**The Stochastic Solute Transport Model in 2-Dimensions**

### **7.1 Introduction**

Computational Modelling of Multi-Scale Non-Fickian Dispersion

104

? To answer this question, we

would be changed to

value.

=0.1 best fit to the experimental data. In other words, by

and the mean flow velocity, and the mean values of

, we can obtain the dispersivity for any length of the flow by using the

can be hypothesised to indicate the type of media (e.g. fractured, porous

would be sufficient to characterise the dispersivity at different flow

for equation (6.6.2).

194 in Porous Media - An Approach Based on Stochastic Calculus

 **0.0001 0.001 0.01 0.1 0.25 0.4 0.6 0.8 1.0**  m2 0.589 0.897 1.067 1.150 1.148 1.148 1.148 1.148 1.144 *C* <sup>2</sup> 0.0122 0.0078 0.0103 0.0168 0.0242 0.0311 0.0409 0.0535 0.0725

Experimental Data

2 =0.1

102

L

Figure 6.9. Mean dispersivity from the SSTM and experimental dispersivity vs flow length

We can estimate the approximate dispersivity values either from Figures 6.7 and 6.8, or from equations (6.6.1) and (6.6.2). It is quite logical to ask the question whether we can characterise

resort to the published dispersivity values for aquifers. We use the dispersivity data first published by Gelhar et al. (1992) and reported to Batu (2006). We extracted the tracer tests data related to porous aquifers in 59 different locations characterised by different geologic materials. The longest flow length was less than 10000 m. We then plotted the experimental

SSTM. We can also assume that each experimental data point represents the mean dispersivity for any length of the flow by using the SSTM. We can also assume that each experimental data point represents the mean dispersivity at a particular flow length. If that is the case, Figure 6.9 can be interpreted as follows: by using the SSTM, we can obtain sufficiently large number of

the dispersivities estimated for those concentration realisations do represent the experimental

etc.). These findings support the hypothesis that the dimensionless SSTM is scale-independent,

lengths. It is important to note that the role of the mean velocity in these calculations. We used 0.5 m/day to represent an indicative value in real aquifers, but the character of solutions do

data and overlaid the plot with the dispersivity vs *L* curves from the SSTM for each <sup>2</sup>

2 

(log10 scale).

Table 6.6. *m2* and *C* <sup>2</sup> values for different <sup>2</sup>

100

the large scale aquifer dispersivities using a single value of <sup>2</sup>

not change, if we assume a different value; only the specific values of <sup>2</sup>

10-2

100

Dispersivity

Figure 6.9 shows the plots, and <sup>2</sup>

realisations for particular values of <sup>2</sup>

represent a given flow situation.

using one value of <sup>2</sup>

dispersivities. <sup>2</sup>

i.e., one value of <sup>2</sup>

102

<sup>104</sup>

Mean

In Chapter 6, we developed the generalised Stochastic Solute Transport Model (SSTM) in 1 dimension and showed that it can model the hydrodynamic dispersion in porous media for the flow lengths ranging from 1 to 10000 m. For computational efficiency, we have employed one of the fastest converging kernels tested in Chapter 6 for illustrative purposes, but, in principle, the SSTM should provide scale independent behaviour for any other velocity covariance kernel. If the kernel is developed based on the field data, then the SSTM based on that particular kernel should give realistic outputs from the model for that particular porous medium. In the development of the SSTM, we assumed that the hydrodynamic dispersion is one dimensional but by its very nature, the dispersion lateral to the flow direction occurs. We intend to explore this aspect in this chapter.

First, we solve the integral equation with the covariance kernel in two dimensions, and use the eigen values and functions thus obtained in developing the two dimensional stochastic solute transport model (SSTM2d). Then we solve the SSTM2d numerically using a finite difference scheme. In the last section of the chapter, we illustrate the behaviours of the SSTM2d graphically to show the robustness of the solution.

### **7.2 Solving the Integral Equation**

We consider the flow direction to be *x* and the coordinate perpendicular to *x* to be *y* in the 2 dimensional flow with in the porous matrix saturated with water. Then the distance between the points 1 1 (,) *x y* and 2 2 (,) *x y* , *r*, is given by 1/2 <sup>2</sup> <sup>2</sup> 12 12 ( )( ) *xx yy* . We can then define a velocity covariance kernel as follows:

$$q(\mathbf{x}\_{1'}, y\_{1'}, \mathbf{x}\_{2'}, y\_2) = \sigma^2 \exp\left[-\frac{r^2}{b}\right],\tag{7.2.1}$$

where <sup>2</sup> is a constant. <sup>2</sup> is the variance at a given point, i.e., when 1 2 *x x* and 1 2 *y y* . The covariance can be written as,

$$\begin{split} q(\mathbf{x}\_1, y\_1, \mathbf{x}\_2, y\_2) &= \sigma^2 \exp\left[ -\frac{\left[ \left( \mathbf{x}\_1 - \mathbf{x}\_2 \right)^2 + \left( y\_1 - y\_2 \right)^2 \right]}{b} \right] \\ &= \sigma^2 \exp\left[ -\frac{\left( \mathbf{x}\_1 - \mathbf{x}\_2 \right)^2}{b} \right] \exp\left[ -\frac{\left( y\_1 - y\_2 \right)^2}{b} \right]. \end{split} \tag{7.2.2}$$

and , , *y yy x xx C xyt J J J J*

indicated by a subscript. We can expand *J* using Taylor expansions as follows:

 <sup>2</sup> <sup>3</sup> <sup>2</sup> <sup>3</sup> 2 3

 <sup>2</sup> <sup>3</sup> 2 3

2 3

1 1 1 1! 2! 3!

1 1 1 1! 2! 3!

 

*J J <sup>J</sup> JJ y <sup>y</sup> <sup>y</sup> y y y*

*J J <sup>J</sup> JJ x <sup>x</sup> <sup>x</sup> x x x*

*xx x*

*yy y*

the series,

*x x x*

*y y y*

*t x y* 

where *C*(*x*,*y*,*t*) is the solute concentration and *J* represents the solute flux at the location

The Stochastic Solute Transport Model in 2-Dimensions 197

Lumping the higher order terms greater than 2, and denoting *Rx* and *Ry* as the remainders of

2

2

2

Figure 7.1. Two dimensional infinitesimal volume element with a depth *l* and porosity

*ne* . *x* and *y* are side lengths in *x* and *y* directions, respectively.

higher order terms, and

higher order terms.

<sup>2</sup> <sup>2</sup>

1 2! *x x xx x x J J JJ x x R x x*

1 2! *y y yy y y J J JJ y y R y y*

 

 

, (7.3.1)

2

, and (7.3.2a)

. (7.3.2b)

Then the integral equation can be written for 2 dimensions,

$$\sigma^{2}\left[\left(\exp\left[-\frac{\left(\mathbf{x}\_{1}-\mathbf{x}\_{2}\right)^{2}}{b}\right]\exp\left[-\frac{\left(\mathbf{y}\_{1}-\mathbf{y}\_{2}\right)^{2}}{b}\right]f\left(\mathbf{x}\_{2},\mathbf{y}\_{2}\right)d\mathbf{x}\_{2}d\mathbf{y}\_{2}=\lambda f\left(\mathbf{x}\_{1},\mathbf{y}\_{1}\right),\tag{7.2.3}$$

where *f xy* , and are eigen functions and corresponding eigen values, respectively.

The covariance kernel is the multiplication of a function of *x* and a function of *y* , and from the symmetry of equation (7.2.3), we can assume that the eigen function is the multiplication of a function of *x* and a function of *y*:

$$f(\mathbf{x}, y) = f\_{\mathbf{x}}(\mathbf{x}) f\_{y}(y) \,. \tag{7.2.4}$$

Then the integral equation can be written as,

$$
\sigma^2 \prod\_{0=0}^{1} \left| f\_x e^{-\frac{\left(y\_1 - y\_2\right)^2}{b}} dx\_2 \right| \left| f\_y e^{-\frac{\left(y\_1 - y\_2\right)^2}{b}} dy\_2 \right| = \lambda f\_x f\_y
$$

$$
\text{and}
$$

$$
\left| \left\{ \left| f\_x \, e^{-\frac{\left(y\_1 - y\_2\right)^2}{b}} dx\_2 \right| \right\} \left| \left\{ \left| f\_y \, e^{-\frac{\left(y\_1 - y\_2\right)^2}{b}} dy\_2 \right| \right\} \right| = \frac{\lambda}{\sigma^2} f\_x f\_y \, . \tag{7.2.5}
$$

Therefore, if

$$\begin{aligned} \int\_0^1 f\_\chi e^{-\frac{\left(\mathbf{x}\_1 - \mathbf{x}\_2\right)^2}{b}} d\mathbf{x}\_2 &= \mathcal{A}\_\chi f\_\chi \left(\mathbf{x}\_1\right), \text{ and} \\\\ \int\_0^1 f\_\chi e^{-\frac{\left(\mathbf{y}\_1 - \mathbf{y}\_2\right)^2}{b}} d\mathbf{y}\_2 &= \mathcal{A}\_\chi f\_\chi \left(\mathbf{y}\_1\right) \end{aligned}$$

Then we can see, , *<sup>x</sup> <sup>y</sup> f xy f f* , and <sup>2</sup> *<sup>x</sup> <sup>y</sup>* .

This shows that we can use the eigen functions and eigen values obtained for 1-dimensional covariance kernels in Chapter 4 can be used in constructing the eigen functions and eigen values for two dimensional covariance kernel given in equation (7.2.2). Once we have obtained eigen functions and eigen values as solutions of the integral equation, we can derive the two dimensional mass conservation equation for solutes.

### **7.3 Derivation of Mass Conservation Equation**

Consider the two dimensional infinitesimal volume element depicted in Figure 7.1. We can write the mass balance for solutes with in the element as,

$$\begin{aligned} \Delta \mathsf{C} \left( \mathbf{x}, \mathbf{y}, t \right) n\_{\varepsilon} l \, \Delta \mathbf{x} \, \Delta \mathbf{y} &= \left\{ J\_{\mathbf{x}} \left( \mathbf{x}, \mathbf{y}, t \right) - J\_{\mathbf{x}} \left( \mathbf{x} + \Delta \mathbf{x}, \mathbf{y}, t \right) \right\} l \, \Delta \mathbf{y} \, n\_{\varepsilon} \Delta t \\ &+ \left\{ J\_{\mathbf{y}} \left( \mathbf{x}, \mathbf{y}, t \right) - J\_{\mathbf{y}} \left( \mathbf{x}\_{\prime \prime}, \mathbf{y} + \Delta \mathbf{y}, t \right) \right\} l \, \Delta \mathbf{x} \, n\_{\varepsilon} \Delta t \end{aligned}$$

22 22 1 1

*f xy f x f y* , *x y* . (7.2.4)

, and

. (7.2.5)

2 1

are eigen functions and corresponding eigen values, respectively.

function

, (7.2.3)

196 in Porous Media - An Approach Based on Stochastic Calculus

<sup>2</sup> <sup>2</sup> 1 1 <sup>2</sup> 1 2 1 2

<sup>2</sup> <sup>2</sup> 1 1 1 2 1 2

<sup>2</sup> <sup>2</sup> <sup>1</sup> 1 2 <sup>1</sup> 1 2

*x x y y b b*

*<sup>b</sup> <sup>x</sup> x x f e dx f x*

*y y*

*x x*

<sup>2</sup> <sup>1</sup> 1 2

 *<sup>x</sup> <sup>y</sup>* .

<sup>2</sup> <sup>1</sup> 1 2

0 0

0

0

derive the two dimensional mass conservation equation for solutes.

*C x y t n l x y J x y t J x x y t l yn t*

, , , , , ,

*x x y y b b*

*f e dx f e dy f f*

() () exp exp , , *x x y y <sup>f</sup> x y dx dy f x y <sup>b</sup> <sup>b</sup>*

The covariance kernel is the multiplication of a function of *x* and a function of *y* , and from the symmetry of equation (7.2.3), we can assume that the eigen function is the multiplication

2 2

2 2 2

2 1

, and

*<sup>b</sup> <sup>y</sup> y y f e dy f y*

.

This shows that we can use the eigen functions and eigen values obtained for 1-dimensional covariance kernels in Chapter 4 can be used in constructing the eigen functions and eigen values for two dimensional covariance kernel given in equation (7.2.2). Once we have obtained eigen functions and eigen values as solutions of the integral equation, we can

Consider the two dimensional infinitesimal volume element depicted in Figure 7.1. We can

*e x x e*

, , ,, ,

*y y e*

*J x y t J x y y t l xn t*

*x y x y*

*<sup>x</sup> <sup>y</sup> <sup>x</sup> <sup>y</sup> f e dx f e dy f f*

Then the integral equation can be written for 2 dimensions,

0 0

Then the integral equation can be written as,

2

Then we can see, , *<sup>x</sup> <sup>y</sup> f xy f f* , and <sup>2</sup>

**7.3 Derivation of Mass Conservation Equation** 

write the mass balance for solutes with in the element as,

0 0

of a function of *x* and a function of *y*:

where *f xy* , and

Therefore, if

$$\text{and} \quad \frac{\Delta C \left(x, y, t\right)}{\Delta t} = \frac{\left(f\_x - f\_{x \star \Delta x}\right)}{\Delta x} + \frac{\left(f\_y - f\_{y \star \Delta y}\right)}{\Delta y},\tag{7.3.1}$$

where *C*(*x*,*y*,*t*) is the solute concentration and *J* represents the solute flux at the location indicated by a subscript. We can expand *J* using Taylor expansions as follows:

$$J\_{\boldsymbol{x}\ast\boldsymbol{\Delta\boldsymbol{x}}} - J\_{\boldsymbol{x}} = \frac{1}{1!} \frac{\partial J\_{\boldsymbol{x}}}{\partial\boldsymbol{\alpha}} \boldsymbol{\Delta\boldsymbol{x}} + \frac{1}{2!} \frac{\partial^{2} J\_{\boldsymbol{x}}}{\partial\boldsymbol{\alpha}^{2}} \left(\boldsymbol{\Delta\boldsymbol{x}}\right)^{2} + \frac{1}{3!} \frac{\partial^{3} J\_{\boldsymbol{x}}}{\partial\boldsymbol{\alpha}^{3}} \left(\boldsymbol{\Delta\boldsymbol{x}}\right)^{3} + \text{ higher order terms, and}$$

$$J\_{\boldsymbol{y}\ast\boldsymbol{\Delta\boldsymbol{y}}} - J\_{\boldsymbol{y}} = \frac{1}{1!} \frac{\partial J\_{\boldsymbol{y}}}{\partial\boldsymbol{y}} \boldsymbol{\Delta\boldsymbol{y}} + \frac{1}{2!} \frac{\partial^{2} J\_{\boldsymbol{y}}}{\partial\boldsymbol{y}^{2}} \left(\boldsymbol{\Delta\boldsymbol{y}}\right)^{2} + \frac{1}{3!} \frac{\partial^{3} J\_{\boldsymbol{y}}}{\partial\boldsymbol{y}^{3}} \left(\boldsymbol{\Delta\boldsymbol{y}}\right)^{3} + \text{ higher order terms.}$$

Lumping the higher order terms greater than 2, and denoting *Rx* and *Ry* as the remainders of the series,

$$f\_{\mathbf{x} \star \mathbf{A} \mathbf{x}} - f\_{\mathbf{x}} = \frac{\partial f\_{\mathbf{x}}}{\partial \mathbf{x}} \Delta \mathbf{x} + \frac{1}{2!} \frac{\partial^2 f\_{\mathbf{x}}}{\partial \mathbf{x}^2} (\Delta \mathbf{x})^2 + R\_{\mathbf{x}} \left(\boldsymbol{\varepsilon}\right), \text{ and} \tag{7.3.2a}$$

$$J\_{y\*\Delta y} - J\_y = \frac{\partial J\_y}{\partial y} \Delta y + \frac{1}{2!} \frac{\partial^2 J\_y}{\partial y^2} \left(\Delta y\right)^2 + R\_y\left(\varepsilon\right). \tag{7.3.2b}$$

Figure 7.1. Two dimensional infinitesimal volume element with a depth *l* and porosity *ne* . *x* and *y* are side lengths in *x* and *y* directions, respectively.

