51
2. Analytical solutions of one-dimensional contaminant
transport with source production-decay in a
semi-finite domain
The main part of my Ph.D. work was devoted to developing an 1D solution for an 1D- contaminant transport model with source production and decay.
The results of this work are under review for publication in a scientific journal.
Introduction
The advection-dispersion equation (ADE) well describes the concentration distribution in space and time of a reactant pollutant in groundwater. The physical process governing the advective transport, chemical reaction in the liquid phase, molecular diffusion and hydrodynamic dispersion, contaminant decay or production, and adsorption/desorption processes with the solid phase are present in the advection-dispersion model.
52
If the problem to solve takes into account complex chemical processes: multi-component reaction chains or non-linear sorption, analytical solutions are rarely obtainable. Only for simple first order degradation or decay, zero order production and linear equilibrium sorption, two- or three- dimensional model were solved analytically. Some solutions taking into account non-linear adsorption/desorption have been developed for one-dimensional problems [1,2].
Numerous open and closed analytical solutions of the ADE are present in the literature. Analytical solutions are useful from pollution scenarios point of view: in risk analysis, they are useful to investigate chemical-physical parameters on pollutant transport and to validate numerical models. Fourier analysis, Laplace transform and Bessel or Hankel expansions are the most common methods used to find analytical solutions [3–6].
Multi-dimensional exact analytical solutions in semi-finite and finite domain under first- or third- type boundary conditions are usually expressed in integral open form. Integrals have to be evaluated numerically. Also infinite series can be contained in their expressions [7–12].
Multi-dimensional analytical solutions using the Green Function Method (GFM) were provided by [13–15] for various plane and linear sources. The Green Function Method can be used to derive more complex solutions, starting from 1D-solutions, under some particular initial and boundary conditions. For this reason, the study and the development of one-dimensional analytical solutions in closed form is of high interest.
One-dimensional exact analytical solutions, in closed form, have been developed in the last decades. A growing attention has been dedicated to solutions having time-dependent sources. A library of one-dimensional analytical models that encloses some solutions with source decay was proposed by [16]. [17] gave some 1D - solutions in closed form for exponential source decay by solving the ADE model with Duhamel theorem.
53
A closed form analytical solution of the one-dimensional ADE in semi-finite domain for a reacting solute under first order decay and linear sorption described by a retardation factor is here proposed. The source of contaminant is defined as third-type time-dependent boundary condition, with a combined source production and decay.
54
2.1. Governing equations
The one-dimensional advection-dispersion equation for contaminant transport with first order decay and linear equilibrium sorption in a semi-finite or a finite domain can be expressed as:
) , ( ) , ( ) , ( ) , ( 2 2 t x C x t x C v x t x C D t t x C R (2.1)
Subject to the following initial and boundary conditions: 0 ) 0 , (x C (2.2) ) ( ) , 0 ( t h t C or (0, ) ( ) ) , 0 ( t vg t vC x t C D (2.3) 0 ) , ( 0 x t L C or 0 ) , ( x t C (2.4) Where:
C(x,t) is the contaminant concentration [ML-3]; D is the dispersion coefficient [L2T-1];
v is the average space velocity [LT-1]; λ is the first order decay constant [T-1]; R is the retardation factor;
h(t) and g(t) are general time-dependent functions [ML-3].
Eqns (2.1-2.4) are valid for a saturated homogeneous porous medium, constant and uniform fluid flow and dispersion coefficient.
55
By applying the Green Function Method (GFM) it is possible to derive multidimensional solutions for contaminant transport starting from a 1D – solution. \
It is possible to use the Green Function Method when eqn. (2.1) can be expressed as a linear operator on concentration and the following hypothesis are valid:
homogeneous and anisotropic porous medium, constant and uniform space velocity in X direction, horizontally infinite or semi-finite domain;
vertically semi-finite or finite domain, initial condition set as c(x,y,z,0)0, point, linear, plane, volumetric source.
The source of contaminant can be put as a generation inside the domain or expressed as Dirichelet or Robin boundary conditions.
