Introduction

Carbonate rocks are important reservoirs hosting 50–60% of the hydrocarbon reserves around the world (e.g., Burchette 2012). They are characterized by complex fabric and pore geometry which together represent major challenges in the interpretation of seismic data and fluid transport properties. A comprehensive understanding of the relationships between carbonate rock fabric, their porosity and acoustic velocity is essential for interpreting seismic sections and borehole acoustic logs (e.g., Verwer et al. 2008; Regnet et al. 2015; Soete et al. 2015). Assessment of the impact of the storage capacity properties such as porosity, permeability, and rock density on acoustic velocity helps to analyze the origin of reflections on seismic sections and to calculate the impedance models to compute seismic synthetic sections (Biddle et al. 1992; Campbell and Stafleu 1992; Verwer et al. 2008). The nature of the interface between wetting and non-wetting fluids (e.g., water and oil, respectively), which is valuable for hydrocarbon recovery, is a very complex task in carbonate strata and is controlled by the rock microstructures (David et al. 2008, 2011; Pons et al. 2011). The obtained information on the pore microstructures is quantitatively valuable for the modeling of fluid flow processes in porous geomaterials (David et al. 2015).

Many factors control the acoustic properties of carbonate rocks such as pore shape and type, total porosity, pore fluid nature, and rock fabric, making it extremely difficult to attribute the observed variations in the measured seismic velocity to any of these controlling parameters (Wang et al. 1991; Anselmetti and Eberli 1993, 1999; Anselmetti et al. 1997; Kenter et al. 1997, 2002, 2007; Dürrast and Siegesmund 1999; Brigaud et al. 2010; Casteleyn et al. 2010, 2011; Regnet et al. 2015). Moreover, carbonate rocks have, by virtue of their nature, very high diagenetic potential which can easily alter basic rock parameters such as crystal morphology, porosity, and rock rigidity (Braaksma et al. 2003; Verwer et al. 2008). The major diagenetic processes affecting carbonate rocks are compaction, dissolution, cementation, recrystallization, and dolomitization, which may enhance or diminish the rock porosity and, in extreme cases, can totally alter the rock mineralogy and pore fabric (Eberli et al. 2003; Salah et al. 2016). In addition, cementation of micrite and the subsequent changes in fluid saturation have strong impacts on the response of both electrical resistivity and sonic borehole logging in carbonate reservoirs (Fabricius 2003; Fabricius et al. 2007).

Many limestones have different velocities at a given porosity (Sun et al. 2006), and even different porosities at a certain velocity, which are difficult to predict from published models of the porosity-velocity (φ-V) or velocity/density correlations. Therefore, unlike clastic reservoir rocks, theories which describe the propagation of seismic waves in porous media (Gassmann 1951; Biot 1956a, 1956b) are inapplicable to carbonate rocks because of their complex pore network and unique diagenetic texture with unpredictable seismic properties. Simple theories and relationships, as those of Wyllie et al. (1956, 1958), and Raymer et al. (1980), which describe the propagation of seismic wave velocities in sandstones are difficult to apply for pure limestones (Brigaud et al. 2010). Most of the previous studies on the φ-V relationship have been focusing on grain-supported carbonate rocks having a higher proportion of macropores and thus more permeable and much more significant fluid flow and hydrocarbon recovery. In the giant oil fields of the Middle East, microporous carbonate reservoirs are common and represent a significant component of reservoir rocks in the region (Nurmi and Standen 1997; Ehrenberg et al. 2008). These rocks were not adequately studied because, compared with granular limestones, they are less permeable, ineffectively contributing to fluid flow, and hinder the extraction of their hydrocarbon content (Ehrenberg et al. 2008; Vanorio and Mavko 2011). These rocks, although porous with sometimes porosities between 8 and 30%, are characterized generally by low (rarely moderate) permeability (Kaldi 1989; Moshier 1989; Deville de Periere et al. 2011). The intercrystalline micropores in these carbonate rocks occur usually in a lithified matrix of microcrystalline calcite which is known as micritie (Folk 1966).

In addition to the value of limestones as common reservoir rocks all over the world, the carbonate rocks of Lebanon are good karst aquifers and are highly significant engineering and construction materials (e.g., Nader and Swennen 2004). In this study, we are concerned with the analysis of fifteen rock samples collected from the Paleogene strata exposed in Zahle, central Lebanon to investigate their petrography, nannofossil content, as well as other thin-section-derived parameters. In addition, we measure the basic fluid transport and elastic properties including porosity, permeability, density, electrical conductivity, capillarity, and the acoustic wave velocities and then interpret the observed measurements utilizing the results of the petrographic and scanning electron microscope (SEM) investigations. The study of the spontaneous imbibition of selected porous samples provides valuable information on the fluid flow across reservoir rocks. The present study provides also important clues on the factors affecting the propagation of elastic waves and the physical characteristics of microporous limestones, hence upgrading our current understanding on the interconnections between the petrophysical and acoustic properties as well as the microstructural features of carbonate rocks.

Geologic setting and stratigraphy

During the Paleogene, the Arabian Plate was located between the Tropic of Cancer and the Equator (Beydoun 1991; Alsharhan and Nairn 1995; Sharland et al. 2001). During this time, the opening of both the Atlantic and the Indian Oceans has resumed, resulting in a continuous northward movement of Arabia toward Eurasia and rapidly closing the Neo-Tethys Ocean, an event that inherited a deep foreland basin near the plate margin (Sharland et al. 2001). Thus, the early Paleogene times of Lebanon are dominated by deep carbonate platform of a monotonous chalky, fine-grained character, which is partially rich in organic matter.

