### 1. Introduction

### 2. Governing Equations

##### (1)

$$\frac{\partial c}{\partial t}=-v\frac{\partial c}{\partial x}+D\frac{{\partial}^{2}c}{\partial {x}^{2}}+f$$*c*is the concentration of the contaminant,

*x*indicates the dimension,

*t*is the time,

*D*is the hydrodynamic dispersion coefficient,

*v*is the velocity of the fluid, and

*f*is the reaction rate.

##### (2)

$$\frac{\partial c}{\partial t}=-v\frac{\partial c}{\partial x}+D\frac{{\partial}^{2}c}{\partial {x}^{2}}-kc$$*k*is the first order reaction rate constant. The above equation can be extended to describe the multispecies contaminant transport in porous media. For example, the chlorinated solvent contaminants such as PCE degrades to produce daughter products such as TCE, dichloroethylene (DCE) and vinyl chloride (VC) [5]

*c*

*is the species concentration in the*

_{i}*i*th generation. Species

*i*, which is produced from species i-1, also reacts to produce species i+1 and this further reacts to produce more species. The generalized form of such a sequential transport system can be described using the following equations.

##### (4)

$$\frac{\partial {c}_{i}}{\partial t}=-v\frac{\partial {c}_{i}}{\partial x}+D\frac{{\partial}^{2}{c}_{i}}{\partial {x}^{2}}-{k}_{i}{c}_{i}+{y}_{i-1}{k}_{i-1}{c}_{i-1},\mathrm{\hspace{0.17em}\u200a\u200a}\text{i}=1,\hspace{0.17em}2,\hspace{0.17em}3,\dots .\text{n}$$*i*is the index of the species,

*k*

*is the first order reaction rates, k*

_{i}_{0}= 0,

*y*is the yield coefficient and

*n*is the number of species.

##### (5)

$$\frac{\partial {c}_{1}}{\partial t}=-v\frac{\partial {c}_{1}}{\partial x}+D\frac{{\partial}^{2}{c}_{1}}{\partial {x}^{2}}-{k}_{1}{c}_{1}$$##### (6)

$$\frac{\partial {c}_{2}}{{\partial}_{t}}=-v\frac{\partial {c}_{2}}{\partial x}+D\frac{{\partial}^{2}{c}_{2}}{\partial {x}^{2}}-{k}_{2}{c}_{2}+{y}_{1}{k}_{1}{c}_{1}$$##### (7)

$$\frac{\partial {c}_{3}}{\partial t}=-v\frac{\partial {c}_{3}}{\partial x}+D\frac{{\partial}^{2}{c}_{3}}{\partial {x}^{2}}-{k}_{3}{c}_{3}+{y}_{2}{k}_{2}{c}_{2}$$##### (8)

$${C}_{1}(x,\hspace{0.17em}0)={C}_{2}(x,\hspace{0.17em}0)={C}_{3}(x,\hspace{0.17em}0)=0\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}x\ge 0$$##### (9)

$${C}_{1}(0,\hspace{0.17em}t)=1\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}t>0$$##### (10)

$${C}_{2}(0,\hspace{0.17em}t)={C}_{3}(0,\hspace{0.17em}t)=0\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}t>0$$##### (11)

$${C}_{1}(L,\hspace{0.17em}t)={C}_{2}(L,\hspace{0.17em}t)={C}_{3}(L,\hspace{0.17em}t)=0\mathrm{\hspace{0.17em}\u200a\u200a}t>0$$*C*

_{1}*, C*

_{2}*, C*

*are the concentration of the first, second and third species;*

_{3}*k*

_{1}*, k*

_{2}*, k*

*are the first order reaction rates,*

_{3}*y*

*and*

_{1}*y*

*are the stochiometric yield coefficients and*

_{2}*L*is the length of the domain. The parameters used for this model is shown in Table 1.

### 3. Numerical Method and Spatial Moment Analysis

_{0}) is proportional to the total mass of the fluid in the high permeability fracture. The first spatial moment (M

_{1}) describes the displacement of the centre of the mass and the second spatial moment (M

_{2}) describes the spread of the deviation from the centre of mass. The expressions for the evaluation of the zeroth moment, first moment and second moment are given below.

##### (15)

$$V(t)=\frac{d\left\{{X}_{1}(t)\right\}}{dt},\mathrm{\hspace{0.17em}\u200a\u200a}D(t)=\frac{1}{2}\frac{d\left\{{x}_{11}(t)\right\}}{dt}$$### 4. Results and Discussion

### 4.1. Validation of the Numerical Model

### 4.2. Moments for Different Mean Fluid Velocities

### 4.3. Moments for Different Dispersion Coefficients

^{2}/d) dispersion coefficient, while there is a significant loss of solute mass at a later stage (after 10 d) with an increased dispersion coefficient (D = 4 & 6 m

^{2}/d). Thus, a higher dispersion coefficient is associated with a higher loss of solute mass. Thus, a larger decaying of solute mass can be expected from a system with a relatively higher dispersion coefficient, i.e., with a relatively heterogeneous system, as against the conventional homogeneous system. It can be concluded from Fig. 5 that the decaying of solute mass is higher in a heterogeneous system than a homogeneous system.

^{2}/d is lesser than that with D = 4 m

^{2}/d, and greater than D = 6 m

^{2}/d. It can be concluded from Fig. 7 that the dispersion coefficient is highly time dependent under decaying of solutes with a complex behavior of mixing of solutes.

### 4.4. Moments for Different Decay Rate Coefficients

_{1}= 0.2/d), while there is nearly zero loss of solute mass for relatively higher decay rate of species 1 (k

_{1}= 0.4 & 0.6/d). Thus, it can be concluded from Fig. 8 that the profiles with a relatively low decay rate of species 1 decays into nothing much faster than the rest of the cases implying that the profiles with relatively high decay rate of species 1 paves way for decaying of other species.

### 4.5. Effective Dispersion Coefficient for Different Fluid Velocities, Dispersion Coefficients and Decay Rate Coefficients

^{th}day is reached. The same phenomenon is observed for all the fluid velocities considered. Thus spatial moment analysis helps in determining the effective dispersion coefficient of the solute, which otherwise cannot be known.

^{2}/d, but the increment is sharp for other dispersion coefficients. Then, the dispersion coefficients decreases with time. The effective dispersion coefficient reaches zero early for high dispersion coefficients and later for low dispersion coefficients.

^{th}day is reached in all the cases.

### 5. Conclusions

The decaying of solute mass increases as the magnitude of mean fluid velocity increases.

The transportation of solutes with lower mean fluid velocity experiences the maximum decaying of solutes at the earliest.

The mixing of solutes is significantly reduced by the decaying of solutes and that the intensity of reduction in mixing is a function of mean fluid velocity.

The decaying of solute mass is higher in a heterogeneous system than a homogeneous system.

The dispersion coefficient is highly time dependent under decaying of solutes with a complex behavior of mixing of solutes.

Loss of solute mass is relatively higher with lower decay rate of species 1.

The solute mobility and mixing varies non-linearly with time during its initial period, while the same ceases with higher decay rates of species 1 much faster.

The effective dispersion coefficient increases during the initial time period and then decreases with time duration for different fluid velocities, dispersion coefficients and decay rate coefficients.