2.2. Analytical solutions for one-dimensional problems
Some one-dimensional analytical solutions in closed form and semi-finite domain are summarized here.. These solution can be found in [16]. Here the governing ADE contains an additional term with respect to eqn. (2.1) and is written as:
) , ( ) , ( ) , ( ) , ( 2 2 t x C x t x C v x t x C D t t x C R (2.5)
56
2.2.1. Constant inlet distribution subject to first-type boundary conditions and semi-finite domain
Eq. (2.5) is solved with the following initial and boundary conditions: i C x C( ,0) (2.6) 0 0 0 0 (0, ) 0 C t t C t t t (2.7) 0 ) , ( x t C (2.8)
This solution corresponds to case C5[16].
2.2.2. Constant inlet distribution subject to third-type boundary conditions and semi-finite domain
By considering again eqn. (2.5) and substituting equation (2.6) with:
0 0 0 0 (0, ) (0, ) 0 vC t t C t D vC t t t x (2.9)
57
2.2.3. Exponential inlet distribution subject to first-type boundary conditions and semi-finite domain
By considering again eqn. (2.5) with the following initial and boundary conditions: i C x C( ,0) (2.10) t b a s e C C t C(0, ) (2.11) 0 ) , ( x t C (2.12)
λs is the source decay constant, it is different from the first order decay constant of the solution.
This solution is case C13 [16].
When the zero order production term is null and initial and boundary conditions are as follows:
0 ) 0 , (x C (2.13) t s e C t C(0, ) 0 (2.14) 0 ) , ( x t C (2.15)
58
2.2.4. Exponential inlet distribution subject to third-type boundary conditions and semi-finite domain
By considering again eqn. (2.5) with initial and boundary conditions as follows: i C x C( ,0) (2.16)
t
b a s e C C v t vC x t C D (0, ) (0, ) (2.17) 0 ) , ( x t C (2.18)Ca is the concentration of the contaminant A and Cb is the concentration of the contaminant B and Ca + Cb= C0.
The solution refers to as case C14 [16] .
2.3. The proposed analytical solution with simultaneous source
production and decay
This one-dimensional analytical solution can be used for contaminant release due to failures in underground tanks or pipelines. In fact, it models the source as a function of time with combined production and consumption. In case of underground tanks failures an initial small leakage is observed, follows a concentration increase at the source due to the enlargement of the failure, and a further concentration decrease due to the tank emptying.
NAPL pool degradation as PCE to TCE or radioactive first order decay series at the source with s can be also described by this model.
59
2.3.1. General inlet distribution subject to third-type boundary conditions and semi-finite domain
By considering eqn. (2.1) subject to the following initial and boundary conditions: 0 ) 0 , (x C (2.19) ) ( ) , 0 ( ) , 0 ( t vg t vC x t C D (2.20) 0 ) , ( x t C (2.21)
The new variable c(x, t) can be introduced by this change:
( , ) exp exp ) , ( t c x t D vx t x C (2.22)With this step, it is possible to shift from Robin condition to Newmann condition.
Moreover, eqn. (2.1) becomes:
x t x c v x t x c D t t x c ( , ) ( , ) ( , ) 2 2 (2.23)
60
x t x c t D vx D t x c t D vx v x t x C ( , ) exp exp ) , ( exp exp ) , ( (2.24)By making the substitution of eqn. (2.23) in the first term of the right-hand side of eqn. (2.24) this equation can be expressed as:
x t x c t D vx D t x vC x t x C ( , ) exp exp ) , ( ) , ( (2.25) Eqn. (2.25) in x=0 becomes:
x t c t D t vC x t C (0, ) exp ) , 0 ( ) , 0 ( (2.26)Since the following is valid: ) ( ) , 0 ( ) , 0 ( t vg t vC x t C D (2.27)
Doing some passages, eqn. (2.27) becomes:
D t vg D t vC x t C(0, ) (0, ) ( ) (2.28) By setting ( ) f(t) D t vg we obtain:
61 ) ( ) , 0 ( ) , 0 ( t f D t vC x t C (2.29)
By Eqn. (2.26) and Eqn. (2.29) comparison we obtain:
x t c t t f( ) exp (0, ) (2.30) And finally:
t t f x t c exp ) ( ) , 0 ( (2.31)The final general problem to be studied is eqn. (2.23) with boundary conditions given by eqn. (2.31). It is a one-dimensional problem subject to second-type boundary conditions.