The lower and middle Eocene rocks in central Lebanon overly the thick chalky Paleocene successions and are represented mainly by two facies: one is a chalky limestone, often with chert concretions which is common in basin deposits, while the other is a nummulitic limestone which is found on the basin margins. The chalky limestone facies is geographically limited due to the influence of the Syrian Arc event on the region (Alqudah et al. 2019). This rock sequence has a total thickness of 400 m in Zahle (Fig. 1) where it is well-exposed (Walley 1997). The encountered facies reflect clearly the influence of tectonics on the area during the Paleogene (Walley 1997; Alqudah et al. 2019).

Fig. 1
figure 1

Geological map of Lebanon in the Eastern Mediterranean region (a), with an inset (b) representing the details of the studied rock exposures in Zahle area which are shown by the red rectangle in (a)

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Tectonic events, mainly the collision between Arabia and Eurasia, affected the sedimentation pattern in the region causing a regional uplift, which resulted in the formation of the paleolake of Zahle (Beydoun 1999; Sharland et al. 2001). Neogene rocks of the Bekaa valley include deposits of non-marine origin containing non-marine microfossils such as charophytes, gastropods, and ostracods (Sanjuan and Alqudah 2018). The preservation and diversity of these microfossils imply that the paleolake of Zahle was well-established and characterized by a permanent, shallow oligotrophic freshwater.

Materials and methods

The studied samples were collected from two closely spaced exposures in Zahle, the first of which is the early and middle Eocene sequence where the chalky limestones enriched in planktonic foraminifera and calcareous nannofossils (Fig. 2 and Fig. 3a–c) prograde to the middle Eocene shallow marine platform rocks which are represented by a nummulitic bank (Fig. 3d). The second location is about 1 km away from the first one, the sediments of which belong mainly to the Miocene (Fig. 1). The exposed rocks in this location are composed essentially of mixtures of marl to marly limestones with some intercalations of ash and lignites (Fig. 3 e and f).

Fig. 2
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Columnar section of the Paleogene succession exposed in Zahle area

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Fig. 3
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Field photographs representing the exposed strata in the studied area: a, b, and c belong to the early Eocene rocks; d represents middle Eocene outcrops, whereas e and f represent the Neogene Zahle Paleolake deposits

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Fifteen rock samples were collected from the Paleogene and Neogene deposits in Zahle. The number of samples that were obtained from the chalky limestones, the nummulitic limestones, and from the Neogene deposits are ten, two, and three, respectively. Core samples were extracted from fresh outcrops using a portable, gasoline-powered, core-driller (Model D261-C). Thin sections were prepared from all collected samples and then examined under the polarized microscope to identify the standard microfacies, to differentiate between the primary and secondary porosity, to investigate the different post-depositional diagenetic processes, and to relate these features to the prevailing paleoenvironmental conditions. The procedures described in Salah et al. (2018), and references therein, were followed her to characterize porosity, its origin, grain types, cement and matrix nature, sorting, and any observed diagenetic process. We also examined the calcareous nannofossil content of the rock samples, which provides additional constraints on the paleoecology. The obtained results are listed in Table 1.

Table 1 Summary of the main results of the petrographic investigation of the studied rocks
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Field-drilled plugs were subsequently slivered in the lab to optimum lengths of 2–5-cm long, cleaned with distilled water, and then dried at a temperature of 80 °C for 24 h (Mavko et al. 2009). Porosity (φ) was determined by the triple weight method using water as the saturating liquid. This method gives the effective porosity of a rock sample since it is only the connected pore spaces that are occupied by the saturating fluid.

The bulk density (ρb, g/cm3) is calculated from the dry mass of the core sample divided by its bulk volume obtained during the porosity measurement, whereas the grain density (ρg, g/cm3) is calculated from the porosity and bulk density using the relation: ρg = ρb/(1 − φ).

Permeability is measured by applying Darcy’s law using nitrogen gas and converted automatically to the liquid permeability (KL) as proposed by Klinkenberg (1941). The electrical conductivity was measured twice, first after samples were saturated with tap water (water conductivity; σw = 0.66 mS/m) and then with saline water (σw = 58 mS/m). The formation resistivity factor (F), at the two different brine concentrations, was calculated from the electrical conductivity of the rock sample (σr) and the water conductivity as F = σw/σr. The obtained petrophysical and electrical conductivity data are listed in Table 2.

Table 2 Petrophysical and elastic properties measured for the dry and water-saturated samples
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Many methods are available to study the capillary imbibition and to monitor the moving capillary front in rocks such as the standard weight increase measurement, X-ray imaging, and ultrasonic monitoring (David et al. 2015). In this study, we used the routine mass increase method due to the water intake with time to determine the coefficient of capillarity and examine its relationship with permeability and the rock microtexture. This was accomplished through continuously weighing a sample while monitoring the height of the water front during imbibition (Siebold et al. 1997). The resulting data are presented by plotting the water absorption as a function of the radical of time. In this way, a straight line would be obtained if the porous rock is replaced by one pipe of constant radius r. Because Earth materials are often inhomogeneous, this linear relationship is never discerned for normal rocks. However, the capillarity (C) parameter can be estimated from the average slope of the plot between the mass of water versus the square root of time plot: C = 1/πR2.dm/d(t)0.5, with R being the sample radius, m is the water mass intake, and t is time.