To simplify the notation,

where

*x x* 

be expressed as,

*x*

*y*

We can now write,

 *dt* and *y y* 

The resultant noise term is given by,

*S*

*<sup>h</sup> <sup>S</sup>*

*V V x xx* 

the resulting equations in to equation (7.3.4), we obtain,

, and

.

*dC S V C S V C dt S C dt S C dt xx yy x xx*

in the third and fourth terms of the right hand side,

As in the one dimensional case, we can define,

*y yy*

*dC S V C S V C dt S C dt S C dt xx yy xx yy*

*dt* , and these are the components of a noise vector,

operates in a Hilbert space having eigen functions as co-ordinates. Equation (7.3.12) can now

. *xx yy x xy y dC S V C S V C dt S Cd S Cd*

1 *m*

 

cos *<sup>x</sup> d d* 

> *d d <sup>y</sup>*

*j*

*d* 

where *x j* , *f* eigen functions in *x* direction, and

Now we can express the components in *x* and *y* directions,

*y j* , *f* eigen functions in *y* direction.

,,,,

*f f db t*

*xj yj xj yj j*

*x x* 

*h*

*y y*  *V V y yy* 

*dC S V C dt S V C dt S C dt S C dt x x y y x xx*

By substituting these equations in to equations (7.3.5a) and (7.3.5b), and then substituting

The Stochastic Solute Transport Model in 2-Dimensions 199

*y yy*

, and (7.3.9)

. (7.3.10)

, and bringing *dt* in to the parenthesis

 

, and (7.3.15)

sin . (7.3.16)

, (7.3.14)

, (7.3.11)

. (7.3.12)

, which

(7.3.13)

Substituting equations (7.3.2a) and (7.3.2b) back to equation (7.3.1) and taking the limit *t* 0 ,

$$\begin{split} \frac{\partial \mathbb{C}(\mathbf{x}, \mathbf{y}, t)}{\partial t} &= -\frac{\partial I\_x}{\partial \mathbf{x}} - \frac{1}{2} \frac{\partial^2 I\_x}{\partial \mathbf{x}^2} \Delta \mathbf{x} - \frac{\partial I\_y}{\partial y} - \frac{1}{2} \frac{\partial^2 I\_y}{\partial y^2} \Delta \mathbf{y} + R\_x^\dagger(\boldsymbol{\varepsilon}) \\ &= -\left( \frac{\partial I\_x}{\partial \mathbf{x}} + \frac{\partial I\_y}{\partial y} \right) - \frac{h\_x}{2} \left( \frac{\partial^2 I\_x}{\partial \mathbf{x}^2} \right) - \frac{h\_y}{2} \left( \frac{\partial^2 I\_y}{\partial y^2} \right) + R\_x^\dagger(\boldsymbol{\varepsilon}) + R\_y^\dagger(\boldsymbol{\varepsilon}) \end{split} \tag{7.3.3}$$

where *<sup>x</sup> h x* and *<sup>y</sup> h y* .

$$d\mathbf{C} = -\left[\frac{\partial \mathbf{J}\_x}{\partial \mathbf{x}} + \frac{\hbar\_x}{2} \frac{\partial^2 \mathbf{J}\_x}{\partial \mathbf{x}^2}\right] dt - \left[\frac{\partial \mathbf{J}\_y}{\partial y} + \frac{\hbar\_y}{2} \frac{\partial^2 \mathbf{J}\_y}{\partial y^2}\right] dt + \left(R\_x(\boldsymbol{\varepsilon}) + R\_y(\boldsymbol{\varepsilon})\right) dt \tag{1}$$

Assuming ' ' 0 *R R dt <sup>x</sup> <sup>y</sup>* ,

$$d\mathbf{C}\left(\mathbf{x},\mathbf{y},t\right) = -\left[\frac{\partial I\_x}{\partial \mathbf{x}} + \frac{\hbar\_x}{2}\frac{\partial^2 I\_x}{\partial \mathbf{x}^2}\right]dt - \left[\frac{\partial I\_y}{\partial y} + \frac{\hbar\_y}{2}\frac{\partial^2 I\_y}{\partial y^2}\right]dt\,\tag{7.3.4}$$

Now we can express the solute flux in terms of solute concentration and velocity,

$$J\_x(\mathbf{x}, y, t) = V\_x(\mathbf{x}, y, t) \mathbf{C}(\mathbf{x}, y, t), \text{ and }\tag{7.3.5a}$$

$$J\_y(\mathbf{x}, y, t) = V\_y(\mathbf{x}, y, t) \mathbf{C}(\mathbf{x}, y, t) \,. \tag{7.3.5b}$$

We can express the velocity in terms of the mean velocity vector and a noise vector,

$$
\underline{V}(\mathbf{x}, \mathbf{y}, t) = \underline{\overline{V}}(\mathbf{x}, \mathbf{y}, t) + \underline{\underline{\xi}}(\mathbf{x}, \mathbf{y}, t) \, \, \, \, \, \tag{7.3.6}
$$

where *V xyt* , , , *V xyt* , , and *xyt* , , are velocity, mean velocity and noise vectors respectively. Instantaneous velocity vector can now be expressed as,

$$
\underline{V}\left(\mathbf{x},\mathbf{y},t\right) = V\_x\left(\mathbf{x},\mathbf{y},t\right)\underline{\mathbf{i}} + V\_y\left(\mathbf{x},\mathbf{y},t\right)\underline{\mathbf{j}}\\_{\underline{\mathbf{i}}}\tag{7.3.7}
$$

where *i* and *j* are unit vectors in *x* and *y* directions, respectively; and, *V xyt <sup>x</sup>* , , and , , *V xyt <sup>y</sup>* are the magnitudes of the velocities in *x* and *y* directions. By substituting the vector components in equation (7.3.6) in to equation (7.3.7), we obtain,

$$\begin{split} \underline{V} \begin{pmatrix} \mathbf{x}, \mathbf{y}, t \end{pmatrix} &= \left( \overline{V}\_{\times} \begin{pmatrix} \mathbf{x}, \mathbf{y}, t \end{pmatrix} + \underline{\boldsymbol{\varepsilon}}\_{\times} \begin{pmatrix} \mathbf{x}, \mathbf{y}, t \end{pmatrix} \right) \underline{\mathbf{i}} + \left( \overline{V}\_{\times} \begin{pmatrix} \mathbf{x}, \mathbf{y}, t \end{pmatrix} + \underline{\boldsymbol{\varepsilon}}\_{\times} \begin{pmatrix} \mathbf{x}, \mathbf{y}, t \end{pmatrix} \right) \underline{\mathbf{j}} \\ &= \left( \overline{V}\_{\times} \underline{\mathbf{i}} + \overline{V}\_{\times} \underline{\mathbf{j}} \right) + \left( \underline{\boldsymbol{\varepsilon}}\_{\times} \begin{pmatrix} \mathbf{x}, \mathbf{y}, t \end{pmatrix} \underline{\mathbf{i}} + \underline{\boldsymbol{\varepsilon}}\_{\times} \begin{pmatrix} \mathbf{x}, \mathbf{y}, t \end{pmatrix} \underline{\mathbf{j}} \end{split} \tag{7.3.8}$$

where *<sup>x</sup>* and *<sup>y</sup>* are the noise components in *x* and *y* directions. We can see the noise term appearing as, *<sup>x</sup> xyt i xyt j xyt* ,, ,, ,, *<sup>y</sup>* .

To simplify the notation,

Computational Modelling of Multi-Scale Non-Fickian Dispersion

'

*x*

''

*J xyt V xyt Cxyt <sup>x</sup>* ,, ,, ,, *<sup>x</sup>* , and (7.3.5a)

*J xyt V xytCxyt <sup>y</sup>* ,, ,, ,, *<sup>y</sup>* . (7.3.5b)

*xyt* , , are velocity, mean velocity and noise vectors

*V xyt V xyt i V xyt j* ,, ,, ,, *<sup>x</sup> <sup>y</sup>* , (7.3.7)

*x y*

''

 

, (7.3.6)

, (7.3.3)

. (7.3.4)

, (7.3.8)

*x y*

198 in Porous Media - An Approach Based on Stochastic Calculus

Substituting equations (7.3.2a) and (7.3.2b) back to equation (7.3.1) and taking the limit

2 2

*y y y x x x*

*JhJ JhJ dC dt dt R R dt*

 <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> , , <sup>2</sup> <sup>2</sup> *yyy xxx JhJ JhJ dC x y t dt dt*

.

*xx yy* 

*xy x y*

 

2 2

2 2

2 2

*J hJ <sup>J</sup> h J R R*

2 2

<sup>2</sup> <sup>2</sup>

*<sup>x</sup> y R t xx yy*

<sup>2</sup> <sup>2</sup> 2 2 *yyy xxx*

Now we can express the solute flux in terms of solute concentration and velocity,

We can express the velocity in terms of the mean velocity vector and a noise vector,

respectively. Instantaneous velocity vector can now be expressed as,

vector components in equation (7.3.6) in to equation (7.3.7), we obtain,

*V xyt V xyt xyt* ,, ,, ,,

where *i* and *j* are unit vectors in *x* and *y* directions, respectively; and, *V xyt <sup>x</sup>* , , and , , *V xyt <sup>y</sup>* are the magnitudes of the velocities in *x* and *y* directions. By substituting the

*x x y y*

.

,, ,, ,, ,, ,,

*V x y t Vx y t x y t i Vx y t x y t j Vi V j xyt i xyt j*

*xy x y*

 *<sup>x</sup> xyt i xyt j xyt* ,, ,, ,, 

*<sup>y</sup>*

,, ,,

are the noise components in *x* and *y* directions. We can see the noise

*xx yy*

*y y x x*

2 2

, , 1 1

*C xyt J J J J*

*t* 0 ,

where

where *<sup>x</sup> h x* and *<sup>y</sup> h y* .

Assuming ' ' 0 *R R dt <sup>x</sup> <sup>y</sup>* 

where *V xyt* , , , *V xyt* , , and

where *<sup>x</sup>* 

 and *<sup>y</sup>* 

term appearing as,

 ,

$$V\_{\chi} = \overline{V}\_{\chi} + \underline{\xi}\_{\chi} \text{ and } \tag{7.3.9}$$

$$V\_y = \overline{V}\_y + \underline{\xi}\_y \,. \tag{7.3.10}$$

By substituting these equations in to equations (7.3.5a) and (7.3.5b), and then substituting the resulting equations in to equation (7.3.4), we obtain,

$$d\mathbf{C} = \mathcal{S}\_x \left( \overline{V}\_x \mathbf{C} \right) dt + \mathcal{S}\_y \left( \overline{V}\_y \mathbf{C} \right) dt + \mathcal{S}\_x \left( \mathcal{C}\_x \underline{\xi}\_x \right) dt + \mathcal{S}\_y \left( \mathcal{C}\_y \underline{\xi}\_y \right) dt \tag{7.3.11}$$

where 2 <sup>2</sup> 2 *x x <sup>h</sup> <sup>S</sup> x x* , and

$$S\_y = -\left(\frac{\partial}{\partial y} + \frac{h\_y}{2} \frac{\partial^2}{\partial y^2}\right).$$

We can now write,

*dC S V C S V C dt S C dt S C dt xx yy x xx y yy* , and bringing *dt* in to the parenthesis in the third and fourth terms of the right hand side,

$$d\mathbf{C} = \left(\mathbf{S}\_x\left(\overline{V}\_x\mathbf{C}\right) + \mathbf{S}\_y\left(\overline{V}\_y\mathbf{C}\right)\right)dt + \mathbf{S}\_x\left(\mathbf{C}\xi\_x dt\right) + \mathbf{S}\_y\left(\mathbf{C}\xi\_y dt\right). \tag{7.3.12}$$

As in the one dimensional case, we can define,

*x x dt* and *y y dt* , and these are the components of a noise vector, , which operates in a Hilbert space having eigen functions as co-ordinates. Equation (7.3.12) can now be expressed as,

$$d\mathbf{C} = \left( S\_x \left( \overline{V}\_x \mathbf{C} \right) + S\_y \left( \overline{V}\_y \mathbf{C} \right) \right) dt + S\_x \left( \mathbf{C} d\mathcal{J}\_x \right) + S\_y \left( \mathbf{C} d\mathcal{J}\_y \right). \tag{7.3.13}$$

The resultant noise term is given by,

$$d\mathcal{J} = \sigma \sum\_{j=1}^{m} \sqrt{\mathcal{k}\_{\mathbf{x}\_{\cdot j}} \mathcal{k}\_{\mathbf{y}\_{\cdot j}}} f\_{\mathbf{x}\_{\cdot j}} f\_{\mathbf{y}\_{\cdot j}} db\_{\rangle}(\mathbf{t}) \,, \tag{7.3.14}$$

where *x j* , *f* eigen functions in *x* direction, and

*y j* , *f* eigen functions in *y* direction.

Now we can express the components in *x* and *y* directions,

$$d\mathcal{J}\_x = d\beta \cos \theta \text{ , and }\tag{7.3.15}$$

$$d\mathcal{J}\_y = d\mathcal{J}\sin\theta \,. \tag{7.3.16}$$

Now,

Then,

cos

 <sup>2</sup> 2 , , , cos 2 *x x j*

cos

,

*x xj*

*S Cf*

cos

,

*y yj*

*S Cf*

2

Simplifying, we obtain,

*C xyt <sup>h</sup> <sup>f</sup> x*

2 ,

, , cos 2

*x x j*

 . 

