## 1 Introduction  
Among many geophysical methods, electrical conductivity imaging and streaming potential (SP) measurements attract an increasing interest to better understand and monitor water distribution and dynamics in hydrosystems, aquifers, or reservoirs, e.g., [1, 2, 3]. These geo-electrical methods are used to detect and monitor groundwater flow, e.g., [4, 5, 6, 7, 8, 9], geothermal areas, e.g., [10, 11, 12], detection of contaminant plumes, e.g., [13, 14], monitoring flow of water in the vadose zone, e.g., [9, 15, 16] or eco-hydroology, e.g., [17, 18]. Thanks to accurate petrophysical relationships and advanced hydrogeophysical inversion approaches, these geo-electrical methods can be exploited to estimate hydrogeological parameters of the aquifer, e.g., [19, 20]. Nevertheless, complex environments such as fractured media and karstic formations remain a challenge for geo-electrical methods. Even if fractured media constitute a large fraction of hydrosystems, it remains largely under studied. The interest of geo-electrical methods to study fractured media is clearly established in the literature using electrical conductivity, e.g., [21, 22, 23] or streaming potential, e.g., [5, 24, 25], but the quantitative use of these methods remains scarce in the literature. The nature of fractured media makes it a challenge for integrative methods such as electrical conductivity imaging and streaming potential [26]. Indeed, fractures usually present a huge contrast in terms of properties with the surrounding matrix making it difficult to be clearly identified through diffusion-based methods such as electrical resistivity tomography, e.g., [27, 28]. Different numerical methods have been proven useful for both electrical conductivity and self-potential simulations, among which the very fine discretization in finite-element modelling, e.g., [29; 30] or discrete-dual-porosity, e.g., [23; 31]. Nevertheless, analytical models and petrophysical relationships can be beneficial to the development and use of geo-electrical methods for faster and simpler approaches. Petrophysical relationships relating electrical conductivity and structural properties in fractured media are not abundant in the literature. Among other works, [PERSON] [32] uses crack networks and percolation to study the transport properties, while [PERSON] et al. [33] revisited the classic model of [PERSON] [34] to relate topological and electrical properties. To the best of the authors knowledge, no analytical petrophysical relationships for fractured media exist in the literature. Regarding the modeling of electrokinetic coupling, the most crucial parameter is the streaming potential coupling coefficient (SPCC). That parameter can be obtained through either the Helmholtz-Smoluchoski (HS) equation or the effective excess charge density approach. The HS equation is commonly used to determine the SPCC via the zeta potential, properties of pore water for a single cylindrical capillary, e.g., [35; 36]. Applying a volume averaging procedure, [PERSON] [37] showed that the HS equation is still valid for porous media in case of negligible surface electrical conductivity, see also, [38]. Modified HS equations have been proposed in the literature when the surface conductivity cannot be neglected, e.g., [39; 40; 41]. Then, an alternative approach allows the determination of the SPCC via the excess charge that is effectively dragged by the water flow in porous media, e.g., [42; 43; 44]. Capillary-based models have been successfully applied to describe SP in porous media, i.e., the porous medium is described as a bundle of parallel cylindrical tubes, e.g., [45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56]. However, microstructure of porous media is normally very complex with tortuous and non-circular pores. Recently, [PERSON] et al. (2018) have theoretically studied the dependence of dynamic electrokinetic coupling coefficient on the electrical double layer thickness for a single cylindrical capillary and a single capillary fracture [57].