Primary and secondary wave velocities (Vp, and Vs) were first measured for the dried plugs at ambient conditions using Panametrics Pulser-Receiver (Model 5058PR) and an Agilent DSO-X-2014A Digital Storage Oscilloscope (100 MHz). Two transducers are attached firmly to the plug ends, and the travel time of the acoustic pulse across the core is read on a screen with an accuracy of 0.01 μs and then used, along with the core length, to calculate the corresponding seismic velocity. Three measurements of velocity at perpendicular directions have been taken for each core, and the results averaged in order to account for any possible anisotropy and to have a more representative velocity for each rock sample (e.g., Soete et al. 2015). Acoustic velocities are strongly affected by fluids due to scattering, rock-fluid interaction, and local grain-scale flow effects (Mavko and Jizba 1991). Fluid saturation causes also velocity dispersion where the measured velocity increases with the measurement frequency, whereas the seismic velocity in dry rocks is insensitive to the frequency (Mavko et al. 1998; Khazanehdari and Sothcott 2003). Therefore, both Vp and Vs have been re-measured for the water-saturated samples. The obtained bulk density and acoustic wave velocities were used to compute the acoustic impedance and the Poisson’s (ν) ratio of the studied rocks. The ν ratio is the ratio between transverse contractions over longitudinal stretching of a stressed sample and controls the speed and reflection of seismic waves and, therefore, is substantial for anticipating the quality of rocks (e.g., Rafavich et al. 1984; Soete et al. 2015). It is calculated from Vp and Vs as follows (e.g., Mavko et al. 1998):

ν = (Vp2 − 2Vs2)/2(Vp2 − Vs2).

In seismology, the ν ratio is used as an indicator of the physical properties and the fluid content in rocks, as well as the seismogenic behavior of the Earth’s crust (Zhao et al. 1996; Zhao and Negishi 1998; Kearey et al. 2002; Salah and Zhao 2003; Salah et al. 2011). The elastic moduli and the ν ratio relate stresses applied to rocks to resulting strains and, hence, are good measures of the rock stiffness and mechanical properties (Siegesmund and Dürrast 2014; Yu et al. 2016). The ν ratio and Young’s modulus, in particular, are used to assess the mechanical behavior of rocks used in construction purposes (Gerçek 2007; Hudson et al. 2011). The combinations of either the shear and bulk moduli or the ν ratio and the Young’s modulus are enough to fully characterize the mechanical properties of Earth’s materials. The measured and calculated petrophysical and elastic parameters are listed in Table 2.

Results and discussion

Petrography

Based on the petrography of the collected rocks, the Eocene limestones of Zahle area can be divided into two facies, fine-grained limestones (samples E1–E10) varying from wack- to packstones and coarse-grained limestones (samples E11 and E12). The collected Miocene rocks (samples M1-M3) are mainly packstones with minor wackstone intercalations (Table 1; Fig. 2).

Wackstone

Samples E1, E9, and E10 are dominantly matrix-supported with about 10% bioclasts of planktonic and benthic foraminifera which are generally well-preserved (Fig. 4 a and b; Fig. 5 d–i), in addition to some shark bone remains (Fig. 5g–h). Non-bioclasts in these samples are composed mainly of calcite; matrix is almost calcitic although it contains some amount of iron oxides. Calcium carbonate sparite either partially or completely fills the intragranular pores (Fig. 4 a and b). In the same time, secondary calcium carbonate precipitations fill the spaces in the fractures (Fig. 4c).

Fig. 4
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A plate showing the different types of microfacies encountered in the studied samples of the early Eocene

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Fig. 5
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A plate showing the different types of microfacies encountered in the studied samples of the late Eocene and early Miocene

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Wack-/packstone

This microfacies is detected in samples E4–E7 where the percentage of bioclasts increased slightly. Both well- and poorly-preserved planktonic and benthic foraminifera are detected, reflecting shallower marine settings during the deposition of these rocks (Fig. 4 l, m, n, p, q, r, and s). Non-bioclasts are represented mainly by calcium carbonate crystals of irregular shapes. All bioclasts and non-bioclasts are floating in a groundmass of micrite composed mainly of calcium carbonate as well as other components like iron oxides (Fig. 4 l and m).

Porosity resulting from the intraparticle pores inside the foraminifera is rare in this case since they have been filled with sparite (Fig. 4l). Intergranular pores are also recognized (Fig. 4m). Although the secondary precipitation of sparite diminished the primary porosity in some cases, the pore space volume has increased in others due to the high dissolution of calcium carbonate producing sometimes very large pores (Fig. 4o). Therefore, porosity varies from moderate to high according to the degree of dissolution.

Packstone

This microfacies was detected in the majority of samples and can be divided into three, slightly different, subfacies: the first includes samples E2, E3, and E8; the second comprises samples E11 and E12; whereas the third one includes samples M1, M2, and M3.

Samples of the first subfacies contain both bio- and non-bioclasts with a good percentage of matrix and micro-sparite in their texture. Bioclasts contain benthic and planktic foraminifera, gastropods, and bone fragments (Fig. 4d–k and Fig. 5a–c). Non-bioclasts are represented by well-sorted microcalcium carbonate crystals (Fig. 5a–c). These micro-crystals resulted probably from secondary dissolution and re-precipitation of calcium carbonate as sparite crystals. Porosity is represented by intragranular spaces (Fig. 5c), with clear dissolution effects in the samples (Fig. 5 a and b). The SEM imaging helped to check some processes that might have affected porosity. In this regard, two features have been observed which comprise the free formation of calcium carbonate crystals inside the foraminifera chambers as a secondary precipitation (Fig. 6-1) and the development of fenestral porosity (Fig. 6-5). These two processes have contrasting effects on porosity; the first diminishes porosity, while the second would enhance it.