*x xj*

*y y j*

, , sin . 2

*f*

2 ,

*C xyt h*

*y*

2

Similarly,

*C xyt <sup>h</sup> <sup>f</sup> x*

<sup>2</sup> <sup>2</sup> , , , ,

*x j x j x j x j x*

<sup>2</sup> <sup>2</sup> , , ,

*x j x j x j x*

2

cos cos

,

*y j y*

*x j*

, ,

*x x*

*x j x j x*

 

  

, , 2 2

cos cos cos

cos cos cos , , cos <sup>2</sup> cos 2

*<sup>f</sup> <sup>h</sup> <sup>f</sup> <sup>f</sup> C xyt f <sup>f</sup>*

, 2

cos , , <sup>2</sup>

*<sup>f</sup> <sup>h</sup> <sup>f</sup> S Cf C x y t*

*x x j*

sin sin , , sin , , sin

, , , 2 ,

*y y y y*

*y j y j y j <sup>x</sup>*

, , cos . 2

2 ,

*C xyt <sup>h</sup> <sup>f</sup>*

*<sup>f</sup> <sup>h</sup> <sup>f</sup> C xyt <sup>f</sup> C xyt f h*

*x*

,

*C xyt <sup>f</sup> f h x x*

, , cos cos

*x j x*

*x j x*

cos cos cos cos , , cos cos

The Stochastic Solute Transport Model in 2-Dimensions 201

*<sup>f</sup> <sup>h</sup> <sup>f</sup> <sup>f</sup> <sup>f</sup> C xyt f <sup>f</sup>*

, , ,

*C xyt <sup>f</sup> <sup>h</sup> <sup>f</sup> <sup>f</sup> <sup>f</sup> <sup>f</sup> <sup>f</sup> x x x x x x*

*x j x j x j*

, , cos cos

2

, ,

*x j x j*

, , cos cos 2 2 cos 2

*C xyt <sup>h</sup> <sup>f</sup> <sup>f</sup> <sup>f</sup> x x x*

*x j x j*

2

2

2

*x j x j*

 2

, , ,

*x x x xx x xx*

, , 2 2

*x x x xx x*

,

*x j x j x j x*

,

*x xj*

*S Cf*

We make an assumption that *q* is defined by

$$\cos \theta = \frac{\mathbf{x}}{\sqrt{\left|\mathbf{x}\right|^2 + \left|y\right|^2}}; \quad \sin \theta = \frac{y}{\sqrt{\left|\mathbf{x}\right|^2 + \left|y\right|^2}}. \text{ This is a simplifiedying approximation which makes } \theta = \frac{\mathbf{x}}{\sqrt{\left|\mathbf{x}\right|^2 + \left|y\right|^2}}$$

the modelling more tractable; as the noise term is quite random, this approximation does not make significant difference to final results.

Then

$$d\mathbf{C} = \left( S\_x \left( \overline{V}\_x \mathbf{C} \right) + S\_y \left( \overline{V}\_y \mathbf{C} \right) \right) dt + S\_x \left( \mathbf{C} \left( \mathbf{x}, y, t \right) d\boldsymbol{\beta} \cos \theta \right) + S\_y \left( \mathbf{C} \left( \mathbf{x}, y, t \right) d\boldsymbol{\beta} \sin \theta \right). \tag{7.3.17}$$

Analogous to equation (4.2.4),

$$\begin{split} S\_{\boldsymbol{x}} \left( \mathbb{C} \left( \mathbf{x}, \boldsymbol{y}, t \right) d\boldsymbol{\beta} \cos \theta \right) &= S\_{\boldsymbol{x}} \left( \mathbb{C} \left( \mathbf{x}, \boldsymbol{y}, t \right) \left\{ \sigma \sum\_{j=1}^{m} \sqrt{\lambda\_{\boldsymbol{x},j} \lambda\_{\boldsymbol{y},j}} f\_{\boldsymbol{x},j} f\_{\boldsymbol{y},j} db\_{j} \left( t \right) \right\} \cos \theta \right) . \\ &- S\_{\boldsymbol{x}} \left( \mathbb{C} d\boldsymbol{\beta} \cos \theta \right) = -\sigma \sum\_{j=1}^{m} \sqrt{\lambda\_{\boldsymbol{x},j} \lambda\_{\boldsymbol{y},j}} f\_{\boldsymbol{y},j} S\_{\boldsymbol{x}} \left( \mathbb{C} f\_{\boldsymbol{x},j} \cos \theta \right) db\_{j} \left( t \right) \\ &= \sigma \sum\_{j=1}^{m} \sqrt{\lambda\_{\boldsymbol{x},j} \lambda\_{\boldsymbol{y},j}} f\_{\boldsymbol{y},j} \left\{ -S\_{\boldsymbol{x}} \left( \mathbb{C} f\_{\boldsymbol{x},j} \cos \theta \right) \right\} db\_{j} \left( t \right) \end{split} . \tag{7.3.18}$$

Now we can expand the terms in the brackets in equation (7.3.18),

$$-S\_{\chi} \left( \mathbf{C} f\_{\chi, f} \cos \theta \right) = \left( \frac{\partial}{\partial \chi} + \frac{h\_{\chi}}{2} \frac{\partial^2}{\partial \chi^2} \right) \left( \mathbf{C} f\_{\chi, f} \cos \theta \right) \dots$$

We see that,

$$\frac{\partial}{\partial \mathbf{x}} \left( \mathbb{C} f\_{\mathbf{x}, \boldsymbol{\zeta}} \cos \theta \right) = \mathbb{C} f\_{\boldsymbol{\chi}, \boldsymbol{\zeta}} \frac{\partial \cos \theta}{\partial \mathbf{x}} + \mathbb{C} \cos \theta \frac{\partial f\_{\boldsymbol{\chi}, \boldsymbol{\zeta}}}{\partial \mathbf{x}} + f\_{\boldsymbol{\chi}, \boldsymbol{\zeta}} \cos \theta \frac{\partial \mathbb{C}}{\partial \mathbf{x}} \text{, and } \mathbf{x}$$

$$\begin{split} \frac{\hat{\mathcal{O}}^{2}}{\partial\mathbf{x}^{2}} \Big( \mathbb{C}f\_{\boldsymbol{x},j}\cos\theta \Big) &= \Big( \mathbb{C}f\_{\boldsymbol{x},j}^{\prime}\frac{\hat{\mathcal{O}}^{2}\cos\theta}{\partial\mathbf{x}^{2}} + \mathbb{C}\frac{\partial\cos\theta}{\partial\mathbf{x}}\frac{\partial\mathbf{f}^{\prime}\_{\boldsymbol{x},j}}{\partial\mathbf{x}} + f\_{\boldsymbol{x},j}\frac{\partial\cos\theta}{\partial\mathbf{x}}\frac{\partial\mathbf{C}}{\partial\mathbf{x}} \Big) \\ &+ \Big( \mathbb{C}\cos\theta\frac{\partial^{2}f\_{\boldsymbol{x},j}}{\partial\mathbf{x}^{2}} + \mathbb{C}\frac{\partial\cos\theta}{\partial\mathbf{x}}\frac{\partial f\_{\boldsymbol{x},j}}{\partial\mathbf{x}} + \frac{\partial f\_{\boldsymbol{x},j}}{\partial\mathbf{x}}\cos\theta\frac{\partial\mathbf{C}}{\partial\mathbf{x}} \Big) \\ &+ \Big( f\_{\boldsymbol{x},j}\cos\theta\frac{\partial^{2}\mathbf{C}}{\partial\mathbf{x}^{2}} + f\_{\boldsymbol{x},j}\frac{\partial\cos\theta}{\partial\mathbf{x}}\frac{\partial\mathbf{C}}{\partial\mathbf{x}} + \cos\theta\frac{\partial f\_{\boldsymbol{x},j}}{\partial\mathbf{x}}\frac{\partial\mathbf{C}}{\partial\mathbf{x}} \Big). \end{split}$$

Now,

Computational Modelling of Multi-Scale Non-Fickian Dispersion

. This is a simplifying approximation which makes

*<sup>y</sup>* , , sin

*f f db t*

 

. (7.3.18)

. (7.3.17)

200 in Porous Media - An Approach Based on Stochastic Calculus

the modelling more tractable; as the noise term is quite random, this approximation does

1 , , cos , , cos *m*

*xj yj yj x xj j*

 

.

*f S Cf db t*

cos

.

*x j*

*j*

,,, ,

,,, ,

 <sup>2</sup> , <sup>2</sup> , cos cos 2 *x*

*x x*

, , ,

*x j xj xj*

*<sup>f</sup> f f <sup>C</sup> <sup>C</sup> <sup>C</sup>*

 

> cos cos cos

 

*<sup>C</sup> <sup>C</sup> <sup>f</sup> <sup>C</sup> <sup>f</sup> <sup>f</sup>*

cos cos cos

cos cos cos

 

*x j*

<sup>2</sup> ,

*x xx x x*

*x x x xx*

*x xj x j <sup>h</sup> S Cf Cf*

, , , cos

 , and

2

*x j x j x j*

*<sup>f</sup> <sup>C</sup> Cf Cf <sup>C</sup> <sup>f</sup> x x x x xx*

, 2 ,

*x j x j*

cos cos cos *x j x j x j x j <sup>f</sup> <sup>C</sup> Cf Cf C <sup>f</sup> x x x x* 

*dC S V C S V C dt S C x y t d S C x y t d xx yy <sup>x</sup>* , , cos

,,,,

cos cos

*S Cd f S Cf db t*

*x xj yj yj x xj j*

 

*x x xj yj xj yj j*

1

 

,

 <sup>2</sup> <sup>2</sup> , 2 , , 2 , 2

*j m*

*m*

1

*j*

Now we can expand the terms in the brackets in equation (7.3.18),

cos

 

We make an assumption that *q* is defined by

*x y x y*

not make significant difference to final results.

 

*S C xyt d S C xyt* 

 

<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> cos ; sin *<sup>x</sup> <sup>y</sup>*

Analogous to equation (4.2.4),

Then

We see that,

$$\begin{split} & -S\_{x} \{ \mathbf{C}\_{x,j}^{\prime} \cos \theta \} \\ &= \mathbb{C} \{ \mathbf{x}, y, t \} \Big[ f\_{x,j} \frac{\partial \cos \theta}{\partial \mathbf{x}} + \cos \theta \frac{\partial f\_{x,j}}{\partial \mathbf{x}} + \frac{h\_{x}}{2} \Big[ f\_{x,j} \frac{\partial^{2} \cos \theta}{\partial \mathbf{x}^{2}} + \frac{\partial \cos \theta}{\partial \mathbf{x}} \frac{\partial f\_{x,j}}{\partial \mathbf{x}} + \cos \theta \frac{\partial^{2} f\_{x,j}}{\partial \mathbf{x}^{2}} + \frac{\partial \cos \theta}{\partial \mathbf{x}} \frac{\partial f\_{x,j}}{\partial \mathbf{x}} \Big] \Big] \\ &+ \frac{\partial \mathbf{C} \{ \mathbf{x}, y, t \} }{\partial \mathbf{x}} \Big[ f\_{x,j} \cos \theta \frac{\partial f\_{x,j}}{\partial \mathbf{x}} + \frac{h\_{x}}{2} \Big( f\_{x,j} \frac{\partial \cos \theta}{\partial \mathbf{x}} + \frac{\partial f\_{x,j}}{\partial \mathbf{x}} \cos \theta + f\_{x,j} \frac{\partial \cos \theta}{\partial \mathbf{x}} + \cos \theta \frac{\partial f\_{x,j}}{\partial \mathbf{x}} \Big) \Big] \\ &+ \frac{\partial^{2} \mathbf{C} \{ \mathbf{x}, y, t \} }{\partial \mathbf{x}} \Big[ h\_{x} \Big( f\_{x,j} \cos \theta \Big) \Big] \end{split}$$

Then,

$$\begin{split} & -S\_x \left( \mathcal{G}f\_{x,j} \cos \theta \right) \\ &= \mathcal{C} \left( x, y, t \right) \left| \left( f\_{x,j} \frac{\partial \cos \theta}{\partial x} + \cos \theta \frac{\partial f\_{x,j}}{\partial x} \right) + \frac{h\_x}{2} \left( f\_{x,j} \frac{\partial^2 \cos \theta}{\partial x^2} + 2 \frac{\partial \cos \theta}{\partial x} \frac{\partial f\_{x,j}}{\partial x} + \cos \theta \frac{\partial^2 f\_{x,j}}{\partial x^2} \right) \right|^2 \\ &+ \frac{\partial \mathcal{C} \left( x, y, t \right)}{\partial x} \left| f\_{x,j} \cos \theta + \frac{h\_x}{2} \left( 2 f\_{x,j} \frac{\partial \cos \theta}{\partial x} + 2 \cos \theta \frac{\partial f\_{x,j}}{\partial x} \right) \right| \\ &+ \frac{\partial^2 \mathcal{C} \left( x, y, t \right)}{\partial x^2} \left| \frac{h\_x}{2} f\_{x,j} \cos \theta \right|. \end{split}$$

Simplifying, we obtain,

$$\begin{split}-S\_{x}\left(\mathbb{C}f\_{\boldsymbol{x},\boldsymbol{j}}\cos\theta\right) &= \mathbb{C}\left(\mathbf{x},\boldsymbol{y},\boldsymbol{t}\right)\bigg|\frac{\partial\Big(f\_{\boldsymbol{x},\boldsymbol{j}}\cos\theta\Big)}{\partial\mathbf{x}} + \frac{\mathbf{h}\_{\boldsymbol{x}}}{2}\frac{\partial^{2}\Big(f\_{\boldsymbol{x},\boldsymbol{j}}\cos\theta\Big)}{\partial\mathbf{x}^{2}}\bigg|\frac{\partial\Big(f\_{\boldsymbol{x},\boldsymbol{j}}\cos\theta\Big)}{\partial\mathbf{x}} \\ &+\frac{\partial\Big(\mathbf{C}\left(\mathbf{x},\boldsymbol{y},\boldsymbol{t}\right)}{\partial\mathbf{x}}\bigg|f\_{\boldsymbol{x},\boldsymbol{j}}\cos\theta + \mathbf{h}\_{\boldsymbol{x}}\frac{\partial\Big(f\_{\boldsymbol{x},\boldsymbol{j}}\cos\theta\Big)}{\partial\mathbf{x}}\bigg) \\ &+\frac{\partial^{2}\mathbf{C}\left(\mathbf{x},\boldsymbol{y},\boldsymbol{t}\right)}{\partial\mathbf{x}^{2}}\bigg|\frac{\mathbf{h}\_{\boldsymbol{x}}}{2}\bigg(f\_{\boldsymbol{x},\boldsymbol{j}}\cos\theta\Big)\bigg|.\end{split}$$