Solving with Laplace transform in time, the general solution is the following:
t x n D t v D vx D t x vn t x c , 4 exp 2 exp 2 , ) , ( 2 2 1 (2.32) Where: 62
dq D q t x q t v erfc q q f t x n t
0 1 2 exp ) ( , (2.33)
dq q t q t D x D qv q q f t x n t
0 2 2 2 ) ( 4 exp 4 exp exp ) ( , (2.34)Eqn. (2.33) and eqn. (2.34) are related by the following equality:
D t x n D vx D tv x t x n , 2 exp 4 exp , 2 2 1 (2.35)The integral solution becomes:
x t x n D t x vn t x c , 2 , ) , ( 1 1 (2.36) 63
2.3.2. Models comparison
In the last decades, a great number of multispecies transport or convective-dispersive transport of solutes with first-order reactions models have been developed.
[18] formulated one of the first models with analytical solutions about multi-species transport. The Laplace Transform technique was used; the model describes a three-species nitrogen decay chain. Van Genuchten [19] proposed a technique suitable to solve a four-species decay chain.
Clement [20]developed a generalized methodology for models involving multispecies transport with serial, parallel, multiparent, converging, diverging and reversible first – order reaction. This method involves a system of linear transformations and it is applicable to one-, two- and three-dimensional problems. Anyway, it has a limitation: it is not possible to solve multispecies problems with different retardation factors.
Quezada et al. [21] extended Clement’s methodology [20] by making possible to solve a three species, multi-dimensional transport problem subject to first-order reaction, non-sequential reaction network and different retardation factors. The solution involves Laplace transformation and linear transformations steps that are fundamentals to uncouple the system of governing partial differential equations. Quezada et al. [21] gave an analytical solution for a two-species reactive transport problem subject to Dirichelet pulse type boundary conditions. For a higher number of species, a numerical evaluation of Laplace Transform is necessary.
A mathematical model for a multi-species reactive transport equations coupled with sorption and sequential first-order reactions subject to Bateman boundary conditions was elaborated by Srinivasan et al. [22]. The analytical solutions are developed under Dirichelet or Cauchy boundary conditions. The mathematical construction is based on the linear transformation used by Clement [20]. Here again, there is the same limitation highlighted by Clement [20]: the system can be expressed
64
as a set of separate equations if and only if the retardation factor is the same for each component.
Decomposition techniques and matrix linear transformations restricts the field of solutions and for more complex models are difficult to solve. Furthermore, they are limited to infinite or semi-finite spatial domain [23]
A new method for solving analytically multi-species, advective-dispersive transport equations coupled with first-order decay reactions was proposed by [23]. Laplace Transform and integral transform are involved. The set of partial differential equations is converted into an algebraic equation system. A transport problem with four-species in finite domain is given as example with Cauchy boundary conditions.
The one-dimension advective-dispersive problem presented in this chapter, is solved by Laplace Transform for a general boundary condition function and a number of analytical solutions are given with different Cauchy and Dirichelet boundary conditions.
Exponential production and decay are taken into account at the source while a first order decay is present in the domain.
By applying the Green’s Function Method to the one-dimensional model subject to Diricheret boundary conditions, it is possible to extend the solution to the three-dimensional geometry.
65
2.4. Some possible solutions
2.4.1. Linear combination of exponential inlet distribution subject to third-type boundary conditions and semi-finite domain
By setting: t t s p
e
C
e
y
C
t
g
(
)
0(
1
)
0 (2.37)where C0 is the initial concentration [ML-3] in the source and 0 0 C C
y
is the ratio between the residual source concentration at the steady state and the initial concentration; p is the production constant [T-1] and s is the decay constant [T-1]. Eqns. (33) and (34) become:
dq D q t x q t v erfc q e C e y C D v t x n t q q s p
0 0 0 1 2 exp ) 1 ( , (2.38)
dq q t q t D x D qv q e C e y C D v t x n t q q s p
0 2 2 0 0 2 ) ( 4 exp 4 exp exp ) 1 ( , (2.39)by solving eqn. (2.36) with eqns. (2.38) and (2.39), after the substitution of eqn. (2.22), the final solution is:
x t C y
A
x t B
x t
C E
x t 66
2 ( ) ( , ) exp 2 2 ( ) exp 2 2 exp exp 2 2 x x x x x x x v v u x x tu A x t erfc v u D D t v v u x x tu erfc v u D D t v vx x tv t erfc D D D t (2.41)
2 ( ) exp 2 2 ( , ) exp ( ) exp 2 2 exp exp 2 2 x x p x x x p x x v v w x x tw erfc v w D D t B x t t v v w x x tw erfc v w D D t v vx x tv t erfc D D D t (2.42)
2 ( ) exp 2 2 ( , ) exp ( ) exp 2 2 exp exp 2 2 x x s x x s x x x v v W x x tW erfc v W D D t E x t t v v W x x tW erfc v W D D t v vx x tv t erfc D D D t (2.43) 2 4D v u x (2.44) 2 ) ( 4D v w x p (2.45) 2 ) ( 4D v W x s (2.46)That is the new one-dimensional analytical solution here proposed in semi-finite domain.
