Multiscale, Generalised Stochastic Solute Transport Model in One Dimension

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 3 and 4, we have developed a stochastic solute transport model in 1-D without rosorting to simplifying Fickian assumptions, but by using the idea that the fluctuations in velocity are influenced by the nature of porous medium. We model these fluctuations through the velocity covariance kernel. We have also estimated the dispersivity by taking the realisations of the solution of the SSTM and using them as the observations in the stochastic inverse method (SIM) based on the maximum likelihood estimation procedure for the stochastic partial differential equation obtained by adding a noise term to the advectiondispersion equation. We have confined the estimation of dispersitivities to a flow length of 1 m (i.e,   0,1 x  ) except in Chapter 3, section 3.10, where we have estimated the dispersitivities up to 10 km using the SIM by simplifying the SSTM. This approach was proven to be computationally expensive and the approximation of the SSTM we have developed was based on the spatial average of the variance of the fluctuation term over the flow length. Further, the solution is based on a specific kernel. This development in Chapter 3 is inadequate to examine the scale dependence of the dispersitivity. Therefore, we set out to develop a dimensionless model for any given arbitrary flow length, L , in this Chapter for any given velocity kernel provided that we have the eigen functions in the form given by equation ( where the coefficients ,  and x , respectively. ( ) dw t are the standard Wiener increments with zero-mean and dt variance. As discussed in Chapter 4, equation (6.1.1) has to be interpreted carefully to understand it better. Equation (6.1.1) is a SDE and also an Ito diffusion with the coefficients depending on the functions of space variables. It gives us the time evolution of the concentration of solute at a given point x which is denoted by subscript x . Obviously, the computation of x C also depends on how the spatial www.intechopen.com   One should note that the Langevin equation for any system reflect the role of external noise to the system under consideration (van Kampen, 1992). Even though we have derived equation (6.1.1) starting from the mass conservation of solute particles, the fluctuations associate with hydrodynamics dispersion are a result of dissipation of energy of particles due to momentum changes associated near to the surfaces of porous medium. For a physical ensemble of solute particles, porous medium through which it flows act as an external source of noise. From this point of review, the Langevin type equation for solute concentration is justified. As a SDE, equation (6.1.1) is a Wiener process with stochastic, at best nonlinear, time-dependent coefficients, and it is also an Ito diffusion which should be interpreted locally, i.e., for a given x and t , equation (6.1.1) is valid only for short time intervals beyond t . This naturally leads us to evaluate the associated spatial derivatives at the previous time, which is valid according to Ito's interpretation of stochastic integral. In terms of discretized times, 0 1 1 , ,..., , ,..., i i t t t t  equation (6.1.1) can be written as, where the drift coefficient, x  , and the diffusion coefficient, x  , are evaluated at time i t  . This restrictive nature of equation (6.1.14) in evaluating the coefficient has to be taken in to account in developing numerical algorithms to solve it.

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Computational Modelling of Multi-Scale Non-Fickian Dispersion in Porous Media -An Approach Based on Stochastic Calculus 180 6.2 Partially Dimensionless SSTM with Flow Length L We start the derivation of partially dimensionless SSTM by defining the dimensionless distance, Z , as: where L is the total flow length.
x and t , then for any t and x . We can define dimensional concentration As the domain of x is the generalized SSTM is from 0 to 1, we can replace x with Z in the dimensionless generalized SSTM. For example, Now we can write equation (6.1.1) in the following manner: Therefore, the Langevin form of the generalized SSTM is given by Using equation (6.2.14) we can compute the time course of the dimensionless concentration for any given L .
  Z C t is proportioned to the number of solute moles within a unit volume of porous/water matrix, and 0 C is proportional to the maximum possible number of solute moles within the same matrix. Therefore, can be interpreted as the likelihood (probability) of finding solute moles within the matrix.
It should be noted that time, t , is not a dimensionless quantity and therefore, equation (6.2.14) is partially dimensionless equation. We will explore the dispersivity using equation (6.2.14) first before discussing a completely dimensionless equation.
www.intechopen.com , which has a follow-on affect on subsequent calculation. In other words, the affects of porous media, which 2  and the covariance kernel signify, can not be ignored as they affect the flow velocities significantly in making them stochastic. . It should be noted that as L is increased, the time duration for the numerical solution of equation (6.2.14) should be increased. For example, when 10 L m  , the model was run for 25 days to obtain Figures 6.3a and 6.3b. However, the order of magnitude for Z  and Z  has not changed as we change L in an order of magnitude.
www.intechopen.com  is quite dramatic, and this shows that equation (6.2.14) can display very complex behaviour patterns albeit its simplicity. It should be noted however that 2  plays major role in delimiting the nature of realizations; 2  values high than 0.25 in these situations produces highly irregular concentration realizations which could occur in highly heterogeneous porous formations such as fractured formations.

Dispersivities Based on the Langevin Form of SSTM for
10 L  m One of the advantages of the partially dimensionless Langevin equation for the SSTM (equation 6.2.14) is that we can use it to compute the solute concentration profiles when the travel length   L is large. Equation (6.2.14) allows us to compute the dispersitivities using the stochastic inverse method (SIM) by estimating dispersivity for each realization of   Z  . For the SIM, we need to modify the deterministic-advection and dispersion equation into a partially dimensionless one. We start with the deterministic advection-dispersion equation with additive Gaussian noise, The partially dimensionless form of equation ( Table 6.1. Mean dispersivities for the data in Figure 6.5.

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As mentioned previously, the partially dimensionless equation (6.2.14) still requires us to compute for a large number of days when L is large. While the computational times are still manageable, we would like to develop a completely dimensionless Langevin equation for the SSTM. This could be especially useful and insightful when the mean velocity V could be considered as a constant.

Dimensionless Time
We introduce dimensionless time,  , as, The completely dimentionless Langevin form of the SSTM is therefore, As we can see we may not gain much computational advantage with a completely dimensionless Langevin form of the SSTM.

Estimation of Field Scale Dispersivities
We have estimated the longitudinal dispersivities using SSTM for two different boundary conditions: R t is taken to be 1/3 of the total time (T) of the computational experiment. We have also computed the dispersivities for larger scales up to 10,000 m, and Table 6.  (1 and 5). Therefore, we develop the following statistical nonlinear regression models for these significant relationships: , and (6.6.1) where s D is the dispersivity, and 1 C and 2 C are given in Tables 6.5 and 6.6 , respectively, along with m1 and m2 values. R-square values for equations (6.6.1) and (6.6.2) are 0.96 and 0.94, respectively.     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 the large scale aquifer dispersivities using a single value of 2  ? To answer this question, we 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 data and overlaid the plot with the dispersivity vs L curves from the SSTM for each 2  value. Figure 6.9 shows the plots, and 2  =0.1 best fit to the experimental data. In other words, by using one value of 2  , we can obtain the dispersivity for any length of the flow by using the 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 realisations for particular values of 2 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.