Similarly,

$$\begin{split} &\mathcal{S}\_{y}\left(\mathsf{C}f\_{y,j}\cos\theta\right) \\ &=\mathsf{C}\left(\mathbf{x},y,t\right)\left|\frac{\partial\left(f\_{y,j}\sin\theta\right)}{\partial y} + \frac{h\_{x}}{2}\frac{\partial^{2}\left(f\_{y,j}\sin\theta\right)}{\partial y^{2}}\right| + \frac{\partial\mathsf{C}\left(\mathbf{x},y,t\right)}{\partial y}\left|f\_{y,j}\sin\theta + h\_{y}\frac{\partial\left(f\_{y,j}\sin\theta\right)}{\partial y}\right| \\ &+\frac{\partial^{2}\mathsf{C}\left(\mathbf{x},y,t\right)}{\partial y^{2}}\left|\frac{h\_{y}}{2}\left(f\_{y,j}\sin\theta\right)\right|. \end{split}$$

1 , ,1

*x xx xj yj j j j*

 

*dI dt Q db t*

*<sup>V</sup> dI V h dt P db t*

The Stochastic Solute Transport Model in 2-Dimensions 203

*m*

 2

1 , ,1

*V*

*y*

<sup>2</sup> , ,2 <sup>2</sup> <sup>1</sup> *<sup>m</sup> <sup>y</sup>*

*y y xj yj j j j*

Equations (7.3.25) - (7.3.31) constitute the SSTM2d with the definitions for *P* s and *Q* s given by equations (7.3.19) to (7.3.24). The SSTM2d has similar Ito diffusions for velocities as in the one dimensional case. Equation (7.3.19) shows an elegant extension of SSTM into 2 dimensions. It should be noted that the eigen values for both directions are the same for the

The development of the SSTM2d is based on the fact that any kernel can be expressed as a multiplication of two kernels, for example, as in equation (7.2.2); and we know the methodology of obtaining the eigen values and eigen functions for any kernel. Therefore, we can solve the SSTM2d for any kernel. However, for the illustrative purposes, we only focus

To understand the behaviour of the SSTM2d, we need to solve the equations numerically by using a finite difference scheme developed for the purpose. We only highlight the pertinent

*n t*

[ , ] [ , ] [ , ] 0, 1, <sup>2</sup> 2, [ , ] 0, 1, <sup>2</sup> 2, *<sup>n</sup> i j i i i j j j*

We use the forward difference to calculate the first first-order derivatives with respect to time (*t*), the backward difference to calculate the first-order derivative in x and *y* directions

*t n n i j i j n i j i j x y i j i j x x x i j y y y*

and the central difference to calculate the second-order derivatives, i.e.,

*dc dc c dI dI dI c dI dI dI*

*i j*

*dx dx dy dy* .

2 2 [,] [,] [,] [,]

*n n n n*

*dc d c dc d c*

*i i j j*

*<sup>i</sup> <sup>j</sup> <sup>n</sup> <sup>i</sup> <sup>j</sup> <sup>x</sup> <sup>y</sup> x i xy j yt n t C C* , Equation (7.3.25) can be redisplayed as ,

<sup>0</sup> <sup>2</sup> , ,0 2 <sup>1</sup> *<sup>m</sup> yy y y xj yj j j*

*<sup>m</sup> <sup>y</sup> y yy xj yj j j j*

*dI V h dt Q db t*

 

*x x xj yj j j j*

*x*

*x*

<sup>2</sup> , ,2 <sup>2</sup> <sup>1</sup>

*VhV*

*y y*

*x*

*h dI V dt*

[0,1] domain, further simplifying the equations.

on the kernel given in equation (7.2.2) in this chapter.

**7.3.1A Summary of the Finite Difference Scheme** 

Now let [,] [ , ] , , , and *<sup>n</sup>*

equations in the algorithm.

*<sup>h</sup> dI V dt*

and

1 *m*

*j*

 

1

 

*P db t*

 

*Q db t*

, (7.3.27)

, (7.3.28)

, (7.3.29)

, (7.3.30)

. (7.3.31)

$$\begin{split} & \quad \vdots \quad S\_{x} \{ \text{Cd} \boldsymbol{\beta} \cos \theta \} \\ &= -\sigma \sum\_{j=1}^{n} \sqrt{\mathcal{k}\_{x,j} \mathcal{k}\_{y,j}} \left| \left\{ P\_{\text{0\\_x},y\_{r}} \text{C} \{ \mathbf{x},y\_{r}t \} + P\_{\text{1\\_x},y\_{r},j} \frac{\text{\mathcal{C}C} \{ \mathbf{x},y\_{r}t \}}{\text{\mathcal{C}x}} + P\_{\text{2\\_x},y\_{r},j} \frac{\text{\mathcal{C}^2C} \{ \mathbf{x},y\_{r}t \}}{\text{\mathcal{C}x}^2} \right\} \right| \end{split} $$

Where

$$\begin{array}{ll} \{\} \\\\ P\_{0,j} = P\_{0,x,y,j} = f\_{y,j} \left[ \frac{\partial \left( f\_{x,j} \cos \theta \right)}{\partial \mathbf{x}} + \frac{h\_x}{2} \frac{\partial^2 \left( f\_{x,j} \cos \theta \right)}{\partial \mathbf{x}^2} \right]; \end{array} \tag{7.3.19}$$
 
$$P\_{1,j} = P\_{1,x,y,j} = f\_{y,j} \left[ f\_{x,j} \cos \theta + h\_x \frac{\partial \left( f\_{x,j} \cos \theta \right)}{\partial \mathbf{x}} \right]; \text{ and} \tag{7.3.20}$$
 
$$P\_{2,j} = P\_{2,x,y,j} = f\_{y,j} \left( \frac{h\_x}{2} \{f\_{x,j} \cos \theta \} \right). \tag{7.3.21}$$
 
$$\begin{array}{ll} \hline \end{array} \tag{7.3.21}$$

$$P\_{1,j} = P\_{1, \mathbf{x}, y, j} = f\_{y, j} \left( f\_{\mathbf{x}, j} \cos \theta + h\_{\mathbf{x}} \frac{\partial \left( f\_{\mathbf{x}, j} \cos \theta \right)}{\partial \mathbf{x}} \right); \text{ and} \tag{7.3.20}$$

$$P\_{2,j} = P\_{2,x,y,j} = f\_{y,j} \left(\frac{h\_x}{2} (f\_{x,j} \cos \theta) \right). \tag{7.3.21}$$

Similarly,

 <sup>2</sup> , , 0, 1, 2, 2 1 , , , , sin , , *m y xj yj j j j j C xyt C xyt S Cd Q C xyt Q <sup>Q</sup> y y* , <sup>2</sup> , , 0, , 2 sin sin ; <sup>2</sup> *y j y y j j xj <sup>f</sup> <sup>h</sup> <sup>f</sup> Q f y y* (7.3.22) 

$$Q\_{1,j} = f\_{x,j} \left( f\_{y,j} \sin \theta + h\_y \frac{\partial \left( f\_{y,j} \sin \theta \right)}{\partial y} \right); \text{ and} \tag{7.3.23}$$

$$Q\_{2,j} = f\_{x,j} \left( \frac{h\_y}{2} (f\_{y,j} \sin \theta) \right). \tag{7.3.24}$$

$$\begin{split} & \quad \text{\dots C.} \left( \text{Cd} \beta\_{x} \right) + \text{S}\_{y} \left( \text{Cd} \beta\_{y} \right) \\ &= -\sigma \sum\_{j=1}^{n} \sqrt{\lambda\_{x,j} \lambda\_{y,j}} \left\{ \left( P\_{0j} + Q\_{0,j} \right) \text{C} \left( \text{x}, t \right) + \left( P\_{1,j} \frac{\partial \text{C}}{\partial \text{x}} + Q\_{1,j} \frac{\partial \text{C}}{\partial y} \right) + \left( P\_{2,j} \frac{\partial^{2} \text{C}}{\partial \text{x}^{2}} + Q\_{2,j} \frac{\partial^{2} \text{C}}{\partial y^{2}} \right) \right\} db\_{\uparrow}(t) \end{split}$$

Therefore,

$$\begin{split} d\mathbf{C} &= -\mathbf{C} \left( \mathbf{x}, \mathbf{y}, t \right) d\mathbf{I}\_{0\mathbf{x}} - \frac{\partial \mathbf{C}}{\partial \mathbf{x}} d\mathbf{I}\_{1\mathbf{x}} - \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} d\mathbf{I}\_{2\mathbf{x}} \\ &- \mathbf{C} \left( \mathbf{x}, \mathbf{y}, t \right) d\mathbf{I}\_{0\mathbf{y}} - \frac{\partial \mathbf{C}}{\partial \mathbf{y}} d\mathbf{I}\_{1\mathbf{y}} - \frac{\partial^2 \mathbf{C}}{\partial \mathbf{y}^2} d\mathbf{I}\_{2\mathbf{y}} \end{split} \tag{7.3.25}$$

$$\text{where}\quad dl\_{0x} = \left(\frac{\partial \overline{V}\_x}{\partial \mathbf{x}} + \frac{h\_x}{2} \frac{\partial^2 \overline{V}\_x}{\partial \mathbf{x}^2}\right) dt + \sigma \sum\_{j=1}^m \sqrt{\lambda\_{x,j} \lambda\_{y,j}} P\_{0j} db\_j(t) \,\,\,\,\tag{7.3.26}$$

$$dI\_{1x} = \left(\overline{V}\_x + h\_x \frac{\partial \overline{V}\_x}{\partial \mathbf{x}}\right) dt + \sigma \sum\_{j=1}^m \sqrt{\lambda\_{x,j} \lambda\_{y,j}} P\_{1j} db\_j(t),\tag{7.3.27}$$

$$dI\_{2x} = \left(\frac{h\_x}{2}\overline{V}\_x\right)dt + \sigma\sum\_{j=1}^m \sqrt{\lambda\_{x,j}\lambda\_{y,j}}P\_{2j}db\_j\left(t\right),\tag{7.3.28}$$

$$dI\_{0y} = \left(\frac{\partial \overline{V}\_y}{\partial y} + \frac{h\_y}{2} \frac{\partial^2 \overline{V}\_y}{\partial y^2}\right) dt + \sigma \sum\_{j=1}^m \sqrt{\lambda\_{x,j} \lambda\_{y,j}} Q\_{0j} db\_j(t) \,\,\,\,\tag{7.3.29}$$

$$dI\_{1y} = \left(\overline{V}\_y + h\_y \frac{\partial \overline{V}\_y}{\partial y}\right) dt + \sigma \sum\_{j=1}^m \sqrt{\lambda\_{\mathbf{x},j} \lambda\_{\mathbf{y},j}} Q\_1 d b\_j(t) \,,\tag{7.3.30}$$

and

Computational Modelling of Multi-Scale Non-Fickian Dispersion

*x x*

(7.3.19)

and (7.3.20)

(7.3.21)

(7.3.22)

and (7.3.23)

, (7.3.25)

(7.3.24)

*y y*

<sup>2</sup>

*C xyt C xyt P C xyt P <sup>P</sup>*

 <sup>2</sup> , ,

*x x*

cos ; *x j*

*x j x j x*

cos cos

,

*x*

, , 0, 1, 2, 2

,

*y*

<sup>2</sup> <sup>2</sup>

*CC C C P Q C x t P Q P Q db t*

2

2

 

, (7.3.26)

sin

; <sup>2</sup>

*xy x y*

,

 <sup>2</sup> , ,

*y y*

sin ; *y j*

*y j y y j*

 

sin sin

cos

, , , , , ,

; <sup>2</sup>

202 in Porous Media - An Approach Based on Stochastic Calculus

*xj yj xyj xyj xyj*

0, 0, , , , 2

*<sup>f</sup> <sup>h</sup> <sup>f</sup> PP f*

*j xyj yj xj x <sup>f</sup> PP ff h*

*j xyj yj*

1, 1, , , , ,

1

 

*j xj*

1, , ,

*j xj yj y <sup>f</sup> Q ff h*

,

, ,

, ,

*xx x*

*x x*

 

*j*

  *m*

, , 0 ,, 1 ,, 2 ,, 2

,

2, 2, , , , , cos . 2 *x*

<sup>2</sup>

*C xyt C xyt S Cd Q C xyt Q <sup>Q</sup>*

0, , 2

2, , , sin . 2 *y j xj yj h Qf f*

*C C dC C x y t dI dI dI*

where <sup>2</sup> <sup>0</sup> <sup>2</sup> , ,0 2 <sup>1</sup>

*C C C x y t dI dI dI*

*x xj yj j j*

*VhV dI dt P db t*

, , 0 0, 1, 1, 2, 2 2, 2

*xj yj j j j j j j j*

0 1 2 2

*x x x*

*x x*

0 1 2 2

*y y y*

*m*

*j*

*y y*

, , , , sin , ,

*j xyj yj xj <sup>h</sup> PP f f*

*y xj yj j j j*

*<sup>f</sup> <sup>h</sup> <sup>f</sup> Q f*

*x*

Where

Similarly,

*S Cd*

cos

 

1

*j*

 

 

*x xy y*

 

*S Cd S Cd*

1

*j*

Therefore,

*m*

*m*

 

$$dI\_{2y} = \left(\frac{h\_y}{2}\overline{V}\_y\right)dt + \sigma\sum\_{j=1}^{m}\sqrt{\mathcal{k}\_{x,j}\mathcal{k}\_{y,j}}Q\_{2j}db\_j\left(t\right). \tag{7.3.31}$$

Equations (7.3.25) - (7.3.31) constitute the SSTM2d with the definitions for *P* s and *Q* s given by equations (7.3.19) to (7.3.24). The SSTM2d has similar Ito diffusions for velocities as in the one dimensional case. Equation (7.3.19) shows an elegant extension of SSTM into 2 dimensions. It should be noted that the eigen values for both directions are the same for the [0,1] domain, further simplifying the equations.

The development of the SSTM2d is based on the fact that any kernel can be expressed as a multiplication of two kernels, for example, as in equation (7.2.2); and we know the methodology of obtaining the eigen values and eigen functions for any kernel. Therefore, we can solve the SSTM2d for any kernel. However, for the illustrative purposes, we only focus on the kernel given in equation (7.2.2) in this chapter.

### **7.3.1A Summary of the Finite Difference Scheme**

To understand the behaviour of the SSTM2d, we need to solve the equations numerically by using a finite difference scheme developed for the purpose. We only highlight the pertinent equations in the algorithm. developed

Now let [,] [ , ] , , , and *<sup>n</sup> i j n t <sup>i</sup> <sup>j</sup> <sup>n</sup> <sup>i</sup> <sup>j</sup> <sup>x</sup> <sup>y</sup> x i xy j yt n t C C* , Equation (7.3.25) can be redisplayed as ,

2 2 [,] [,] [,] [,] [ , ] [ , ] [ , ] 0, 1, <sup>2</sup> 2, [ , ] 0, 1, <sup>2</sup> 2, *<sup>n</sup> i j i i i j j j n n n n t n n i j i j n i j i j x y i j i j x x x i j y y y i i j j dc d c dc d c dc dc c dI dI dI c dI dI dI dx dx dy dy* .

We use the forward difference to calculate the first first-order derivatives with respect to time (*t*), the backward difference to calculate the first-order derivative in x and *y* directions and the central difference to calculate the second-order derivatives, i.e.,

Figure 7.3. A realisation of concentration at y=0.5 m when <sup>2</sup>

Figure 7.4. A realisation of concentration at y=0.5 m when <sup>2</sup>

=0.001.

