1
1.
Pollutants propagation: mathematical modeling
Introduction
Contaminant propagation is a complex phenomenon that depends on different processes such as hydrologic phenomena as advection, dispersion, and molecular diffusion; chemical and physical phenomena such as radioactive decay, hydrolysis, dissolution, adsorption and volatilization; biological phenomena i.e. degradation of pollutants due to bacteria or biotic agents.
Because of its complexity, the comprehension and description of contaminant propagation requires complex mathematical models. The first step in developing such models is to describe mathematically each type of process that can happen in the aquifer depending on porosity, soil type, presence of groundwater flow, geometry of the aquifer and type of source.
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1.1. Propagation mechanisms of pollutants in groundwater
The propagation of a contaminant in groundwater depends on different factors covered below.
1.1.1.
AdvectionAdvection describes the way groundwater transports the contaminant with a velocity equal to the mean velocity of the system:
e e e v i v n n K (1.1)
In a orthogonal system x, y,z with vx v ve, y 0,vz the mass flow of solute per unit 0 of area of aquifer jA,x is:
,
A x e e
j v n C (1.2)
Where C is the solute concentration in [ML-3].
1.1.2. Molecular diffusion
Molecular refers to the net flux of molecules in response to concentration gradients. It results a flux oriented in the same direction of the concentration gradient where the mass flux per unit of area is expressed by Fick’s law:
,i M x d i C j D x (1.3)
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Where jM,xi is expressed in [MT-1L-1], C is the pollutant concentration in [ML-3], and the molecular diffusion coefficient Dd is expressed as [L2T-1]
Molecular diffusion values are very low and around 10-9 m2/s. Molecular diffusion varies depending on temperature and type of solute. In a porous medium, molecular diffusion is slowed down by the tortuosity, so it changes depending on size of grains. The effective molecular diffusion is then multiplied by the tortuosity in order to take into account this effect:
0 d
D D (1.4)
Figure 1.1: Non-uniform velocity in the porous media: a) distribution into a single pore; b) pores different
diameter effect; c) tortuosity due to different pores size (from [1])
1.1.3.
Kinematic dispersionThe heterogeneity of soils results in non-uniform flow directions and velocities among pores This phenomena is named kinematic dispersion. The transverse component of velocity causes the enlargement of the contaminated area with flow distance.
Since the result of dispersion is an irreversible spreading of contaminants similar to diffusion, Fick’s law is used to kinematic dispersion. However, instead of a diffusion coefficient, a kinematic dispersion coefficient is used.
4 , , K x K L C j D x (1.4) , , K y K T C j D y (1.5) , , K z K T C j D z (1.6)
Where DK,L and DK,T are the longitudinal and the transverse dispersion coefficient
respectively. They are expressed as [L2T-1].
From experimental experiences, it is known that the proportionality between kinematic dispersion and velocity can be described by:
, K L L e D v (1.7) , K T T e D v (1.8)
Where αL and αT are the longitudinal and the transverse dispersivities respectively and
are dimensionally measured as length.
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Figure 1.3: one-dimensional solute transport: advective-diffusive and dispersive contribute (from [1])
1.1.4.
Hydrodynamic dispersionMolecular diffusion and kinematic dispersion can be expressed as a single process named hydrodynamic dispersion. The hydrodynamic dispersion expresses the solute mass propagation per unit of area along the three principal dispersion axes:
, H x L C j D x (1.9) , H y T C j D y (1.10) , H z T C j D z (1.11)
Where DL and DT [L2T-1] are the longitudinal and transverse hydrodynamic dispersion
coefficient respectively. It follows that: 0 , 0 L K L L e D D D D v (1.12) 0 , 0 T K T T e D D D D v (1.13)
6 1.1.4.1. Peclet number
Peclet number is a non-dimensional number and it is the ratio between advective phenomenon and the diffusive one:
e d v d Pe D (1.14)
Where d is a characteristic length of the porous medium.
For low values of Peclet number, the ratio DL/Dd is constant and independent from flow velocity. In that case molecular diffusion is the predominant phenomenon while dispersion is negligible. For higher flow velocities, over a transition zone, dispersion is the principal phenomenon.
Figure 1.4: longitudinal dispersion coefficient and diffusion coefficient ratio as a function of Peclet number
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Figure 1.5: transverse dispersion coefficient and diffusion coefficient ratio as a function of Peclet number
(from [1])
For usual groundwater values of flow velocity, molecular diffusion is negligible; in consequence, longitudinal and transverse hydrodynamic dispersion coefficients can be approximated to the respectively kinematic dispersion coefficients:
L L e
D v (1.15)
T T e
D v (1.16)
Dispersivity values are strongly dependent on the scale of flow; the longer is the scale, the higher the value of dispersivity. This effect is in part due to the increase of heterogeneities encountered the longer the flow path.
1.1.5.
Reaction modelsIn groundwater, a contaminant can react with a substance and form another compound. The general reversible reaction is as follows:
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Where a, b, c are moles numbers. At the equilibrium,
eq C K A B (1.18)In the aquifer, point distribution of products and reactants is due to transport mechanisms. If the reaction velocity is higher than the transport velocity, the system suddenly reaches the equilibrium in each point.
The local equilibrium hypothesis is not valid for irreversible reactions or low reaction velocity. In this case, the kinetic model is used.
