Low enthalpy geothermal energy is a renewable resource that is still underexploited nowadays in relation to its potential for development in society worldwide. Most of its applications have already been investigated, such as heating and cooling of private and public buildings, road defrosting, cooling of industrial processes, food drying systems or desalination.

Geothermal power development is a long, risky and expensive process. It basically consists of successive development stages aimed at locating the resources (exploration), confirming the power generating capacity of the reservoir (confirmation) and building the power plant and associated structures (site development). Different factors intervene in influencing the length, difficulty and materials required for these phases, thereby affecting their cost.

One of the major limitations related to the installation of low enthalpy geothermal power plants regards the initial development steps that are risky and the upfront capital costs that are huge.

Most of the total cost of geothermal power is related to the reimbursement of invested capital and associated returns.

In order to increase the optimal efficiency of installations which use groundwater as a geothermal resource, flow and heat transport dynamics in aquifers need to be well characterized. Especially in fractured rock aquifers these processes represent critical elements that are not well known. Therefore there is a tendency to oversize geothermal plants.

In the literature there are very few studies on heat transport, especially on fractured media.

This study is aimed at deepening the understanding of this topic through heat transport experiments in fractured networks and their interpretation.

Heat transfer tests have been carried out on the experimental apparatus previously employed to perform flow and tracer transport experiments, which has been modified in order to analyze heat transport dynamics in a network of fractures. In order to model the obtained thermal breakthrough curves, the Explicit Network Model (ENM) has been used, which is based on an adaptation of Tang's solution for the transport of the solutes in a semi-infinite single fracture embedded in a porous matrix.

Parameter estimation, time moment analysis, tailing character and other dimensionless parameters have permitted a better understanding of the dynamics of heat transport and the efficiency of heat exchange between the fractures and the matrix. The results have been compared with the previous experimental studies on solute transport.

An important role in transport of natural resources or contaminant transport through subsurface systems is given by fractured rocks. Interest in the study of dynamics of heat transport in fractured media has grown in recent years because of the development of a wide range of applications, including geothermal energy harvesting (Gisladottir et al., 2016).

Quantitative geothermal reservoir characterization using tracers is based on different approaches to predicting thermal breakthrough curves in fractured reservoirs (Shook, 2001; Kocabas, 2005; Read et al., 2013).

The characterization and modeling of heat transfer in fractured media are particularly challenging as open and well-connected fractures can induce highly localized pathways which are orders of magnitude more permeable than the rock matrix (Klepikova et al., 2016; Cherubini and Pastore, 2011).

The study of solute transport in fractured media has recently become a widespread research topic in hydrogeology (Cherubini, 2008; Cherubini et al., 2008, 2009, 2013d; Masciopinto et al., 2010), whereas the literature about heat transfer in fractured media is somewhat limited.

Hao et al. (2013) developed a dual continuum model for the representation of discrete fractures and the interaction with the surrounding rock matrix in order to give a reliable prediction of the impacts of fracture–matrix interaction on heat transfer in fractured geothermal formations.

Moonen et al. (2011) introduced the concept of a cohesive zone which represents a transition zone between the fracture and undamaged material. They proposed a model to adequately represent the influences of fractures or partially damaged material interfaces on heat transfer phenomena.

Geiger and Emmanuel (2010) found that matrix permeability plays an important role in thermal retardations and attenuation of thermal signals. At high matrix permeability, poorly connected fractures can contribute to the heat transport, resulting in heterogeneous heat distributions in the whole matrix block. For lower matrix permeability heat transport occurs mainly through fractures that form a fully connected pathway between the inflow and outflow boundaries, which results in highly non-Fourier behavior characterized by early breakthrough and long tailing.

Numerous field observations (Tsang and Neretnieks, 1998) show that flow in fractures is being organized in channels due to the small-scale variations in the fracture aperture. Flow channeling causes dispersion in fractures. Such channels will have a strong influence on the transport characteristics of a fracture, such as, for instance, its thermal exchange area, crucial for geothermal applications (Auradou et al., 2006). Highly channelized flow in fractured geologic systems has been credited with early thermal breakthrough and poor performance of geothermal circulation systems (Hawkins et al., 2012).