The Stochastic Solute Transport Model in 2-Dimensions 205

=0.01.

$$\begin{split} \frac{d\boldsymbol{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{u}}}{dt} &= \frac{\mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}+1} - \mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}}}{\Delta t}, \quad \frac{d\boldsymbol{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}}}{d\mathbf{x}} = \frac{\mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}} - \mathbf{c}\_{[\boldsymbol{l}-1,\boldsymbol{\cdot},]}^{\boldsymbol{n}}}{\Delta \mathbf{x}}, \quad \frac{d\boldsymbol{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}}}{d\boldsymbol{y}} = \frac{\mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}} - \mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}}}{\Delta \mathbf{y}}, \\\ \frac{d^{2}\mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}}}{d\mathbf{x}^{2}} &= \frac{\mathbf{c}\_{[\boldsymbol{l}+1,\boldsymbol{\cdot}]}^{\boldsymbol{n}} - 2\mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}} + \mathbf{c}\_{[\boldsymbol{l}-1,\boldsymbol{\cdot}]}^{\boldsymbol{n}}}{\Delta \mathbf{x}^{2}}, \quad \frac{d^{2}\mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}}}{d\mathbf{x}^{2}} = \frac{\mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}} - 2\mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}} + \mathbf{c}\_{[\boldsymbol{l},\boldsymbol{\cdot}]}^{\boldsymbol{n}}}{\Delta \mathbf{y}^{2}}. \end{split} \tag{7.3.32}$$

We can develop the finite difference scheme to solve the SSTM2d based on the following equation:

$$\begin{split} c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u+1} - c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} &= -c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} d\boldsymbol{I}\_{0,\boldsymbol{x}\_{\boldsymbol{j}}} - \left(\frac{c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} - c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u}}{\Delta\mathbf{x}}\right) d\boldsymbol{I}\_{1,\boldsymbol{x}\_{\boldsymbol{j}}} - \left(\frac{c\_{[l+1,\boldsymbol{j}]}^{u} - 2c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} + c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u}}{\Delta\mathbf{x}^{2}}\right) d\boldsymbol{I}\_{2,\boldsymbol{x}\_{\boldsymbol{j}}} \\ &- c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} d\boldsymbol{I}\_{0,\boldsymbol{y}\_{\boldsymbol{j}}} - \left(\frac{c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} - c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} - 1}{\Delta\mathbf{y}}\right) d\boldsymbol{I}\_{1,\boldsymbol{y}\_{\boldsymbol{j}}} - \left(\frac{c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} - 2c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u} + c\_{[l\_{\boldsymbol{l}},\boldsymbol{j}]}^{u}}{\Delta\mathbf{y}^{2}}\right) d\boldsymbol{I}\_{2,\boldsymbol{y}\_{\boldsymbol{j}}} \end{split} .\tag{7.3.33}$$

We illustrate some realisations of the solutions graphically in the next section.

### **7.3.2Graphical Depictions of Realisations**

In the following figures, we present a sample of solution realisations of the SSTM2d to illustrate the behaviours of the model under different parameter values for the boundary condition: *C*(*t*, *x*, *y*)=1.0 at ( *x*=0.0 and *y*=0.0) for any given *t*. The value of *b* is kept at 0.1for all computations.

Figure 7.2. A realisation of concentration at y=0.5 m when <sup>2</sup> =0.0001.

(7.3.32)

. (7.3.33)

2

2

=0.0001.

204 in Porous Media - An Approach Based on Stochastic Calculus

[ , ] [ , ] [ , ] [ , ] [ , ] [ 1, ] [ , ] [ , ] [ , 1]

*n nn n nn n nn ij ij ij ij ij i j ij ij ij*

*dc c c dc c c dc c c dt t dx x dy y dc c c c dc c c c dx x dx y*

*n n n n n n n n ij i j ij i j ij ij ij ij*

1 [ , ] [ 1, ] [ 1, ] [ , ] [ 1, ] [ , ] [ , ] [ , ] 0, 1, 2 2,

*c c c dI dI dI*

*n n n ij i j i j ij i j i j i j i j x x x*

[ , ] 0, 1, 2

*n ij ij i j ij ij*

In the following figures, we present a sample of solution realisations of the SSTM2d to illustrate the behaviours of the model under different parameter values for the boundary condition: *C*(*t*, *x*, *y*)=1.0 at ( *x*=0.0 and *y*=0.0) for any given *t*. The value of *b* is kept at 0.1for

*j j*

We illustrate some realisations of the solutions graphically in the next section.

*ij y y*

*c dI dI*

Figure 7.2. A realisation of concentration at y=0.5 m when <sup>2</sup>

, , ,

[ , ] [ , 1] [ , 1] [ , ] [ , 1]

*i i i*

*n n n n n*

*c c c cc*

*n n n n n*

*y y*

2, *<sup>j</sup> <sup>y</sup> dI*

*x x c c c cc*

<sup>2</sup> <sup>2</sup> , .

[ , ] [ 1, ] [ , ] [ 1, ] [ , ] [ , 1] [ , ] [ , 1] 2 2 2 2

We can develop the finite difference scheme to solve the SSTM2d based on the following

1

**7.3.2Graphical Depictions of Realisations** 

equation:

all computations.

2 2

Figure 7.3. A realisation of concentration at y=0.5 m when <sup>2</sup> =0.001.

Figure 7.4. A realisation of concentration at y=0.5 m when <sup>2</sup> =0.01.

Figure 7.7. A realisation of concentration at *t*=1 day when <sup>2</sup>

Figure 7.8. A realisation of concentration at *t*=1 day when <sup>2</sup>

=0.001.

The Stochastic Solute Transport Model in 2-Dimensions 207

=0.01.

Figure 7.5. A realisation of concentration at y=0.5 m when <sup>2</sup> =0.1

Figure 7.6. A realisation of concentration at *t*=1 day when <sup>2</sup> =0.0001.

=0.1

=0.0001.

206 in Porous Media - An Approach Based on Stochastic Calculus

Figure 7.5. A realisation of concentration at y=0.5 m when <sup>2</sup>

Figure 7.6. A realisation of concentration at *t*=1 day when <sup>2</sup>

Figure 7.7. A realisation of concentration at *t*=1 day when <sup>2</sup> =0.001.

Figure 7.8. A realisation of concentration at *t*=1 day when <sup>2</sup> =0.01.

Figure 7.11. A realisation of concentration at *t*=3 days when <sup>2</sup>

The Stochastic Solute Transport Model in 2-Dimensions 209

Figure 7.12. A realisation of concentration at *t*=3 days when <sup>2</sup>

=0.001.

=0.01.

Figure 7.9. A realisation of concentration at *t*=1 day when <sup>2</sup> =0.1.

Figure 7.10. A realisation of concentration at *t*=3 days when <sup>2</sup> =0.0001.

=0.1.

=0.0001.

208 in Porous Media - An Approach Based on Stochastic Calculus

Figure 7.9. A realisation of concentration at *t*=1 day when <sup>2</sup>

Figure 7.10. A realisation of concentration at *t*=3 days when <sup>2</sup>

Figure 7.11. A realisation of concentration at *t*=3 days when <sup>2</sup> =0.001.

Figure 7.12. A realisation of concentration at *t*=3 days when <sup>2</sup> =0.01.

The randomness of heterogeneous groundwater systems can be accounted for by adding a

The Stochastic Solute Transport Model in 2-Dimensions 211

<sup>2</sup> <sup>2</sup> ( ,) *<sup>L</sup> <sup>T</sup> <sup>x</sup> C CCC D D v xt*

<sup>2</sup> <sup>2</sup> ( ) *<sup>L</sup> <sup>T</sup> <sup>x</sup> C C <sup>C</sup> dC D D dt v dt t x y x*

The two parameters to be estimated are *DL* and *DT* (while 0.5 *<sup>x</sup> v* in this case). For the two

 

2

*x* ;

> <sup>2</sup> *DT* .

<sup>1</sup> <sup>2</sup> ( ,) *<sup>C</sup> a Ct*

0 1 1 2 2

*i i i i*

*i i i*

*a C t a C t a C t dt*

 

 

 

 

<sup>1</sup> { ( , ) ( , ) ( , )} <sup>2</sup>

If we have values for *Cxyt* ( , ,) at *M* discrete points in (*x*, *y*) coordinate space for a period of

( , ) ( ) { ( , ) ( , ) ( , )} ( , ) 0

( , ) ( ) { ( , ) ( , ) ( , )} ( , ) 0

2

, (7.4.2)

. (7.4.3)

2

*y* ;

. (7.4.5)

1 and <sup>2</sup> ,

. (7.4.6)

( ,) *x t dt* by ζ(*t*). We

, (7.4.4)

<sup>2</sup> <sup>2</sup> ( ,) *<sup>C</sup> a Ct*

2 2

*tx y x*

stochastic component to equation (7.4.1), and it can be given by

( ,) *x t* is described by a zero-mean stochastic process.

We multiple equation (7.4.2) by *dt* throughout and, formally replace

2 2

parameter case, we can write the right hand side of equation (7.4.3) as follows:

<sup>1</sup> *DL* ; and

1 2 0 1 1 2 2

( , ) { ( , ) ( , ) ( , )} ( )

*l a C t a C t a C t dC t*

time *t* (where 0 *t T* ), then differentiating equation (7.4.5) with respect to

1 0 1 1 2 2 1

*i i i i i i*

*a C t dC t a C t a C t a C t a C t dt*

*i i i i i i*

*a C t dC t a C t a C t a C t a C t dt*

2 0 1 1 2 2 2

 

12 0 1 1 2 2 *f* (, , , ) ( ,) ( ,) ( ,) *tC a Ct a Ct a Ct*

 

can now obtain the stochastic partial differential equation as follows,

 

*x x* ;

The log-likelihood function can be written as (see Chapter 1),

1 0

*i*

*M T*

1 0

*i*

respectively, we get the following two simultaneous equations:

*M T*

<sup>0</sup> ( ,) 0.5 *<sup>x</sup> C C a Ct v*

> 

1 0 1 0

*i i M T M T*

*i i*

*M T M T*

1 0 1 0

where

where,

Figure 7.13. A realisation of concentration at *t*=3 days under <sup>2</sup> =0.1.

The figures above shows that the numerical scheme is robust to obtain the concentration realisations for a range of values of <sup>2</sup> . As <sup>2</sup> increases the stochasticity of the realisations increases.

### **7.4 Longitudinal and Transverse Dispersivity according to SSTM2D**

To estimate the longitudinal and transverse dispersivities, we start with the partial differential equation for advection and dispersion, taking *x* axis to be the direction of the flow.

The two-dimensional advection-dispersion equation can be written as,

$$\frac{\partial \mathbf{C}}{\partial t} = \left\{ D\_{\perp} \left( \frac{\partial^{2} \mathbf{C}}{\partial \mathbf{x}^{2}} \right) + D\_{\Gamma} \left( \frac{\partial^{2} \mathbf{C}}{\partial \mathbf{y}^{2}} \right) \right\} - v\_{\mathbf{x}} \left( \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \right) \tag{7.4.1}$$

where *C* = solution concentration (mg/l),

*t* = time (day),


The randomness of heterogeneous groundwater systems can be accounted for by adding a stochastic component to equation (7.4.1), and it can be given by

$$\frac{\partial \mathbf{C}}{\partial t} = \left\{ D\_{\perp} \left( \frac{\partial^{2} \mathbf{C}}{\partial \mathbf{x}^{2}} \right) + D\_{\mathbf{r}} \left( \frac{\partial^{2} \mathbf{C}}{\partial y^{2}} \right) \right\} - \upsilon\_{\mathbf{x}} \left( \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \right) + \xi(\mathbf{x}, t) \quad \text{,} \tag{7.4.2}$$

where ( ,) *x t* is described by a zero-mean stochastic process.

We multiple equation (7.4.2) by *dt* throughout and, formally replace ( ,) *x t dt* by ζ(*t*). We can now obtain the stochastic partial differential equation as follows,

$$d\mathbf{C} = \left\{ D\_{\perp} \left( \frac{\partial^{2} \mathbf{C}}{\partial \mathbf{x}^{2}} \right) + D\_{r} \left( \frac{\partial^{2} \mathbf{C}}{\partial y^{2}} \right) \right\} dt - v\_{x} \left( \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \right) dt + \boldsymbol{\xi}(\mathbf{t}) \,. \tag{7.4.3}$$

The two parameters to be estimated are *DL* and *DT* (while 0.5 *<sup>x</sup> v* in this case). For the two parameter case, we can write the right hand side of equation (7.4.3) as follows:

$$f(t, \mathcal{C}, \theta\_1, \theta\_2) = a\_0(\mathcal{C}, t) + \theta\_1 a\_1(\mathcal{C}, t) + \theta\_2 a\_2(\mathcal{C}, t) \, , \tag{7.4.4}$$

where,

Computational Modelling of Multi-Scale Non-Fickian Dispersion

=0.1.

increases the stochasticity of the

(7.4.1)

210 in Porous Media - An Approach Based on Stochastic Calculus

The figures above shows that the numerical scheme is robust to obtain the concentration

To estimate the longitudinal and transverse dispersivities, we start with the partial differential equation for advection and dispersion, taking *x* axis to be the direction of the

> 2 2 *L* 2 *T* 2 *x C C CC DD v tx y x*

*DL* = hydrodynamic dispersion coefficient parallel to the principal direction of flow

*DT* = hydrodynamic dispersion coefficient perpendicular to the principal direction of

 . As <sup>2</sup> 

Figure 7.13. A realisation of concentration at *t*=3 days under <sup>2</sup>

**7.4 Longitudinal and Transverse Dispersivity according to SSTM2D** 

The two-dimensional advection-dispersion equation can be written as,

realisations for a range of values of <sup>2</sup>

where *C* = solution concentration (mg/l),

(longitudinal) (m2/day),

*<sup>x</sup> v* = average linear velocity (m/day).

flow (transverse) (m2/day), and

realisations increases.

*t* = time (day),

flow.

<sup>0</sup> ( ,) 0.5 *<sup>x</sup> C C a Ct v x x* ; 2 <sup>1</sup> <sup>2</sup> ( ,) *<sup>C</sup> a Ct x* ; 2 <sup>2</sup> <sup>2</sup> ( ,) *<sup>C</sup> a Ct y* ; <sup>1</sup> *DL* ; and <sup>2</sup> *DT* . 

