Introduction

Solid waste disposal sites like open dumps represent a significant source of metals released into the environment1,2,3,4,5. Soil contaminated by heavy metals from solid waste disposal sites is a serious problem because soils are regarded as the ultimate sink for heavy metals discharged into the environment, as many heavy metals are bound to soils6. The soil may be contaminated with heavy metals such as lead, copper, zinc, iron, manganese, chromium and cadmium and these heavy metals in solid waste led to serious problem because they cannot be biodegraded. According to Freeze and Cherry7 leachate from a solid waste disposal site generally contain major elements like calcium, magnesium, potassium, nitrogen and ammonia, trace metals like iron, copper, manganese, chromium, nickel, lead and organic compounds like phenols, polyaromatic hydrocarbons, acetone, benzene, toluene and chloroform. According to Ahaneku and Sadiq8, heavy metal uptake in agricultural soils is of great concern due to food safety issues and potential health implications. MSW leachate varies widely in composition, contains both dissolved and suspended materials depending on the age of the dumpsite and the type of solid waste. Leachate that escapes from the MSW dumpsite may migrate through the unsaturated zone and eventually reach the groundwater table and then get transported through the saturated zone to a point of discharge (i.e., a pumping well, a stream, a lake, etc.) thereby causing contamination.

Modeling is the process by which scientists represent ideas about the natural world to each other, and then collaboratively make changes to these representations over time in response to new evidence and understandings9,10. A model can come in many shapes, sizes, and styles. It is important to emphasize that a model is not the real world but merely a human construct to help us better understand real world systems. In general, all models have an information input, an information processor, and an output of expected results. Models do not just reflect reasoning, they also stimulate new ideas11,12. According to Ndirika and Onwualu13, model is a representation of the construction and working of some system of interest; model is similar to but simpler than the system it represents; one purpose of a model is to enable the analyst or the researcher to predict the effect of changes to the system. Pachepsky et al.14 developed a generalized Richards’ equation to simulate water transport in unsaturated soils. Simulations of water transport in soil are ubiquitous for experiments on water transport in soil horizontal columns, Richards’ equation predicts that volumetric water contents should depend solely on the ratio (distance)/(time)q where q ¼ 0:5: Substantial experimental evidence shows that value of q is significantly less than 0.5 in some cases. Nielsen et al.15 related values of q, 0.5 to ‘jerky movements’ of the wetting front, i.e. occurrences of rare large movements. The corresponding mathematical model is a generalized Richards’ equation in which the derivative of water content on time is a fractional one with the order equal or less than one. The equation was first solved numerically and then fitted the solution to data on horizontal water transport. For such systems, Richards’ equation reduces to the mathematical expression presented in Eq. (1).

$$ \frac{\partial \theta }{{\partial {\text{t}}}} = { }\frac{\partial }{{\partial {\text{x}}}}\left[ {{\text{D}}\left( \theta \right)\frac{\partial \theta }{{\partial {\text{x}}}}} \right] $$
(1)

where \(\theta\) is the volumetric soil water content, D is the soil water diffusivity, x is the distance from one of the ends of the column, t is time. Soil bulk density changes and soil water hysteresis are ignored in this formulation16. Introduction of the Boltzmann variable transforms Eq. (1) into an ordinary differential equation presented in Eqs. (2), (3).

$$ {\uplambda } = { }\frac{{\text{x}}}{{{\text{t}}^{0.5} }} $$
(2)
$$ - \frac{\lambda }{2}\frac{{{\text{d}}\theta }}{{{\text{d}}\lambda }} = { }\frac{{\text{d}}}{{{\text{d}}\lambda }}\left[ {{\text{D}}\left( \theta \right)\frac{\partial \theta }{{\partial \lambda }}} \right] $$
(3)

which has been used to find analytical solutions for soil water flow problems and also to find the dependence of the diffusivity D on soil water content \(\theta\)17. If Eq. (3) is applicable then soil water content is a function of the Boltzmann variable l, and, for the same values of soil water content, one should expect the same values of the Boltzmann variable. Validity of Eq. (3) can be tested with experimental data consisting of observed soil moisture changes during infiltration in horizontal soil columns with initially uniform soil water content. Distances and times at which the same values of water content have been observed must be noted as shown in Eqs. (4), (5).

$$\frac{{\mathrm{x}}_{1}}{{{\mathrm{t}}_{1}}^{0.5}}= \frac{{\mathrm{x}}_{2}}{{{\mathrm{t}}_{2}}^{0.5}}=\frac{{\mathrm{x}}_{3}}{{{\mathrm{t}}_{3}}^{0.5}}$$
(4)

Meanwhile,

$$\mathrm{x}=\mathrm{ A}{\mathrm{t}}^{0.5}$$
(5)

where the multiplier A depends only on water content. This equation means that the dependence between log(x) and log(t) plotted in log–log coordinates is linear and the slope of this dependence is 0.5 whereas the intercept depends on the water content. Significant deviations from Eq. (5) have been observed in many published experiments. Gardner and Widtsoe18 and Nielsen et al.9 recorded the progress of the wetting front in air-dry soil uniformly packed in horizontal columns and a negative pressure head was held at one end of the columns. The largest distance where the wetting front was observed was 50 cm. A linear dependence can be traced in mathematical relationship expressed in Eq. (6).

$$\mathrm{logx}=\mathrm{ LogA}+\mathrm{qlogt}$$
(6)

As a key to the management of groundwater, Patil and Chore19 demonstrated the usefulness of mathematical models in the study of the movement of fluids and contaminants in the subsurface environment. They compared experimental, analytical solution and numerical method in the evaluation of soil and groundwater contaminants transport. Similarly, Islam et al.20 applied the governing equation for contaminant transport involving advection–diffusion and using the explicit finite difference method to evaluate groundwater contaminants. Several researchers have studied pollutant migration in solid waste dumpsite. However, very few literatures on application of finite element method to predict contaminant transport exist.