Lu et al. (2012) conducted experiments of saturated water flow and heat transfer in a regularly fractured granite at meter scale. The experiments indicated that the heat advection due to water flow in vertical fractures nearest to the heat sources played a major role in influencing the spatial distributions and temporal variations of the temperature, impeding heat conduction in the transverse direction; such an effect increased with larger water fluxes in the fractures and decreased with a higher heat source and/or a larger distance of the fracture from the heat source.

Neuville et al. (2010) showed that fracture–matrix thermal exchange is highly affected by the fracture wall roughness. Natarajan et al. (2010) conducted numerical simulation of thermal transport in a sinusoidal fracture–matrix coupled system. They affirmed that this model presents a different behavior with respect to the classical parallel plate fracture–matrix coupled system. The sinusoidal curvature of the fracture provides high thermal diffusion into the rock matrix.

Ouyang (2014) developed a three-equation local thermal non-equilibrium model to predict the effective solid-to-fluid heat transfer coefficient in geothermal system reservoirs. They affirmed that due to the high rock-to-fracture size ratio, the solid thermal resistance effect in the internal rocks cannot be neglected in the effective solid-to-fluid heat transfer coefficient. Furthermore the results of this study show that it is not efficient to extract the thermal energy from the rocks if fracture density is not large enough.

Analytical and semi-analytical approaches have been developed to describe the dynamics of heat transfer in fractured rocks. Such approaches are amenable to the same mathematical treatment as their counterparts developed for mass transport (Martinez et al., 2014). One of these is the analytical solution derived by Tang et al. (1981).

While the equations of solute and thermal transport have the same basic form, the fundamental difference between mass and heat transport is that (1) solutes are transported through the fractures only, whereas heat is transported through both fractures and matrix, and (2) the fracture–matrix exchange is large compared with molecular diffusion. This means that the fracture–matrix exchange is more relevant for heat transport than for mass transport. Thus, matrix thermal diffusivity strongly influences the thermal breakthrough curves (BTCs) (Becker and Shapiro, 2003).

Contrarily, since the heat capacity of the solids will retard the advance of the thermal front, the advective transport for heat is slower than for solute transport (Rau et al., 2012).

The quantification of thermal dispersivity in terms of heat transport and its relationship with velocity has not been properly addressed experimentally and has conflicting descriptions in the literature (Ma et al., 2012).

Most studies neglect the hydrodynamic component of thermal dispersion because of thermal diffusion being more efficient than molecular diffusion by several orders of magnitude (Bear, 1972). Analysis of heat transport under natural gradients has commonly neglected hydrodynamic dispersion (e.g., Bredehoeft and Papadopulos, 1965; Domenico and Palciauskas, 1973; Taniguchi et al., 1999; Reiter, 2001; Ferguson et al., 2006). Dispersive heat transport is often assumed to be represented by thermal conductivity and/or to have little influence in models of relatively large systems and modest fluid flow rates (Bear, 1972; Woodbury and Smith, 1985).

Some authors suggest that thermal dispersivity enhances the spreading of thermal energy and should therefore be part of the mathematical description of heat transfer in analogy to solute dispersivity (de Marsily, 1986), and have incorporated this term into their models (e.g., Smith and Chapman, 1983; Hopmans et al., 2002; Niswonger and Prudic, 2003). In the same way, other researchers (e.g., Smith and Chapman, 1983; Ronan et al., 1998; Constanz et al., 2002; Su et al., 2004) have included the thermomechanical dispersion tensor representing mechanical mixing caused by unspecified heterogeneities within the porous medium.

By contrast, some other researchers argue that the enhanced thermal spreading is either negligible or can be described simply by increasing the effective diffusivity; thus, the hydrodynamic dispersivity mechanism is inappropriate (Bear, 1972; Bravo et al., 2002; Ingebritsen and Sanford, 1998; Keery et al., 2007). Constantz et al. (2003) and Vandenbohede et al. (2009) found that thermal dispersivity was significantly smaller than the solute dispersivity. Others (de Marsily, 1986; Molina-Giraldo et al., 2011) found that thermal and solute dispersivity was on the same order of magnitude.

Tracer tests of both solute and heat were carried out at Bonnaud, Jura, France (de Marsily, 1986), and the thermal dispersivity and solute dispersivity were found to be of the same order of magnitude.