The log-likelihood function can be written as (see Chapter 1),

$$\begin{split} l(\theta\_1, \theta\_2) &= \sum\_{i=1}^{M} \Big[ a\_0(\mathbf{C}\_i, t) + \theta\_1 a\_1(\mathbf{C}\_i, t) + \theta\_2 a\_2(\mathbf{C}\_i, t) \big] d\mathbf{C}\_i(t) \\ &- \frac{1}{2} \sum\_{i=1}^{M} \Big[ \left\{ a\_0(\mathbf{C}\_i, t) + \theta\_1 a\_1(\mathbf{C}\_i, t) + \theta\_2 a\_2(\mathbf{C}\_i, t) \right\}^2 dt \end{split} \tag{7.4.5}$$

If we have values for *Cxyt* ( , ,) at *M* discrete points in (*x*, *y*) coordinate space for a period of time *t* (where 0 *t T* ), then differentiating equation (7.4.5) with respect to 1 and <sup>2</sup> , respectively, we get the following two simultaneous equations:

$$\begin{aligned} \sum\_{i=1}^{M} \int\_{0}^{T} a\_{1}(\mathbf{C}\_{i},t) d\mathbf{C}\_{i}(t) - \sum\_{i=1}^{M} \int \left\{ a\_{0}(\mathbf{C}\_{i},t) + \theta\_{1} a\_{1}(\mathbf{C}\_{i},t) + \theta\_{2} a\_{2}(\mathbf{C}\_{i},t) \right\} \left\{ a\_{1}(\mathbf{C}\_{i},t) \right\} dt &= 0 \\ \sum\_{i=1}^{M} \int\_{0}^{T} a\_{2}(\mathbf{C}\_{i},t) d\mathbf{C}\_{i}(t) - \sum\_{i=1}^{M} \int \left\{ a\_{0}(\mathbf{C}\_{i},t) + \theta\_{1} a\_{1}(\mathbf{C}\_{i},t) + \theta\_{2} a\_{2}(\mathbf{C}\_{i},t) \right\} \left\{ a\_{2}(\mathbf{C}\_{i},t) \right\} dt &= 0 \end{aligned} \tag{7.4.6}$$

of equations (7.4.9) are,

for each of <sup>2</sup>

the longitudinal direction.

2 

coefficient for the flow length [0,1] when <sup>2</sup>

smaller in 2 dimensions especially when <sup>2</sup>

0.5 of longitudinal dispersion coefficient when <sup>2</sup>

concentration realisations from SSTM2d for each of <sup>2</sup>

<sup>2</sup> <sup>2</sup>

*i*

. (7.4.14)

(7.4.15)

is very small but approaches approximately

>0.01 . This needs to be expected as the lateral

increases (Figure 7.12). Comparing Table

2 2 1 0 *M T*

*d C <sup>l</sup> dt dy*

The Stochastic Solute Transport Model in 2-Dimensions 213

The two simultaneous equations in (7.4.9) can be solved to obtain the estimates of the unknown parameters, *DL* and *DT*, for a two-dimensional groundwater system. The solutions

> *ml ml <sup>D</sup> kl l*

*ml mk <sup>D</sup> l kl*

We have estimated the longitudinal and lateral dispersion coefficients for 100 realisations

The transverse dispersion coefficient is significantly less than the longitudinal dispersion

7.1 with Table 4.9, we see that the dispersion coefficient, therefore, the dispersivity, is

dispersion provides another mechanism of energy dissipation, thwarting the dispersion in

*DL DT*

0.001 0.0251 0.0003 0.005 0.0258 0.0012 0.01 0.0264 0.0017 0.02 0.0273 0.0027 0.04 0.0293 0.0053 0.05 0.0304 0.0072 0.06 0.0314 0.0089 0.08 0.0332 0.012 0.1 0.0354 0.0145 0.15 0.04 0.0197

Table 7.1. Estimated mean longitudinal and transverse dispersion coefficients using 100

0.06 0.0314 0.0089 of

value.

value chosen, and their mean values are given in Table 7.1 .

12 21 2 12 1

,

.

11 2 1 2 1 12

*i t*

and

*T*

*L*

We simplify equation (7.4.6) to

 1 0 1 1 0 1 0 2 1 1 2 1 2 1 0 1 0 2 0 2 1 0 1 0 1 1 2 2 2 ( ,) () ( ,) ( ,) ( ,) ( ,) ( ,) 0 ( ,) () ( ,) ( ,) ( ,) ( ,) ( *M T M T i i i i i i M T M T i i i i i M T M T i i i i i i i i i a C t dC t a C t a C t dt a C t dt a C t a C t dt a C t dC t a C t a C t dt a C t a C t dt a C* <sup>2</sup> 1 0 1 0 ,) 0 *M T M T i i t dt* . (7.4.7) *dy <sup>C</sup>*

Now we substitute <sup>0</sup> ( ,) *<sup>i</sup> aCt* , 1 *a Ct* ( ,) , 2 *a Ct* ( ,), 1 and 2 in equations (7.4.7) to obtain the following set of equations:

$$\begin{aligned} & \left\{ \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbf{C}\_i}{dx^2} \right\} \mathrm{dC}\_i(t) + 0.5 \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d \mathbf{C}\_i}{dx} \right\} \left\{ \frac{d^2 \mathbf{C}\_i}{dx^2} \right\} dt \right\} \\ & - \theta\_1 \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbf{C}\_i}{dx^2} \right\}^2 dt - \theta\_2 \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbf{C}\_i}{dx^2} \right\} \left\{ \frac{d^2 \mathbf{C}\_i}{dy^2} \right\} dt = 0 \\ & \left\{ \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbf{C}\_i}{dy^2} \right\} \mathrm{dC}\_i(t) + 0.5 \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d \mathbf{C}\_i}{dx} \right\} \left\{ \frac{d^2 \mathbf{C}\_i}{dy^2} \right\} dt \right\} \\ & - \theta\_1 \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbf{C}\_i}{dx^2} \right\} \left\{ \frac{d^2 \mathbf{C}\_i}{dy^2} \right\} dt - \theta\_2 \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbf{C}\_i}{dy^2} \right\}^2 dt = 0 \end{aligned} \tag{7.4.8}$$

We can rewrite equations (7.4.8) as,

$$\begin{aligned} m\_1 - D\_1 k\_1 - D\_1 l\_1 &= 0\\ m\_2 - D\_1 k\_2 - D\_1 l\_2 &= 0 \end{aligned} \tag{7.4.9}$$

$$\text{Where} \quad m\_1 = \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbf{C}\_i}{d\mathbf{x}^2} \right\} d\mathbf{C}\_i(\mathbf{t}) + 0.5 \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d\mathbf{C}\_i}{d\mathbf{x}} \right\} \left\{ \frac{d^2 \mathbf{C}\_i}{d\mathbf{x}^2} \right\} dt \,\tag{7.4.10}$$

$$k\_1 = \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbf{C}\_i}{d\mathbf{x}^2} \right\}^2 dt \qquad , \tag{7.4.11}$$

$$l\_1 = k\_2 = \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathcal{C}\_i}{d\mathbf{x}^2} \right\} \left\{ \frac{d^2 \mathcal{C}\_i}{dy^2} \right\} dt \qquad , \tag{7.4.12}$$

$$m\_2 = \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathcal{C}\_i}{dy^2} \right\} d\mathcal{C}\_i(t) + 0.5 \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d\mathcal{C}\_i}{dx} \right\} \left\{ \frac{d^2 \mathcal{C}\_i}{dy^2} \right\} dt \text{ , and} \tag{7.4.13}$$

Computational Modelling of Multi-Scale Non-Fickian Dispersion

212 in Porous Media - An Approach Based on Stochastic Calculus

*i i i*

*a C t dt a C t a C t dt*

( ,) ( ,) ( ,) 0

1 and

<sup>2</sup>

2

*i*

*d C dt dy*

,) 0

0

0

. (7.4.9)

, (7.4.11)

, (7.4.12)

2 in equations (7.4.7) to obtain the

. (7.4.7)

. (7.4.8)

*t dt*

*i i*

1 0 1 0

*i i t i t M T M T*

*M T M T*

1 0 1 0

*i*

*i t i t*

2 2

1 0

*i t*

We can rewrite equations (7.4.8) as,

Where

*M T*

2 1 1 2 1 2

*M T M T*

1 0 1 0

1 0 1

( ,) () ( ,) ( ,)

*a C t dC t a C t a C t dt*

*i i i i*

( ,) () ( ,) ( ,)

*a C t dC t a C t a C t dt*

*i i i i*

( ,) ( ,) (

*a C t a C t dt a C*

*i i i*

2 2 2 2

 

*i i i*

*d C dC d C dC t dt dx dx dx*

*i i i*

*d C dC dC dt dt dx dx dy*

1 0 <sup>2</sup> 2 2 <sup>2</sup>

*i t*

*i i i*

*dC dC d C dt dt dx dy dy*

0 0

, (7.4.10)

<sup>2</sup> <sup>2</sup>

*i*

2 2

*i i*

, and (7.4.13)

*M T*

<sup>2</sup> <sup>2</sup> 2 2

1 2 2 2 2

( ) 0.5

<sup>2</sup>

*M T M T*

*i t i t*

2 2

*d C dC d C m dC t dt dx dx dx*

*i i i*

1 2 2 1 0 1 0

*M T M T*

*i i t i t*

( ) 0.5

*d C dC dC t dy dx*

1 2 2 2 2 1 0 1 0

*i i*

 

> 1 11 2 22

*m Dk Dl m Dk Dl* 

*L T L T*

1 2 1 0 *M T*

1 2 2 2 1 0 *M T*

*dC dC l k dt dx dy*

2 2

*d C dC d C m dC t dt dy dx dy*

*i i i*

*i t*

*M T M T*

*i i t i t*

2 2 2 1 0 1 0

( ) 0.5

*d C <sup>k</sup> dt dx*

*i t*

( ) 0.5

2 0 2

*M T M T*

1 1 2 2 2

1 0 1 0

1 0 1 0

*i i M T M T*

*i i*

*M T M T*

1 0 1 0

*i i*

We simplify equation (7.4.6) to

following set of equations:

Now we substitute <sup>0</sup> ( ,) *<sup>i</sup> aCt* , 1 *a Ct* ( ,) , 2 *a Ct* ( ,),

$$d\_2 = \sum\_{i=1}^{M} \sum\_{t=0}^{T} \left\{ \frac{d^2 \mathbb{C}\_i}{dy^2} \right\}^2 dt \quad . \tag{7.4.14}$$

The two simultaneous equations in (7.4.9) can be solved to obtain the estimates of the unknown parameters, *DL* and *DT*, for a two-dimensional groundwater system. The solutions of equations (7.4.9) are,

$$\begin{aligned} D\_L &= \frac{m\_1 l\_2 - m\_2 l\_1}{k\_1 l\_2 - l\_1^2}, \\ \text{and} \\ D\_T &= \frac{m\_1 l\_1 - m\_2 k\_1}{l\_1^2 - k\_1 l\_2}. \end{aligned} \tag{7.4.15}$$

We have estimated the longitudinal and lateral dispersion coefficients for 100 realisations for each of <sup>2</sup> value chosen, and their mean values are given in Table 7.1 .

The transverse dispersion coefficient is significantly less than the longitudinal dispersion coefficient for the flow length [0,1] when <sup>2</sup> is very small but approaches approximately 0.5 of longitudinal dispersion coefficient when <sup>2</sup> increases (Figure 7.12). Comparing Table 7.1 with Table 4.9, we see that the dispersion coefficient, therefore, the dispersivity, is smaller in 2 dimensions especially when <sup>2</sup> >0.01 . This needs to be expected as the lateral dispersion provides another mechanism of energy dissipation, thwarting the dispersion in the longitudinal direction. 


Table 7.1. Estimated mean longitudinal and transverse dispersion coefficients using 100 concentration realisations from SSTM2d for each of <sup>2</sup> value.

**Multiscale Dispersion in 2 Dimensions** 

In Chapter 7, we have developed the 2 dimensional solute transport model and estimated the dispersion coefficients in both longitudinal and transverse directions using the stochastic inverse method (SIM), which is based on the maximum likelihood method. We have seen that transverse dispersion coefficient relative to longitudinal dispersion coefficient increases

 increases when the flow length is confined to 1.0. In this chapter, we extend the SSTM2d into a partially dimensional form as we did for 1 dimension, so that we can explore the larger scale behaviours of the model. However, the experimental data on transverse dispersion is scarce in laboratory and field scales limiting our ability to validate the multiscale dispersion model. In this chapter, we briefly outline the dimensionless form of SSTM2d and illustrate the numerical solution for a particular value of flow length. We also

estimate the dispersion coefficients using the SIM for the same flow length.

As in the one dimensional case, we define dimensionless distances to start with:

*x x z z <sup>L</sup>*

*y <sup>y</sup> <sup>z</sup> <sup>z</sup> <sup>L</sup>*

As in Chapter 6, we derive the following partial derivatives:

2 2 0 2 22 ; *x x*

*C C x Lz* 

<sup>0</sup> ; *x x*

6), we define the cosine and sine of the angle as follows,

*C C x Lz*  , 0 1, *<sup>x</sup> <sup>x</sup>*

, 0 1. *<sup>y</sup> <sup>y</sup>*

We also define dimensionless concentration with respect to the maximum concentration, *C*<sup>0</sup> :

0 ( , ,) ( , , ) , 0 1. *Cxyt xyt <sup>C</sup>* 

> *C C y Lz*

As we have developed the SSTM2d for [0,1] domains in both x and y directions (see Chapter

<sup>0</sup> ; and *y y*

2 2 0 2 22 . *y y*

*C C y L z* 

**8.1 Introduction** 

**8.2 Basic Equations** 

as <sup>2</sup> 

and

Figure 7.14. The ratio of the transverse dispersivity to the longitudinal dispersivity vs <sup>2</sup> .

### **7.5 Summary**

In this chapter, we developed the 2 dimensional version of SSTM for the flow length of [0,1], and estimated the transverse dispersivity using the Stochastic Inverse Method (SIM) adopted for the purpose. The SSTM2d has mathematically similar form to SSTM but computationally more involved. However, the numerical routines developed are robust. We will extend SSTM2d in a dimensionless form to understand multi-scale behaviours of SSTM2d in the next chapter.

## **Multiscale Dispersion in 2 Dimensions**

### **8.1 Introduction**

Computational Modelling of Multi-Scale Non-Fickian Dispersion

.

214 in Porous Media - An Approach Based on Stochastic Calculus

Figure 7.14. The ratio of the transverse dispersivity to the longitudinal dispersivity vs <sup>2</sup>

In this chapter, we developed the 2 dimensional version of SSTM for the flow length of [0,1], and estimated the transverse dispersivity using the Stochastic Inverse Method (SIM) adopted for the purpose. The SSTM2d has mathematically similar form to SSTM but computationally more involved. However, the numerical routines developed are robust. We will extend SSTM2d in a dimensionless form to understand multi-scale behaviours of

**7.5 Summary** 

SSTM2d in the next chapter.

