Non-Fickian mass transport in fractured porous media
Research highlights
► Mass flux in a fractal porous medium is derived in a form of fractional derivative. ► An order of this derivative depends on a fractal dimension of the medium. ► A fractional mass transport equation for a fractured porous medium is derived. ► A temporal fractional derivative models the fracture-blocks interaction. ► Analytical solutions of this equation are obtained.
Introduction
The common feature of all porous media is presence of two phases: solid phase (solid matrix) that occupies a part of domain and fluid phase, which occupies a void space in the rest of the domain. For example, the pore space of terrigenous rocks is a complex irregular system of communicating (but sometimes isolated) intergranular voids (pores) with sizes in the range from micrometers to tens of micrometers. Besides the granular porous medium which contains fluid in the intergranular pores space, in fractured rocks fluid flow takes place in a network of interconnecting fractures. A fractured porous medium is made up of blocks of an ordinary porous medium, possessing the nonzero porosity of blocks and high hydraulic conductivity of the network of fissures. Water flow and solute transport by the seeping groundwater are relatively slow and it is not possible to make experiments over the thousands of years and hundreds of meters of interest. Instead one has to rely on models that describe the processes and mechanisms that will be dominant over long times.
In order to deal with the flow and transport problems in fractured porous systems a number of modeling approaches were developed. These modeling approaches are traditionally divided into two major classes: discrete fracture models and continuum models. The first class is based on depicting the rock as a network of discrete fractures and another as a non-uniform, single, dual or multiple continuum. A third way of modeling is to combine these into a hybrid model of a non-uniform continuum containing a relatively small number of discrete dominant fractures. Extensive reviews of various approaches for modeling transport in fractured porous media (discrete models, continuum models and hybrid models) can be found in [43], [44], [8]. Although the discrete fracture network models are considered to be conceptually appealing approach (see, for example, [12], [18], [41], [46], [48], [51], [56]), in many situations a continuum concept is sufficient. For example, if fracture density is sufficiently high so that statistical methods can be employed, the system is likely to behave like a continuum. Furthermore, even for rather sparse fracture distribution, the continuum model with proper calibration can provide rather accurate tool for modeling the aquifer as it is shown in [19]. It is worth noting that continuum models are more convenient for practical applications, since these models demand less field data for calibration than discrete fracture network models [8], [19]. Continuum models can consist of single continuum, double continuum, or multiple interacting continua. The double-porosity models (double continua), which have been suggested by Barenblatt et al. [3] and later, independently, by Warren and Root [57], are often used for modeling flow and mass transport in the fractured systems [40], [59], [23]. These models account for two interacting systems (fractures and porous blocks) wherein each is conceptualized as a continuum occupying the entire domain.
Numerous field experiments for the solute transport in highly heterogeneous media (see, for example, [5], [27], [50], [54], [58], [20]), demonstrate that solute concentration profiles exhibited anomalous non-Fickian growth rates, skewness, sharp leading edges and so-called “heavy tails”. These effects cannot be predicted by the conventional mass transport equations. In [9], [10], [11], [33] a continuous time random walk (CTRW) formalism has been applied to quantify the anomalous character of the chemical transport within porous and fractured geological formations. These studies validate the relevance and effectiveness of the CTRW approach by analyzing numerical simulations and laboratory and field measurements. Hilfer and Anton [30] and Barkai et al. [4] have shown that in asymptotic case (large time and/or distances) the CTRW converges to the fractional-order differential equations. Fractional differential equations, which may be viewed as long-time and long-space limit of CTRW, were successfully applied to describe anomalous transport phenomena in many areas [6], [7], [22], [28], [31], [35], [36], [37], [38], [39], [49], [52]. In [15] it was pointed out that “fractional differential equations have two advantages over a random walk approach: first, they allow one to explore various boundary conditions and, second, to study diffusion and/or relaxation phenomena in external fields. Both possibilities are difficult to realize in the framework of CTRW”. As it was demonstrated in a number of publications [1], [7], [54], [58], [20], [21], [29], [34], fractional differential equations is an effective tool for simulating the anomalous character of solute transport in highly heterogeneous media. Recently, Schumer et al. [54] suggested a relatively simple mobile/immobile model, with fractal retention times, capable of simulating the anomalous character of solute concentration distributions for the flows in heterogeneous media. Suggesting inclusion a fractional time derivative into the mass transport equation (in addition to the conventional derivative with respect to time), Schumer et al. [54] referred to the conceptual model of multi-rate diffusion into immobile zones that had been described by Cunningham et al. [16], Haggerty and Gorelick [26], Carrera et al. [13]. On the basis of the model suggested by Schumer et al. [54], Fomin et al. [21] studied the effect of retardation of the contaminant transport caused by diffusion into the confining rocks and porous blocks within the fractured porous aquifer. However, in both cited above publications dealing with mass transport fractured porous