## 1 Introduction  
Carbonate formations are considered to be useful reservoirs for human development. They are often used for their resources such as water, oil and gas, but also for their storage capacity or their ability to heat, such as in geothermal energy [1, 2]. It is essential to comprehend the properties of these reservoirs in order to operate them in a sustainable and responsible way. Carbonate rocks are known to be highly heterogeneous, with varying properties at different scales. They are known as a complex medium, due to their formation of component particles in matrixes composed of cement and/or limestone mud [3, 4, 5, 6]. Variabilities in microstructure lead to difficulties in reservoir characterization. Studying pore scale is need to understand carbonate systems. Rock structure is adequately characterized by measuring and calculating properties at pore scale [7, 8, 9, 10]. Furthermore, once the structural-property relationships are identified at pore scale they can be replicated at large scale, therefore facilitating large scale reservoir characterization. Two factors are essential in rock characterization: solid structure [11] and porosity [3]. Indeed, geometry and arrangements of pores are crucial features in the comprehension of reservoir structure and its characterization [5, 12]. Different techniques are used to characterize pore arrangement and associated petrophysical properties. Seismic properties are known to be strongly affected by microstructure and grain contacts. Measurements of P and S waves velocities by current injection in rock are therefore very interesting for structure characterization [7]. Indeed, porosity is considered the most important parameter in controlling elastic wave velocities, as it increases when velocities decrease [10]. Otherwise,hydrodynamic and transport properties are controlled by the pores' geometry, connectivity (i.e., general availability of pathways for transport [13]) and network architecture. Indeed, the electric current is conducted by the saturating fluid present in the connected poral phase of the rock. Then, properties can be deduced from electrical resistivity measurements, using empirical resistivity-porosity [14] and porosity-permeability relationships ([PERSON]'s equations, [15; 16]). [PERSON]'s law is used at various investigation scales and relates porosity to rock resistivity through the formation factor and the cementation index. Even if this equation is proven to be more efficient on sandstones rocks, there is no better law to characterize carbonate rocks. Tortuosity, which is assimilated to connectivity of the poral phase, is also deduced from rock resistivity [17; 18; 19]. Constrictivity (i.e., parameter related to bottleneck effect in pores [20]) is also one of the related parameter which is mostly neglected as it is very complicated to evaluate. Nevertheless, a recent study [21] demonstrates that constrictivity can be used to better estimate the tortuosity. Permeability is also given to strongly control hydrodynamic properties, and can be estimated from the combination of several parameters, such as grain diameters, critical pore size and specific surface [15; 16; 22; 23; 24]. On multiple discrete carbonate samples from two different areas, it has been shown that acoustic velocity depends on porosity and predominant pore type [25]. P and S waves velocities are known to be affected by the age of carbonate rocks, as well as by the porosity type [7]. For example, higher velocities are found in samples with dominant moldic and interfossil porosity than with dominant interparticle and microporosity [7]. Analyses of texture, mineralogy and acoustic properties carried out on dry and saturated samples show that in granular rocks, the shear modulus decreases with saturation, contrary to crystalline and cemented carbonates [26]. The combining effect of microporosity, pore network complexity and pore size are also considered to be very influent for acoustic velocities [27] and electrical resistivity [8; 28].