Bear (1972), Ingebritsen and Sanford (1998), and Hopmans et al. (2002), among others, concluded that the effects of thermal dispersion are negligible compared to conduction and set the former to zero.

However, Hopmans et al. (2002) showed that dispersivity is increasingly important at higher flow water velocities, since it is only then that the thermal dispersion term is of the same order of magnitude or larger than the conductive term.

Sauty et al. (1982) suggested that there was a correlation between the apparent thermal conductivity and Darcy velocity; thus, they included the hydrodynamic dispersion term in the advective–conductive modeling.

Other similar formulations of this concept are present in the literature (e.g., Papadopulos and Larson, 1978; Smith and Chapman, 1983; Molson et al., 1992). Such treatments have not explicitly distinguished between macrodispersion, which occurs due to variations in permeability over larger scales, and the components of hydrodynamic dispersion that occur due to variations in velocity at the pore scale.

One group of authors have utilized a linear relationship to describe the thermal dispersivity and the relationship between thermal dispersivity and fluid velocity (e.g., de Marsily, 1986; Anderson, 2005; Hatch et al., 2006; Keery et al., 2007; Vandenbohede et al., 2009; Vandenbohede and Lebbe, 2010; Rau et al., 2012), while others have identified the possibility of a nonlinear relationship (Green et al., 1964).

The present study is aimed at providing a better understanding of heat transfer mechanisms in fractured rocks. Laboratory experiments on mass and heat transport in a fractured rock sample have been carried out in order to analyze the contribution of thermal dispersion in heat propagation processes, the influence of nonlinear flow dynamics on the enhancement of thermal matrix diffusion and finally the optimal conditions for thermal exchange in a fractured network.

Section 1 shows a short review of mass and heat transport in fractured media highlighting what is still unresolved or contrasting in the literature.

In Sect. 2 the theoretical background related to nonlinear flow and solute and heat transport behavior in fractured media has been reported.

A better development of the Explicit Network Model (ENM), based on Tang's solution developed for solute transport in a single semi-infinite fracture inside a porous matrix, has been used for the fitting of the thermal BTCs. The ENM model explicitly takes the fracture network geometry into account and therefore permits one to understand the physical meaning of mass and heat transfer phenomena and to obtain a more accurate estimation of the related parameters. In an analogous way, the ENM has been used in order to fit the observed BTCs obtained from previous experiments on mass transport.

Section 3 shows the thermal tracer tests carried out on an artificially created fractured rock sample that has been used in previous studies to analyze nonlinear flow and non-Fickian transport dynamics in fractured formations (Cherubini et al., 2012, 2013a, b, c, 2014).

In Sect. 4 have been reported the interpretation of flow and transport experiments together with the fitting of BTCs and interpretation of estimated model parameters. In particular, the obtained thermal BTCs show more enhanced early arrival and long tailing than solute BTCs.

The travel time for solute transport is an order of magnitude lower than for heat transport experiments. Thermal convective velocity is thus more delayed with respect to solute transport. The thermal dispersion mechanism dominates heat propagation in the fractured medium in the carried out experiments and thus cannot be neglected.

For mass transport the presence of the secondary path and the nonlinear flow regime are the main factors affecting non-Fickian behavior observed in experimental BTCs, whereas for heat transport the non-Fickian nature of the experimental BTCs is governed mainly by the heat exchange mechanism between the fracture network and the surrounding matrix. The presence of a nonlinear flow regime gives rise to a weak growth on heat transfer phenomena.

Section 5 reports some practical applications of the knowledge acquired from this study on the convective heat transport in fractured media for exploiting heat recovery and heat dissipation. Furthermore the estimation of the average effective thermal conductivity suggests that there is a solid thermal resistance in the fluid-to-solid heat transfer processes due to the rock–fracture size ratio. This result matches previous analyses (Pastore et al., 2015) in which a lower heat dissipation with respect to Tang's solution in correspondence to the single fracture surrounded by a matrix with more limited heat capacity has been found.

With few exceptions, any fracture can be envisioned as two rough surfaces in contact. In cross section the solid areas representing asperities might be considered the grains of porous media.

