such strange dissemination in soils, leaves, roots, etc., and then crushed the soil with the Fractional Derivative had turned into an auspi- a grinder, like a 0.3 mm diameter sand screen, sieving cious methodology as of late. The FADE attempts that after the soil is less than 0.3 mm particle size of soil dispersion of solute complies with a Levy conveyance samples. The experiments were carried out by using whilst substituting the Gaussian dissemination as in a Plexiglas column with a length of 50 cm and an inner ADE, (Benson, 1998) suggested to utilize Lévy move- diameter of 10 cm. Experimental devices consisted of ment hypothesis to portray the non-Gaussian trans- an organic glass column, Ma bottle, pressure plate, salt portation and displayed the transient and spatial sensor, measuring cylinder, water tank, and other com- circulation of focus of solute utilizing the one- ponents (Figure 1). The concentration of NaCl solution dimensional spatial Fractional Advection-Dispersion was 240 mmol /L, and the conductivity of the solution Equation (FADE). −2 was 13.95 ms/cm measured by the conductivity FADE is an equation that solves the Fickian and meter. In this experiment, one soil column was Non-Fickian conduct of a tracer shipping in low and involved. The total length of the loading medium excessive permeability porous media. Non-Fickian (clay) was 50 cm. An electrical conductivity sensor styles of transportation come from heterogeneity, was placed at after 10 cm in the middle of soil column which may be diagnosed with the aid of using “anom- for monitoring during the filling process. After the soil alous” early method and recently tailed instances in column is filled, the medium will be saturated and filled BTCs. This dispersion depending on scales (addition- with water until the water reaches a steady state. After ally termed as either “non-Gaussian” or “anomalous”) that experiment replaces the contaminated water in the is referred to with the aid of using us as “non-Fickian” bottle with 240 mmol/L of NaCl solution. The volume transportation. This Non-Fickian behavior often con- of the liquid flowing out was recorded, and the elec- tends because the aftereffect of heterogeneities, in any trical conductivity at each position was measured. respect degrees which cannot be neglected. After the experiment the relative concentration Concentrating at the stated contention, endeavors (C/C ) data was obtained at 50 cm depth of clay soil have been undertaken to make use of estimations as column under saturated conditions. The data includes better a desire as possible to define houses of aqui- time (sec) and relative concentration (NaCl +H O). fers (common conductivity of hydraulics). This is The schematic diagram of experiment is shown in conflicting to the incentive in the back of FADE in Figure 1. The transport behavior in soil media was mild of the reality that making use of FADE is to investigated at a depth of 50 cm and was divided into make use of an increasing number of large abnormal five equal parts, each part having a depth of 10 cm. motions to symbolize the heterogeneity of the media 4 Y. V. VAGANOV ET AL. transition probability of solute particle. The probabil- ity in the close interval 1 � γ � 0 is for backward transition, whereas the probability in the closed inter- val 0 � γ � 1 is for forward transitioning, however, when γ = 0, solute particle dispersion and transition in FADE becomes symmetric(Guangyao et al., 2009). Any semi-infinite system i.e initially free from the solute x ¼ 0 with C as concentration, the analytical solution of Equation (1) will be discussed and estab- lished as given by (Griffioen et al., 1998): 2 0 13 x vt 4 @ A5 Cðx; tÞ ¼ C 1 F (2) 0 λ � �� � 1 πλ λ � � 0 cos D t In equation (2) F ð yÞ is known as the symmetric λ - stable probability function (Chakraborty, Meerschaert et al., 2009): � � signð1 λÞ 1 λ 1 λ F ð yÞ ¼ CðλÞþ ∫ exp y U ðxÞ dx λ λ (3) Where in equation (3) integration is taken w.r.t ðxÞ and CðλÞ; U ðxÞ could be stated as: � � CðλÞ ¼ 1λ> 1 λ< 1 (4) � � λ 1 λ πλx sin ð Þ U ðxÞ ¼ λ πλ cos ð Þ � � The relative concentration of solutes data in porous media of clayey soils at 50 cm depths. The analytical study of solute transportation criteria ðv; D; λÞ will be analyzed by applying Fraction Advection Dispersion Equation (Guanhua et. al, 2005). Figure1. Schematic sketch of fluid flow experiment. 