Model output

Equivalent porous medium (EPM) model

The EPM model is the simplest of the considered models and is widely used in practice. Compared to the dual-continuum model or the discrete-fracture model, it has low computational costs. It can be used for a rough approximation of the transport behavior of a substance, but must be used with care. This becomes obvious when trying to simulate the tracer tests, which were performed at the Akacievej site. The homogeneous equivalent porous medium model could only be fitted to the fast arrival observed in the tracer tests by lowering the (effective) porosity. Very low porosity values (below 1 %) had to be chosen to fit the fast tracer arrival. Then, the peak concentrations were higher than those observed, because diffusion of tracer into the matrix is neglected by the model. This also leads to an earlier decrease of the tracer concentrations, and the tailing observed in measurements due to fracture-matrix interaction cannot be reproduced.

It is possible to obtain a tailing in the simulated breakthrough curve if a heterogeneous parameter distribution in the porous medium is included (Figure 6.9), introducing very conductive structures and less conductive structures that are less penetrated by flow (Pedretti et al. 2013). However, the variation of the hydraulic conductivity must be of a similar order of magnitude to the contrast between fracture and matrix conductivity and have a similar connectivity of the highly conductive zones (fractures). Note that a different vertical placement of the screens for injection and extraction can also lead to a tailing in the breakthrough curves. However, this does not account for fracture-matrix interaction and the storage effect of the matrix.

It is preferable from the point of view of the actual physics to describe fractures as discrete features. However, a very high grid resolution is required to account for thin fractures, so many smaller fractures cannot be included in a model. This can be accounted for by increasing the effective diffusion coefficient.

The use of an EPM model for the simulation of the migration of a contaminant is not recommended, because exchange processes between fractures and matrix are generally neglected and model results may be misleading for risk assessment or remedial planning. For example, remediation times can be greatly underestimated because the effect of back-diffusion from the matrix cannot be reproduced.

Dual continuum model (DCM)

The dual-continuum model matches the observed tracer breakthrough curves from the Akacievej tracer tests better than the EPM model. The second continuum represents the matrix and the coupling of the two continua allows an exchange of tracer between fractures and matrix. With this model, the breakthrough behavior can be reproduced by fitting the fracture and matrix porosities and conductivities and the parameters governing the exchange behavior between fractures and matrix (Dpm, α, a). However, this concept has many degrees of freedom, and it is not clear how to determine the required parameters governing the fracture-matrix exchange experimentally. It is also reported in the literature to be a “black-box” model (Riley et al. 2001). The computational effort required to run this model type is usually moderate. It is not known whether the model is capable of consistently simulating contaminant plume behavior at both small (site) and larger (catchment) scales.

Discrete fracture model (DFM)

The discrete-fracture model aims at representing the actual physics and yields the best results. Drawbacks are the often-limited knowledge of the fracture geometry and parameters and the large computational costs. The computational costs limit the amount of fractures that can be included. The diffusion coefficient must be augmented to account for neglected fractures and channeling within the fissures, increasing the diffusive exchange between fractures and matrix. This was already reported in the literature (Riley, Ward, and Greswell, 2001, DeDreuzy et al., 2013).

When setting up the model for the tracer tests in Geo18s, Geo19d and Geo5, it was sufficient to include just a few horizontal fractures to provide preferential flow paths for the tracer. The fractures were located to match flow log observations in the boreholes. The real medium is likely to have many more fractures on different scales, providing a bigger specific surface area for the exchange between fractures and matrix. This can be accounted for by adjusting the matrix diffusivity, which controls the exchange between fractures and matrix. In the simulations presented here, it had to be increased by a factor of 100 to 1000 in order to fit the observed breakthrough behavior.

Discussion and comparison

The tracer tests and model applications have clearly shown that a crucial aspect of the transport of a substance in fractured limestone cannot be reproduced with a simple equivalent porous medium model: the diffusion and back-diffusion of a substance between fractures with strong flow and low-conductive matrix. In a fractured aquifer, this should be accounted for, or the propagation of a substance will not be realistically simulated. Hence, the use of a traditional equivalent porous medium model is not recommended for fractured limestone aquifers.

The dual-continuum model can describe the exchange between fractures and matrix while keeping computational efforts low. However, the specification of the exchange terms between fracture and matrix continuum has a crucial influence on the modeling results and a physically-based choice is often challenging. It is not clear how to determine the exchange parameters by measurements. Further, it is questionable if a model, once calibrated, can be employed at a different scale without modifying the used parameters. A requirement for the use of a dual-continuum model is a fracture network with many connected fractures with a relatively uniform distribution, since the fractures are represented as averaged quantities in the fracture continuum.

The use of a discrete-fracture model comes with the cost of being the most complex and numerically demanding model described here. However, it represents the actual physics best and can, depending on the knowledge of the fracture system, lead to the best results. Usually, only few details about the fracture network are available. In this study, the information provided by flow logs could successfully be used to setup a representative network containing the major horizontal flow paths. But such a model does not contain smaller fractures and to compensate for that the diffusion coefficient governing the exchange between fractures and matrix had to be greatly increased. With that, the measured breakthrough curves from the tracer tests could be reproduced. The tracer test in Geo18s before pumping clearly shows that the discrete-fracture model best reproduced the observed data.

Since matrix and fractures are both included in the model and the exchange between the two happens naturally (continuity of fluxes, concentrations and heads at the fracture-matrix interface), the discrete-fracture model is the recommended approach in cases were fractures dominate the transport behavior. Even a simple analytical tool (such as Chambon et al. 2011) or a dual-continuum model should be preferred to an equivalent porous medium model, which completely neglects the influence of the fractures.

Return to Content