67
2.4.1.1. Limit cases
The proposed analytical solution reduces to known solutions for certain values of source constants.
Case 1: For p 0and s 0we have: 0 ) (t C g (2.47) u w W (2.48)
x t F
x t E , , (2.49)The solution, for this particular case, is:
x t C A
x t C , 0 , (2.50) Where:
2 ( ) ( , ) exp 2 2 ( ) exp 2 2 exp exp 2 2 x x x x x x x v v u x x tu A x t erfc v u D D t v v u x x tu erfc v u D D t v vx x tv t erfc D D D t (2.51)It corresponds to case C6, [16]by setting C i 0 and 0.
68 t s e C y C t g( ) 0 0 (2.52)
The solution can be obtained by performing the following limit:
lim
pC
(
x
,
t
)
2 ( ) exp 2 2 lim lim ( , ) exp ( ) exp 2 2 lim exp exp 2 2 x x p p p x x p p x x x v v w x x tw erfc v w D D t B x t t v v w x x tw erfc v w D D t v vx x tv t erfc D D D t (2.53)The limit is null so the solution becomes:
x t C yA
x t C E
x t C , 0 , 0 , (2.54) Where:
2 ( ) ( , ) exp 2 2 ( ) exp 2 2 exp exp 2 2 x x x x x x x v v u x x tu A x t erfc v u D D t v v u x x tu erfc v u D D t v vx x tv t erfc D D D t (2.55) 69
2 ( ) exp 2 2 ( , ) exp ( ) exp 2 2 exp exp 2 2 x x s x x s x x x v v W x x tW erfc v W D D t E x t t v v W x x tW erfc v W D D t v vx x tv t erfc D D D t (2.56)That is identical to case C14, [16] by setting Ci 0 and 0.
2.4.2. Linear combination of exponential inlet distribution subject to first-type boundary conditions and semi-finite domain
The model presented in par. 2.2. is subject to third-type boundary conditions. Even if Cauchy boundary conditions well describe mass conservation, Dirichelet boundary conditions are more common and easy to use since they need the knowledge of the concentration at the source instead of flux concentration. Moreover, a large number of software are implemented by using first-type or Dirichelet boundary conditions model based.
From a physical point of view, first-type boundary conditions solutions do not conserve mass if concentrations are taken as volume-average i.e. the amount of solute per unit of volume.
Systems subject to Dirichelet boundary conditions can be mass conservative when concentrations are taken as flux-average i.e. the quantity of solute per fluid unit volume that passes the cross-sectional area in a unit of time. ([24])
It is then important, as mentioned by several papers, to make a distinction between the resident or volume-average concentration Cr and the flowing or flux-averaged concentration Cf in one-dimensional solute transport. [25–27].
70
The above-mentioned studies gave the following correlation between Cr and
Cf: x C v D C C x r r f (2.57)
This formulation leaves one-dimensional transport equation unchanged Transport equation can be, then, written in term of Cr or Cf indipedently[28,29].
By considering eqn.(2.1) with the following boundary and initial conditions:
0 ) 0 , (x C (2.58) ) ( ) , 0 ( t h t C (2.59) 0 ) , ( x t C (2.60) t t s p
e
C
e
y
C
t
h
(
)
0(
1
)
0 (2.61)it is possible to obtain a new solution following passages in Appendix A.
