## 1. Introduction

Most near-surface carbonate karst systems host groundwater reservoirs that supply freshwater to 20–25% of the global population [

1]. Deeper carbonate formations contain around 60% of the world’s conventional petroleum [

2]. Despite increasing pressure on resources stored in karst reservoirs and the consequent need for sustainable management tools, modelling fluid dynamics in karst systems continues to be a challenge.

Specific karst features, especially conduit networks, are difficult to consider explicitly in models. In addition to their high heterogeneity and anisotropy at all scales that they overprint to the medium, karst conduits may undermine the hypothesis of the Darcian flows that are classically assumed for underground flows. Additionally, the high level of contrast between the hydraulic properties of the different media combined with the size and continuity of karst features makes it difficult to identify a representative elementary volume (REV) for the characterization of properties and upscaling. Moreover, the hierarchical organisation of the conduit network concentrate flows to one or a few outlets [

1]. These outlets are usually river springs whose hydrographs (spring discharge versus time,

Figure 1a) integrate all hydrologic processes occurring in the reservoir to varying degrees and with various delays. For instance, the characteristics of the conduits, their density and connectivity and the structure of the conduit network affect the system response and the hydrograph shape [

3,

4,

5,

6,

7]. The distribution of the recharge between diffuse and concentrated flows and the exchanges between the matrix and conduits also control the characteristics of the hydrograph, such as peak discharge and base flow [

8,

9]. This integrative role of hydrographs combined with good accessibility make springs favourable monitoring points. Consequently, spring hydrographs constitute primary variables to study karst systems [

10,

11,

12,

13,

14,

15,

16] or calibrate numerical models [

17,

18,

19,

20]. Such discharge rate time series nevertheless differ from the usual piezometric monitoring approaches used to constrain groundwater models.

The importance of the vertical structuration of karst on flow properties and processes at the reservoir scale is widely acknowledged [

1,

21,

22,

23,

24]. Notably, the epikarst and transmission zone constitute very different subsystems, whose petrophysical properties differ enough to be distinguished in the models [

25]. The epikarst is the near-surface weathered zone of the karst system [

1]. Its porosity may reach 10%, while its hydraulic conductivity is generally higher than 10

^{−5} m·s

^{−1} and tends to be isotropic due to alteration processes [

26,

27]. The transmission zone constitutes the relatively unaltered part of the unsaturated zone, where water mainly flows vertically towards the saturated zone. In the transmission and saturated zones, at the scale of the flow unit, the matrix porosity and hydraulic conductivity are usually less than 2% and 10

^{−4} m·s

^{−1} respectively [

24]. Flow processes in the unsaturated zone (soil, epikarst and transmission zone) can vary greatly in time and space [

28,

29,

30,

31]. Variable connectivity inside the flow path network controls the infiltration processes [

19,

29,

32,

33,

34]. Flows in the unsaturated zone can be either direct through conduits or delayed because they slowly circulate in the matrix [

32]. The karst unsaturated zone may therefore act as a main storage reservoir [

35,

36], whose complex functioning largely affects the shape of hydrographs [

26,

36,

37,

38,

39,

40,

41,

42,

43]. However, the unsaturated zone is rarely represented explicitly in models of karst hydrodynamics [

17,

44,

45,

46,

47,

48,

49]. Most modelling studies only consider the saturated zone of the aquifer [

18,

20,

50,

51,

52,

53,

54,

55].

Introducing all these karst specificities into numerical models is difficult. Considering only physically-based 3D models, to date, aquifer-scale karst hydrodynamics have mostly been modelled using equivalent porous medium approaches [

18,

50,

56]. These modelling methods represent the entire karst aquifer (matrix, fractures and karst conduits) as a single equivalent porous medium in which only Darcy’s law applies. This simplification corrupts the simulated global response [

5]. The relevance of such models is therefore dependent of the scale of the problem studied and that of reservoir heterogeneity [

18,

50,

57]. In an opposite way, other modelling techniques enable the explicit representation of discrete channel networks. They allow the simulation of turbulent flow in karst conduit networks with complex geometry while neglecting the storage and flows in the matrix. These models are thus mostly dedicated to fractured reservoirs or conduit flow-dominated karst systems [

58]. Taking into account both a mature karst conduit network and highly capacitive matrix requires a dual media approach [

52,

59]. In double continuum models, matrix and karst conduits are considered as two equivalent porous media linked by exchange terms. Such a dual representation does not solve all the difficulties as, in most cases, karst conduits are represented through an equivalent porous medium with Darcy flow. Moreover, the exchange term between the matrix and conduits cannot be measured and may be difficult to calibrate [

47].