Reaction velocities referring to eqn. (1.17) can be written as follows:
p q A A A A B dC r k C C dt (1.19) p q B B B A B dC r k C C dt (1.20) u C C C C dC r k C dt (1.21)
For high concentrations of pollutants reaction velocity can be independent from concentration, this means zero order kinetic:
A A dC k dt (1.22)
If kinetic depends only on concentration of reactant A, reaction velocity can be expressed by a first order differential equation:
9 A A A dC k C dt (1.23)
Depending on the type of reaction considered, an equilibrium model or a kinetic one can be chosen. Irreversible reactions can be correctly simulated by kinetic models and radioactive decay. If a reaction is reversible, it is necessary to compare flux velocity and reaction velocity in order to assume a local equilibrium.
1.1.6.
BiodegradationBiodegradation is a process by which microorganisms degrade pollutants. Reactions involved in this phenomenon are redox reactions.
In the last decades, scientists have demonstrated how microorganisms, such as cyanobacteria and prokaryotes, living in the ground, can transform a lot of contaminants.
The biodegradation of an organic compound in water can be expressed by a first order kinetic equation:
dC
C dt
(1.24)
A more rigorous model for biodegradation is the Monod model, here the reaction velocity of the contaminant is not constant and it depends on the contaminant concentration:
max C dC C C C dt K C (1.25) 10 Where:
μ is the microorganisms specific growth rate [T-1]
μmax is the microorganisms maximum specific growth rate [T-1] C is the contaminant concentration [ML-3]
KC is the half-velocity constant[ML-3]
If C>>KC the term (KC + C) is approximated to C and microorganism growth has a zero order kinetic:
max
(1.26)
If C<<KC the reaction velocity of the contaminant reduces to a first order kinetic:
max
C
dC C
dt K
(1.27)
The half-velocity constant KC represents the substrate concentration that lets microorganisms to growth with a velocity μ equal to μmax/2
1.1.7.
Adsorption/DesorptionAdsorption/Desorption is a reversible process where by contaminants adsorb and desorb from solid surfaces.
For low concentrations of pollutants, there is a linear relation between adsorbed mass of solute per unit of soil mass and the concentration in the groundwater, referred to as linear isothermal adsorption:
d
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Where Kd is the solid-liquid partitioning (distribution) coefficient [L3M-1].
Figure 1.6: linear isothermal adsorption (from [1])
For higher concentrations, where a solid matrix saturation occurs, the linear model does not describe the system in the correct way, the partitioning is then described by a Langmuir model: 1 C S C (1.29)
Where α and β are empirical coefficients.
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If a certain number of pollutants compete for the adsorption on the solid matrix, a Freundlich model better describes the partitioning between soil and water:
1/ N
S KC (1.30)
K and N are empirical constants.
Figure 1.8: Freundlich isothermal model (from [1])
Contaminant adsorption on the solid matrix and the consequently partitioning of the pollutant causes a delay of the advection front. This delay is expressed by a retardation factor R. It is defined as the ratio between the effective water flow velocity and the effective contaminant velocity:
w c v R v (1.31)
For a contaminant subject to a linear equilibrium sorption R depends on aquifer and contaminant characteristics: 1 b d e R K n (1.32)
13 Where:
ρb is the dry bulk density ( Ms/Vt) ne is the effective porosity
Kd is the partitioning coefficient [L3 M-1]
1.2. Differential equation fundamentals for mass transport
Transport of solutes is influenced by chemical, physical and biochemical processes. This phenomenon is subject to different interactions between the aquifer and the solute.
Differential equations applied to the solute can describe these processes by changing boundary and initial conditions.
1.2.1.
Mass balance for the contaminantIn order to describe solute propagation in groundwater, the mass balance equation is applied to the REV in a certain time interval dt. For a general contaminant:
o i
M M M (1.33)
Where Mo and Mi are outlet and inlet solute mass respectively while M is the accumulation in time within the REV.
Solutes can be divided in two categories:
Conservative solutes: affected by hydrological phenomena only
Non-conservative solutes: subject to hydrological and chemical, physical and biological phenomena
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1.2.2.
Conservative and non-conservative solutesFor conservative solutes, advection and hydrodynamic dispersion are the only two phenomena that have to be taken into account.
By setting the reference system x, y, z as the one constituted by principal axes of dispersivity with x the direction of groundwater flow so vx=v, vy=vz=0
The application of mass balance equation let to define the advection-dispersion equation for the solute in the following form:
( e ) ( ) x e y e z e n C C C C vC D n D n D n t x x y y z z x (1.34)
By neglecting porosity change and by assuming an homogeneous media eq. 1.34 becomes: ( ) x y z C C C C vC D D D t x x y y z z x (1.35)
By considering also constant in space values of dispersion coefficient and the motion field i.e. v=constant, the differential equation for conservative solute is the following:
2 2 2 2 2 2 x y z C C C C C D D D v t x y z x (1.36) Where:
15 0 x x D D v (1.37) 0 y y D D v (1.38) 0 z z D D v (1.39)
Non-conservative solutes are subject not only to advective and dispersive phenomena but also to chemical, physical and biological phenomena. For these reasons, each pollutant has to be studied individually.
Reaction can be described through an equilibrium kinetic model: in groundwater it is possible to have:
chemical equilibrium in heterogeneous phase, i.e. the adsorption of the pollutant on aquifer grains. This process makes the pollutant motion slower with respect to groundwater flow.
Reaction of pollutant in the field with production and consumption of the solute. This phenomena varies pollutant concentration in the plume.