The new solution is:
x,t C0y
a(x,t) b(x,t)
C0e(x,t) 71 Where: 2 2 1 exp 4 1 1 1 ( , ) exp 2 2 2 1 exp 4 1 1 1 exp 2 2 2 exp exp e 1 2 x x x x x x x x x tu x D D t x v u tu x a x t erfc D D t t u v tu x D D t x u v tu x erfc D D t t v u vx t D 2 1 xp 4 x x tv x v D t t D (2.63)
2 2 1 exp 4 1 1 1 ( ) ( , ) exp exp 2 2 2 1 exp 4 1 1 1 exp exp 2 2 2 ex 1 2 x x p x x x x p x x tw x D D t tw x x v w b x t erfc t D D t t v w tw x D D t x v w tw x erfc t D D t t v w
2 1 p exp exp 4 x x p x tv x vx t v D D t t D (2.64) 72 2 2 1 exp 4 1 1 1 ( ) ( , ) exp exp 2 2 2 1 exp 4 1 1 1 exp exp 2 2 2 e 1 2 x x s x x x x s x x tW x D D t tW x x v W e x t erfc t D D t t v w tW x D D t x v W tW x erfc t D D t t v W 2 1 xp exp exp 4 x x s x tv x vx t v D D t t D (2.65) 2.4.2.1. Limit cases
This analytical solution reduces to known solutions for particular values of the production / decay constants.
Case 1: For p 0and s 0we have:
0 ) (t C h (2.66) u w W (2.67)
x t C a
x t C , 0 , (2.68) Where:
1 1 1 1 1 ( , ) exp exp 2 2 x 2 x 2 x 2 x x v u x u v tu x tu x a x t erfc erfc D D D t D t (2.69) 73 It is equal to case C5, [16] with Ci 0and 0.
0 exp 2 2 ( , ) exp 2 2 exp 2 x x x x x ux x ut erfc D D t C vx C x t D ux x ut erfc D D t (2.70)
Case 2: For p we have: t s
e
C
y
C
t
h
(
)
0
0 (2.71)and the solution can be obtained by computing
lim
pC
(
x
,
t
)
that is:
p
x x x p x x x x p x x x x D t v t D x tv t D vx t D w v x w v t t D x tw D t D x tw erfc t D w v x w v t t D x tw D t D x tw erfc t x b p p 2 2 2 4 1 exp exp exp 2 1 exp 2 1 exp 4 1 exp 2 1 2 1 exp ) ( 2 1 exp 4 1 exp 2 1 2 1 lim ) , ( lim (2.72)is equal to zero, hence the solution becomes:
x t C ya
x t C e
x t 74 Where: 1 1 1 1 1 ( , ) exp exp 2 2 x 2 x 2 x 2 x x v u x u v tu x tu x a x t erfc erfc D D D t D t (2.74)
1 1 1 ( ) ( , ) exp exp 2 2 2 1 1 1 exp exp 2 2 2 s x x s x x tW x x v W e x t erfc t D D t x v W tW x erfc t D D t (2.75)That is identical to case C13, [16] by setting Ci 0 and 0.
2.4.3. Consecutive reactions at the source subject to third-type boundary conditions in semi-finite domain
Consecutive reactions at the source can be expressed as follows:
C B
A
With the proposed model, it is possible to simulate the behaviour of contaminant B produced by the decay of contaminant A, B decays then into product C.
This solution can be meaningful for radioactive waste decay at the source in groundwater or to study PCE to TCE degradation.
pt st
e
e
K
C
t
g
(
)
0
(2.76) 75
C0 is the initial concentration [ML-3] and 2 1
1 k k k K
is the global chemical kinetic rate.
x t C K
A
x t B x t
C , 0 , , (2.77)
2 ( ) exp 2 2 ( , ) exp ( ) exp 2 2 exp exp 2 2 x x p x x x p x x v v w x x tw erfc v w D D t A x t t v v w x x tw erfc v w D D t v vx x tv t erfc D D D t (2.78)
2 ( ) exp 2 2 ( , ) exp ( ) exp 2 2 exp exp 2 2 x x s x x s x x x v v W x x tW erfc v W D D t B x t t v v W x x tW erfc v W D D t v vx x tv t erfc D D D t (2.79) 2 ) ( 4D v w x
p (2.80) 2 ) ( 4D v W x s (2.81) 76
2.4.4. Consecutive reactions at the source subject to first-type boundary conditions in semi-finite domain
The solution for consecutive reactions in semi-finite domain subject to first-type boundary conditions can be derived by applying steps presented in Appendix A.