Fig. 6
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A plate showing the calcareous nannofossil assemblages found in the studied Paleocene and Eocene strata

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The second subfacies is found in samples E11 and E12, in which the Nummulites sp. is the major bioclastic component. The Nummulites mark the margin between the continent and ocean setting pointing to a very shallow marine deposition. In such setting, the bioclasts, i.e., Nummulites sp., are dominant, while micrite is barely found (Fig. 5 k, l, and m). Porosity is very high and is represented by intergranular primary pores and fenestral secondary pores which are filled partially with cementation followed by later dissolution (Fig. 5 j and l).

The third type of packstone subfacies is detected in the Miocene lacustrine sediments, specifically in samples M1, M2, and M3, where a significant amount of gastropods and ostracods is encountered (Fig. 5 n and o). These bioclasts are surrounded by fine lime particles partially filling the interior of the bioclasts (Fig. 5 p and q). The sediments are barely affected by cementation and other diagenetic processes where most of the bioclasts are not filled with precipitates of calcium carbonate although some of the grains suffered dissolution (Fig. 5q). Some samples, however, show an evidence of fracturing which enhanced the porosity initially; later on, a secondary calcium carbonate precipitated in the fracture space (Fig. 5r). In the same way, a secondary calcium carbonate precipitated also inside the gastropod chambers, thus reducing the porosity of the rock sample (Fig. 5s).

Calcareous nannofossil assemblages and porosity

Calcareous nannofossils are more common in the pelagic settings rather than in the neritic zone. Due to this fact, calcareous nannofossil assemblages are common in the majority of the Eocene samples from E1 to E10, whereas samples E11 and E12 and those of the Miocene (M1 to M3) are barren. Marker calcareous nannofossil species recognized in the samples such as T. serotinus and D. barbadiensis assign an early Eocene age for these sediments (Fig. 6-3, and 8).

On a microscale, both intergranular and intragranular pores were observed (Fig. 6-2, 6, 9, and 10), due to the good preservation of microfossils in the deep pelagic settings. Nannofossils, which are defined traditionally by their rims and central area, are commonly empty or have a sieve, cross, or bar shapes. In all aforementioned cases, microporosity still exists if the assemblages are quiet well-preserved or where there is no clay intrusion in the central area (Fig. 6-4, 7, 9, and 11). In few cases, like in Discoaster spp., the central area is closed, meaning that no microporosity can be afforded (Fig. 6-8, 10, and 12).

Transport and elastic properties

Measured values of φ, ρb, ρg, the Klinkenberg permeability, F1 and F2, Vp, Vs, the ν ratio, and the calculated elastic moduli are all listed in Table 2. The measured porosity varies widely from 0.01 to 0.40, with a mean value of 0.23. The lowest-porosity samples (E11 and E12) are grain-supported packstones (nummulitic) enriched in benthic foraminifera with around 20% microcalcite crystals. Although the measured Klinkenberg permeability of the studied samples varies over three orders of magnitude (from 0.005 to 9.013 mD), it is generally low with an average of 0.94 mD. The two low-porosity samples have also the lowest permeability (0.005 and 0.007 mD, respectively). The low permeability originates mainly from the large proportion of micropores and the existence of micritic and microcalcite crystals, which do not contribute effectively to the flow of fluids. The highest porosity was observed in the matrix-supported wackstone facies (Tables 1 and 2).

The bulk and grain densities vary between 1.62 and 2.68, and 2.62 and 2.72, with averages of 2.07 and 2.69 g/cm3, respectively. The unified lithology of the studied rocks is reflected in the very narrow range of the grain density data, whereas the complex diagenetic processes affecting these samples induce wide variations in their porosity and accordingly in their bulk densities. The electrical conductivity data vary over more than three orders of magnitude for a given measurement cycle (Table 2). The calculated values of the formation resistivity factor (F1 and F2) are generally small because of the high conductivity of most rock samples induced by their generally moderate/high porosity.

Acoustic wave velocities in the dry samples vary widely from 2281 to 6449 m/s and from 1438 to 3292 m/s for Vp and Vs, respectively (Table 2). The average Vp and Vs are 3878 and 2213 m/s; whereas the average Vp/Vs and ν ratios are 1.71, and 0.23, respectively (Table 2). The ν ratio varies, however, between 0.16 and 0.33, whereas the Vp/Vs ratio varies from 1.56 to 1.98 (Table 2). The acoustic impedance (AI) of a medium through which the seismic waves are propagating is the product of Vp and ρb. It controls the reflectivity of rocks and is an essential parameter in the synthesis of seismic sections and their correlation with the observed seismic profiles (e.g., Soete et al. 2015). It is affected by many parameters such as lithology, porosity, fluids, and pore geometry (Zhang et al. 2012). The Vp and ρb data of the dry and water-saturated samples listed in Table 2 result in AI values that vary, respectively, from 3.7 to 17.25 × 106 kg/m2.s, and from 3.38 to 17.3 × 106 kg/m2.s.