To the classical Eq. 1.36, the accumulation term has to be rewritten as:
deg 1 b d e C C K n t t (1.40) Where: 1 b d e R K n (1.41)
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If a first order kinetic is considered for pollutant chemical degradation, the second term in Eq. 1.40 can be expressed as:
deg C C t (1.42)
Where is the degradation coefficient.
The advection-dispersion equation for a reactive solute can then be expressed as follows, by considering dispersive coefficients constant in space and uniform fluid motion field: 2 2 2 2 2 2 x y z C C C C C R D D D v C t x y z x (1.43)
To solve the advection-dispersion equation, initial and boundary conditions are necessary: initial conditions specify flux domain and initial concentration of pollutants; boundary conditions describe interactions between the system and the domain.
Three types of boundary conditions are usually adopted: Dirichelet or first type boundary conditions, Neumann or second type boundary conditions and Cauchy or third type boundary conditions.
The advection-dispersion equation can be analytically solved only for a specific hypothesis on source geometry, boundary conditions, flux regime and way of pollutant release:
Constant dispersion coefficients Darcy’s Law validity
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Chemical reaction following simplified equilibrium laws Isotropic, homogenous and saturated medium
Constant water density and viscosity and concentration independent
For more complex cases the differential problem can be solved numerically only.
1.3. Theoretical models for one-dimensional problems
1.3.1.
Conservative solutesFor one-dimensional problem, the simplified advection-dispersion equation for conservative solutes is the following:
2 2 ( , ) ( , ) ( , ) x C x t C x t C x t D v t x x (1.44)
1.3.1.1. Continuous inlet distribution
Ogata and Banks [2] gave the analytical solution for the pure advection, dispersion one-dimensional problem with constant inlet concentration; for Eq. 1.44 with the following initial and boundary conditions:
0 ) 0 , (x C (1.45) 0 (0, ) C t C (1.46) ( , ) 0 C t (1.47)
18 t D vt x erfc D vx t D vt x erfc C t x C x x x 2 exp 2 2 ) , ( 0 (1.48)
1.3.1.2. Impulsive inlet distribution
Analytical solution of Eq. 1.44 for impulsive inlet distribution was given by Sauty [1]:
2 ( , ) exp 4 4 x x x vt M C x t D t D t (1.49)Where M is the mass injected at time t0 per unit of transversal section.
1.3.2. Non-conservative solutes
For non-conservative solutes chemical, physical and biological phenomena cannot be neglected
With the following hypothesis:
Constant dispersion coefficients Darcy’s Law validity
Chemical reaction following simplified equilibrium laws Isotropic, homogenous and saturated medium
Constant water density and viscosity and concentration independent
Different types of ADE equations can be expressed depending on which phenomena we are considering.
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1.3.2.1. Solutes subject to chemical degradation
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2 , , , , x C x t C x t C x t D v C x t t x x (1.50)Bear [3] investigated the one-dimensional problem with advection, dispersion, first-order decay and constant inlet concentration in an infinite domain.
By considering eq. (1.50) with initial and boundary conditions expressed in Eq. (1.45), (1.46), (1.47):
The analytical solution is:
t D ut x erfc D ux t D ut x erfc D ux D vx C t x C x x x x x 2 exp 2 2 exp 2 exp 2 ) , ( 0 (1.51) Where: x D v u 24 (1.52)
1.3.2.2. Solutes subject to adsorption
For solutes subject to only adsorption, the governing equation is the following:
2
2 , , , x C x t C x t C x t R D v t x x (1.53) 20 0 ( , ) exp 2 2 2 c c c c c c C x v t v x x v t C x t erfc erfc D D t D t (1.54)
Where vc=v/R and Dc=Dx/R are the velocity and the dispersion coefficient for a solute subject to retardation effects.
Eq. (1.54) equals Eq. (1.48) for vc=v and Dc=Dx.
1.3.2.3.
Solutes subject to chemical degradation and adsorption The governing equation for this case is the following:) , ( ) , ( ) , ( ) , ( 2 2 t x C x t x C v x t x C D t t x C R (1.55)
The analytical solution of Eq. (1.55) subject to boundary and initial conditions as Eq. (1.45), (1.46), (1.47) is:
0
( , ) exp exp exp
2 2 2 2 2 c x x x x x C v x ux Rx ut ux Rx ut C x t erfc erfc D D D Rt D D Rt (1.56) With: 2 4 c x u v D (1.57)
Pérez Guerrero [4] proposed a new set of solutions based on Eq. (1.55) subject to time-dependent boundary conditions: eq. (1.55) is applied for contaminant transport in homogeneous porous media with an average transport velocity and first order decay for a semi-infinite or a finite domain of length L0.
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C(x,t) is the contaminant concentration [ML-3], D is the dispersion coefficient [L2T
-1], v is the constant average velocity [LT-1], λ is the first order decay constant [T-1]
and R is the retardation coefficient.
Boundary and initial conditions:
) ( ) , 0 ( t g t C (1.58) or ) ( ) , 0 ( ) , 0 ( t vg t vC x t C D (1.59) 0 ) , ( 0 x t L C (1.60) or 0 ) , ( x t C (1.61)
g(t) is a general time-dependent function for the inlet concentration [ML-3].
By using the Duhamel’s Theorem, the solutions in some special cases are the following.
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Instantaneous pulse subject to first-type boundary conditions and semi-finite domain In this case g(t) is written as g(t)m0(x) where m0 is the quantity of mass injected
over the cross-sectional area divided by the volumetric water flux passing through the same area [MTL-3].