2.1. Fractional Advection-Dispersion Equation 2.2. Numerical modeling (FADE) The numerical modeling of solute transport is per- For one dimensional and non-reactive tracer transpor- formed by utilizing FADEMain (Guanhua et. al, tation the spatial Fractional Advection-Dispersion 2005). For using this software, it needs some of the Equation (FADE) is stated as (Pachepsky et al., 2000): parameters to be known. After having those parameter values, we can run the program and gets the results. � � @C @C 1 γ @ C These parameters are: Average pore-water velocity ¼ v þ þ D @t @x 2 2 @x which is measured by v ¼ , where q is Darcy’s velo- � � 1 γ @ C city and n is effective porosity. Dispersion coefficient þ D (1) that is measured by: 2 2 @ð xÞ D ¼ ðt t Þ (5) In equation (2) t represents the time, C is the concen- L 0:84 0:16 8t −3 −1 0:5 tration of resident solute (ML ), v (LT ) is average λ −1 flow velocity, D (L T ) is the coefficient of disper- Where v is the average groundwater velocity, the sion, the order of fractional differentiation is λ values of t , t and t can be measured at the 0.16 0.5, 0.84 ð1< λ � 2Þ moreover, if λ equals 2 then FADE C/C =0.5, C/C =0.84 and C/C =0.16 of the soil 0 0 0 becomes ADE. (x) is the representation of the spatial column experiment. Hydrodynamic dispersion coeffi- coordinate, γ over the close interval 1 � γ � 1 cient expresses the ability of a dissolved substance to gives the relative weight for forwarding vs backward mix with water in the soil medium. GEOLOGY, ECOLOGY, AND LANDSCAPES 5 FADEMain isbased on FORTRAN (Huang et al., ðr Þ is less than 0.96 then the fitted values are not 2006), to estimate the parameters of Fractional good. Advection Equation (FADE) including fractional dif- ferentiation (λ), the dispersive coefficient (D), and the average pore-water velocity (v). FADEMain is built on 3. Results and discussion the nonlinear least-square fitting algorithm and occu- The simulation of low permeability homogenous satu- pied the analytical solution of FADE. For FADE simu- rated soil is done by FADEMain FORTRAN-based lation BTCs curves in a homogeneous soil, we used software. Implementing all the required values in the FADEMain Fortran code and three parameters (v, D, software generates a graph, which is known as λ) were estimated. a Breakthrough curve (BTC). Parameters were esti- Additionally, to best fitting approach, it was also mated at depth of (10 cm, 20 cm, 30 cm, 40 cm, calculated the determination of coefficient (r ) and 50 cm). The generated breakthrough curves by Root Mean Squared Error (RMSE) to analyze the FADE main are shown below. It was found that the sensitivity of fitting results. The r and RMSE are estimated fitted dispersion coefficient (D) was increas- a useful tool to estimate overall fit and have been ing and was greater than the measured, which is con- used to estimate solute transport parameters from sistent with the previous studies (Huang et al., 2006). BTCs. However, because the larger r and RMSE dis- Table 1 shows the different estimated parameters proportionally weigh very large differences between of low permeability homogenous saturated soil of model and measured data the smaller values of r 50 cm column. The pore water velocity is between and larger values of RMSE values may not provide 0.0007106 to 0.1094 tells us about some sort of the best fit to the observed tailing behavior. We, there- heterogeneity in the soil column. This heterogene- fore, determine the r and calculating the RMSE, ity is maybe due to non-uniformity in the packing examining the data over the entire simulation. For of the homogeneous soil column. The velocity the assessment fitting, Evaluation of Root Mean ranges are in direct accordance with the results Square Error (RMSE) and Coefficient ðr Þ will be evaluated by (Huang et al.,). Dispersion coefficient used to find the quality of fit which could be stated (Dʹ) increases with the transport distance from as (Gao et al., 2012): 10 cm to 5 cm, this scale-dependent dispersion is P another evidence of non-Fickian transport. � � ðC C ÞðC C Þ ic ic im im 2 i¼1 However, using FADE to simulate the data of r ¼ (6) P P 2 2 N N � � ðC C Þ ðC C Þ ic ic im im small-scale column test showed that the D` of i¼1 i¼1 FADE does not change significantly with the depths, and the scale effect is reflected by the rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N order of the fractional differentiation (λ) and was RMSE ¼ ðC C Þ (7) ic im i¼1 less scale dependent. As we know that if the λ value (the order of fractional differentiation) is equal to 2 Where N represents the unit number of concentra- or greater than 2 it will