0 ) 0 , (x C (2.82) ) ( ) , 0 ( t h t C (2.83) 0 ) , ( x t C (2.84)
pt st
e
e
K
C
t
h
(
)
0
(2.85)
x,t C0K
a(x,t) b(x,t)
Cf (2.86) 2 2 1 exp 4 1 1 1 ( ) ( , ) exp exp 2 2 2 1 exp 4 1 1 1 exp exp 2 2 2 ex 1 2 x x p x x x x p x x tw x D D t tw x x v w a x t erfc t D D t t v w tw x D D t x v w tw x erfc t D D t t v w 2 1 p exp exp 4 x x p x tv x vx t v D D t t D (2.87) 77 2 2 1 exp 4 1 1 1 ( ) ( , ) exp exp 2 2 2 1 exp 4 1 1 1 exp exp 2 2 2 e 1 2 x x s x x x x s x x tW x D D t tW x x v W b x t erfc t D D t t v w tW x D D t x v W tW x erfc t D D t t v W 2 1 xp exp exp 4 x x s x tv x vx t v D D t t D (2.88)
2.4.5. Finite release at the source subject to third-type boundary conditions in semi-finite domain
By setting:
( ) ( ) t
g t H t C e (2.89)
Where H( t) is the Heaviside function, and Cis the residual source concentration, is the first order reaction constant.
The solution we get is:
(2.90)
, x
,
,
vx D t
78
( ) ( , ) 2 exp 2 2 ( ) exp 4 2 2 2 2 x x x x x x x D C tu x D C v u x A x t erfc t v u D t v u D C v u u v x x tu t erfc D D t vC x tv erfc D t (2.91)
( ) ( , ) exp exp 2 2 2 ( ) 2 exp 2 2 2 2 x x x x x x x x t v x C D C v C v u v x B x t erfc t v u D D t x t u x tu C D u v x erfc erfc t u v D D t D t x t u erfc erf D t 2 2 2 x 2 x vC x tu x tv c erfc D t D t (2.92)2.5. Some simulations and discussion
In order to test the proposed model, some simulations of concentration fields are reported.
Some values are taken from [30]; dispersion coefficients are expressed as the product of dispersivities and velocity and are summarized in Table 2.1.
79
Parameter Value
Longitudinal dispersivity (αx) 42.58 m
Space velocity (v) 0.2151 m/d
Initial source concentration (C0) 850 mg/l Residual source concentration (C0∞) 350 mg/l
Simulation time (tm) 5110 d
First order reaction constant (λ) 0.001 d-1
Table 2.1: Simulation data (from [31])
Simulations test model behaviour at changing values of exponential source production and decay constants λp and λs. Values of λp , λs and λ are chosen in order to test model behaviour when it tends to the well-known limit cases given by eqn. (2.50) and eqn. (2.54).
The effective velocities w in eqn. (2.42) and W in eqn. (2.43), can be negative for certain values of λp and λs. This problem is also present in the one-dimensional solution proposed by [16]. The result is anyway a real number because the overall expression is part of an integral whose solution is geometrically the area under the integral function. In this case, a tool able to manage complex exponential and complex error function is necessary in order to achieve the result.
80
2.5.1. Linear combination of exponential inlet distribution subject to third-type boundary conditions and semi-finite domain
Parameters λp [s-1] λs [s-1] g(t) C0 C0∞ Case 0.0022 0.00015 pt st
e
C
e
y
C
t
g
(
)
0(
1
)
0 850 350 a 0.0022 0.00008 pt ste
C
e
y
C
t
g
(
)
0(
1
)
0 850 350 0.0022 0.000012 t t s pe
C
e
y
C
t
g
(
)
0(
1
)
0 850 350 0 0.0008 g t C est 0 ) ( 850 350 b 0 0.0012 g t C est 0 ) ( 850 350 0 0.0022 g t C est 0 ) ( 850 350 0.0008 0 0 0(
1
)
)
(
t
C
y
e
C
g
pt
850 350 c 0.0012 0 0 0(
1
)
)
(
t
C
y
e
C
g
pt
850 350 0.0022 0 0 0(
1
)
)
(
t
C
y
e
C
g
pt
850 350Table 2.2: Simulation cases
81
Figure 2.1: Initial condition function g(t) for third type boundary conditions and concentration profile at X=0 -
Case a
Figure 2.2: Initial condition function g(t) for third type boundary conditions and concentration profile at X=0 -
Case b
Figure 2.3: Initial condition function g(t) for third type boundary conditions and concentration profile at X=0 -
82
In figure 2.1, case a is showed: behaviour of the boundary condition on the left and concentration at x=0 curve on the right.