2 2 3/2 0 ( ) exp 4 ( , ) 2 tv Rx t DR x DRt R C x t C DRt (1.62)Instantaneous pulse subject to third-type boundary conditions and semi-finite domain
1/2 2 2 / 1 2 0 2 2 exp 2 4 ) ( exp ) , ( DRt Rx tv erfc DR R t D vx v DRt R t DRt Rx tv v C t x C (1.63)Instantaneous pulse subject to first-type boundary conditions and finite domain
1 2 0 2 2 2 0 2 2 12 0 2 4 exp 4 ) , ( ) , ( m m m m R L Dt DR t v R t d vx R L D DR v x E C t x C (1.64) D vL D vL L x x E m m m m 2 2 sin 2 ) , ( 0 2 0 2 0 12 (1.64a) 23
Where are the solutions of the transcendental equation: m
0 2 ) cot( 0 D vL m m (1.64b)
Instantaneous pulse subject to third-type boundary conditions and finite domain
1 2 0 2 2 2 0 2 2 0 2 4 exp 4 ) , ( 3 ) , ( m m m m R L Dt DR t v R t d vx R L D DR v x E C t x C (1.65) 2 0 2 0 2 0 2 0 0 0 0 32 2 2 sin 2 cos 2 ) , ( D vL D vL D vL L x D vL L x D vL x E m m m m m m m (1.65a)Where is obtained by solving the following equation: m 0 4 ) cot( 0 0 2 D vL vL D m m m (1.65b)
Exponential inlet distribution subject to first-type boundary conditions and infinite domain
The exact analytical solution for the one-dimensional problem concerning advection, dispersion, sorption, first-order decay and exponential source decay of the contaminant concentration in time was investigated firstly by Van Genuchten and Alves [5] in the most general case and, subsequently, was investigated by Williams and Tomasko [6] for a particular case.
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By considering the following governing equation with the following initial conditions and first-type boundary conditions:
) , ( ) , ( ) , ( ) , ( 2 2 t x C x t x C v x t x C D t t x C R (1.66) i C x C( ,0) (1.67) t b a s e C C t C(0, ) (1.68) 0 ) , ( t C (1.69)
Where γ represents the rate constant for zero-order production in two solid phases [5] Ci is the existing initial concentration and λs is the source term decay constant, this
value can be different from the first order decay constant.
The solution is:
) , ( ) , ( ) ( ) , ( ) ( ) , (x t C A x t C B x t C E x t C i a b (1.70) DRt vt Rx erfc e DRt vt Rx erfc e t x A D vx R t s 2 2 1 2 2 1 1 ) , ( (1.71) DRt ut Rx erfc e DRt ut Rx erfc e t x B D x u v D x u v 2 2 1 2 2 1 ) , ( 2 ) ( 2 ) ( (1.72) DRt wt Rx erfc e DRt wt Rx erfc e e t x E D x w v D x w v t s 2 2 1 2 2 1 ) , ( 2 ) ( 2 ) ( (1.73)
25 With: 2 / 1 2 ) 4 1 ( v D v u (1.74) 2 / 1 2 ( ) 4 1 R v D v w s (1.75)
By considering eqn. (1.50), with the following initial condition and first-type boundary conditions: 0 ) 0 , (x C (1.76) t s e C t C(0, ) 0 (1.78) 0 ) , ( t C (1.79)
By applying Laplace transforms and the inversion technique, the computation yields the following solution [6]:
0 1 exp 2 exp ( , ) exp 2 2 1 exp 2 src src src src src erfc t t t vx C x t C D erfc t t (1.80) Where x D x2 ; x D v 4 2 26 This solution can be also rewritten as:
0 exp 2 2 ( , ) exp 2 2 exp 2 x x s x x x ux x ut erfc D D t C vx C x t t D ux x ut erfc D D t (1.81) Where: s x x D D v u 2 4 4 (1.82)
For which the corresponding integral solution the following:
D d x t D x C t x C x t x
4 ) ' ( exp )) ( exp(-) ( 2 ' ) , ( 2 0 s 3 / 2 0 (1.83)By comparing eqn. (1.81) with the previous general solution eqn. (1.70) [5] it is possible to proof that the former is a particular case of the latter by making some simplification:.
By setting R=1 and γ=0, Ci=0, Ca=0and Cb=C0 eqn.(1.80) becomes:
( ) 2 0 0 ( ) 2 1 2 2 ( , ) ( , ) 1 2 2 s v w x D t v w x D Rx wt e erfc DRt C x t C E x t C e Rx wt e erfc DRt (1.84)
27 0 exp 2 2 ( , ) exp 2 2 exp 2 2 s x wx Rx wt erfc D DRt C vx C x t t D wx Rx wt erfc D DRt (1.85) And s x x sR v D D v D v w 1 4 ( ) 2 4 4 2 / 1 2
So it is clear that the solution given by Williams and Tomasko is a particular case of the Van Genuchten and Alves analytical solution.