be fiction transport if less −3 tions, C represents estimated concentrations (ML ), ic than 2 it will be non-Fickian transport. As the −3 C represents the determined concentrations (ML ), values are between 1.85–1.99 which are less than im � � C and C , respectively represents the mean values 2, so the transport is non-Fickian and well found ic im for C and C . by FADEMain. ic im The parameters Determination of Coefficient ðr Þ Figure 2 at depth shows that it is well fitted, as and Root Mean Square Error (RMSE) are the checks time increases the concentration also increases. At for determining whether the work done is good or has a time between 550 sec to 650 sec the measured some flows. If the value of the Determination of line and the estimated line are deviating, this might Coefficient ðr Þ is greater than 0.96 then the fitted be because of soil dispersion. It is well known that values are good if the Determination of Coefficient if the λ value (the order of fractional Table 1. Estimated parameters of transport process for FADE of low permeability homogenous soil media. FADE v Dʹ λ 2 Depth (cm) (cm/sec) (cm /sec) λ R RMSE 10 0.0007106 0.00048 1.85 0.9889 0.02684 20 0.0339 0.00182 1.98 0.9935 0.023548 30 0.0484 0.00299 1.94 0.99670 0.022049943 40 0.0625 0.00570 1.96 0.9987 0.014946572 50 0.0757 0.00973 1.89 0.99948 0.007976841 6 Y. V. VAGANOV ET AL. 1.2 0.8 0.6 Measured FADE 0.4 0.2 time (sec) Figure 2. A measured breakthrough curve fitted with FADE, for the homogenous saturated soil column having a length of 10 cm. differentiation) is equal to 2 it will be Fickian the concentration begins from this point. At the transport if less than 2 it will be non Fickian start, the estimated concentration is high and devi- transport. Furthermore, the estimated λ value (the ates between 644sec and 772sec, this could be due order of fractional differentiation) in FADE is 1.85 to soil dispersion. The value of pore water velocity which is less than 2. The pore water velocity is is 0.0484 and of dispersion is 0.00299, as appeared 0.0007106 which is less due to low heterogeneity in Table 1. The assessed λ value (the order of and the value of dispersion is 0.00048, as shown in fractional differentiation) in FADE is 1.94 which Table 1. is less than 2. Figure 3 at depth 20 cm shows the relative con- Figure 5 at depth 40 cm shows the relative concen- centration plotted at the y-axis whereas time in tration plotted at y-axis and time in seconds marked at (seconds) is shown at the x-axis. The graph starts x- axis. The graph is starting at 455sec because the at 550 sec because the concentration begins from concentration begins from this point. It is well-fitted this point. It is well fitted from the start but devi- graph of measured and estimated values. The value of ates between 600 sec and 734 sec, this could be due pore water velocity is 0.0625 and of dispersion is to soil dispersion. The value of pore water velocity 0.00570, as appeared in Table 1. The assessed λ value is 0.0339 and of dispersion is 0.00182, as appeared (the order of fractional differentiation) in FADE in Table 1. The assessed λ value (the order of is 1.96. fractional differentiation) in FADE is 1.98 which Figure 6 at depth 50 cm shows the relative concen- is less than 2. tration plotted at the y-axis and time in seconds Figure 4 at depth 30 cm shows the relative con- marked at the x-axis. The graph starts at 552sec centration plotted at the y-axis and time in seconds because the concentration begins from this point. It at the x-axis. The graph starts at 504 sec because is a well-fitted graph of measured and estimated values. The value of pore water velocity is 0.0757 and of dispersion is 0.00973, as presented in Table 1. The assessed λ value (the order of fractional differentia- 1.2 tion) in FADE is 1.89. The estimated values of dispersion (D) and order 0.8 of fractional differentiation (λ) of FADE, respec- 0.6 tively, provide a suitable clarification for Fickian Measured and non-Fickian transport. It was observed that 0.4 FADE the FADE is based on clear heterogeneities which 0.2 lead to the Lévy flight patterns of solute transport. The FADE is successful at modeling saturated and unsaturated transport in porous media. (Pachepsky Time(sec) et al., 2000; Zhang et al., 2005; Huang et al., 2006; Chakraborty, Meerschaert et al., 2009), and is very Figure 3. A measured Breakthrough curve fitted with FADE, for useful for solute transport in soils for the the homogenous saturated soil media at depth of 20 cm. C/C C/C 