In order to predict reliably and effectively the pollutant contaminant transport in soil layers, it is necessary to develop a comprehensive mathematical modelling of the heavy metal concentration in the soil column, and to design an efficient simulation procedure. In this study, a general approach which combines a mathematical modelling method and Finite Element Method (FEM) simulation is proposed to investigate the concentration of selected heavy metals passing through saturated soil layers. An approach to modelling and computation of the spread of the heavy metal due to the leachate immigration into the soil layers and groundwater at the actual municipal solid waste landfill site was implemented to validate the proposed novel mathematical modeling and FEM-based algorithms using COMSOL Multiphysics, in which the initial value of the selected metal concentrations were calculated by using the data from the experiments which are analyzed from the soil samples collected from the actual municipal solid waste dumpsite. COMSOL Multiphysics is easy, seamless interface between solute transport and other physics fields. It has the ability to modify governing equations, and flexibility in solver selection in the form of direct or fully coupled solutions21,22,23 which makes it unique from other applications. The newly introduced Multiphysics method for modelling the spread of the metal concentration in the leachate immigration in the 2D space of soil layers, and the new procedure for simulating the pollutant transport in the Multiphysics interface are the main contributions of this study.

Methodology

Study area

The study was conducted at Uyo Municipal Solid Waste Dumpsite in Uyo Local Government Area, Akwa Ibom State, Nigeria. Uyo, the capital of Akwa Ibom State lies between latitude 4°30″ and 5°30″ N and longitudes 7°30″ and 8°30″ E. Uyo is within the equatorial region characterized by wet and dry seasons. According to Robert24, the most outstanding feature of the equatorial climate is its uniformity of temperature throughout the year. Rainfall begins about March and ends around October with a little dry spell called “the August break” occurring in August25,26,27,28.

With its location within the tropical rainforest and dense population, Uyo, like other major cities in Nigeria generates enormous municipal solid waste which is not adequately managed. The site is used by Environmental Protection and Waste Management Agency for waste disposal. Most of the wastes disposed are domestic and household wastes. This dumpsite is operated as an open dumpsite. The waste dumpsite has function efficiently for about twenty years and there is surface water at the vicinity of the waste dumpsite.

Collection and analysis of sample

Nine core samples were collected at each of the three leachate monitoring stations constructed within the waste dumpsite after infiltration runs were conducted on each of the three spots using synthetic container (Fig. 1). Core samples were collected from the nine designated depth zones with core cylinders measuring 10, 20, 30, 40, 50, 60, 70, 80 and 90 cm in height and 4.5 cm internal diameter for the determination of leachate flux (q). The cylinders were driven into each of the soil by hitting a piece of wood placed on top of the cylinder making sure that the cylinder did not tilt at any point. The soil was excavated from around the cylinder and the soil beneath the cylinder bottom was cut. Excess soil from the cylinder end was trimmed. The bottom of the cylinder was covered with a piece of cloth hold in place with a rubber band and then labeled and taken to laboratory. Heavy metals namely Fe, Cu, Zn, Pb, Mn, Ni and Cd were determined using double beam DW-AA320N Atomic absorption spectrometer. Infiltration tests were made using Gilson HMA-635 double ring infiltrometer apparatus.

Figure 1
figure 1

(Source: AISMLS, 2020).

Map of Uyo Local Government Area showing the waste dumpsite and sample collection points

Full size image

Infiltration measurement

Infiltration runs were carried out beside each leachate monitoring stations. Infiltration tests were made using the double ring infiltrometer method29. The infiltrometer, consisting of an outer ring with diameter 50 cm and an inner ring with diameter 30 cm were placed concentrically on the soil surface and carefully driven 10 cm into the soil using a driving plate and impact absorbing hammer. The soil surface within the rings was covered with leaves to protect it from forming the kinetic force of water and to prevent direct splash. Sediment free water was first poured into the outer ring and allowed to infiltrate. Immediately afterwards, water was added to a depth of 15 cm in the inner ring. The outer ring acted as a buffer to discourage lateral flow of the water in the inner ring. Infiltration rate was measured from the inner ring for 2 h using a collaborated float hold in place by a bridge. The intake was recorded and the data was used to determine the infiltration rate of initial and final infiltration rate as well as other infiltration characteristics with soils.

Computation of sorptivity and transmissivity

Soil water sorptivity (S) which represents the ability of a soil to absorb or desorb water by capillary processes, and transmissivity (A) a measure of the ability of the soil to conduct the flow of water were determined graphically by fitting the approximate Philip30 algebraic infiltration equation into the field data. The primary data are measured values of cumulative infiltration I expressed in cm, as a function of time. The values represent the total amount of water infiltrated into the soil surface from the beginning of the infiltration test. Philip30 showed that the cumulative, one-dimensional infiltration commonly encountered in the field, can easily be expressed as a function of time, I (t), thus:

$$\mathrm{I}={\mathrm{St}}^{1/2}+\mathrm{At}$$
(7)

where I (cm) is cumulative infiltration at time t (sec.), S (cm sec−1) is the soil water Sorptivity obtained as the slope of I versus √t, A (cm sec−1) is the soil water transmissivity related to soil’s hydraulic conductivity and is the intercept, and the infiltration rate is:

$$\mathrm{I}=\frac{1}{2}{\mathrm{St}}^\frac{1}{2}+\mathrm{A}$$
(8)