Discussion

Porosity versus other petrophysical properties

Porosity and bulk density are two closely related parameters which are inversely correlated to each other (e.g., Ali et al. 2018). Their robust negative correlation (Fig. 7a) is a good indicator of the reliability of the measurements. On the other hand, porosity is directly proportional to permeability (the poro-perm relationship) with a coefficient of correlation (R2) of 0.78 (Fig. 7b). The observed slight scatter indicates that, in addition to porosity, there are other factors affecting permeability in carbonate rocks including the grain size, pore volume, pore throat size, and pore geometry. Few samples have different permeabilities at a given porosity such as samples E2 and M1, which have different permeabilities of 0.02 and 0.572 mD, but similar porosities of 0.16 and 0.17, respectively (Table 2, Fig.7b). Another fact is that although most studied samples are porous, with sometimes porosity of up to 0.40, permeability is generally very low (few fractions of a millidarcy), perhaps because of the small size and the poor connectivity of many pores, as well as the dominance of foraminifera with its cul-de-sac (pores closed at one end) porosity. Carbonate rocks have primarily complex pore fabrics and are prone to diagenetic alterations such as dissolution, recrystallization, cementation, and dolomitization, which additionally complicate the poro-perm relationship (Nurmi and Standen 1997; Lapponi et al. 2011; Dewit et al. 2012).

Fig. 7
figure 7

Crossplots of porosity versus bulk density (a), permeability (b), and the formation resistivity factor (F1 and F2) calculated from electrical conductivity measurements at two different brine concentrations (c and d)

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Porosity is strongly correlated with the formation resistivity factor (F1 and F2 data) by power relationships and high correlation coefficients (Fig. 7 c, and d). The greater improvement in the electrical conductivity of brine-saturated carbonate rocks (the second run of measurements) is expected if they have negligible clay content or any other electricity-conducting minerals (Casteleyn et al. 2011; Regnet et al. 2015). The obtained average values of the tortuosity (a) factor in the equation F = a.φ-m (Archie 1942; Guéguen and Palciauskas 1992) at the two measurement runs are 0.56 and 1.83 while those of the cementation (m) exponent are 1.72 and 1.89, respectively. Regnet et al. (2015) found, in his log-log plot of porosity versus the formation resistivity factor, that the data are not clustered by texture because transmitting properties are greatly affected by the pore diversity even within the same setting. They, however, found a clear clustering by micrite microtextures revealing a strong relationship between micrite properties and micrite morphology. In our study, it is difficult to show such correlation because of the small number of samples.

Permeability and other rock properties

Porosity is inversely correlated with bulk density but directly correlated with permeability, respectively (Fig. 7 a and b). Therefore, an inverse relationship is expected between bulk density and permeability (Fig. 8a). Some scatter in the points is seen, and the correlation coefficient is fair (R2 = 0.72). The difficulty to measure permeability in the field and the relative easiness of measuring the electrical properties have encouraged many researchers to study the relationship between permeability and the formation resistivity factor. The formation resistivity factor is inversely related to permeability usually with a power relationship (e.g., Ravalec et al. 1996; Regnet et al. 2015). According to Walsh and Brace (1984), the two parameters are related as K α F-p, where 1 ≤ p ≤ 3. The values obtained in our study are 0.75 and 0.76, which are very low, indicating the poor correlation between permeability and formation resistivity factor and that estimating permeability from electrical measurements is subject to large possible errors (Regnet et al. 2015). The improvement in the coefficient of correlation from 0.58 at the first run of brine concentration to 0.73 at the second run (Fig. 8 b and c) points to the effect of factors other than lithology, such as the type and geometry of the pore spaces as well as the fluid nature which dominate the electrical properties of the investigated samples at the higher brine concentration.

Fig. 8
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Crossplots of bulk density versus permeability (a); formation resistivity factors versus permeability (b and c); permeability versus acoustic wave velocities (d and e); and between Vp and Vs of the dry (solid circles) and water-saturated (colored squares) samples (f)

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Permeability is inversely correlated with the acoustic wave velocities but with fair to moderate correlations (Fig. 8 d and e). Because permeability is sensitive to many other parameters rather than to the total volume of pores only, the relationships between permeability and acoustic wave velocities exhibit higher scatter and lower correlations compared with the φ-V relationship. The measured Vp and Vs for the dry and water-saturated samples are, however, strongly related to each other (Fig. 8f) because they are basically influenced equally by most rock parameters such as lithology, porosity, pore types, and shapes although they may respond differently to the fluid type and degree of saturation. Both relationships have higher correlations indicating the consistency of the measurements. For a given dry rock sample, the Vs is 57% of the corresponding Vp value but decreased to only 48% of the corresponding Vp after saturation (Table 2; Fig. 8F). This is induced mainly by the stronger reduction of shear wave velocities under saturation conditions (Anselmetti et al. 1997; Kearey et al. 2002; Verwer et al. 2008).

Capillary imbibition

It is not easy to achieve complete saturation when conducting capillary imbibition tests because the driving mechanism of the capillary front in substances with pores of variable sizes induces air trapping. Thus, estimating the exact amount of saturation (by water) in the middle phases is not that easy. In contrast, it is much easier to obtain the overall saturation when the front of water reaches the topmost end of the investigated core (David et al. 2015). As stated before, we measured the mass increase (due to water intake) with time for eight samples then plot the mass intake versus radical time data to calculate the coefficient of capillarity C. The imbibition curves of five plugs are illustrated in Fig. 9a, revealing the direct proportionality between the mass intake and the square root of time. The obtained coefficient of capillarity C, the strength of the fit judged from the value of the coefficient of correlation, porosity, and permeability of the tested samples are listed in Table 3. The final saturation at the end of the experiment is also reported. In their study of the X-ray imaging of water motion during capillary imbibition in porous media, David et al. (2011) predicted a positive relationship between the capillarity coefficient (C) and permeability (K) because both of them are controlled by the motion of fluids in the connected pore network of the material. The obtained relationship is illustrated in Fig. 9b where a direct trend is observed between the two parameters which are related to each other as K = 165.97C1.6 (R2 = 0.82). The gradually rising water front is monitored during the experiment as shown for sample 9 in Fig. 9c–f. Recently, Benavente et al. (2015) have arrived to an analogous relationship, which uses both the capillary coefficient and the rock microstructural parameters to calculate permeability as:

Fig. 9
figure 9

(a) Water mass intake evolution of five carbonate samples as a function of the square root of time and (b) the relationship between the coefficient of capillarity and permeability of the studied eight samples. Height evolution of the water front across sample E9 at different time intervals (c–f). The experiment started at 09:19:31 am and ended at 13:31:51 (i.e., lasted for 4 h, 12 min, and 20 s)

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Table 3 Petrophysical properties and coefficient of capillarity “C” of 8 selected samples obtained with the mass intake method
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K = C2.ηr/4φρ2γcosθ.