Therefore, in most studies examining hydrodynamic processes in fractured media, the general equations describing flow and transport in porous media are applied, such as Darcy's law, which depicts a linear relationship between the pressure gradient and fluid velocity (Whitaker, 1986; Cherubini and Pastore, 2010).

However, this linearity has been demonstrated to be valid in low flow regimes
(

When

It is possible to express the Forchheimer law in terms of hydraulic head

Fluid flow and heat transfer in a single fracture (SF) undergo advective, diffusive and dispersive phenomena. Dispersion is caused by small-scale fracture aperture variations. Flow channeling is one example of macrodispersion caused by preferred flow paths, in that mass and heat tend to migrate through the portions of a fracture with the largest apertures.

In fractured media another process is represented by diffusion into the surrounding rock matrix. Matrix diffusion attenuates the mass and heat propagation in the fractures.

According to the boundary layer theory (Fahien, 1983), solute mass transfer

In an analogous manner, the specific heat transfer flux

The continuity conditions at the fracture–matrix interface require a balance
between mass transfer rate and mass diffused into the matrix described as

In the same way, the specific heat flux must be balanced by heat diffused
into the matrix described as

The effective terms (

According to Bodin (2007) the governing equation for the
1-D advective–dispersive transport along the axis of a semi-infinite
fracture with 1-D diffusion in the rock matrix, in perpendicular direction to
the axis of the fracture, is

Assuming that fluid flow velocity in the surrounding rock matrix is equal to
zero, the equation for the conservation of heat in the matrix is given by

Tang et al. (1981) presented an analytical solution for solute transport in a
semi-infinite single fracture embedded in a porous rock matrix with a
constant concentration at the fracture inlet (

The heat transport conservation equation in the matrix is expressed as
follows:

In terms of heat transport, the coefficients

The relative effect of dispersion, convection and matrix diffusion on mass or heat propagation in the fracture can be evaluated by comparing the corresponding timescale.

Peclet number

Another useful dimensionless number, generally applied in chemical
engineering, is the Damköhler number that can be used in order to
evaluate the influence of matrix diffusion on convection phenomena.

When

When

The product between

The 2-D Explicit Network Model (ENM) depicts the fractures as 1-D pipe elements forming a 2-D pipe network, and therefore expressly takes the fracture network geometry into account. The ENM permits one to understand the physical meaning of flow and transport phenomena and therefore to obtain a more accurate estimation of flow and transport parameters.

With the assumption that a

In case of steady-state conditions and for a simple 2-D fracture network geometry, a straightforward manner can be applied to obtain the solution of a flow field by applying the first and second Kirchhoff laws.

In a 2-D fracture network, fractures can be arranged in series and/or in
parallel. Specifically, in a network in which fractures are set in a chain,
the total resistance to flow is calculated by simply adding up the resistance
values of each single fracture. The flow in a parallel fracture network
breaks up, with some flowing along each parallel branch and re-combining when
the branches meet again. In order to estimate the total resistance to flow,
the reciprocals of the resistance values have to be added up, and then the
reciprocal of the total has to be calculated. The flow rate

Once the flow field in the fracture network is known, to obtain the PDF at a generic node, the PDFs of each elementary path that reaches the node have to be summed up. They can be calculated as the convolution product of the PDFs of each single fracture composing the elementary path.

Definitely, the BTC describing the concentration in the fracture as a
function of time at the generic node, using the convolution theorem, can be
obtained as follows:

In the same way the BTC

The heat transfer tests have been carried out on the experimental apparatus
previously employed to perform flow and tracer transport experiments at bench
scale (Cherubini et al. 2012, 2013a, b, c, 2014). However, the apparatus has
been modified in order to analyze heat transport dynamics. Two thermocouples
have been placed at the inlet and the outlet of a selected fracture path of
the limestone block with parallelepiped shape
(0.6

Schematic diagram of the experimental setup.

The average flow rate through the selected path can be evaluated as

Two-dimensional pipe network conceptualization of the fracture
network of the fractured rock block in Fig. 1.

Fitting of BTCs at different injection flow rates using the ENM with Tang's solution for mass transport. The green square curve is the observed specific mass flux at the outlet port; the continuous black line is the simulated specific mass flux.