912 GEOLOGY, ECOLOGY, AND LANDSCAPES 7 1.2 0.8 0.6 measured FADE 0.4 0.2 Time(sec) Figure 4. A measured Breakthrough curve fitted with FADE, for the homogenous saturated soil column having a length of 30 cm. 1.2 0.8 0.6 measured 0.4 FADE 0.2 TIME(SEC) Figure 5. A measured Breakthrough curve fitted with FADE, for the homogenous saturated soil column having a length of 40 cm. 1.2 0.8 0.6 measured FADE 0.4 0.2 TIME(SEC) Figure 6. A measured Breakthrough curve fitted with FADE, for the homogenous saturated soil column having a length of 50 cm. C/C C/C C/C 0 0 0 552 436 497 571 483 590 502 535 609 521 628 540 573 647 559 666 578 611 685 597 704 616 649 723 635 742 654 687 761 673 780 692 725 799 711 818 730 763 837 749 856 768 801 875 787 894 806 839 913 825 932 844 877 951 863 896 8 Y. V. VAGANOV ET AL. separation of scale effects from the values of trans- “Low Pressure Gas Extraction Technologies of the Cenomanian Production Complex” (project No. FEWN- port coefficient. 2020-0013 for 2020−2022). 4. Conclusion Disclosure statement We used the Factional Advection-Dispersion Equation (FADE) framework to characterize the No potential conflict of interest was reported by the transport process at a depth of 50 cm. FADEMain author(s). based on FORTRAN, to estimate the parameters of Fractional Advection Equation (FADE) including was References fractional differentiation (λ), the dispersive coefficient (D), and the average pore-water velocity (v). Relative Benson. (1998). The fractional advection-dispersion concentration (C/C ) data (NaCl +H O) was com- 0 2 eqution. Deveploment and application, Doctoral disserta- tion, Virginia Tech. posed and analyzed in one-dimensional low perme- Bergaya, F., & Lagaly, G. (2013). 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For the assessment of fitting, the FADE fits the 1029/97WR02027 breakthrough curves (BTCs) have lower values of root Guangyao, G., Zhan, H., Feng, S., Huang, G., & Mao, X. mean square error (RMSE) and higher values of deter- (2009). Comparison of alternative models for simulating mination coefficient ðR Þ. It was found that numerical anomalous solute transport in a large heterogeneous soil simulation turns out to be in good agreement for any column. Journal of Hydrology, 377(3–4), 391–404. https:// doi.org/10.1016/j.jhydrol.2009.08.036 optimal of the parameters in the soil column. Proper Guanhua, H., Quanzhong, H., Hongbin, Z., Jing, C., Yunwu, characterization and accurate prediction of contami- X., & Shaoyuan, F. (2005). Modeling contaminant trans- nants in the soil column system would help to risk port in homogeneous porous media with fractional assessment of pollution in different sources of all advection-dispersion equation. Science China-Earth water bodies. The dispersion coefficient of FADE (D ) Sciences, 48, 295–295. https://doi.org/10.1360/YD2005- 48-S2-295 can be quantified with an exponential function (e.g., λ Hewett, T. A. (1986). Fractal distributions of reservoir het- in FADE). The FADE can better describe the late time erogeneity and their influence on fluid transport. In SPE tailing non-Fickian transport in small homogeneous Annual Technical Conference and Exhibition. Society of soil columns. The numerical forms of the FADE will Petroleum Engineers. predict contaminated concentrations in downstream Huang, H., Huang, Zhan, Q., & Zhan, H. (2006). Evidence aquifers far earlier than other transport models. of one-dimensional scale-dependent fractional advection-dispersion. Journal of Contaminant Hydrology, 85(1), 53–71. https://doi.org/10.1016/j.jcon Acknowledgments hyd.2005.12.007 Nagar, R., & Raju, S. (2003). Women, NGOs and the The research was carried out using instruments provided by Contradictions of Empowerment and Disempowerment: the Centre for Advanced Research and Innovation (Tyumen A Conversation. Antipode, 35, 1–13. Industrial University, Tyumen city) and Centre of the Neuman, S. P. (1990). Universal scaling of hydraulic con- Shared Facilities (Kazan Federal University). This work ductivities and dispersivities in geologic media. Water was supported by the Ministry of Science and Higher Resources Research, 26(8), 1749–1758. https://doi.org/10. Education of the Russian Federation under project titled 1029/WR026i008p01749 GEOLOGY, ECOLOGY, AND LANDSCAPES 9 Neuzil. (1986). Groundwater flow in low–permeability Šimůnek. (2006). Models of water flow and solute transport environments. Water Resources Research, 22(8), in the unsaturated zone. 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