In this case, λp is always greater than λs, i.e. the production is always greater than the source decay. Here the behaviour of both g(t) and C(t,0) has a rapid rise, proportional to the source production, and then a decrease. The two variables tends to a specific base value.
In figure 2.2, case b is proposed: λp is null for this case. This is then a case of sole decay. This is also visible from the curve shape of the boundary condition that decreases starting from the initial concentration value C0.
Because of the third type boundary condition, the concentration has again a raise and a decrease.
In figure 2.3 case c is reported; here only source production is simulated since the source decay constant is set equal to zero. The concentration and the boundary condition functions start from the value of C0 and then they rise up asymptotically to an upper limit value.
In Figure 2.4, concentration profile for t=5110 days are showed for each case.
83
Figure 2.5: flux profiles at t= 5110 [d] cases a – c
2.5.1.1. Limit cases
Parameters Cases Note
λp [s-1] λs [s-1]
0 0 1 Limit case of C6 [16]
1000 0 2 Limit case of C14 [16]
1000 0.00008 3 Limit case of C14 [16]
84
Figure 2.6:Initial condition function g(t) for third type boundary conditions and concentration profile at X=0 –
Table 2.3, limit case 1
Figure 2.6 shows a comparison of the proposed model with the solution case C6, [16]. The numerical simulation confirms that, , by setting λpand λs equal to zero, the new model equals the solution C6 proposed by Van Genuchten and Alves [16].
85
Figure 2.7:Initial condition function g(t) for third type boundary conditions and concentration profile at X=0 –
86
Figure 2.8: Initial condition function g(t) for third type boundary conditions and concentration profile at X=0 –
Table 2.3, limit case 3
87
Figure 2.10: flux profiles at t=5110 [s] - Table 2.3, limit cases 1-3
Figures 2.7 and 2.8 show comparisons between the proposed model and case C14, [16] when λp equals a value much more greater than λs or λs is equal to zero. Again, a numerical simulation confirms that the new model has the same behavior of the solution C14 proposed by Van Genuchten and Alves [16].
Concentration profiles at t= 5110 days for cases 1-3 are reported in figure 2.9, it can be seen that, for each case, the particular solution proposed by Van Genuchten and Alves and the limit cases of the new solution have the same behavior.
88
2.5.2. Linear combination of exponential inlet distribution subject to first-type boundary conditions and semi-finite domain
A simulation of solution obtained by the passage from third-type to first-type boundary conditions was done on the base of values reported in Table 2.1 and of the following cases. Parameters λp [s-1] λs [s-1] g(t) C0 C0∞ Case 0.0022 0.00015 pt st
e
C
e
y
C
t
h
(
)
0(
1
)
0 850 350 a 0.0022 0.00008 pt ste
C
e
y
C
t
h
(
)
0(
1
)
0 850 350 0.0022 0.000012 pt ste
C
e
y
C
t
h
(
)
0(
1
)
0 850 350 0 0.0008 h t C est 0 ) ( 850 350 b 0 0.0012 h t C est 0 ) ( 850 350 0 0.0022 h t C est 0 ) ( 850 350 0.0008 0 0 0(
1
)
)
(
t
C
y
e
C
h
pt
850 350 c 0.0012 0 0 0(
1
)
)
(
t
C
y
e
C
h
pt
850 350 0.0022 0 0 0(
1
)
)
(
t
C
y
e
C
h
pt
850 350 89
Figure 2.11:initial condition g(t) for first type boundary conditions - case a
90
Figure 2.13:initial condition g(t) for first type boundary conditions - case c
91
Figure 2.15:flux profiles at t=5110 [s] cases a – c
2.5.2.1. Limit cases
Parameters Cases Note
λp [s-1] λs [s-1]
0 0 1 Limit case of C5 [16]
1000 0 2 Limit case of C13 [16]
1000 0.00008 3 Limit case of C13[16]
92
Figure 2.16:concentration profile at X=0 - Table 2.5,limit case 1
93
Figure 2.18: concentration profile at X=0 – Table 2.5, limit case 3
94
Figure 2.20:flux profiles at t=5110 [s] limit cases 1-3
2.5.3. Finite release at the source subject to third-type boundary conditions in semi-finite domain
By considering values in Table 2.6, the behavior of the boundary condition and of the solution are in the following:
Parameter Value
Longitudinal dispersivity (αx) 42.58 m
Space velocity (v) 0.2151 m/d
Residual source concentration (C∞) 350 mg/l First order reaction constant (λ) 0.001 d-1
Delta () 500 d
95
Figure 2.21:finite release at the source
96
Figure 2.23:concentration profile at t = 1000 days
2.6. Example of application
Multi-species contaminants with sequential decay problems are of great importance. A lot of researchers have developed models with different complexity levels in order to simulate behaviour of chemicals such as nitrogen, chlorinated solvents and radionuclides [18,19].