In Chapter 7, we have developed the 2 dimensional solute transport model and estimated the dispersion coefficients in both longitudinal and transverse directions using the stochastic inverse method (SIM), which is based on the maximum likelihood method. We have seen that transverse dispersion coefficient relative to longitudinal dispersion coefficient increases as <sup>2</sup> increases when the flow length is confined to 1.0. In this chapter, we extend the SSTM2d into a partially dimensional form as we did for 1 dimension, so that we can explore the larger scale behaviours of the model. However, the experimental data on transverse dispersion is scarce in laboratory and field scales limiting our ability to validate the multiscale dispersion model. In this chapter, we briefly outline the dimensionless form of SSTM2d and illustrate the numerical solution for a particular value of flow length. We also estimate the dispersion coefficients using the SIM for the same flow length. directions

### **8.2 Basic Equations**

As in the one dimensional case, we define dimensionless distances to start with:

$$z\_{\chi} = \frac{\chi}{L\_{\chi}}, \qquad 0 \le z\_{\chi} \le 1,$$

and

$$z\_y = \frac{y}{L\_y}, \qquad 0 \le z\_y \le 1.$$

We also define dimensionless concentration with respect to the maximum concentration, *C*<sup>0</sup> :

$$
\Gamma(\mathbf{x}, \mathbf{y}, t) = \frac{\mathbf{C}(\mathbf{x}, \mathbf{y}, t)}{\mathbf{C}\_0}, \qquad 0 \le \Gamma \le 1.
$$

As in Chapter 6, we derive the following partial derivatives: following

$$\frac{\partial \mathbf{C}}{\partial \mathbf{x}} = \frac{\mathbf{C}\_{0}}{\mathbf{L}\_{\times}} \frac{\partial \Gamma}{\partial \mathbf{z}\_{\times}}; \quad \frac{\partial^{2} \mathbf{C}}{\partial \mathbf{x}^{2}} = \frac{\mathbf{C}\_{0}}{\mathbf{L}\_{\times}^{2}} \frac{\partial^{2} \Gamma}{\partial \mathbf{z}\_{\times}^{2}}; \frac{\partial \mathbf{C}}{\partial \mathbf{y}} = \frac{\mathbf{C}\_{0}}{\mathbf{L}\_{\times}} \frac{\partial \Gamma}{\partial \mathbf{z}\_{y}}; \text{ and } \frac{\partial^{2} \mathbf{C}}{\partial \mathbf{y}^{2}} = \frac{\mathbf{C}\_{0}}{\mathbf{L}\_{\times}^{2}} \frac{\partial^{2} \Gamma}{\partial \mathbf{z}\_{y}^{2}}.$$

As we have developed the SSTM2d for [0,1] domains in both x and y directions (see Chapter 6), we define the cosine and sine of the angle as follows,

We can also express the partial derivatives of the mean velocities in both x and y directions in terms of dimensionless space variables:

1 ; *x x <sup>x</sup> <sup>x</sup> x x x v vz v x z x Lz* 2 2 2 2 2 11 1 ; *<sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>x</sup> x xx x x x x v vz v v x z x x LzLz L z* <sup>1</sup> ;and, *<sup>y</sup> <sup>y</sup> y y v v y Lz* 2 2 2 22 <sup>1</sup> . *<sup>y</sup> <sup>y</sup> y y v v y L z* 

Similarly, we can express the derivatives related to solute concentration in terms of dimensionless variables:

$$\begin{aligned} \frac{\partial \Gamma}{\partial y^{2}} &= \frac{1}{L\_{y}^{2}} \frac{\partial \Gamma}{\partial x\_{y}^{2}}, \\\\ \text{divaves related to solute concentration in terms of} \\\\ \frac{\partial \Gamma}{\partial \bar{t}} &= \mathcal{C}\_{0} \frac{\partial \Gamma}{\partial \bar{t}}; \\\\ \frac{\partial^{2} \Gamma}{\partial x^{2}} &= \frac{\mathcal{C}\_{0}}{L\_{x}^{2}} \frac{\partial^{2} \Gamma}{\partial x\_{x}^{2}}; \text{and}, \\\\ \frac{\partial^{2} \Gamma}{\partial y^{2}} &= \frac{\mathcal{C}\_{0}}{L\_{y}^{2}} \frac{\partial^{2} \Gamma}{\partial x\_{y}^{2}}. \end{aligned}$$

We recall the SSTM2d in *x* and *y* co-ordinates,

$$\begin{aligned} \text{dC} &= -\text{C} \text{dI}\_{0,\text{x}} - \frac{\partial \text{C}}{\partial \text{x}} \text{dI}\_{1,\text{x}} - \frac{\partial^2 \text{C}}{\partial \text{x}^2} \text{dI}\_{2,\text{x}} - \text{C} \text{dI}\_{0,\text{y}} - \frac{\partial \text{C}}{\partial \text{y}} \text{dI}\_{1,\text{y}} - \frac{\partial^2 \text{C}}{\partial \text{y}^2} \text{dI}\_{2,\text{y}}; \vdots \\\\ &\text{where} \quad \text{dI}\_{0,\text{x}} = \left( \frac{\partial \overline{\text{w}}\_{\text{x}}}{\partial \text{x}} + \frac{\text{h}\_{\text{x}}}{2} \frac{\partial^2 \overline{\text{w}}\_{\text{x}}}{\partial \text{x}^2} \right) \text{d}t + \sigma \sum\_{j=1}^m \sqrt{\text{A}\_{\text{x}j} \text{A}\_{\text{y}}} \text{P}\_{0j} \text{d}b\_j(t); \vdots \\\\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \$$

2, 2

*<sup>m</sup> <sup>y</sup> <sup>y</sup> <sup>y</sup> y xj yj j j*

2 0, 2 0

1, 1

*y*

*v v h*

 

*y y*

 

*h dI v dt*

Because 012 0 1 <sup>2</sup> , , , , , , , , , and *xj yj <sup>x</sup> yj j j j j <sup>j</sup>*

symbols: 012 0 1 <sup>2</sup> , , , , , , , , , and *x y xy <sup>z</sup> <sup>j</sup> <sup>z</sup> <sup>j</sup> z z jjj j j <sup>j</sup>*

transformed partially dimensional governing equation:

*x*

*L z*

2, 2

*y*

 

2, 2

the direction perpendicular to the main direction is 25 m.

*y x y <sup>m</sup> <sup>z</sup> y z zj zj j j j*

 

*x x y <sup>m</sup> <sup>z</sup> x z zj zj j j j*

 

*Lz L z*

 

*x*

*y*

 

*h dI v dt*

 

*h dI v dt*

2 0, 2 2 0

*<sup>v</sup> <sup>h</sup> <sup>v</sup> dI dt P db t*

<sup>1</sup> ( ); *<sup>x</sup> <sup>x</sup> x y <sup>m</sup> <sup>x</sup> x z z zj zj j j x x j <sup>v</sup> dI v h dt P db t*

*x x y <sup>m</sup> <sup>x</sup> <sup>z</sup> <sup>x</sup> z zj zj j j x x x x j*

1, 1

1 ( ); <sup>2</sup>

 

2 0, 2 2 0

*y x y <sup>m</sup> <sup>y</sup> <sup>z</sup> <sup>y</sup> z zj zj j j*

*y y y y j <sup>h</sup> <sup>v</sup> <sup>v</sup> dI dt Q db t Lz L z*

1, 1

1 ( ). <sup>2</sup>

 

 

where

 

Multiscale Dispersion in 2 Dimensions 217

*dI dt Q db t*

*<sup>m</sup> <sup>y</sup> y y y xj yj j j j*

2, 2

*d dI dI dI dI dI dI*

1

 

1 1 ( ); <sup>2</sup>

1

 

*P db t*

1 1 ( ); <sup>2</sup>

1 <sup>1</sup> ( ); and, *<sup>y</sup> <sup>y</sup> x y <sup>m</sup> <sup>y</sup> y z z zj zj j j y y j <sup>v</sup> I vh t Q db t L z* 

 

*Q db t*

1

 

The above equations constitute the multiscale SSTM2d and we developed the numerical solutions when the flow length along the main flow direction is 100 m and the flow length in

*<sup>m</sup> <sup>y</sup> y y xj yj j j j*

use the same values and functions but we use the following

*<sup>v</sup> dI v h dt Q db t*

*<sup>h</sup> dI v dt*

1 ( ); <sup>2</sup> *m <sup>x</sup> <sup>x</sup> <sup>x</sup> xj yj j j j*

 

1 ( ); <sup>2</sup>

 

*j*

 

 

*h hP P P Q Q Q* are calculated for the domain [0, 1], we

*h h PPPQQ Q* . Now we calculate *d* based on the

2 2

1

1 ( ). <sup>2</sup>

0, 1, 2 2 2, 0, 1, 2 2 2, 1 1 1 1 ; *<sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>y</sup> <sup>y</sup> <sup>y</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> x x x x y y y y*

*Lz Lz Lz Lz* (8.2.1)

*P db t*

*Q db t*

( );and,

$$
\cos \theta = \frac{z\_x}{\sqrt{z\_x^2 + z\_y^2}}, \text{and}$$

$$
\sin \theta = \frac{z\_y}{\sqrt{z\_x^2 + z\_y^2}}.
$$

216 in Porous Media - An Approach Based on Stochastic Calculus

2 2 cos ,and *<sup>x</sup> x y*

2 2 sin . *<sup>y</sup>*

We can also express the partial derivatives of the mean velocities in both x and y directions

*v vz v x z x Lz* 

2 2 2 2 2

> *v v y Lz*

> > 2 2 2 22 <sup>1</sup> . *<sup>y</sup> <sup>y</sup> y y*

*v v y L z* 

Similarly, we can express the derivatives related to solute concentration in terms of

*<sup>C</sup> <sup>C</sup> t t* 

2 2 0 2 22 ; and, *x x*

> 2 2 0 2 22 . *y y*

*C C y L z* 

*C C C C dC CdI dI dI CdI dI dI*

2 0, 2 0

1, 1

*x x*

 

 

*<sup>m</sup> <sup>x</sup> <sup>x</sup> <sup>x</sup> x xj yj j j*

*<sup>m</sup> <sup>x</sup> x x x xj yj j j j*

*<sup>v</sup> dI v h dt P db t x*

*vhv dI dt P db t*

*C C x Lz* 

<sup>0</sup> ;

2 2

1 ( ); <sup>2</sup>

 

( );

*j*

 

1

0, 1, <sup>2</sup> 2, 0, 1, <sup>2</sup> 2, ; *<sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>y</sup> <sup>y</sup> <sup>y</sup>*

*x x y y*

*v vz v v x z x x LzLz L z* 

; *<sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>x</sup> x xx x x x x*

> <sup>1</sup> ;and, *<sup>y</sup> <sup>y</sup> y y*

in terms of dimensionless space variables:

dimensionless variables:

We recall the SSTM2d in *x* and *y* co-ordinates,

where

*z z z*

*z z z*

*x y*

1 ; *x x <sup>x</sup> <sup>x</sup> x x x*

11 1

$$dI\_{2,x} = \left(\frac{\hbar\_x}{2}\overline{\boldsymbol{\upsilon}}\_x\right)dt + \sigma\sum\_{j=1}^m \sqrt{\mathbb{A}\_{ij}\overline{\mathbb{A}\_{jj}}}P\_{2,j}db\_j(t);$$

$$dI\_{0,y} = \left(\frac{\partial\overline{\boldsymbol{\upsilon}}\_y}{\partial y} + \frac{\hbar\_y}{2}\frac{\partial^2\overline{\boldsymbol{\upsilon}}\_y}{\partial y^2}\right)dt + \sigma\sum\_{j=1}^m \sqrt{\mathbb{A}\_{jj}\overline{\mathbb{A}\_{jj}}}Q\_{0j}db\_j(t);$$

$$dI\_{1,y} = \left(\overline{\boldsymbol{\upsilon}}\_y + \boldsymbol{h}\_y\frac{\partial\overline{\boldsymbol{\upsilon}}\_y}{\partial y}\right)dt + \sigma\sum\_{j=1}^m \sqrt{\mathbb{A}\_{jj}\overline{\mathbb{A}\_{jj}}}Q\_{1j}db\_j(t); \text{and},$$

$$dI\_{2,y} = \left(\frac{\boldsymbol{h}\_y}{2}\overline{\boldsymbol{\upsilon}}\_y\right)dt + \sigma\sum\_{j=1}^m \sqrt{\mathbb{A}\_{jj}\overline{\mathbb{A}\_{jj}}}Q\_{2,j}db\_j(t).$$

Because 012 0 1 <sup>2</sup> , , , , , , , , , and *xj yj <sup>x</sup> yj j j j j <sup>j</sup> h hP P P Q Q Q* are calculated for the domain [0, 1], we use the same values and functions but we use the following symbols: 012 0 1 <sup>2</sup> , , , , , , , , , and *x y xy <sup>z</sup> <sup>j</sup> <sup>z</sup> <sup>j</sup> z z jjj j j <sup>j</sup> h h PPPQQ Q* . Now we calculate *d* based on the transformed partially dimensional governing equation:

$$d\Gamma = -\Gamma d\boldsymbol{I}\_{0,\boldsymbol{z}\_{\boldsymbol{z}}} - \frac{1}{\mathcal{L}\_{\boldsymbol{x}}} \frac{\partial \Gamma}{\partial \boldsymbol{z}\_{\boldsymbol{x}}} d\boldsymbol{I}\_{1,\boldsymbol{z}\_{\boldsymbol{z}}} - \frac{1}{\mathcal{L}\_{\boldsymbol{x}}^{2}} \frac{\partial^{2} \Gamma}{\partial \boldsymbol{z}\_{\boldsymbol{x}}^{2}} d\boldsymbol{I}\_{2,\boldsymbol{z}\_{\boldsymbol{x}}} - \Gamma d\boldsymbol{I}\_{0,\boldsymbol{z}\_{\boldsymbol{y}}} - \frac{1}{\mathcal{L}\_{\boldsymbol{y}}} \frac{\partial \Gamma}{\partial \boldsymbol{z}\_{\boldsymbol{y}}} d\boldsymbol{I}\_{1,\boldsymbol{z}\_{\boldsymbol{y}}} - \frac{1}{\mathcal{L}\_{\boldsymbol{y}}^{2}} \frac{\partial^{2} \Gamma}{\partial \boldsymbol{z}\_{\boldsymbol{y}}^{2}} d\boldsymbol{I}\_{2,\boldsymbol{z}\_{\boldsymbol{y}}};\tag{8.2.1}$$

where 2 0, 2 2 0 1 1 1 ( ); <sup>2</sup> *x x x y <sup>m</sup> <sup>x</sup> <sup>z</sup> <sup>x</sup> z zj zj j j x x x x j <sup>v</sup> <sup>h</sup> <sup>v</sup> dI dt P db t Lz L z* 1, 1 1 <sup>1</sup> ( ); *<sup>x</sup> <sup>x</sup> x y <sup>m</sup> <sup>x</sup> x z z zj zj j j x x j <sup>v</sup> dI v h dt P db t L z* 2, 2 1 ( ); <sup>2</sup> *x x x y <sup>m</sup> <sup>z</sup> x z zj zj j j j h dI v dt P db t* 2 0, 2 2 0 1 1 1 ( ); <sup>2</sup> *y y x y <sup>m</sup> <sup>y</sup> <sup>z</sup> <sup>y</sup> z zj zj j j y y y y j <sup>h</sup> <sup>v</sup> <sup>v</sup> dI dt Q db t Lz L z* 1, 1 1 <sup>1</sup> ( ); and, *<sup>y</sup> <sup>y</sup> x y <sup>m</sup> <sup>y</sup> y z z zj zj j j y y j <sup>v</sup> I vh t Q db t L z* 2, 2 1 ( ). <sup>2</sup> *y y x y <sup>m</sup> <sup>z</sup> y z zj zj j j j h dI v dt Q db t j*

The above equations constitute the multiscale SSTM2d and we developed the numerical solutions when the flow length along the main flow direction is 100 m and the flow length in the direction perpendicular to the main direction is 25 m.