Subsequently, the parameters S and A which were obtained graphically were fitted into Eq. (8) to calculate the infiltration rate. The Philip30 equation is a mechanistic infiltration model because it is derived from the physically-based water flow equation, i.e. from sound principles of physics. The model was chosen because the parameters S and A give considerable insight into the hydraulic conditions which prevail in the soil profile prior to and during the infiltration process, and, excluding the development of infiltration inhibiting factors, aid the understanding of the decaying infiltration rate with elapsed time. However, the curve-fitting procedure was adopted which converted the equation into an empirical rather than a physical model.

Components of the model

There are four major components arising from the different forces that influence leachate and its potential in the soil. These include:

  1. a)

    The matric force: This refers to attraction of leachate to solid particle surfaces and capillary attraction in pore spaces. This force gives rise to the matric potential, M, which is always negative.

  2. b)

    Osmotic force: This refers to attraction of leachate to ions, giving rise to the osmotic or solute potential, o which is always negative too.

  3. c)

    Gravitational force: This refers to attraction of leachate to center of the earth. It is a downward pull on leachate. It gives rise to the gravitational potential, ψg which may be positive or negative depending on elevation of that point relative to a reference elevation. If the point is higher than the reference elevation the ψg is positive but if lower then ψg is negative.

  4. d)

    Pressure of leachate molecules: This gives rise to pressure potential p, which is always positive. Hence, total leachate potential at a point along the soil column (ψT) is given by the formula in Eq. (9):

    $${\uppsi }_{\mathrm{T}}={-\uppsi }_{\mathrm{m}}+ {\uppsi }_{\mathrm{p}} + {\uppsi }_{\mathrm{m }}+ \left(-{\uppsi }_{\mathrm{o}}\right)+\left(-{\uppsi }_{\mathrm{g}}\right)$$
    (9)

where \({\uppsi }_{\mathrm{T}}\) = Total soil leachate potential, ψm, ψp, ψo and ψg are the matric, pressure, osmotic and gravitational potentials respectively. In terms of head presented in Eq. (10):

$${\mathrm{H}}_{\mathrm{T}}={-\mathrm{H}}_{\mathrm{m}}+ {\mathrm{H}}_{\mathrm{p}} + {\mathrm{H}}_{\mathrm{m }}+ \left(-{\mathrm{H}}_{\mathrm{o}}\right)+\left(-{\mathrm{H}}_{\mathrm{g}}\right)$$
(10)

where \({\mathrm{H}}_{\mathrm{T}}\) is the total head (cm), \({\mathrm{H}}_{\mathrm{m}}\), \({\mathrm{H}}_{\mathrm{p}}\), \({\mathrm{H}}_{\mathrm{o}}\) and \({\mathrm{H}}_{\mathrm{g}}\) are the matric, pressure, osmotic and gravitational heads respectively.

Generally, leachate is produced on the soil surface but its flow into the soil occurs when there is a difference in total soil leachate potential (TSLP) between the point of production to point of deposition. Hence, the driving force is the gradient (the change) in the total leachate potential between the two points. For instance, considering two points in the soil A and B; at point A, TSLP is higher than at point B. Leachate will then flow from point A to pint B. The driving force is therefore the gradient given as expressed in Eq. (11):

$$\mathrm{DF}=\frac{{\uppsi }_{\mathrm{A}}- {\uppsi }_{\mathrm{B}}}{\mathrm{A B}}$$
(11)

where \(\mathrm{DF}\) is the driving force \({\uppsi }_{\mathrm{A}}\mathrm{ and }{\uppsi }_{\mathrm{B}}\) are the TSLP at points A and B while LA B is the distance between point A and B in the soil. Therefore, leachate flow will always occur in the direction of decreasing TSLP.

Basic principles of the model

There are three basic principles covering this Soil Leachate Transport Model (SLTM):

  1. i)

    The driving force of leachate for the transport of leachate from the point of production (soil surface) down the soil is the gradient in the intensity of the components of Total Soil Leachate Potential (TSLP).

  2. ii)

    Flow of leachate down the soil is directly proportional to the intensity gradient

  3. iii)

    Flow is in the direction of decreasing intensity

Mathematical expression of the model

The driving force (DF) is the gradient in total hydraulic head (intensity) between the two points within which flow is occurring. It should be recalled that hydraulic head, H, is the leachate potential ψ, expressed as weight or distance. Hence, Total hydraulic head, H, is defined as presented in Eq. (12):

$$\mathrm{H}=\mathrm{h}+\mathrm{ z}$$
(12)

where h is the pressure head of water and z is the gravitational head, all expressed in cm.

Consider a vertical soil column, L (cm) long with a cross sectional area, A (cm2). The top of the column, T is ponded with water to a height of h cm and leachate is just dripping from the bottom, B, of the column as shown in Fig. 2.

Figure 2
figure 2

Hypothetical description of soil leachate transport in a unit soil area.