This equation includes information on water flow rate (the coefficient of capillarity C), water properties (viscosity η and density ρ), wetting condition (interfacial tension γ and the angle of contact θ), and pore structure (porosity φ and radius r of the pore). If these parameters are accurately known, a better correlation between permeability and the capillarity coefficient could be obtained. The scatter in our plot (Fig. 9b) and the moderate correlation coefficient are indeed induced by the variability of these parameters especially porosity and the pore radii. This possibility of predicting important rock parameters from simple experiments justifies the new interest in introducing the inexpensive capillary imbibition tests in the lab (David et al. 2015).

Acoustic velocities and porosity

The relationship between Vp and the Vp/Vs ratio is shown in Fig. 10a. The obtained Vp/Vs and the ν ratios (Table 2) are in general lower than the commonly observed values for limestones (ν = 0.3; Schlumberger 2013) and may result from the existence of crystalline, rather than granular, textures in the studied carbonate rocks (Verwer et al. 2008). This inference is supported by the petrographic study which indicates the dominance of crystalline textures in the studied samples (Figs. 4, and 5). Upon saturation, a great decrease in the shear wave velocity was observed, where the average Vp and Vs are 3723 and 1801 m/s (Fig. 8f) and averages Vp/Vs and ν ratios of 2.10 and 0.35, respectively. In this case, the Vs value is only 48% of the Vp value of the water-saturated sample. The Vp/Vs ratios of the water-saturated samples vary widely between 1.70 and 2.57 (Fig. 10b) compared with only from 1.56 to 1.98 for the dry samples. The higher Vp/Vs and ν ratios of the water-saturated rocks are induced mainly by the greater drop in the shear wave velocity. Although the Vp of the water-saturated samples are generally higher than those of the dry rocks, some previous results revealed a mixed behavior whereby the Vp may increase or decrease with saturation (e.g., Verwer et al. 2008). Such behavior was also observed for our samples (Table 2).

Fig. 10
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Crossplots of Vp versus the Vp/Vs ratio of the dry (a) and water-saturated (b) samples; ρb versus Vp and Vs (c and d); and φ versus the acoustic impedance (AI) of the dry (e) and water-saturated (f) samples

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Acoustic velocities are strongly dependent on bulk densities (Kearey et al. 2002) where velocities generally display linear increase with increasing density and decreasing porosity (e.g., Verwer et al. 2008; Ali et al. 2018), a fact that is clearly supported by the strong positive correlation between ρb and both the Vp and Vs of the presently investigated samples (Fig. 10 c and d). On the other hand, the AI, calculated for the dry and water-saturated rocks, is inversely related to porosity with strong correlations (Fig. 10 e and f).

Total porosity plays the major role in controlling the seismic wave velocity in rocks of similar mineralogical composition where both Vp and Vs decrease with increasing porosity (e.g., Verwer et al. 2008; Regnet et al. 2015). Similar to the usually observed concave-upward trends between porosity and acoustic velocity (Neto et al. 2015), our seismic velocity data display inverse polynomial trends with porosity (Fig. 11 a and b). Velocity variation at a given porosity is higher for the water-saturated samples than for the dry rocks. The good fit of the V-φ relationship and the low/moderate dispersion of velocity data at a given porosity have also been observed in previous studies (e.g., Baechle et al. 2008; Fournier et al. 2011), although the exact pattern may differ for a larger number of samples. Few samples, especially the water saturated, display different seismic velocities at a given porosity. For example, samples E2 and M1 have similar porosities of 0.162 and 0.169 but differ largely in their Vp which are 3903 and 4654 m/s, respectively (Table 2, Fig. 11a) even though both are grain-supported packstones (Table 1). This behavior has also been observed by other researchers and arises possibly due to different pore types in the carbonate rock (Eberli et al. 2003). Pores occurring in a rigid rock framework induce higher velocities than interparticle pores, for example.

Fig. 11
figure 11

Crossplots of φ versus Vp of the dry (solid circles) and water-saturated (colored squares) samples (a); φ versus Vs of the dry (solid circles) and water-saturated (colored squares) samples (b); φ versus ν ratio of the dry (solid circles) and water-saturated (colored squares) samples (c); φ versus Young’s modulus of the dry (solid circles) and water-saturated (colored squares) samples (d); φ versus bulk modulus of the dry (solid circles) and water-saturated (colored squares) samples (e); and φ versus shear modulus of the dry (solid circles) and water-saturated (colored squares) samples (f)

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Tightly cemented marine carbonate grainstones with significant amounts of micropores (Fournier et al. 2011, 2014) show lower seismic wave velocities at a given porosity compared with travertines (Soete et al. 2015), which are deposited in thin water films under greatly variable sub-aerial conditions. Previous studies revealed that not only porosity, but also pore types and shapes, affect the seismic velocity in carbonate rocks, and any increase in the proportion of micropores reduces the velocity at a given porosity. On the other hand, stiff (such as mouldic) pores would induce a more rigid framework and thus higher seismic velocity at a given porosity (Anselmetti et al. 1997; Eberli et al. 2003; Weger et al. 2009; Xu et al. 2009).