Exponential inlet distribution subject to third-type boundary conditions and semi-finite domain [5]
By considering eqn.(1.59) and by setting g(t)Ca Cbest
it is possible to derive the following solution: ) , ( ) , ( ) , ( ) , ( ) , ( 0A x t C E x t C t x B C t x A C t x C b b a R R s s (1.86) 2 ( ) ( , ) exp 2 2 ( ) exp 2 2 exp 2 2 v v u x Rx ut A x t erfc u v D DRt v v u x Rx ut erfc u v D DRt v vx t Rx vt erfc D D R DRt (1.87)
28 2 ( ) exp 2 2 ( ) ( , ) exp 2 2 exp 2 ( ) 2 st s v v w x Rx wt erfc w v D DRt v v w x Rx wt B x t e erfc w v D DRt v vx t Rx vt erfc D R D R DRt (1.88) DRt vt Rx erfc D vx DR t v D vx DRt vt Rx DR t v DRt vt Rx erfc e t x E st 2 exp 2 1 2 1 4 ) ( exp 2 2 1 ) , ( 2 2 2 (1.89) Where: x D v u 24 (1.90) s x x D D v w 2 4 4 (1.91)
By considering eqn. (1.51)-(1.52) and eqn. (1.81)-(1.82) it is possible to note that they are formally identical and they differ only for the terms est
and 4Dxλs that take into
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1.4. Comparison of the one-dimensional models
In this section we want to compare some of the one-dimensional models presented before. In particular, we want to point out differences between eqns. (1.48)-(1.51)-(1.81)-(1.83), and show out how eqn. (1.81) is a limit case of eqn.(1.48)-(1.51). Moreover, we discuss the sign of eqn. (1.82) that changes by varying some values of the parameters.
1.4.1. Limit conditions of the Williams & Tomasko
one-dimensional solution
In order to verify if eqn. (1.81) is a limiting case of eqns. (1.48)-(1.51) we consider two cases:
λs=0 (no source decay, i.e. constant concentration at the boundary) λs=0 + λ=0 (no source decay, no first-order reaction, i.e. pure advection)
We used values adopted by [7]
Parameter Value v [md-1] 0.2151 x [m] 41.58 Dx =xv [m2d-1] 9.159 t [d] 5110 C0 [mg/l] 850 [d-1] 0.001 x [m]
0 :2200
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By setting S 0 we obtain u u and eqn.(1.83) becomes:
D d x D x C t x C x t x
4 ) ' ( exp ) ( 2 ' ) , ( 2 0 3 / 2 0 (1.92)In this case eqn.(1.81) must be equal to eqn.(1.51) i.e. William and Tomasko solution [6] must tend to Bear one [3].
Figure 1.9:one-dimensional integral solution with
s=0
Figure 1.10:Williams analytical solution with s=0
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By setting 0 and s 0, we obtain uuv and eqn.(1.83) becomes:
D d x D x C t x C x t x
4 ) ' ( exp ) ( 2 ' ) , ( 2 0 3 / 2 0 (1.93)Eqn. (1.81) and eqn. (1.51) must tend to eqn. (1.48).
Figure 1.12: one- dimensional integral solution
with s=0 and =0
Figure 1.13: Williams analytical solution with s=0
and =0
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1.4.2. Validity of the Williams & Tomasko one dimensional
solution
Williams & Tomasko monodimensional solution, written as eqn. (1.80) or as eqn. (1.81), presents the term (1.82) in which, , by fixing the other parameters, the term under square root is valid till the parameter λs reaches the limit value:
λs*= v /4Dx
2
For values of λs greater than λs* the term under the square root is negative.
In [6] no discussion about this point is faced; in this paper, only cases with λs< λ were
investigated. They corresponds to our case 1.
We checked the validity in these extended different situations; we used the values adopted in [7] with the following values for λs for each case:
Case Value 1. λs< λ 0.0008 2. λs= λ 0.001 3. λ < λs< λs* 0.0018 and 0.0021 4. λs= λs* 0.0023 5. λs> λs* 0.0025 and 0.0028
Table 1.2: s values for each case considered
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Figure 1.16: one-dimensional integral solution for each case of s
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1.5. Theoretical models for three-dimensional problems
Three-dimensional problems of the advection-dispersion equation with different types of initial and boundary conditions well describe contaminant transport in groundwater. Different types of initial and boundary conditions were investigated in the last decades as different types of sources, put inside the domain or on the boundary.
Three-dimensional problems are largely used because of their better precision with respect to one- and two-dimensional problems but they require complex numerical tools to be solved.
Although complex numerical models have been developed,, analytical solutions are still widely used, both in open and closed form.
1.5.1. Conservative solutes
Partial differential equation for three-dimensional transport problems for conservative solutes is: 2 2 2 2 2 2 c c c c c R v D D D x y z t x x y z (1.94)
Where x is assumed as the longitudinal axes coincident with fluid flow.
1.5.1.1. Instantaneous inlet distribution, point source
Baestlé [8] gave the analytical solution for impulsive and point source in infinite 3D domain for a conservative solute:
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2 0 0 3 3 0 0 2 2 0 0 0 0 ( ) ( ) ( , , , ) exp 4 ( ) 8 ( ) ( ) ( ) exp exp 4 ( ) 4 ( ) x x y z y z x x t t M c x y z t D t t D D D t t y y z z D t t D t t (1.95)Baestlé solution is good to analyze a point source with instantaneous release such as accidents i.e. instantaneous release from a tank failure.