Full size image

The above hypothetical case is used to define the driving force used in this model and other concepts associated with leachate flow in the soil according to the outlined assumptions of this model. Driving force (DF) is the gradient in the total hydraulic head between the top (point T) and the bottom (point B). Total hydraulic gradient therefore is defined by the formula in Eq. (13):

$$\frac{\mathrm{\Delta H}}{\mathrm{\Delta X}}= \frac{{\mathrm{H}}_{\mathrm{T}}- {\mathrm{H}}_{\mathrm{B}} }{{\mathrm{X}}_{\mathrm{T }}- {\mathrm{H}}_{\mathrm{B}}}$$
(13)

where HT and HB are the total hydraulic heads at T and B respectively and XT and XB are distances at points T and B, respectively. Substituting Eq. (13), into Eq. (12), it becomes the expression in Eq. (14):

$$\frac{\mathrm{\Delta H}}{\mathrm{\Delta X}}= \frac{{(\mathrm{h}}_{\mathrm{T}}+ {\mathrm{z}}_{\mathrm{T}})-({\mathrm{h}}_{\mathrm{B}}+{\mathrm{z}}_{\mathrm{B }})}{{\mathrm{X}}_{\mathrm{T }}- {\mathrm{X}}_{\mathrm{B}}}$$
(14)

Substituting hT = h cm, ZT = 1 cm, hB = 0 cm, ZB = 0 cm, XT = 0 cm, and XB = − 1 cm into (12) yields the mathematical relationship in Eq. (15):

$$\frac{\mathrm{\Delta H}}{\mathrm{\Delta X}}=\frac{(\mathrm{h}+ 1)-(0+0)}{0-(- 1)}= \frac{\mathrm{h}+ 1}{1}$$
(15)

Thus, driving force (DF) is expressed in Eq. (16):

$$\frac{\mathrm{\Delta H}}{\mathrm{\Delta X}}= \frac{\mathrm{h}+\mathrm{ L}}{\mathrm{L}}$$
(16)

Since, leachate is on the top of the column especially when it is much as in during rainfall event, the top T, is therefore saturated and experiencing hydrostatic pressure or positive pressure potential. The height of the leachate above this point is the numerical value of the leachate pressure head (cm) at point T31. Since leachate is just dripping at the bottom, it is at atmospheric pressure and therefore at this point is zero. The above applies to unsaturated condition. The second principle was based on the second assumption that the flow of leachate down the soil is directly proportional to the intensity gradient. This was used to model the flux which is defined as the amount of leachate flowing through a unit cross sectional area of the soil per unit time. Considering Fig. 2, flux of leachate between T and B is given by the expression in Eq. (17):

$$ {\text{q}} = \frac{{\text{Volume of leachate}}}{{{\text{Area }} \times {\text{ time}}}}{ } = { }\frac{{\text{Q}}}{{{\text{At}}}}{ } = { }\frac{{{\text{cm}}^{3} }}{{{\text{cm}}^{{2{ }}} {\text{sec}}}} $$
(17)

But this flux is proportional to the hydraulic head gradient and this is the Darcy’s law of steady state as expressed in Eq. (18)32:

$$\mathrm{q}= \frac{\mathrm{Q}}{\mathrm{At}} = -\mathrm{K}\frac{\mathrm{\Delta H}}{\mathrm{\Delta X}}$$
(18)

where K is the proportionality constant known as hydraulic conductivity. Hence, Eq. (17) is the SLTM which defined amount of leachate flux as the product of the hydraulic conductivity and hydraulic head gradient. The negative sign indicates that the flux is in the direction of decreasing hydraulic head as stated in the third assumption.

According to Eq. (18), it is possible to model leachate transport rates for any head difference over any distance in a given growing medium. The real difficulty, however, is in finding K. According to Allaire et al.33, K may be found with a series of formulae according to the structure like in Eqs. (17) and (18) as follows in Eq. (19):

$$ {\text{K}} = {\text{ Ksat}}\left( {\frac{{{\text{WEP}}}}{{{\text{TPS}}}}} \right)^{{\text{m}}} $$
(19)

where K is unsaturated hydraulic conductivity, Ksat is the saturated hydraulic conductivity, WEP is the leachate fill pores fraction, TPS is the total pores space (total porosity) while m is the constant defined by type of soil structure as follows:

  1. i)

    Very fine structure: m = 1

  2. ii)

    Fine structure: m = 2

  3. iii)

    Medium structure: m = 3

  4. iv)

    Blocky, Platy, Massive structure: m = 4

The influence of WEP on K is very large as can be seen from the exponential function. The relation WEP over TPS represents the tortuosity of the transport path. The measurement of either Ksat or Kh requires careful pre-treatment of the samples and apparatus so to overcome error in determination of Ksat, Eq. (20) was adopted as used by Allaire et al.33 to determine unsaturated hydraulic conductivity directly as follows in Eq. (20):

$$\mathrm{I}={\mathrm{C}}_{1}\mathrm{t }+ {\mathrm{C}}_{2}\sqrt{\mathrm{t}}$$
(20)

where I is infiltration rate, C1 is the transitivity which is the hydraulic conductivity (unsaturated), C2is the soil Sorptivity. The numerical solution of Eq. (20) produced the soil leachate transport model of the study area referred to as LEATRAM.

$$\mathrm{q}=\mathrm{ C}{\mathrm{x}}^{-\mathrm{n}}$$
(21)

where q is the soil leachate flux (cm hr−1), C is transmissivity coefficient which depends on the overall soil characteristics, x is the thickness of the soil (cm) representing depth of transport while the negative exponent represent the influence of soil structure called tortuosity, indicating that the flux is in the direction of decreasing hydraulic head as stated in the third principle of the model and n is the non linear exponent which varies with volume of leachate and location of the soil.