Fitting of BTCs at different injection flow rates using the ENM with Tang's solution for heat transport. The blue curve is the temperature observed at the inlet port used as the temperature injection function, the red square curve is the observed temperature at the outlet port, and the black continuous curve is the simulated temperature at the outlet port.

Solute and temperature tracer tests have been conducted through the following steps.

As an initial condition, a specific value of the hydraulic head difference
between the upstream tank and the downstream tank has been assigned. At

For the solute tracer test at time

As concerns thermal tracer tests at the time

The ultrasonic velocimeter is used in order to measure the instantaneous flow rate, whereas a multiparametric probe located at the outlet port measures the pressure and the electric conductivity.

The Kirchhoff laws have been used in order to estimate the flow rates flowing in each single fracture. In Fig. 3 a sketch of the 2-D pipe conceptualization of the fracture network is reported.

The resistance to flow of each SF can be evaluated as the square bracket in Eq. (34). For simplicity the linear and nonlinear terms have been considered constant and equal for each SF.

The resistance to flow for the whole fracture network

The flow rate

The linear and nonlinear terms are equal, respectively, to

Because of the nonlinearity of flow, varying the inlet flow rate

The behavior of mass and heat transport has been compared by varying the
injection flow rates. In particular, 21 tests in the range
1.83

The observed heat and mass BTCs for different flow rates have been
individually fitted using the ENM approach presented in Sect. 2.3. For
simplicity, the transport parameters

The experimental BTCs are fitted using Eqs. (36) and (37) for mass and heat
transport, respectively. Note that, for mass transport,

The determination coefficient (

Estimated values of parameters, RMSE, and determination coefficient

Estimated values of parameters, RMSE, and determination coefficient

Tables 1 and 2 show the values of transport parameters, the RMSE and

Velocity

Dispersion

The results presented in Tables 1 and 2 highlight that the estimated
convective velocities

In order to investigate the different behavior between mass and heat
transport, the relationships between injection flow rate and the transport
parameters have been analyzed. In Fig. 6 the relationship between

Regarding mass transport experiments according to previous studies (Cherubini
at al., 2013a, b, c, 2014), Fig. 5 shows that for values of

Instead, Fig. 7 shows a linear relationship between

In the same way as for mass transport, for heat transfer a linear
relationship is evident between dispersion and convective velocity. Even if
heat convective velocity is lower than solute advective velocity, the
longitudinal thermal dispersivity assumes higher values than the longitudinal
solute dispersivity. Also, for heat transport experiments, a linear
relationship between

Figure 8 shows the exchange rate coefficient

Transfer coefficient

Regarding the mass transport, the estimated exchange rate coefficient

A very different behavior is observed for heat transport. Heat convective
velocity does not seem to be influenced by the presence of the inertial
force, whereas

The mean residence time

Mean travel time

In Fig. 9 is reported the residence time versus the injection flow rates. The
figure highlights that

Skewness as a function of the injection flow rate for both mass and heat transport.

Tailing character

Peclet number as a function of the injection flow rate

In the same way the skewness

A different behavior for heat and mass transport is observed for the skewness
coefficient. For heat transfer the skewness shows a growth trend which seems
to decrease after

The tailing character does not exhibit a trend for either mass and heat
transport. In either cases

In order to explain the transport dynamics, the trends of dimensionless
numbers

Figure 12 reports

The convective transport scale is very low with respect to the exchange transport scale; thus, the mass transport in each single fracture can be represented with the classical advection dispersion model.

As regards heat transport,

Heat power exchanged per difference temperature unit

Effective thermal conductivity

These arguments can be explained because the relationships between

Furthermore this effect is evident also in the trend observed in the graph

Note that even if

Figure 14 shows the dimensionless group

In order to find the optimal conditions for heat transfer in the analyzed
fractured medium, the thermal power exchanged per unit temperature difference

Moreover, in a similar way to

In order to estimate the effective thermal conductivity coefficient

This result is coherent with previous analyses on heat transfer carried out on the same rock sample (Pastore et al., 2015). In this study Pastore et al. (2015) found that the ENM model failed to model the behavior of heat transport, in correspondence to parallel branches where the hypothesis of Tang's solution of a single fracture embedded in a porous medium having unlimited capacity cannot be considered valid. In parallel branches the observed BTCs are characterized by less retardation of heat propagation as opposed to the simulated BTCs.