1.5.1.2. Continuous inlet distribution, areal source
Domenico and Robbins [9] gave the following solution for vertical, areal source of width Y and height Z with continuous inlet distribution in a semi-finite domain:
0 1 2 2 ( , , , ) - 8 2 2 2 - 2 2 x x x x x Y Y y y x vt
c x y z t c erfc erf erf
D t D x D x v v z Z z Z erf erf D x D x v v (1.96)
Figure 1.18: 3D concentration [mg/l] plume evolution for an areal contaminant source with continuous release
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1.5.2. Non conservative solutes
The advection-dispersion of a solute, subject to a first order decay, can be represented with a three-dimensional model by the following governing equation:
c z c z D y c y D x c x D x c v t c R 2 2 2 2 2 2 (1.97)
c(x,y,z,t) is the solute concentration [ML-3]; x is the longitudinal coordinate;
y and z are the horizontal transverse and the vertical coordinates;
v is the average pore scale velocity of the fluid, it is taken as unidirectional along x [LT-1];
Dx is the longitudinal dispersion coefficient [L2/T];
Dy is the horizontal transverse dispersion coefficient [L2/T]; Dz is the vertical transverse dispersion coefficient [L2/T]; t is the time [T];
λ is the first order decay constant [T-1];
R is the retardation factor [-];
37 1.5.2.1. Instantaneous point source The general solution was given by Baestlé [8]:
0 3 3 0 2 0 0 0 2 2 0 0 0 0 ( , , , ) exp ( ) 8 ( ) ( ) ( ) exp 4 ( ) ( ) ( ) exp exp 4 ( ) 4 ( ) x y z x y z M c x y z t t t R D D D t t R R x x t t D R t t R y y R z z D t t D t t (1.98)Where M is the total mass inserted by the instantaneous point source.
More complex solutions can be derived by using GFM and by integrating the solution for the instantaneous point source in space and time. In consequence, by integrating eqn. (1.98) in the source domain, it is possible to derive the three-dimensional analytical solution for a volumetric instantaneous source.
By integrating eqn. (1.98) in time, the three-dimensional solution for a continuous point or non-point source can be obtained.
Instantaneous volumetric source
The analytical solution for the instantaneous volumetric source can be obtained by integrating eqn. (1.98) along the source domain.
If: ) z < z < z ; y < y < y ; x < x < (0 = V 1 1 2 1 2 (1.99)
38
dz dy dx ) ( 4 ) ( exp ) ( 4 ) ( exp ) ( 4 ) ( ) ( exp ) ( exp ) ( 8 ) , , , ( 0 0 0 0 2 0 0 2 0 2 0 0 0 0 0 3 0 3 1 2 1 2 1
t t D z z t t D y y t t D t t x x t t t t D D D M t z y x c z y x x z z y y z y x v (1.100)Where Mv is the mass released instantaneously at time t0 per unit source volume
[ML-3].
The solution is:
) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( ) ( 2 ( ) ) ( exp 8 ) , , , ( 0 1 0 2 0 1 0 2 0 0 0 0 1 0 t t D z z erfc t t D z z erfc t t D y y erfc t t D y y erfc t t D t t x erfc t t D t t x x erfc t t M t z y x c z z y y x x v (1.101)And, for a semi-finite domain along z the solution becomes:
) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( ) ( 2 ( ) ) ( exp 8 ) , , , ( 0 1 0 2 0 1 0 2 0 1 0 2 0 0 0 0 1 0 t t D z z erfc t t D z z erfc t t D z z erfc t t D z z erfc t t D y y erfc t t D y y erfc t t D t t x erfc t t D t t x x erfc t t M t z y x c z z z z y y x x v (1.102)
39 1.5.2.2. Continuous volumetric source
The solution for a continuous volumetric source can be obtained by integrating eqn.(1.101) in time. By calling τ=t-t0 the solution is:
d 2 2 2 2 2 2 ) ( exp 8 ) , , , ( 1 2 1 2 1 0 0
z z y y x x t D z z erfc D z z erfc D y y erfc D y y erfc D x erfc D x x erfc r t z y x c (1.103)The solution for a z semi-finite domain is:
d ) ( 2 ) ( 2 ) ( 2 ) ( 2 2 2 2 2 ) ( exp 8 ) , , , ( 0 1 0 2 0 1 0 2 1 2 1 0 0
t t D z z erfc t t D z z erfc t t D z z erfc t t D z z erfc D y y erfc D y y erfc D x erfc D x x erfc r t z y x c z z z z y y x x t (1.104)By considering a volumetric source decay expressed by a general function f(t0), the
solution is:
d 2 2 2 2 2 2 ) ( exp ) ( 8 ) , , , ( 1 2 1 2 1 0 0
z z y y x x t D z z erfc D z z erfc D y y erfc D y y erfc D x erfc D x x erfc t f r t z y x c (1.105) 40
d ) ( 2 ) ( 2 ) ( 2 ) ( 2 2 2 2 2 ) ( exp ) ( 8 ) , , , ( 0 1 0 2 0 1 0 2 1 2 1 0 0
t t D z z erfc t t D z z erfc t t D z z erfc t t D z z erfc D y y erfc D y y erfc D x erfc D x x erfc t f r t z y x c z z z z y y x x t (1.106) 1.5.2.3. Plane sourceSagar, [11] and Wexler [12] derived the analytical solution of the 3D ADE in open form for a plane source by setting the following conditions:
0 ) , , , ( 0 0 c t z y x c y0 y y1; z0 zz1 (1.107) 0 ) , , , ( y z t c (1.108) 0 ) , , , (x tz c (1.109) 0 ) , , , (x y t c (1.110) 0 ) 0 , , , (x y z c (1.111)
The final analytical solution is found by using traditional integration transform methods or GFM: 2 0 0 2 1 2 1 3 2 ' ( ' ) ( , , , ) exp 4 8 2 2 1 d ( ) 2 2 t x x y y z z C x x v c x y z t D D y y y y erfc erfc D D z z z z erfc erfc D D
(1.112) 41 Where τ = t - t0; x'= x - x0.