Contaminant transport model

The bulk motion of the fluid and contaminant transport through the soil column by molecular diffusion and mechanical dispersion is negligible. At the same time, generation of loss of mass takes place due to adsorption and biokinetics of the mass dissolved or suspended in the moving water. In this study only heavy metals namely Fe, Cu, Zn, Pb, Mn, Ni, and Cd were modeled. In general, for steady–state water flow condition, the transport terms for heavy metals are as given in Eq. (22):

$${\mathrm{J}}_{\mathrm{s}}={\mathrm{J}}_{\mathrm{DL}}+ {\mathrm{J}}_{\mathrm{CL}}$$
(22)

where \({\mathrm{J}}_{\mathrm{s}}\) is total heavy metal flux (mg kg−1), \({\mathrm{J}}_{\mathrm{DL}}\) is the diffusion flux in the liquid phase and \({\mathrm{J}}_{\mathrm{CL}}\) is the convection flux in the liquid phase. In the case of diffusion in the liquid phase in a porous media, the equation represented by Fick’s law is given as shown in Eq. (23):

$${\mathrm{J}}_{\mathrm{DL}}={\mathrm{D}}_{\mathrm{m}}\left(\uptheta \right)+ {\mathrm{J}}_{\mathrm{CL}}$$
(23)

where \({\mathrm{D}}_{\mathrm{m}}(\uptheta )\) is the molecular diffusion coefficient. The value of \({\mathrm{D}}_{\mathrm{m}}(\uptheta )\) can be determined through Eq. (23) as given by Kelley34 in Eq. (24):

$${\mathrm{D}}_{\mathrm{m}}\left(\uptheta \right)={\mathrm{D}}_{\mathrm{OL}}.\mathrm{a}.\mathrm{exp}\left(\mathrm{b\theta }\right)$$
(24)

where DOL is the diffusion coefficient in a pure liquid phase and a and b are empirical constants reported by Olsen and Kemper35 to be approximately b = 10 and 0.005 < a < 0.01. The convective flux of respective heavy metal is represented as expressed in Eq. (25)8:

$${\mathrm{J}}_{\mathrm{cL}}={-\mathrm{\theta D}}_{\mathrm{h}}\left(\mathrm{q}\right)+ {\left(\uptheta \right)\frac{{\mathrm{dC}}_{\mathrm{l}}}{\mathrm{dz}}}_{1}+{\mathrm{qC}}_{1}$$
(25)

where q is the water flux, and Dh (q) is the hydrodynamic dispersion coefficient that describes mixing between large and small pore as the result of local variations in mean water flow velocity. Partitioning the heavy metal between absorbed and solution phases, according to Alemi et al.36, adsorption of elements are assumed taken to be nonlinear equilibrium process described by the mathematical relationship in Eq. (26):

$${\mathrm{C}}_{\mathrm{s}}={\mathrm{K}}_{\mathrm{s}}{\mathrm{C}}^{\mathrm{n}}$$
(26)

where Cs is the concentration of the elements absorbed on the soil (mg k Kg−1), Ks is the adsorption coefficient for respective metal (L Kg−1), C is the concentration of the element in soil solution (mg k Kg−1), n is the nonlinear equilibrium adsorption reaction exponent for respective metals. The total concentration of contaminants (CT) contained in the solution and adsorbed phases in a soil volume of one liter is given as presented in Eq. 27:

$${\mathrm{C}}_{\mathrm{T}}={\mathrm{\rho C}}_{\mathrm{s}}+\uptheta {\mathrm{C}}_{\mathrm{l}}$$
(27)

where ρ is the soil bulk density (g cm−3). Substituting Eq. (24) for Cs in Eq. (25) gives the convection–dispersion formula in Eq. (28):

$${\mathrm{C}}_{\mathrm{T}}={\mathrm{C}}_{\mathrm{l}}+\left(\uptheta +\uprho {\mathrm{K}}_{\mathrm{s}}\right)$$
(28)

Heavy metal transports in soil system occur under non-steady (transient) water flow condition and it varies with depth and time as given in Eq. (29):

(29)

Numerical solution

Prediction of the concentration of heavy metal contaminants in the liquid phase were determined through the solution of Eq. (29) for all heavy metals where CT is concentration of respective contaminants (mol L−1), t is the time of transport (m), Js is the total concentration of contaminant in the leachate (mol L−1), Z is the thickness of the soil column (cm) and is the error term. The COMSOL Multiphysics software 6.0 was used to simulate the movement of pollutants in the soil37,38; first, geometry interface was used to design the model of the soil column, then, the transport of diluted species physics was used to create the starting point for the source of heavy metal pollution, and finally, the time-dependent study was used to simulate the change in pollutant concentration with time. The soil in the sampling area was characterized. The depth of the studied soil area is 90; 60 and 30 cm, heavy metal pollutants contained in leachate in the soil can be regarded as a solute in a homogeneous medium according to the law of migration37,39. Therefore, the model can be displayed as a plane. The entire design layout is shown in Fig. 3. The numerical simulation was carried in COMSOL Multiphysics software 6.0, using HP folio 1040 with a processing speed of 2.7 GHz, RAM 8 GB, intel core i5 at a computation time 50 min. The parameters used for heavy metal transport through soil with their respective units are shown in Table 3. Figure 3 depicts the procedure for simulation of numerical model built in COMSOL Multiphysics.

Figure 3
figure 3

(a) Procedure used to implement the numerical model in COMSOL (b) screen shots of software interface.

