Transport Models

As long as a (groundwater) system possesses any homogenisation scale, at least in mathematical terms, a REV can be defined, and the concepts of continuum models can be used. In literature, numerous single, double and multiple interacting continua models can be found (see Chapter 2.3.1).

Obviously, the more phases and processes as well as heterogeneities need to be considered for a simulation, the more comprehensive a model needs to be at the end. Transport in a system consisting just of the water and solid phases can be simulated with a transport model that is developed for saturated porous media such as MT3D (Zheng 1990). But when the gaseous phase takes up a significant portion of the groundwater system, a transport model for unsaturated porous media should be used in order to obtain more appropriate results. Depending upon the existence of heterogeneity structures in model domains, various deterministic and stochastic formulations of equations can be used in transport models. Heterogeneity structures can significantly spatially vary velocity fields.

Such velocity fields may be better quantified with stochastic models such as Monte Carlo, by means of perturbation, and random walk analyses than with the deterministic models which only condition heterogeneity structures in model domains (Berkowitz 2002). The following two subsections discuss model approaches to simulate transport in coupled conduit-continuum and hybrid systems.

2.3.1 Coupled Conduit-Continuum Systems

Homogenisation scales can also be found in groundwater systems with embedded dense networks of highly interconnected conduits or fractures (Berkowitz 2002). Therefore, for such coupled conduit-continuum systems, the REV approach can be used and concepts of continuum models can be applied.

Similar to hybrid systems, two different aspects are combined in coupled conduit-continuum, namely the velocities in conduits are often relatively high compared to the ones in the continuum matrix. Thus, the combination of conductive conduits and considerably less permeable continuum results in two possible very distinct flow and transport regimes, i.e., one in the conduits and one in the continuum (compare Chapter 1.1). However, depending upon the magnitude of

exchange between conduits and porous matrix, migration rates along possible highly conductive conduits can also significantly be reduced (Ghogomu &

Therrien 2000 for fractured systems). Generally, it can be stated that in systems with embedded conduits, the hydrodynamic dispersion arises mainly from advective and diffusive exchange between faster flow in the conduit and slower flow (or possibly even immobile water) in the rock matrix (Xu & Pruess 2001).

Factors such as the relative time scales for global flow and transport in coupled conduit-continuum systems and degree of interaction between the conduit and matrix system to establish a local equilibrium determine the type of (single, double or multiple) continuum models that is to be used (Berkowitz 2002). The basic idea behind all these approaches is to explicitly resolve domains with different advective velocities through appropriate spatial discretisation (gridding) to account for their “hydrodynamic dispersion” (Xu & Pruess 2001).

Single continuum model approach. This is the simplest, maybe also the first and foremost convenient approximation of modelling the contaminant transport in groundwater systems with dense networks of highly interconnected conduits. It uses the concept of a single or an effective continuum - equivalent porous medium - model (ECM). The individual conduits are treated, as would they be individual pores in the porous media with different hydraulic conductivities. In each REV, the ECM assumes that the conduits and matrix have the same state variables such as concentrations (Pruess et al. 1990; Xu & Pruess 2001;

Berkowitz 2002).

Double continuum model approach. For groundwater systems with dense networks of highly interconnected conduits for which the ECM is not valid, it is maybe more appropriate to use the concept of a double porosity or double continua, in which the conduits and matrix are treated as two separate interacting continua. In this methodology, a network of interconnected conduits is embedded in a matrix block of low permeability in each REV. The double-porosity approach considers only global flow and transport through the conduit network, which describes the effective porous continuum (Xu & Pruess 2001). The porous host rock behaves thereby as a storage/release reservoir for solutes. For such flow and transport conditions, various phenomenological “mobile-immobile” models have been developed (Berkowitz 2002). To let the two continua interact, an exchange function is introduced. Fluid is thereby exchanged between the matrix continuum and conduits locally, through so-called “interporosity flow”, which is controlled by the difference in, such as the hydraulic heads between the two continua (Xu & Pruess 2001). When in addition to the conduit network, as the matrix has to take an active part in transporting solutes, the concept of a dual- or double-permeability approach can be used. In this approach, the interaction between conduits and adjacent “host rock” strongly control the flow and transport through the entire system (Berkowitz 2002).