Hybrid models have arisen recently; by coupling a 3D equivalent porous medium representation of the matrix on a grid with networks of discrete 2D fractures or 1D conduits, they hold promise for a realistic representation of karst geometries [

20,

25,

51,

55,

60,

61]. They allow the separate and explicit consideration of some large conductive discontinuities that upscaling rules make it difficult to encompass in the equivalent porous medium representation [

62]. Some hybrid models allow different flow physics in karst conduits to be taken into account [

61,

63,

64,

65]. However, in another paper [

25], we reported the difficulty of considering both turbulent flows in the conduits and unsaturated flows in the matrix. We nevertheless showed the ability of hybrid models to simulate karst hydrodynamics in unsaturated conditions and to reproduce most processes that occur at the conduit scale and that are reported in the literature correctly. Moreover, we highlighted how varying the model parameters affects the flow processes and the exchanges between the matrix and conduits in both the epikarst and the transmission zone. Hybrid models thus seem mature enough, and their availability through market software makes them easy to apply [

66,

67].

Therefore, the question arises of whether approaches taking into account both unsaturated subsystems and explicit karst conduits enhance the simulation of both hydrodynamics at the karst reservoir scale and the hydrograph at the outlet. This paper focuses on the impact of such a configuration and the related parameters on the reservoir scale response, especially the spring hydrograph. First, this approach requires the capacity to distinguish different behaviours in the hydrograph shape, and particularly to determine the key descriptors of this response. Then, we study how these descriptors vary as functions of model parameters, particularly regarding the range of responses that we can expect from models whose parameters are consistent with literature values and how each subsystem, epikarst or transmission zone affects the model response. Based on modelling methods, results and commonly accepted concepts from the literature [

25], we build a 3D hybrid model of a hypothetical karst aquifer; we assess and compare the hydrographs resulting from the simulation of recharge events for various sets of parameters. This work seeks to provide modellers with a range of parameters, guidelines and useful tips to enhance the modelling of field cases.

## 4. Evaluation of Models

The numerical experiments presented in this paper aim to assess the interest and quantify the impact of explicit representations of both karst conduits and unsaturated zones in karst reservoir modelling. We built a single hypothetical model whose geometry and parameters were chosen with the condition of being consistent with the literature. The simulations performed cover a wide range of behaviours, which allows us to highlight the major contributions and limitations of this modelling approach.

#### 4.1. Model Assumptions

Hybrid models are able to reproduce many characteristics of the karst aquifer structure. However, as with any modelling approaches, hybrid flow modelling relies on assumptions and simplifications, which provide a compromise between realism, the ability to provide input data and computational tractability. For example, the conductive discrete features represented in hybrid models are only a small fraction of the actual karst network. Indeed, only the most important drains or an upscaled representation of the preferential flow network can be considered in models because of limitations in both knowledge of the system and numerical capabilities.

In this study, we considered homogeneous recharge and homogeneous hydrodynamic properties for both media, which both minimize preferential pathways and flow hierarchy. Most authors choose an a priori repartition of recharge between the matrix and conduit network to favour concentrated flow [

9,

14]. Here, the flow concentration towards the conduits is enabled by the epikarst subsystem [

25]. Contrasting behaviours are obtained by varying the epikarst flow properties. The effects of the topography and dip are not considered here, although they may play a major role at the reservoir scale in recharge distribution and the concentration of flow towards conduits.

Turbulent flow is characteristic of karst conduits and can be accounted for by using the Manning–Stickler equation [

95,

96]. However, the importance of taking turbulence into account varies with the size and roughness of the simulated conduits; thereby, applying laminar flow equations is sufficient for saturated, mature karst systems with well-developed conduit networks [

65]. In unsaturated flow conditions, recent work successfully coupled variably saturated flow modelling in a matrix with turbulent flow modelling in the conduit [

61]; the scale investigated was nevertheless smaller than in the present case. Here, preliminary tests revealed the difficulty of coupling the Richards equation in the equivalent porous medium and the Manning–Strickler equation in the discrete features. We therefore used Darcy law to simulate conduit flow. Conduits are assumed to be fully conductive whatever their saturation state, which seems to be consistent with the expected properties of the mainly vertical karst conduits in the vadose zone, which never reach saturation.

Only one formula with only one set of parameters was tested regarding the constitutive relationship between the saturation and the relative permeability of the matrix. The thorough assessment of this latter relation would deserve dedicated studies, including datasets of measurements on rock samples, relationship fitting with data and upscaling rules considering small-scale heterogeneity as fractures or vugs. Likewise, assessing the value of the conduit flow capacity is difficult. It is bounded by the concerns of (i) establishing a conductivity contrast between the matrix and conduits, (ii) ensuring sufficient drainage of the recharge for the lower bound, and (iii) avoiding the creation of an overly conductive conduit that would be efficiently replaced by fixed-head boundary conditions for the higher bound. Above all, this parameter must be consistent with the object or the processes it represents.