Full size image

Statistical analysis

The data generated were analyzed using descriptive and inferential statistics. Descriptive statistics employed included mean, standard deviation and standard error of mean while inferential statistics used were two-way Analysis of Variance (ANOVA), Pearson’s Product Moment Correlation and Factor Analysis. Significantly different means were separated using the Duncan multiple range test at 5% level of probability40,41,42.

In addition, to evaluate the numerical model the accuracy and performance of the COMSOL Multiphysics prediction, the root means square error (RMSE) and the mean square error (MSE) were utilized as a function of error43,44 via Eqs. (30) and (31):

$$\mathrm{RMSE}=\sqrt{\frac{1}{\mathrm{N}}\sum_{\mathrm{j}=1}^{\mathrm{N}}{(\mathrm{Experimental}-\mathrm{Predicted})}^{2}}$$
(30)
$$\mathrm{MSE}=\frac{1}{\mathrm{N}}\sum_{\mathrm{j}=1}^{\mathrm{N}}{(\mathrm{Experimental}-\mathrm{Predicted})}^{2}$$
(31)

Results and discussion

Model analysis and validation

Leachate transport model

The volume of leachate transported over a given distance of the soil was modeled based on Eq. (17). Practical determination of the observed and estimated quantity of leachate (leachate flux) requires that the soil hydraulic conductivity (unsaturated) must be known. The real difficulty, therefore, was in finding K. In this study, unsaturated hydraulic conductivity was determined from transmissivity coefficient of the soil using Philip’s infiltration equation. Table 1 shows result of infiltration run conducted in the dumpsite soil.

Table 1 Infiltration characteristics of the dumpsite soil.
Full size table

Infiltration characteristics of the dumpsite shows close similarities among the three points conducted. Generally, initial (Io) and final (If) infiltration rate obtained in the study is low. Values of lo ranged from 0.30 to 0.50 cm/min with mean of 0.40 ± 0.08 cm/min at PT01; 0.20–0.80 with mean of 0.43 ± 0.26 cm/min at PT02 and between 0.30 and 0.60 cm/min with mean of 0.43 ± 0.12 cm/min at PT03. All of them have their means between 0.40 and 0.43 cm/min. Similarly If ranged from 0.08 to 0.10 cm/min with mean of 0.09 ± 0.01 cm/min at PT01; 0.07–0.15 cm/min with mean of 0.10 ± 0.04 cm/min at PT02 while it was between 0.07and 0.20 cm/min with mean of 0.13 ± 0.05 cm/min at PT03. The means suggest low infiltration rate in the dumpsite soil.

Sorptivity and transmissivity are two sisters in relation to soil water flow. While sorptivity relates to absorption and adsorption of water in the soil matrix, transmissivity is the characteristics of the soil which indicates the contribution of the soil to intake of water arising from gravity. In relation to soil water potential, sorptivity arises from matric potential while transmissivity results from gravitational potential. Generally, matrix potential is always low and sometimes negligible as noted by Onofiok45 and Edem31. This can be seen experimentally in value of sorptivity of the soil which is low. Generally, values of sorptivity are lower than that of transmissivity in all the three points for the nine cases obtained in the study. It ranged from 0.510 to 0.710 cm min−1 at PT01 with mean of 0.60 ± 0.08 cm min−1, 0.59–0.75 cm min−1 with mean of 0.66 ± 0.07 cm min−1 at PT02 and between 0.59 and 0.77 with mean of 0.67 ± 0.07 cm min−1 at PT03. On average, mean sorptivity obtained in the study was 0.64 ± 0.09 cm min−1.

Similarly, values of transmissivity (A) obtained in the study are fairly high. It ranged from 2.17 to 2.38 with mean of 2.27 ± 0.08 cm min−1 at PT01, 2.23–2.46 with mean of 2.33 ± 0.10 cm min−1 at PT02 and 1.82–2.31 with mean of 2.12 ± 0.21 cm min−1 at PT03 with average of 2.24 ± 0.18 cm min−1. The average transmissivity of the soil revealed that A which represents the hydraulic conductivity of the soil is almost constantly distributed throughout the experiment across the three points. This could be attributed to several factors such as water content, total porosity, pore size distribution and pore continuity. It is reported that hydraulic conductivity increases with increasing moisture content in the soil. This is why saturated hydraulic conductivity is far greater than unsaturated hydraulic conductivity45. Hydraulic conductivity also increases with increasing total porosity. This is because, as total porosity increases, more pores or conducting spaces are available for water to be conducted. The converse is true as porosity decreases. For this reason, when the porosity of a soil is reduced, for instance by compaction, K decreases. More water is conducted through the larger (macro) pores than the meso or micro pores. The latter is used mainly for retention.

Thus, K of a soil that has predominantly large pores will be larger than that of the same soil when these large pores are lost, say through compaction. So changes in pore size distribution with depth contribute to decrease in hydraulic conductivity. Hence, hydraulic conductivity is higher when the pores are continuous and straight than when they are discontinuous and meandering 8. In a meandering pore water takes a longer time to reach its destination than when it is flowing through a straight pore. In a discontinuous pore water movement stops where the pore is blocked, then the water finds another pore to continue its movement. Consequently flow is reduced, hence reduction in flux with depth. In this study, the soil was unsaturated, hence, water content was the same and the porosity was similar across the three points and at the same depth resulting in constant K. This has affirmed Darcy’s law of steady state water flow that the flux is directly proportional to \(\Delta \mathrm{H}/\Delta \mathrm{x}\) with K as constant of proportionality32.