For the dry samples, the calculated ν ratio is inversely related to porosity with a high coefficient of correlation (Fig. 11c). Verwer et al. (2008) noticed that the highest porosity rocks have Poisson’s ratios lower than 0.28, which is consistent with our observations. Neto et al. (2015) found also an inverse relationship between porosity and the Vp/Vs ratio of a suite of carbonate rocks from Brazil. Samples showing this behavior are characterized by high-porosity recrystallized textures with a frame that is supporting the shear wave propagation. The ν ratio of the water-saturated samples increases significantly (Table 2 and Fig. 11c) because of the large decrease in Vs with saturation especially at higher porosities. A similar behavior has also been observed for the Neogene carbonates and siliciclastics from the subsurface of the Florida Keys by Anselmetti et al. (1997), revealing the higher Vs sensitivity to the induced shear weakening.

Porosity is inversely related to the elastic moduli of the dry and water-saturated samples (Fig. 11 d, e, and f). They are related to each other by polynomial trends that are clearly concave upward. In addition, there is a clear decrease in the values of Young’s and shear moduli of the water-saturated samples (Fig. 11 d and f), although the Gassmann’s theory assumes that the shear modulus remains essentially the same during fluid substitution. Results obtained in this study as well as other previous studies (e.g., Adam et al. 2006) indicate that this does not apply for limestones. The rocks are usually influenced by rigidity changes (up or down), hence restricting the application of Gassmann’s theory to predict the seismic wave speeds in carbonate rocks. The shear modulus decreases with saturation in granular carbonate rocks but increases in crystalline and cemented carbonates (Verwer et al. 2008). Therefore, the elastic properties of a carbonate rock alone may enable a quick estimation of its granular or crystalline texture and can help in predicting the rock nature from seismic data acquired across boreholes. On the other hand, values of the bulk moduli have slightly increased upon saturation (Fig. 11e).

In addition to porosity, acoustic velocities in carbonate rocks are controlled also both by the depositional lithology and the subsequent diagenetic alterations (Verwer et al. 2008). In this way, primary fabrics may be quickly overprinted by diagenetic changes such as recrystallization and cementation. Most of the matrix-supported wackstones have higher porosities and lower seismic wave velocities. On the other hand, samples E5, E11, E12, M1, and M3 have higher-than-average seismic velocities and are either matrix-supported wackstone/packstone or grain-supported packstones. Verwer et al. (2008) used the texture class assignment proposed by Kenter et al. (2007), whether granular or crystalline, to relate particular ranges of diagnostic acoustic velocities for each texture. Granular textures refer to depositional grains where grain-to-grain contacts support the rock framework. Rocks with granular textures have little matrix or cement and result typically in low Vp, relatively low Vs, and higher ν ratios under saturation conditions. Rocks with crystalline textures have been recrystallized or cemented after deposition and have altered their original grain-dominated fabric to a crystal-dominated one. These rocks have generally low to high Vp, high Vs, resulting in low ν ratios. Highly cemented, sparite-forming and intensive secondary crystal-fill samples of low porosity and higher seismic wave velocity such as E5, E11, E12, M1, and M3 belong to this category (Tables 1 and 2).

Mineralogy and the ν ratio in carbonate rocks

The mineral composition has a larger impact on velocity variations in siliciclastic sediments which exhibit wide variations in mineralogy from quartz to clay, than in carbonates (Christensen and Szymanski 1991; Braaksma et al. 2003). Consistent with earlier findings of Anselmetti and Eberli (1993), Verwer et al. (2008) found no significant effect of the dolomite fraction on the measured seismic velocities of carbonates which indicate that the observed variations in seismic velocities are induced mainly by the rock texture rather than variations in mineralogy. Based on the results of Anselmetti and Eberli (1993, 2001) and Kenter et al. (2002), porosity is the main control on the seismic velocity of carbonate rocks, although pore type and the percentage of micropores play also a role and this is why some rocks have different velocities at a given porosity. However, the details of the coupling between pore types and seismic velocity are not straightforward and very difficult to quantify, in contrast to those of the φ-V relationship, because pore type characterization is a qualitative process and because many pores are combinations of different types and their description remains subject to interpretation (Lønøy 2006; Verwer et al. 2008). Increasing quartz content reduces the acoustic velocities of the carbonate rocks because it resists dissolution, hindering cementation and the associated increase of the seismic wave velocities. On the other hand, acoustic wave velocities increase slightly with increasing dolomite content.