Aquifers offer a possibility of exploiting geothermal energy by withdrawing the heat from groundwater by means of a heat pump and subsequently supplying the water back into the aquifer through an injection well. In order to optimize the efficiency of the heat transfer system and minimize the environmental impacts, it is necessary to study the behavior of convective heat transport especially in fractured media, where flow and heat transport processes are not well known.

Laboratory experiments on the observation of mass and heat transport in a fractured rock sample have been carried out in order to analyze the contribution of thermal dispersion in heat propagation processes, the contribution of nonlinear flow dynamics to the enhancement of thermal matrix diffusion, and finally the optimal heat recovery and heat dissipation strategies.

The parameters that control mass and heat transport have been estimated using the ENM model based on Tang's solution.

Heat transport shows a very different behavior compared to mass transport. The estimated transport parameters show differences of several orders of magnitude. Convective thermal velocity is lower than solute velocity, whereas thermal dispersion is higher than solute dispersion, mass transfer rate assumes a very low value, suggesting that fracture–matrix mass exchange can be neglected. Non-Fickian behavior of observed solute BTCs is mainly due to the presence of the secondary path and the nonlinear flow regime. Contrarily, heat transfer rate is comparable with convective thermal velocity giving rise to a retardation effect on heat propagation in the fracture network.

The discrepancies detected in transport parameters are moreover observable through the time moment and tail character analysis which demonstrate that the dual porosity behavior is more evident in the thermal BTCs than in the solute BTCs.

The dimensionless analysis carried out on the transport parameters proves
that, as the injection flow rate increases, thermal convection timescale
decreases more rapidly than the thermal exchange timescale, explaining the
reason why the relationship

Thermal dispersion dominates heat transport dynamics and the Peclet number, and the product between the Peclet number and the Damköhler number is almost always less than the unit.

The optimal conditions for thermal exchange in a fracture network have been
investigated. The power exchanged increases in a potential way as

The Explicit Network Model is an efficient computation methodology to represent flow, mass and heat transport in fractured media, as 2-D and/or 3-D problems are reduced to resolving a network of 1-D pipe elements. Unfortunately, in field case studies, it is difficult to obtain full knowledge of the geometry and parameters such as the orientations and aperture distributions of the fractures needed by the ENM, even by means of field investigation methods. However, in real case studies the ENM can be coupled with continuum models in order to represent greater discontinuities with respect to the scale of study, which generally gives rise to preferential pathways for flow, mass and heat transport.

A method to represent the topology of the fracture network is represented by multifractal analysis as discussed in Tijera at al. (2009) and Tarquis at al. (2014).

This study has permitted one to detect the key parameters to design devices for heat recovery and heat dissipation that exploit the convective heat transport in fractured media.

Heat storage and transfer in fractured geological systems is affected by the spatial layout of the discontinuities.

Specifically, the rock–fracture size ratio which determines the matrix block size is a crucial element in determining matrix diffusion on the fracture–matrix surface.

The estimation of the average effective thermal conductivity coefficient shows that it is not efficient to store thermal energy in rocks with high fracture density because the fractures are surrounded by a matrix with more limited capacity for diffusion giving rise to an increase in solid thermal resistance. In fact, if the fractures in the reservoir have a high density and are well connected, such that the matrix blocks are small, the optimal conditions for thermal exchange are not reached, as the matrix blocks have a limited capability to store heat.

On the other hand, isolated permeable fractures will tend to lead to more distribution of heat throughout the matrix.

Therefore, subsurface reservoir formations with large porous matrix blocks will be the optimal geological formations to be exploited for geothermal power development.

The study could help to improve the efficiency and optimization of industrial and environmental systems, and may provide a better understanding of geological processes involving transient heat transfer in the subsurface.

Future developments of the current study will be carrying out investigations and experiments aimed at further deepening of the quantitative understanding of how fracture arrangement and matrix interactions affect the efficiency of storing and dissipating thermal energy in aquifers. This could be achieved by means of using different formations with different fracture density and matrix porosity.

The authors declare that they have no conflict of interest.

Research founded with the regional program in support of the smart specialization and social and environmental sustainability – FutureInResearch. Edited by: J. M. Redondo Reviewed by: three anonymous referees