#### 4.2. Scaling Issues

Providing realistic values for model parameters is a concern when dealing with scaling issues. Upscaling, which should be a key issue in such systems, is surprisingly often neglected when property values are proposed. Laboratory measurements are generally performed for rock samples whose volume is smaller than the representative elementary volume (REV), if it exists, and whose selection criterion is mainly based on the homogeneity of the sample, leading to the avoidance of specific carbonate features such as fractures, vugs or fossils [

97]. At the larger scale, the equivalent permeability value for a given larger volume strongly depends on the geometric organization of the permeability field within this volume, which often lacks characterization [

98]. Moreover, considering hybrid models requires thresholds in hybrid implicit–explicit representations of fractures and karst features to be partitioned [

62]: smaller drains should be lumped with the rock matrix in the upscaling process to limit the number of discrete features explicitly represented in the model. Finally, dealing with variably saturated flow modelling may raise the most topical scaling issues, with both theoretical [

99] and methodological [

100] unanswered questions.

In this work, parameter values were chosen in a usually admitted range based on the literature review, assuming that the values in the literature—which are generally independent of the support and not actually measured—are effectively representative of the volumes to be quantified for the model grids.

#### 4.3. Evaluation of Models Outputs

#### 4.3.1. The Need of Hydrographs Descriptors

The effect of varying parameters has been quantified on the simulated hydrographs. In order to assess the differences between the hydrographs resulting from the various simulations, we defined some characteristics of interest: the peak flow, time after the event until peak flow, discharge duration, skewness and kurtosis. Moreover, we proposed a hydrograph classification based on inflections points and—more generally—slope changes.

Only four of the five proposed types of hydrographs were obtained with the model. As type 3 and type 5 occur, the absence of the intermediate type 4, which includes an early peak followed by an inflection point and corresponds to a common observed shape of hydrographs, is probably related to the need for a delicate parametrization to produce it, but may also highlight some flaws in the model setup. For instance, a matrix area distant from the karst network would have poor drainage due to the use of uniform parameters, with the consequence of giving an important weight to the diffuse flow component and the possible over-sensitivity of the related parameters, which should therefore be finely controlled to produce a type 4 inflection point. This simplification also contributes to explaining the high number of type 2 hydrographs including a wide distribution of the diffuse flow component. These considerations highlight the impact on the hydrograph shapes of large-scale heterogeneity in the karst conduit distribution.

#### 4.3.2. Matching Model Outputs with Field Measurements

Even if the modelled aquifer is hypothetical, the resulting hydrograph characteristics seem to be realistic in terms of some aspects for an aquifer with a catchment area of 100 km

^{2} and a uniform thickness of 250 m: the peak flow value varies between 597 and 1063 L·s

^{−1}, the peak flow time varies between 4 and 204 days and the discharge duration varies between 912 and 3464 days. We use skewness and kurtosis descriptors for the shape of the hydrographs.

Figure 3 shows kurtosis as a function of skewness for all the simulations and for hydrographs from the well-known karst systems described in

Section 4.3.2. The values from simulations are consistent with the values from field sites. They cover the same ranges, and the reference simulation is almost centred. The long discharge durations could possibly be questioned, but these can probably be related to the huge uncertainty related to the upscaling issue. This is likely accentuated here by the model structure, with a poorly karstified area far from the represented karst network. These results nevertheless highlight the important delaying effect of the unsaturated zone.

## 5. Conclusions

This work focuses on the consideration of several karst zones and explicit conduits in the reservoir modelling of a karst aquifer at a large scale. Together with the saturated zone, the models include the unsaturated zone, in which a distinction is made between the epikarst and the transmission zone. More generally, the paper addresses the issue of performing realistic simulations of flows in complex media such as a karst. Based on numerous flow simulations on a hypothetical karst aquifer model, we investigated the ability of hybrid models to simulate spring hydrographs that are usual observations in karst studies. Moreover, we explored the relationships between model parameters and the relevant hydrograph characteristics.

In addition to classical characteristics such as the maximum discharge value and corresponding time, we have considered other key features, such as inflections, but also the overall hydrograph shapes through parameters such as skewness and kurtosis or the proposed classification. All these features are definitively useful for both the study of hydrographs and the analysis of flow simulation results.

At the reservoir scale, the hydrograph incorporates the hydrodynamics of the entire system and therefore constitutes a primary output to assess or calibrate a model. Varying parameters affect pathways distribution and transit times to various extents, which results in a large variety of hydrograph shapes. The relationships between model parameters and hydrograph characteristics are not all linear: some of them have local extrema (e.g., peak flow time vs thickness of epikarst) or threshold limits (e.g., all characteristics vs thickness of the transmission zone). The numerous simulations help to assess the sensitivity of hydrograph characteristics to the different parameters. For instance, the discharge duration is more sensitive to the storage capacity (porosity and thickness) of the epikarst than to its conductivity. More generally, the storage capacity appears to be at least as important a feature as hydraulic conductivity in flow distribution. Therefore, this study should help researchers involved in modelling to identify the key parameters to modify to reproduce observations from actual sites.

Finally, the hybrid models are able not only to reproduce flow processes at the interface between the matrix and conduit [

25] but also to simulate the overall response of complex karst aquifers. Several avenues for improvement nevertheless arise, in particular with regard to the problems of flow physics up-scaling in both unsaturated porous media and conduits.