Verwer et al. (2008) evaluated the specific effects of rock texture on a crossplot of Vp and the ν ratio. Samples which plot in the lower left of the plot (with Vp < 4500 m/s and ν ratio < 0.28) have granular textures, whereas samples located on the remaining space of the plot (Vp > 4500 m/s and ν ratio > 0.28 or Vp < 4500 m/s and ν ratio > 0.28) have crystalline textures. In this way, we can distinguish the samples of granular textures (E1, E2, E3, E4, E6–E10, and M2) from those of crystalline textures (E5, E11, E12, M1, and M3) on the plot of Vp versus the ν ratio (Fig. 12a). Moreover, the crossplot of Vp versus ν ratio for the saturated samples displays a V-shaped cluster in which the granular textures occupy the upper part of the left wing, whereas the crystalline textures occupy the lower left and the right wing of the crossplot (Fig. 12b). This means that rocks of granular textures show large increase of ν ratio upon saturation with a mean value of 0.15, whereas rocks of the crystalline textures show only slight ν ratio increase (Verwer et al. 2008). The transition between the granular and crystalline textures occurs at a Vp of 4500 m/s and a ν ratio of 0.28. Early diagenesis and compaction of rocks having granular textures increase their Vp and decrease their ν ratio (e.g., Wilkens et al. 1984). With further maturation and cementation, the Vp will increase accompanied by a slight increase of ν ratio. Toward zero porosity (high cementation and high Vp), the ν ratio of carbonate rocks is approximated by the ν ratio of pure calcite and dolomite at 0.31 and 0.29, respectively (Simmons and Wang 1971). Therefore, samples E11 and E12 (grain-supported packstone) have very low porosity (1.1%), high Vp (above 6300 m/s), and high ν ratios of 0.32. The processes of dolomitization, recrystallization, dissolution, and cementation have differential impacts on the Vp and Vs and, accordingly, on the ν ratio as explained in detail by Verwer et al. (2008). Any diagenetic process may act on a rock whose Vp and ν ratio have a certain position on the plot, but each process has its specific trend on the path (Anselmetti and Eberli 2001).

Fig. 12
figure 12

Crossplots of Vp versus the ν ratio of the dry (a) and water-saturated (b) samples. Plots of φ versus Vp (c) and ρb versus Vp (d) are shown with the models of WTA, RHG, GRD, and ANEB superimposed (see text for details)

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In order to compare the presently obtained trend between porosity and velocity with previous results, we plot our data superimposed on the trends obtained by the Raymer-Hunt-Gardner (RHG) model (Raymer et al. 1980) and the Wyllie’s time average (WTA) equation (Wyllie et al. 1956, 1958) in Fig. 12c. In addition, the density-velocity trends of Gardner et al. (1974) (GRD) and Anselmetti and Eberli (1993) (ANEB) equations are also superimposed on the crossplot of bulk density versus velocity (Fig. 12d). At a given porosity, velocities obtained in our study are lower than those of the RHG model but close to the results of the WTA equation. On the other hand, the obtained velocities are higher than velocities of the GRD model but closer the results of ANEB at a given bulk density. In general, depositional conditions, composition, the sequence of diagenetic alterations, and the subsequent deformation are all factors leading to the observed differences (Anselmetti and Eberli 1993, 2001; Eberli et al. 2003; Verwer et al. 2008).

Conclusions

Transport and elastic properties (porosity, density, permeability, electrical conductivity, and acoustic velocities) coupled with thin sections, calcareous nannofossil, and SEM investigations were conducted on fifteen carbonate samples collected from the Paleogene strata exposed at Zahle, central Lebanon. The electrical conductivity was tested in two successive runs of different brine concentrations, whereas the acoustic wave velocities were measured for the dry and water-saturated samples. We also examined the self-imbibition of selected samples by monitoring the moving capillary front across a sample. From the measured properties, we further calculated other petrophysical and elastic parameters such as the formation resistivity factor, the cementation exponent, the Vp/Vs and ν ratios, the acoustic impedance, as well as the rocks’ elastic moduli. This integrated analysis provides a deep insight into the different parameters controlling the petrophysical and elastic behavior of the investigated rocks.

Porosity is negatively and positively correlated with bulk density and permeability, respectively. Imbibition experiments showed good linear fits between the mass increase and the square root of time. The resulting capillary coefficients exhibit good correlation with permeability. Although porosity has the major control on both the transport (permeability and electrical conductivity) and elastic properties of the studied rocks, the amount of microporosity plays a role as well. Acoustic velocities decrease with the increase in porosity because the ultrasonic wave propagation is impeded in porous media. On average, the shear wave velocity is 57% and 48% of the primary wave velocity of a sample under dry and water-saturation condition, respectively, revealing the great drop in the values of the shear modulus of rocks with saturation and the consequent increase of the Vp/Vs and ν ratios.

Some samples in this study display different seismic wave velocities at equal porosities indicating that in addition to porosity, pore types, sizes, and shapes as well as the rock diagenetic history control also the acoustic wave velocities. Higher velocities may result not only from lower porosity but also from the development of stiff and cohesive micrite microstructures. In addition, velocity increase may also result from increase of grain size and growing contact cementation. Grain size, packing, matrix, cement, and the crystal shapes are therefore additional factors controlling the propagation velocity of seismic waves in carbonate rocks. Acoustic impedance data of both the dry and water-saturated samples vary widely because of the wide variations in the dry ρb and the Vp. Therefore, seismic reflection sections in carbonate sequences may contain reflections produced from different pore types and shapes rather than the existence of intercalations of other non-carbonate minerals. The present study revealed also that seismic measurements such as the changes in the shear modulus and hence the Vp/Vs and ν ratios with saturation are useful tools for discriminating between the different textures of carbonate rocks. The ability to extract the rock texture from seismic measurements allows the prediction of porosity types and its evolution from sonic logs acquired for similar sediments.

Results obtained in this study give a deeper insight into the understanding and interpretation of the various petrophysical properties of microporous rocks which are valuable in seismic-based exploration and evaluation of marine platform carbonate rocks. However, more detailed petrographic and pore fabric analyses, coupled with petrophysical and acoustic measurements on a large number of dry and water-saturated core samples, are required for deeper understanding of the nature of these rocks and for extending the obtained results to the interpretation of the seismic profiles acquired above the corresponding subsurface occurrences of the studied formations.