**8.4 Estimation of Dispersion Coefficients** 

Based on 60 realisations for each value of <sup>2</sup>

dispersivities based on a properly validated model.

dispersion coefficients for the same boundary and initial conditions.

Table 8.1. The estimated mean dispersion coefficients for two different <sup>2</sup>

The SPDE becomes,

note the following relations:

2 

**8.5 Summary** 

We use the same methodology as in Chapter 7 with a slight modification to the advection-

Multiscale Dispersion in 2 Dimensions 219

*x x y y xx DD v z t*

We can use the SIM to estimate the parameters but to obtain the dispersion coefficients, we

22 22 ( , ). *<sup>L</sup> <sup>T</sup> <sup>x</sup>*

<sup>2</sup> estimated ;and *<sup>L</sup> L x*

*L*

*x*

<sup>2</sup> estimated . *<sup>T</sup> T y*

*y*

*L*

2

2

, Table 8.1 shows the estimated mean

values (*b*=0.1).

(8.4.1)

dispersion stochastic partial differential equation (SPDE) to make it dimensionless.

2 2

*t L z L z Lz*

*<sup>D</sup> <sup>D</sup> <sup>L</sup>*

 

*<sup>D</sup> <sup>D</sup> <sup>L</sup>*

*DL DT*

0.01 5.445969667 0.259079583 0.1 6.853118 1.043493717

In this brief chapter, we have given sufficient details of development of the multiscale SSTM2d and a sample of its realisations. We also have adopted SIM to estimate dispersion coefficients in both longitudinal and lateral directions. The computational experiments we have done with the SSTM2d show realistic solutions under variety of boundary and initial conditions, even for larger scales such 10000 m. However, it is not important to illustrate the results, as we have discussed the one dimensional SSTM in detail in Chapter 6. If there are reliable dispersivity data in different scales, both in longitudinal and transverse directions, then one can develop much more meaningful relations between longitudinal and lateral

### **8.3 A Sample of Realisations of Multiscale SSTM2d**

For the illustrative purposes, we plot three realisations of concentration when *C*<sup>0</sup> =1.0 at (*x*=0; and *y*=0) when time is 20 days for two different <sup>2</sup> values, 0.01 and 0.1. These are shown in Figures 8.1 and 8.2.

Figure 8.1. A concentration realisation when time is 20 days for <sup>2</sup> =0.01. Mean velocity in *x* direction is 0.5 m/day and, in *y* direction is 0.0.

Figure 8.2. A concentration realisation when time is 20 days for <sup>2</sup> =0.1. (Same conditions as in Figure 8.1.)

### **8.4 Estimation of Dispersion Coefficients**

We use the same methodology as in Chapter 7 with a slight modification to the advectiondispersion stochastic partial differential equation (SPDE) to make it dimensionless.

The SPDE becomes,

Computational Modelling of Multi-Scale Non-Fickian Dispersion

=0.01. Mean velocity in *x*

=0.1. (Same conditions as

values, 0.01 and 0.1. These are

218 in Porous Media - An Approach Based on Stochastic Calculus

For the illustrative purposes, we plot three realisations of concentration when *C*<sup>0</sup> =1.0 at

**8.3 A Sample of Realisations of Multiscale SSTM2d**

shown in Figures 8.1 and 8.2.

(*x*=0; and *y*=0) when time is 20 days for two different <sup>2</sup>

Figure 8.1. A concentration realisation when time is 20 days for <sup>2</sup>

Figure 8.2. A concentration realisation when time is 20 days for <sup>2</sup>

in Figure 8.1.)

direction is 0.5 m/day and, in *y* direction is 0.0.

$$\frac{\partial \Gamma}{\partial t} = \left| \frac{D\_L}{L\_x^2} \frac{\partial^2 \Gamma}{\partial z\_x^2} + \frac{D\_T}{L\_y^2} \frac{\partial^2 \Gamma}{\partial z\_y^2} \right| - \frac{\overline{\upsilon}\_x}{L\_x} \frac{\partial \Gamma}{\partial z\_x} + \xi(z, t). \tag{8.4.1}$$

We can use the SIM to estimate the parameters but to obtain the dispersion coefficients, we note the following relations: but

$$D\_{\perp} = \left[ \text{estimated} \left( \frac{D\_{\perp}}{L\_{\times}^{2}} \right) \right] \times L\_{\times}^{2}; \text{and}$$

$$D\_{\parallel} = \left[ \text{estimated} \left( \frac{D\_{\perp}}{L\_{\times}^{2}} \right) \right] \times L\_{\times}^{2}.$$

Based on 60 realisations for each value of <sup>2</sup> , Table 8.1 shows the estimated mean dispersion coefficients for the same boundary and initial conditions.


Table 8.1. The estimated mean dispersion coefficients for two different <sup>2</sup> values (*b*=0.1).

### **8.5 Summary**

In this brief chapter, we have given sufficient details of development of the multiscale SSTM2d and a sample of its realisations. We also have adopted SIM to estimate dispersion coefficients in both longitudinal and lateral directions. The computational experiments we have done with the SSTM2d show realistic solutions under variety of boundary and initial conditions, even for larger scales such 10000 m. However, it is not important to illustrate the results, as we have discussed the one dimensional SSTM in detail in Chapter 6. If there are reliable dispersivity data in different scales, both in longitudinal and transverse directions, then one can develop much more meaningful relations between longitudinal and lateral dispersivities based on a properly validated model.

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**Index** 

Adapted process, 35, 58

Angular frequency, 29 Anisotropic, 11, 234, 239, 240

Boundary layers, 170 Breakthrough curves, 8

Contamination, 2, 236, 240 Continuity, 1, 5, 23, 26

Cellular, 178

145, 234

Advection, II, 3, 5, 7, 10, 11, 41, 70, 85, 86, 90, 91, 92, 93, 94, 95, 96, 100, 103, 110, Drift, 17, 41, 42, 43, 47, 48, 55, 59, 61, 63, 65, 66, 68, 69, 75, 128, 148, 165, 167, 190, 192 Eigen value, 76, 122, 124, 125, 130, 131, 132, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 151, 153, 206, 207, 215

Fluid, 3, 4, 6, 7, 9, 10, 11, 22, 70, 73, 77, 230,

Flux, II, 4, 5, 6, 11, 70, 71, 72, 81, 174, 208,

Groundwater, 1, 2, 9, 10, 11, 12, 13, 14, 16, 20, 21, 71, 100, 102, 108, 223, 225, 232, 233, 234, 235, 236, 237, 238, 239, 240,

Hilbert space, 15, 74, 77, 80, 84, 85, 211, 242 Hydraulic conductivity, 10, 11, 12, 20, 70,

Hydrodynamic dispersion, IV, 3, 4, 5, 7, 10, 11, 41, 70, 85, 96, 98, 100, 103, 124, 170,

174, 52, 74,

Inverse method, II, 14, 118, 146, 187, 197,

Ito, II, IV, V, 9, 34, 35, 36, 37, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 61, 70, 74, 75, 82, 83, 84, 127, 128, 129, 147, 149, 158, 160, 161, 163, 165, 166, 170, 183, 184, 187, 189, 192, 201, 215

Kernel, V, 34, 73, 74, 75, 78, 80, 90, 91, 116, 123, 124, 125, 130, 134, 135, 137, 138, 139, 140, 142, 144, 145, 146, 147, 150, 169, 170, 187, 189, 193, 206, 207, 215

Markov, 33, 149, 160, 166, 183, 184, 235 Martingale, 32, 33, 34, 36, 157, 166, 183 Maximum likelihood, IV, 16, 96, 98, 116,

73, 97, 98, 103, 108, 109, 128

Event, 15, 16, 20, 35

209, 233, 234 Fourier transform, 29 General Linear SDE, 58

241, 242

192, 206, 222 Indicator, 63

Isometry, 36

Jump, 23, 24, 111

Laminar flow, 7 Left-continuous, 23, 35 Linearity, 26, 35

187, 227, 233

Karhunen-Loeve expansion, 77

226, 227, 232, 235

234, 235, 239, 241

Aquifer, 1, 2, 3, 4, 8, 9, 10, 11, 12, 13, 14, 20, 21, 70, 73, 91, 92, 93, 94, 95, 96, 97, 98, 103, 104, 105 107, 108, 109, 110, 114, 115, 116, 118, 121, 122, 128, 205, 232, 233,

116, 123, 124, 187, 197, 222, 231

234, 235, 236, 238, 239, 240, 241

Computer simulation, IV, 40, 49, 50 Confidence intervals, 89, 145, 146, 147

189, 193, 206, 207, 237, 240

Differential operator, 15, 81, 83, 125

Darcy's law, 3, 73, 170

190, 215, 235 Dirac delta function, 77 Discontinuity, 23, 24

Convergence, 20, 26, 27, 32, 35, 76, 85, 138,

Correlation, 28, 34, 73, 75, 77, 78, 80, 85, 86, 90, 95, 105, 110, 112, 115, 130, 150, 186 Covariance, 11, 28, 30, 36, 73, 74, 75, 76, 77, 116, 123, 124, 125, 130, 131, 134, 135, 137, 138, 139, 142, 143, 144, 145, 146, 149, 169, 170, 176, 177, 183, 186, 188,

Covariation, 25, 26, 31, 37, 38, 44, 45, 46, 149

Diffusion, V, 2, 3, 4, 7, 41, 42, 43, 47, 48, 55, 59, 61, 63, 64, 65, 66, 67, 68, 69, 75, 116, 128, 147, 148, 149, 150, 157, 158, 159, 160, 161, 163, 164, 166, 170, 187, 189,

Dispersion, II. IV, V, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 41, 70, 75, 80, 81, 82, 85, 86, 90, 91, 92, 93, 94, 95, 96, 98, 100, 101, 103, 108, 110, 115, 116, 123, 124, 146, 170, 187, 189, 192, 197, 206, 222, 225, 227, 231, 233, 234, 235, 236, 237, 239, 240, 241

### **Index**


Stochastic Chain Rule, IV, 37, 40


Stochastic Chain Rule, IV, 37, 40

165, 169, 178, 183, 235, 239 Stochastic exponential, 54, 58, 59 Stochastic product rule, 44, 128

Symmetry, 25, 26, 132, 173, 174, 207

Thermodynamics, 171, 174, 179, 236

Tracer, 8, 22, 91, 98, 118, 205, 235, 238, 239,

Transport, II, IV, V, 1, 2, 3, 5, 6, 7, 9, 10, 11, 22, 70, 71, 74, 85, 92, 98, 103, 110, 121, 124, 186, 187, 206, 227, 232, 233, 234,

235, 236, 237, 238, 239, 240, 241 Validation, 60, 61, 62, 63, 64, 110, 111, 115 Variance, 3, 4, 16, 28, 29, 30, 31, 32, 36, 50, 75, 78, 80, 82, 85, 86, 87, 90, 110, 115, 121, 122, 123, 125, 130, 131, 132, 134, 135, 137, 139, 140, 143, 149, 150, 159, 161, 166, 167, 182, 186, 189, 200, 206 Variation, 9, 24, 25, 26, 27, 28, 31, 32, 34, 36, 37, 38, 41, 42, 44, 45, 46, 54, 55, 57, 66,

White noise, 30, 48, 49, 50, 53, 73, 77

Zero Mean Property, 35, 36

Stratonovich, 75, 183

Tortuosity, 3, 6, 7

241

89, 147

Taylor series, 40, 71, 174 Temperature, 170, 171, 173

Stochastic differential, IV, 15, 20, 30, 43, 44, 47, 49, 53, 54, 55, 59, 75, 129, 130, 148,

234 in Porous Media - An Approach Based on Stochastic Calculus

Mean-square, 26 Monitoring wells, 91

Polarization, 25, 26

Pumps, 91

Recurrent, 20, 59

Rhodamine, 91 Riemann, 27

Right-continuous, 23

75, 82, 118, 123 Spectral density, 29, 30

Orthonormal, 15, 74, 76, 77, 80

Population dynamics, 41, 48 Predictable process, 35, 41 Probability space, 15, 22

37, 38, 41, 42, 46, 55

Partial differential equation, IV, 10, 13, 14, 15, 70, 75, 77, 96, 101, 128, 166, 169, 186, 187, 222, 223, 231, 236, 237, 238, 241

Quadratic variation, 25, 26, 27, 311, 32, 36,

Scale dependency, II, IV, 6, 7, 8, 9, 10, 70,

Stochastic calculus, II, IV, 1, 9, 10, 11, 22, 23,

Representative Elementary Volume, 5

Spectral Expansion, II, 76, 80, 81, 170 SSTM, IV, V, 74, 75, 80, 81, 82, 85, 86, 90, 91, 93, 94, 95, 96, 110, 11, 112, 114, 115, 116, 118, 120, 122, 123, 124, 129, 130, 138, 145, 146, 147, 157, 163, 169, 170, 186, 187, 189, 190, 191, 192, 196, 199, 200, 201, 202, 205, 206, 215, 216, 22, 225, 226,

227, 228, 229, 230, 231

24, 25, 37, 49, 70, 77, 127, 237

## *Authored by Don Kulasiri*

This research monograph presents a mathematical approach based on stochastic calculus which tackles the "cutting edge" in porous media science and engineering - prediction of dispersivity from covariance of hydraulic conductivity (velocity). The problem is of extreme importance for tracer analysis, for enhanced recovery by injection of miscible gases, etc. This book explains a generalised mathematical model and effective numerical methods that may highly impact the stochastic porous media hydrodynamics. The book starts with a general overview of the problem of scale dependence of the dispersion coefficient in porous media. Then a review of pertinent topics of stochastic calculus that would be useful in the modeling in the subsequent chapters is succinctly presented. The development of a generalised stochastic solute transport model for any given velocity covariance without resorting to Fickian assumptions from laboratory scale to field scale is discussed in detail. The mathematical approaches presented here may be useful for many other problems related to chemical dispersion in porous media.

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Computational Modelling of Multi-scale Solute Dispersion in Porous Media -

An Approach Based on Stochastic Calculus

Computational Modelling of

Multi-scale Solute Dispersion

in Porous Media

An Approach Based on Stochastic Calculus

*Authored by Don Kulasiri*