By considering a source decay depending on a general function of time f(t0), the
solution for plane source becomes:
2 0 0 2 1 2 1 3 2 ' ( ' ) ( , , , ) ( ) exp 4 8 2 2 1 d ( ) 2 2 t x x y y z z C x x v c x y z t f t D D y y y y erfc erfc D D z z z z erfc erfc D D
(1.113)It is important to stress that this solution does not satisfy the boundary condition expressed in eqn. (21), as already observed by [13], expects concentration equal to c0 at x’=0. This condition is not compatible with the initial condition reported in eqn. (25) so the solution in eqn. (26) is valid for all x'>0.
In Eqn. (25), the integral in time has to be numerically evaluated. [13] introduced a stepwise superposition approach in order to solve eqn.(25) by discretizing the time interval in N steps and by approximating the contribute of Gy×Gz by its weighted average.
However, it is possible to proof that the continuous plane source solution is equivalent to the continuous volumetric source one at x tending to x0.
42
0 0 0 0 2 1 2 1 ( , , , ) lim ( ) exp ( ) 8 2 2 2 2 d 2 2 t x x x x y y z z r c x y z t f t x x x y y y yerfc erfc erfc erfc
D D D D z z z z erfc erfc D D
(1.114)In the solution with a continuous volumetric, the following expression are valid:
0 0 / C A v Q V Q r (1.115)
So, in the volumetric source solution, r0 has to be substituted by:
0 0 0 x v C r (1.116)
This change let us to pass from a solution to the other one. In fact, if we take the limit for x tending to x0 of Gx:
0 0 0 0 2 0 0 2 0 0 2 2 lim lim exp 4 lim exp 4 x x x x x x x x vt x vt C verfc erfc D t D t x x x x vt C v x x D t D t x vt C x x x t D t D t (1.117) 43
That is equal to the solution in x direction that appears in the 3D-plane source solution. Martin-Hayden and Robbins[14] gave a solution in closed form, for eqn. (1.112), with the approximation adopted in the [15], Srinivasan and Clement [7] called it the “modified-Domenico” solution. This solution includes the first order decay by taking over the one-dimensional analytical solution given by [3] and approximates in time to the mean value along y and z directions.
2 2 2 2 2 ' 2 ' ) ( exp 2 ' 2 ' ) ( exp 8 ) , , , ( 1 2 1 2 m z m z m y m y x x x x v t D z z erfc t D z z erfc t D y y erfc t D y y erfc t D ut x erfc D x u t D ut x erfc D x u M t z y x c (1.118) Where u v2 4Dx (1.119)
1.5.2.4. Continuous volumetric source with exponential source decay By considering a source decay as:
)) ( exp( ) ( ) ( 0 0 0 0
r f t r f t r t r s (1.120)and by introducing this law in eqn. (19) we obtain:
d 2 2 2 2 2 2 ) ( exp )) ( exp(-8 ) , , , ( 1 2 1 2 0 0 0
z z y y x x s t D z z erfc D z z erfc D y y erfc D y y erfc D x erfc D x x erfc t r t z y x c (1.121) 44
Where s [s-1] is the exponential source decay coefficient.
For z semi-infinite domain, with the same source time-depending law, we obtain:
0 0 0 2 1 2 1 2 1 ( , , , ) exp[- ( )]exp ( ) 8 2 2 2 2 d 2 2 2 2 t s x x y y z z z z r x x x c x y z t t erfc erfc D D y y y y erfc erfc D D z z z z z z z zerfc erfc erfc erfc
D D D D
(1.122)1.5.2.5. Continuous plane source with exponential source decay
By considering again, the aforementioned source decay expressed in (1.120), it is possible to derive from eqn.(1.112) the solution for the plane source model affected by exponential source decay:
d ) ( 1 2 2 2 2 4 ) ' ( exp )] ( exp[-8 ' ) , , , ( 2 3 1 2 1 2 2 0 0
z z y y x s t x D z z erfc D z z erfc D y y erfc D y y erfc D v x t D x C t z y x c (1.123) 45
1.6. Comparison between plane source and volumetric source
:
n
umerical simulations
In order to test that the solution with plane source is the limit solution of the volumetric source model with x tending to x0, some numerical simulations were done on eqn.
(1.121) and eqn. (1.123) by using the same parameter values adopted in [7]. Dispersion coefficient are expressed as product of dispersivities and velocity.
Integral functions were performed numerically by using adaptive Gauss-Kronrod quadrature method available in GNU Scientific Library, Matlab QUADGK, NAG Numerical Libraries, QUADPACK Library and R).
Parameter Value
Longitudinal dispersivity (αx) 41.58 m
Transverse dispersivity (αy) 8.43 m
Transverse dispersivity (αz) 0.00642 m
velocity (v) 0.2151 m/d
Source width in Y directions (Y) 240.0 m Source width in Z directions (Z) 5.0 m Source concentration (C0) 850 mg/l
Simulation time (tm) 5110 d
First order reaction constant (λ) 0.001 s-1 Table 1.3: Simulation data
Different values of s in the numerical simulations were used: Case s [s-1] a. 0 b. 0.0008 c. 0.0012 d. 0.0018 e. 0.0022
46 a b . c . d .