Again, the higher transmissivity in the soil than sorptivity is an indication that the contribution of gravitational potential in leachate transport through the soil is significantly higher than the contribution of matric potential and this confirms why matric potential is often neglected unlike gravitational potential in soil water transport model. The result agrees perfectly with finding of Ogban46 in Uyo, Edem31 in Uyo and Onofiok45 in Nsukka who, in their different studies, reported higher transmissivity than sorptivity. Substituting values of transmissivity (A) for unsaturated conductivity (K) in Eq. (15), the soil water (leachate) flux was determined for various distances from soil surface (depth). This gives the observed flux values used in modeling leachate transport in the study area (Fig. 4).

Figure 4
figure 4

Observed leachate flux obtained in the study area.

Full size image

Determination of best fit model

The observed leachate flux obtained from Eq. (15) were fitted into four regression functions namely linear function, Logarithmic, power and exponential functions to determine the best fit model for relationship between leachate flux and soil depth. Result shows that the power function is the best fit model in describing the relationship between distance and quantity of leachate entering the soil per unit time (Table 2). This is informed by its R2 value, residual and standard error of the estimate. For samples obtained from PT01, the power function had R2 of 0.996 higher than the Log, linear and exponential functions with 0.979, 0.808 and 0.897 respectively. For PT02 samples, the power function still recorded the highest R2 of 0.959 followed by Logarithmic function with 0.952 to exponential function with 0.878 while the linear Function had the least (0.798). Samples at PT03 still revealed that the power function is the leading model with R2 of 0.988 followed by Log function with 0.985 to exponential (R2 = 0.911) while the linear function had the least (R2 = 0.832). Hence, leachate transport model of Uyo Municipal solid waste dumpsite (LEATRAM) was established using the power model as expressed in Eq. (32):

Table 2 Regression parameters of various estimation functions.
Full size table
$$\mathrm{q}=333.85{\mathrm{x}}^{-0.364}$$
(32)

The model reflects the general flow model originally proposed by Beven and German47 for macro pore flow in soils as macro pore flow law for channel flow in waste dumpsite, where q is the leachate flux density in cm/hr per cross sectional area of the soil, the constant 333.85 can be interpreted as the integral effect of the surface geometry and spatial characteristics of flow path called the sorptivity, x is the thickness of the soil representing depth of transport while the negative exponent represent the influence of soil structure called tortuosity, indicating that the flux is in the direction of decreasing hydraulic head as stated in the third principle of the model. The power function, identified as the best fit model for leachate transport in Uyo dumpsite soil revealed that rate of transport of leachate in soil of the study area slows down with depth (Fig. 5). This also suggests a reduction in the driving force (\(\Delta \mathrm{H}/\Delta \mathrm{x}\)) and distance variability. Initially, when the soil is dry, the driving force is very high and flow is therefore high. With time, (\(\Delta \mathrm{H}/\Delta \mathrm{x}\)) gets smaller and flow decreases as depth of wetting increasing. This follows from the fact that as depth of wetting increases, \(\Delta \mathrm{x}\) will also increase while \(\Delta \mathrm{H}\) remains constant. Hence, the quotient \(\Delta \mathrm{H}/\Delta \mathrm{x}\) decreases resulting in decrease in q.

Figure 5
figure 5

Leachate transport in dumpsite soil of Uyo.

Full size image

The result also shows that the flux is changing with soil depth indicating that water is stored. Thus, the flux entering the soil would not equal the flux leaving it under unsaturated condition. The difference between what is entering and what is leaving is storage; storage can be expressed as a change in volume water content with time (\(\mathrm{\delta \theta v}/\mathrm{\delta t})\) and the difference between inflow and outflow can be expressed as the change in flux along the length of the soil column (\(\mathrm{\delta q}/\mathrm{\delta x}\)), where \(\theta v\) is the volumetric water content of the soil, q the flux transporting through a unit area of the soil at a given time while x is the thickness of the soil or distance of transport from the soil surface. The description given above is termed conservation of mass which states that water is not lost or destroyed: what flows in is either stored or f1ows out from the soil column48.

Contaminant transport model

Leachate entering the soil also carries along various contaminants at varied concentration. In this study, only the heavy metal contaminants were modeled. Leachate flux obtained at respective depth was fitted into contaminant transport Eq. (28) to obtain the estimated values of contaminants. Tables 3 and 4 showed the observed and estimated concentration of contaminants. Result obtained showed a direct relationship between concentration of contaminant and leachate flux. Both revealed a decreasing trend in the concentration of the contaminants as distance from soil surface (depth) increases. This suggests that there is adsorption of the contaminants to the soil surface. It is known that the soil matrix contains negatively charged ions which help in retention of the basic cations in the soil for plant use35. The result showed that some of these metals are being stored as they leached down the soil. As explained in leachate transport, that in non-transient flow, leachate is stored while some is transported along the soil column. Part of the storage contains certain quantity of various contaminants present in the leachate resulting in decrease in concentration of both the leachate and contaminants as depth of the soil increases.

Table 3 Observed concentration of the contaminants in the soils (mol L−1).
Full size table
Table 4 Estimated concentration of the contaminants from leachate flux (mol L−1).
Full size table

Model validation

Leachate transport model obtained in this study was validated by subjecting the observed and estimated leachate flux from the three sample groups into power regression equation to determine the extent to which the observed values can predict the estimated values from the model. Result shows that the observed leachate flux predicted the estimated leachate flux by 96.40, 93.90 and 94.40 for samples from PT01, PT02 and PT03 respectively with residuals of 134.180, 265.82 and 126.68 and standard error (SE) of 11.58, 16.30 and 11.26 respectively (Table 5). Figures 6, 7, 8 and 9 show the observed and estimated leachate flux at PT01, PT02, PT03 and the average. Among the contaminant, observed and estimated values showed significant agreement but with lower R2 than that of leachate except Zn. From the results, Fe predicted the estimated values from the designated depths by 87.10 while Cu, Zn, Pb, Mn, Ni, and Cd predicted their estimated counterparts by 84.80, 94.10, 82.60, 86.40, 82.00 and 89.70% respectively49.