These models are very complex due to reaction-chain laws that originate differential equations systems.
Here is an example of application involving one-dimensional transport of a three-radionuclide decay chain.
238 234 230
97
Input parameters, here used, are kept from [19,32] except for input concentration value and are summed up in Table 2.7.
Parameters 234U
Velocity, v [m year-1] 100 Dispersion coefficient [m2 year-1] 4000 First order decay constant [year-1] 0.0000028 Production constant p [year-1] 0.079 Source decay constant s [year-1] 0.0010028 Initial concentration of 238Pu[mg m3] 5104
Table 2.7:Radionuclide decay parameters(from [32])
A time of 100 years was taken into consideration [19,32,33]. Here simulation involving eqn. (2.77) for 234U:
98
Figure 2.25: concentration profile at X=0 for 234U
99 Here simulation regarding eqn. (2.86):
Figure 2.27: concentration profile at X=0 for 234U
100
Appendix A
Eqn.(A1) links the flux-averaged concentration Cf to the volume-average concentration Cr [25–27]. x C v D C C x r r f (A.1)
By considering eqn. (2.1) rewritten as:
r r r r r r r r C x C v x C D t C R x v D C x C v x C D t C R 2 2 2 2 (A.2)
The term in square brackets is zero.
By doing some passages we get:
x C v D x C D x C v D t C R x v D C x C v x C D t C R r r r r r r r r 2 2 3 3 2 2 2 (A.3) r r r r r r r r C x C v D x C D x C v x C v D x C D x C v D C t R 2 2 3 3 2 2 2 (A.4) x C v D C x C v D C x v x C v D C x D x C v D C t R r r r r r r r r 2 2 (A.5)
By substituting eqn. (A.1) in eqn. (A.5) we get again eqn.(2.1):
f f f f C x C v x C D t C R 2 2 (A.6)
101
The substitution of eqn. (A.1) in third-type boundary conditions let shift to first-type boundary conditions. By considering eqn.(2.20):
) ( ) , 0 ( ) , 0 ( t vg t vC x t C D r r
It follows from eqn. (A.1) that:
x C v D C C x r f r (A.7)
By substituting eqn.(A.7) in eqn.(2.20) we obtain: ) ( ) , 0 ( t g t Cf (A.8)
Eqn. (A.8) is a first-type boundary condition in terms of flux-average concentration.
On the bases of Eqn. (A.1), it is possible to directly compute the solution for first-type boundary conditions starting from the solution subject to third-type boundary conditions and it will be consistent with mass conservation.
By considering eqn. (2.40), the equivalent solution for first boundary conditions is:
C y
A
xt B
xt
C E
xt
v D t x E C t x B t x A y C t x C x f , , , , , , , ' 0 ' ' 0 0 0 (A.9) 102 Where: 2 ' 2 3/ 2 2 1 1 exp exp exp 4 2 1 , 2 1 1 1 ( ) exp ( ) exp 4 2 2 1 2 ( ) exp x x x x x x x x x x tv x tv x vx verfc t D t D t D A x t v D D t tu x tu x x u v u v erfc v D t D t D D v u D t v 2 1 1 1 ( ) ( ) exp 4 2 2 1 2 ( ) x x x x x tu x tu x x v u v u erfc D t D t D D v u D t (A.10) 2 ' 2 3/ 2 2 1 1 exp exp exp 4 2 1 , 2 ( ) 1 1 1 ( ) ( ) exp exp exp 4 2 2 1 2 ( ) x x x x p x p x x x x x tv x tv x vx verfc t D t D t D B x t v D D t tw x tw x x v w v w erfc v t D t D t D D v w D t 1 1 2 1 ( ) ( ) exp exp exp 4 2 2 1 2 ( ) p x x x x x tw x tw x x v w v w erfc v t D t D t D D v w D t (A.11)