Multiple continuum model approach. If the embedded conduit networks have different properties or scales themselves (which may be the case for fractures which were generated by more than one process), but are still dense and the conduits within these networks are highly interconnected, a more accurate representation of the “dispersive processes” in the coupled conduit-continuum system may be achieved with the method of “multiple or overlapping interacting

continua” (Berkowitz 2002). In this approach, the matrix rock is partitioned into several interacting continua that are defined based on the distance from the nearest conduit (Xu & Pruess 2001). Also, to describe more accurately transient interporosity flow and transport by resolving driving gradients such as of concentration at the conduit-matrix interface, the method of multiple interacting continua (MINC) can be utilised. Resolution of gradients near the conduit-matrix interface can be achieved by appropriate subgridding the matrix blocks. The TOUGH family models, for instance, use the MINC concept (Pruess 1983;

Pruess & Narasimhan 1985). According to Xu & Pruess (2001), the MINC method is similar to the “shrinking core” model (Davis & Ritchie 1986; Wunderly et al. 1996). Comparable multiple interacting continua conceptualisations have been used in studies to describe the interplay between fast advective transport in rootholes, wormholes, and cracks, with slow flow and diffusive transport in the soil matrix (Gwo et al. 1996; Xu & Pruess 2001). Apparently, multiple interacting continua approaches require larger computational requirements than other types of continuum models due to the further detailed discretisation of a rock matrix. It is to be noted that multi-continuum model approaches find only validity for heterogeneous media in which regions of higher permeability form spatially extensive correlated structures (Xu & Pruess 2001).

2.3.2 Hybrid Systems

As already discussed in Chapter 1.1, in coupled discrete-continuum or hybrid systems with non-homogenisation scale, the REV approach is not valid for the discrete conduit system. Therefore, hybrid models, which combine aspects of discrete, and continuum models were suggested to simulate (reactive) transport in such domains. As such, the discrete part of a hybrid model could explicitly account for effects of individual conduits on fluid flow and solute transport (Ghogomu & Therrien 2000), which continuum models could not account for.

This is in particular necessary for old mines that are often geometrically complex.

The simplest hybrid model considers flow and transport processes within a single conduit (Berkowitz 2002).

Similarity between simulating flow and transport in underground mines shows that eventually a hybrid approach is necessary to deal with uncertainties and complexity of mine structures. Several methods for estimating the duration of the first flush are briefly described here. The first flush refers essentially to the process of initial displacement of the volume of the mined system through recharged water after completion of rebound or the time that it took for the mine to completely flood following the cessation of pumping (=dr (T)). Prediction of dr is necessary to intercept polluted mine waters before causing damage to the surface environment. dr could theoretically be predicted from mine survey records because it is a function of the void volume of the flooded mined system and the rate of recharge if

• direct measurement of ‘goaf’ areas (collapsed roof strata) in mines were more practical,

• the estimation techniques for that purpose were more reliable and or

• the volume of unmined strata around the mine voids which have been dewatered by mine drainage were more precise known.

However, since, as outlined above, measurement techniques do not or cannot provide precise numbers about the mine void volume, two other types of approximations can be utilised which were in particular acknowledged useful by Younger (2002), namely, the semi-distributed lumped-parameter and the physically-based, distributed models. Semi-distributed lumped-parameter models, also known as ‘pond models’ or ‘box models’, are often sufficiently simple that they can be implemented within a spreadsheet environment. This is possible by reducing the complexity of both mined void volumes and recharge processes through ‘lumping together’ large volumes of extensively-interconnected mine voids as single hydrological units (“ponds”). In general, pond-based rebound models are best applied to relatively large-scale mine systems, such as hundreds to thousands of square kilometre underlying areas (Younger 2002). Physically-based, distributed models regard mines as systems of conduits (representing mine roadway networks / stopes, in which flow may well be turbulent) routed through heterogeneous, variably-saturated porous media (representing the enclosing rockmass, both intact strata and rocks which have fractured in response to mining of voids nearby). However, unlike the ‘pond’

models, these demand numerical methods to find solutions. The mined voids can be mathematically expressed in many ways. One option is to use the Navier-Stokes theory to represent the mine system as a multiple-fracture system. This option is probably the most appropriate when the “conduits” are irregular, planar fractures. Though, in most mined systems, the major conduits are tube-like roadways that are better represented as pipes rather than planar fractures.

Therefore, models like the VSS-NET, a purpose-written code in which a 3-D pipe network formulation (based on the Darcy-Weisbach formula and the Gradient Algorithm network solver) is routed through a 3-D, column-oriented (as opposed to layer-oriented), block-centred finite difference grid that is configured to solve for saturated-unsaturated flows (Younger 2002).