The Stochastic Solute Transport Model in 2-Dimensions

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. following:


Introduction
In Chapter 6, we developed the generalised Stochastic Solute Transport Model (SSTM) in 1dimension 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.

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 exp .
x x y y q x y x y b  Then the integral equation can be written for 2 dimensions,            2  2  1 1  2  1  2  1  2  2  2  2  2  1  1  0 where   , f x y 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: Then the integral equation can be written as,

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

V x y t and  
, , x y t  are velocity, mean velocity and noise vectors respectively. Instantaneous velocity vector can now be expressed as, where i and j are unit vectors in x and y directions, respectively; and,

 
, , x V x y t and   , , y V x y t 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, where x  and y  are the noise components in x and y directions. We can see the noise term appearing as, x y www.intechopen.com The Stochastic Solute Transport Model in 2-Dimensions 199 To simplify the notation, 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, We can now write, , and bringing dt in to the parenthesis in the third and fourth terms of the right hand side, As in the one dimensional case, we can define, The resultant noise term is given by, where , We make an assumption that q is defined by . This is a simplifying approximation which makes the modelling more tractable; as the noise term is quite random, this approximation does not make significant difference to final results.
Now we can expand the terms in the brackets in equation (7.3.18), We see that, , , cos cos cos cos The Stochastic Solute Transport Model in 2-Dimensions 201 Now,   x j x j x x x j x j x j x x S Cf    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.

A 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. 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., dI . (7.3.33) We illustrate some realisations of the solutions graphically in the next section.

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, where C = solution concentration (mg/l), 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 2 2 2 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,

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.
www.intechopen.com 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.