aquifer [21], [54], introduction of fractional derivatives in the advection–diffusion equation, which simulates diffusion into the porous blocks within the fractured porous medium, was not properly justified. Instead, inclusion of this term can be viewed mainly as a conjecture. Furthermore, the magnitude of the coefficient in front of the fractional derivative and the order of this derivative were not clearly determined, i.e. factors that affect the values of these parameters were not fully determined or derived and no solid explanation of the physical meaning of these parameters and limits of their deviation were given. In the present paper, based on double-porosity model, fractional order advection diffusion equation in fractured porous aquifer was derived analytically. The expression for the coefficient in front of the fractional derivative was obtained and all parameters that can affect its value are identified. It is shown that the order of the fractional derivative in the advection–diffusion equation depends on the fractal dimension of the porous medium and the limits within which the order of fractional derivative may vary are also defined. In our modeling the continuum approach is used that leads to the macroscopic description of transport phenomena in porous systems. Reducing the size of the elementary volume containing the chosen point we shall assume that the volume remains large enough to contain a sufficiently large number of pores and grains. At the same time, since the chosen volume centered at a point is to represent what happens at that point and its vicinity, the size of the volume should not be too large. In literature the volume that possesses these features is referred to as representative elementary volume (REV). Note that a very fact of existence of such an intermediate-asymptotic value of porosity is independent from the size of REV, which should be much smaller than the characteristic scale for the entire porous domain but much larger than the characteristic microscale for the porous medium (size of a pore). Averaging of microscale properties over a REV is a powerful and still modern instrument (even though it has a long history of applications) for constructing the macroscale mathematical models for various engineering processes and natural phenomena in porous media. For example, recently, exploiting this approach Gray and Miller [24] developed a thermodynamically constrained averaging theory for modeling flow and transport phenomena in porous medium systems, which can be applied to complex systems involving multiple fluid phases and multiple species. The concept of REV for averaging physical fields defined in the porous medium on the micro-level (pore scale level) can be naturally extended for description of the fractured porous media. The fractured porous medium is defined as a medium in which the void space is composed of two components: an interconnected network of fractures that intersect the porous medium into a set of porous blocks and pores within the blocks of the porous medium. The entire void space is occupied by fluid. Such a domain can be also treated as a continuum, provided that an appropriate REV is defined. It should be noted that very often the fluid behavior in fractures is different from its behavior in the porous blocks. For example, when the fractures apertures are much larger than the sizes of pores in the blocks, the flow takes place mainly through the fractures, while the pore space within the blocks is used for the fluid storage due to the large porosity of the blocks. Under these conditions, it is convenient to consider two ‘apparent fluid phases’ that constitute the fluid in the entire void space: one fluid phase occupies the fracture network and the other occupies the porous blocks, regarding each of these phases as a continuum that occupies the entire fractured porous domain and postulating a possible exchange between these phases (continua). Within this approach, the mass transport in the fractured porous medium can be envisioned as a simultaneous mass transport in the two interpenetrating continua (one represents the fluid within the fracture network and the other - the fluid within the porous blocks) with possible mass exchange between them. Application of the double-continua model, which is also known as the ‘overlapping-continua’ or ‘double-porosity’ model originally proposed in [3], [57], assumes the existence of a common REV for both subsystems.
Section snippets
Derivation of the model of the mass transport in fractured porous medium
Following this approach we should introduce two porosities, one for the fractured m(2) and one for the porous m(3) continuum. Using the double-porosity approach, when porosity does not depend on concentration, the system of governing equations of the mass transport in fractured porous aquifer can be presented in the following form [2], [17], [23], [32]:where c2 and c3 the concentrations in
Mathematical model of the mass transport in the fractured porous aquifer and its solution
A schematic sketch of a fractured porous aquifer is presented in Fig. 1. Cartesian coordinates (x, y) are chosen in such a manner that fluid in the aquifer flows in the x-direction and that the coordinate y is directed upward. It can be readily shown that in the surrounding rocks confining the aquifer from below and above the gradient of solute concentration in the x-direction is much smaller than that in the direction orthogonal to the aquifer. Since the goal of the present study is to estimate
Conclusion
The reliable method for modeling the anomalous contaminant transport by the groundwater in the complex medium of highly heterogeneous fractured porous rocks is presented. This method is based on the principle laws of mass conservation and continuum approach to transport in porous medium, at which the transport problem is transformed from the microscopic level (pore size level) to a macroscopic one, by averaging the microscopic values over the properly defined representative elementary volume
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