Table 5 Regression of observed and estimated leachate flux and contaminants.
Full size table
Figure 6
figure 6

Pattern of observed and estimated leachate flux at PT01 in the dumpsite soil.

Full size image
Figure 7
figure 7

Pattern of observed and estimated leachate flux at PT02 in the dumpsite soil.

Full size image
Figure 8
figure 8

Pattern of observed and estimated leachate flux at PT03 in the dumpsite soil.

Full size image
Figure 9
figure 9

Pattern of observed and estimated leachate flux for average leachate flux in the dumpsite soil.

Full size image

On the whole, there is a strong agreement between the observed and estimated leachate flux and concentration of contaminants from the developed leachate and contaminant transport models in the study area. Figures 10, 11, 12, 13, 14, 15 and 16 show transport of various contaminants in dumpsite soil of the study area. Their model equation revealed that the rate at which each of the contaminant is transported across depth in the dumpsite soil is very low and may not be continuous unlike leachate transport which is continuous and at very high rate. However, both leachate and contaminant transport along the soil column followed the same model but different coefficient and rate. This could be attributed to diffusion and dispersion effect of the contaminants unlike the leachate. Both models can be utilized for various estimations within 0–1 m depth of the soil. Therefore, it is widely recommended for Uyo soil and also in locations with similar physical, chemical and hydrological properties of the soil50.

Figure 10
figure 10

Fe transport in dumpsite soil of the study area.

Full size image
Figure 11
figure 11

Cu transport in dumpsite soil of the study area.

Full size image
Figure 12
figure 12

Zn transport in dumpsite soil of the study area.

Full size image
Figure 13
figure 13

Pb transport in dumpsite soil of the study area.

Full size image
Figure 14
figure 14

Mn transport in dumpsite soil of the study area.

Full size image
Figure 15
figure 15

Ni transport in dumpsite soil of the study area.

Full size image
Figure 16
figure 16

Cd transport in dumpsite soil of the study area.

Full size image

To compare the migration of pollutants over time in a given area, pollution points were selected in the soil column direction from the simulation diagram (geometry). The horizontal distance from the pollution source to the current location is displayed in the positive vertical direction, and the concentration changes with time. The flow field is the same, and the main direction of transport is the vertical direction. The concentration distributions of Fe, Pb, Cu, Mn, Ni, Zn and Cd after 30 days period of infiltration are shown in Figs. 9, 10, 11, 12, 13, 14 and 15. The pollutants clearly have reached steady-state conditions, and it is expected that the heavy metals in the soil will contaminate a large soil area for a long time. Through the soil column simulation model, the simulation was carried using the initial concentrations of the selected heavy metals51. The experimental result of the selected heavy metals is combined with the COMSOL Multiphysics software simulation results as shown in Figs. 8, 9, 10, 11, 12, 13 and 14, it can be seen that regarding the migration of heavy metal pollutants in soil over time, with increasing soil distance, the seepage flow increases with a peak shape in the 2D plot at the top right corner of the graphs. According to the change in color, the concentration gradually decreases from top to bottom; dark blue indicates a lower concentration, and red indicates the maximum concentration. The migration of heavy metal pollutants mainly occurs in the distance range of 1–90 cm (0.1–0.9 m), and their contents are mainly concentrated in this interval. The soil surface was used to measure the total contents of the heavy metals Fe, Pb, Zn, Mn, Ni, Cd, and Cu, as well as the contents of the residual state of heavy metals. The results show that the concentration of each form of Fe, Pb and Cu in the municipal solid waste dumpsite area reached a high degree of pollution. It was observed that the findings of this work agree with that of Xie et al.52 who studied Cu and Cd transport in metal mining area.

However, the summary of error computation for validation of the numerical model from COMSOL Multiphysics is presented in Table 6.

Table 6 Summary of error computation for model validation.
Full size table

Conclusion

The study showed that leachate and contaminants (heavy metals) transport in soil of the study area is in the power functional form. Their model equations revealed that the rate at which various contaminants is transported across depth in the dumpsite soil is very low and may not be continuous but leachate transport is continuous with a very high rate. Validation equation showed that the observed leachate flux predicted the estimated ones perfectly with very high R2 of over 95% justifying the reliability of the model in estimating leachate and contaminants transport in soil of the study area. This study provides a new insight on how the leachate and chemicals from Uyo MSW dumpsite are transported through the soil and causing pollution. The extent of pollution of the soil within Uyo dumpsite can be accurately predicted using leachate transport model of this study. This is one of the main contributions of this study to knowledge which can be used to clearly establish the influence of chemicals from the leachate of MSW dumpsite on the properties of soil for possible control measures to reduce the extent of soil pollution. Also, the study has given an update on the high pollution status of the soil by MSW leachate. Findings from the study has identified level in depth to which leachate from the dumpsite extends; this is highly beneficial to farmers and town planners for agricultural and environmental management policies. The finding has also cleared doubt and fear among the existing borehole owners on the quality of their borehole water due to continuous accumulation of waste in the dumpsite.