47 e
.
Figure 1.19: Concentration surfaces obtained at different value of s wrt table 1, limit of the volumetric source model on the left hand side and plane source model on the right hand side.
Numerical simulations clarify the analytical results obtained by performing the limit expressed in eqn. (1.117); the volumetric model with plane source is equal to the limit solution of the volumetric model with volumetric source for x tending to x0.
In another word, this signify that the plane source, even if it does not satisfy boundary conditions, is equivalent to the volumetric source model in which the thickness along x axes tends to zero.
This represents an important result since, for problem in which the thickness of the source along the x axes is negligible with respect to the other dimensions, it is possible to approximate it to a patch source avoiding the volumetric source model use and now, is proven the equivalency of the two as a limit case. The latter, in fact, is characterized by a more complex analytical expression.
48 Bibliography
[1] A. Di Molfetta, R. Sethi, Ingegneria degli acquiferi, 2011. https://link.springer.com/book/10.1007%2F978-88-470-1851-8.
[2] R.B. Ogata, A., Banks, A solution of differential equation of longitudinal dispersion in porous media, U.S. Geol. Surv. Publ. Warehaose. 411 (1961) A1–A7.
[3] J. Bear, Dynamics of fluids in Porous Media, 1971.
[4] J.S. Pérez Guerrero, E.M. Pontedeiro, M.T. van Genuchten, T.H. Skaggs, Analytical solutions of the one-dimensional advection-dispersion solute
transport equation subject to time-dependent boundary conditions, Chem. Eng. J. 221 (2013) 487–491. doi:10.1016/j.cej.2013.01.095.
[5] M.T. Van Genuchten, W.J. Alves, Analytical solutions of the one-dimensional convective-dispersive solute transport equation, Agric. Res. Serv. Tech. Bull. Number 1661. (1982) 1–149. doi:10.1016/0378-3774(84)90020-9.
[6] G.P. Williams, D. Tomasko, Analytical Solution to the Advective-Dispersive Equation with a Decaying Source and Contaminant, J. Hydrol. Eng. 13 (2008) 1193–1196. doi:10.1061/(ASCE)1084-0699(2008)13:12(1193).
[7] and K.K.L. V. Srinivasan, T.P. Clement, Domenico Solution—Is It Valid?, Ground Water. 45 (2007) 136–146.
[8] L.H. Baestlé, Migration of Radionuclides in Porous Media. Progress in Nuclear Energy, Health Phys. 2 (1969).
[9] P.A. Domenico, G.A. Robbins, A New Method of Contaminant Plume Analysis, Groundwater. 23 (1985) 476–485. doi:10.1111/j.1745-6584.1985.tb01497.x.
[10] V. Zolla, Analisi ei fenomeni di contaminazione d aFe, Mn e Ni nelle acque sotterranee mediante applicazione di modelli numerici, Poltecnico di Torino, 2004.
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49
[12] E.J. Wexler, Analytical solutions for one-, two-, and three-dimensional solute transport in ground-water systems with uniform flow, in: U.S. Geological Survey, Techniques of water –Resources Investigations, 1991.
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doi:10.1016/j.jconhyd.2011.05.003.
[14] J.M. Martin-Hayden, G.A. Robbins, Plume Distortion and Apparent Attenuation Due to Concentration Averaging in Monitoring Wells, Ground Water. 35 (1997) 339–346. doi:10.1111/j.1745-6584.1997.tb00091.x. [15] P.A. Domenico, An analytical model for multidimensional transport of a
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50 List of Tables
Table 1.1: parameters values (from [7]) ... 29
Table 1.2: s values for each case considered ... 32
Table 1.3: Simulation data ... 45
Table 1.4: Values of s for different simulation runs ... 45
List of Figures Figure 1.1: Non-uniform velocity in the porous media: a) distribution into a single pore; b) pores different diameter effect; c) tortuosity due to different pores size (from [1])... 3
Figure 1.2: propagation of a pulse of concentration in advective-dispersive flux (from [1]) . 4 Figure 1.3: one-dimensional solute transport: advective-diffusive and dispersive contribute (from [1]) ... 5
Figure 1.4: longitudinal dispersion coefficient and diffusion coefficient ratio as a function of Peclet number ... 6
Figure 1.5: transverse dispersion coefficient and diffusion coefficient ratio as a function of Peclet number ... 7
Figure 1.6: linear isothermal adsorption (from [1]) ... 11
Figure 1.7: Langmuir isothermal adsorption (from [1]) ... 11
Figure 1.8: Freundlich isothermal model (from [1]) ... 12
Figure 1.9: one-dimensional integral solution with s=0 ... 30
Figure 1.10: Williams analytical solution with s=0 ... 30
Figure 1.11: Bear analytical solution ... 30
Figure 1.12: one- dimensional integral solution with s=0 and =0 ... 31
Figure 1.13: Williams analytical solution with s=0 and =0 ... 31
Figure 1.14: Bear analytical solution with =0 ... 31
Figure 1.15: models comparison ... 31
Figure 1.16: one-dimensional integral solution for each case of s ... 33
Figure 1.17: William one-dimensional solution for each case of s ... 33
Figure 1.18: 3D concentration [mg/l] plume evolution for an areal contaminant source with continuous release (from [10]) ... 35
Figure 1.19: Concentration surfaces obtained at different value of s wrt table 1, limit of the volumetric source model on the left hand side and plane source model on the right hand side. ... 47
