Davide Geremia
email: davide.geremia@cyu.fr

Christian David

Continuous Recording of Viscoelastic Relaxation Processes at Constant Ultrasonic Frequency Due to Wave-Induced Fluid Flow in a Microporous Carbonate Rock

Viscoelastic relaxation processes were continuously recorded vs. time at fixed ultrasonic frequency in fluid substitution tests on chalk • Increasing the water saturation above 80% induces a transition from relaxed to unrelaxed regime • Johnson's model for patchy-saturated porous media accounts well for the observed P wave velocity and attenuation variations with saturation CHAPTER 4 ULTRASONIC MONITORING OF FLUID INJECTION

Introduction

Monitoring of fluid motion in the crust is one of the most discussed topics in oil & gas applications as well as in geothermal systems and CO2 storage. Detection of fluid migration is commonly done using seismic and sonic methods. In laboratory experiments, studies at ultrasonic frequencies remain the most effective way to highlight observations at large scale, because in such studies the boundary conditions are well constrained. However upscaling is an important part of the process for applications at the reservoir scale (Guéguen et al., 2006).

Regardless of their nature, fluids are proven to be associated with several dynamic effects on wave propagation at different scales, which generally depend on parameters such as pore size and structure. Generally, these phenomena can be grouped into the category of socalled wave-induced fluid flow (WIFF) processes (Müller et al., 2010). The presence of water in compliant pores like cracks for example, can produce overpressures responsible for wave dispersion, a mechanism known as squirt or local flow (Biot, 1956;G. Mavko & Jizba, 1991). Biot (1956) computed velocity dispersion based on inertial and viscous interactions between the fluid and the rock matrix (Mavko et al., 2009a). Dvorkin & Nur (1993) combined the latter into a unified Biot-Squirt (BISQ) model. Other approaches consider the increase in stiffness due to the dynamic pressure generated by a passing compressional wave on fluid filled pores and grain contact areas (Murphy et al., 1986;Tutuncu & Sharma, 1992), or the heterogeneity in fluid Wave-Induced Fluid Flow in a Microporous Carbonate Rock [199] distribution in partially saturated media causing flow between patches with different fluids (White, 1975).

Whereas many studies focused on the analytical description, few experimental efforts have been made to understand these phenomena. Recently many works on velocity dispersion and attenuation were conducted by varying the wave frequency over a broad range (Pimienta et al., 2015;Szewczyk et al., 2018) thanks to technological advances. Earlier studies focused on wave velocity measurements at fixed frequency while varying the degree of saturation. Among those, Murphy (1982aMurphy ( , 1982b) ) observed through measurements at seismic and ultrasonic wave frequencies a marked peak in attenuation and velocity variation both in the very low and high saturation ranges. This was attributed to moisture effects and wave induced fluid flow generating viscous dissipation. Le Ravalec et al. (1996) discussed the influence of fluid distribution during imbibition and drying on P-wave velocity: they stated that two parameters, related to how easily fluid patches can drain or saturate, are able to describe such processes. Cadoret et al. (1998), through drying and vacuum saturation tests on limestones, observed the formation of gas patches by CT scan associated with substantial attenuation. Lebedev et al. (2009aLebedev et al. ( , 2009b) ) pointed out that at intermediate saturation, ultrasonic wave velocities are affected by the patch arrangement and size.

In our study, four kinds of experiments have been conducted at in-situ stress or room conditions to understand the mechanism of wave dispersion and attenuation at ultrasonic frequency: (1) Injection test with applied deviatoric stress, (2) injection test in hydrostatic conditions, (3) imbibition test and (4) drying test. In each experiment, we obtained a remarkable continuous recording of relaxation processes showing a transition from a relaxed regime, where the wave induced pressure has enough time to equilibrate, to an unrelaxed regime, where the pressure is not equilibrated and fluid flow can be neglected (Müller et al., 2010), with a substantial increase in P-wave velocity closely linked to a peak in attenuation. We believe that the designed experimental procedure, combined with the characteristics of the tested rock (i.e. grain and pore size) provided favorable conditions for the observation of a WIFF mechanism at fixed frequency, with fluid distribution as the only variable in the system.

Material, methods and results

The rock selected for this study is the Obourg chalk from Mons Basin (Belgium). It is a fine-grained microporous carbonate rock, formed during the Campanian age. It is composedmainly of calcite, with minor (less than 1%) clayey minerals (Faÿ-Gomord et al., 2016;Voake et al., 2019). Petrophysical properties of this rock are summarized as follows (average properties unless indicated): porosity 43%; water and gas permeability 6.4 x 10 -16 and 5 x 10 -15 m 2 , respectively; grain and bulk density 2720 and 1550 kg/m 3 , respectively; peak pore throat radius (mercury injection) 0.291 µm; peak grain radius (statistical image analysis on SEM pictures) 0.74 µm; P and S-wave velocities (dry samples) from 2200 to 2700 m/s for P-wave and from 1400 to 1500 m/s for S-wave (Geremia et al., 2021). The matrix of this chalk is mainly composed of coccoliths fragments with punctic contacts and a large inter and intraparticle porosity, making the pore space open and well connected (Faÿ-Gomord et al., 2016, 2017;Geremia et al., 2021). The degree of cementation is low. The tests were conducted on cylindrical cores 80 x 40 mm, machined from blocks collected in the Harmignes quarry (South of Mons

Injection tests under stress

A detailed description of the experimental setup can be found in the Supporting text S1 and in Geremia et al. (2021). We performed an ultrasonic survey during water injection into rock samples with six P-wave piezoelectric transducers (PZTs) distributed as shown in Figure 1, allowing us to measure the flight time and the first peak amplitude of the recorded signals. We conducted several injection tests at differential stresses between 6.5 and 7.5 MPa, the aim of which was to relate the applied stress and the amount of injected water needed to cause failure through water-induced weakening (Geremia et al., 2021). In the current work, we only focus on the evolution of P-wave velocity (VP) and amplitude (AP) associated with water saturation. The different injection tests are named as follows, OBXX-YY with OB for Obourg chalk, XX for the applied confining pressure and YY for the applied differential stress, both with units in Bar (10 5 Pa). Figure 1 shows the variations of the observed VP and AP normalized to their value at the start of water injection on the three horizontal paths (plane 1, 2 and 3) for test OB15-65.

For both parameters, we observe a sequential change from bottom to top (stage #1) following the upward rise of the water. Such an evolution of the P wave attributes during the rise of the water front has been analyzed in our previous works (David et al., 2015a(David et al., , 2017a(David et al., , 2017b)). In these studies, a decoupling between amplitude and velocity variations was already observed for that stage, with the decrease in AP occurring before the VP one. A review of possible explanations for such behavior has been proposed by Pimienta et al. (2019). Focusing here on stage #2 where the water front is well above the height of all ultrasonic sensors, we observe that VP increases simultaneously in the three planes, while AP goes through a minimum. Indeed the S-shape evolution of P-wave velocity matches a peak in attenuation, a property commonly found in systems where relaxation processes are taking place, such as in viscoelasticity. There is a striking synchronicity in our observations on all sensors planes, showing that the observed variations are linked to a bulk change in the rock rather than to a propagating event, like in stage #1. Examples of recorded waveforms are also displayed in Figure 1, one before the strong increase in attenuation and one exactly at the peak. Similar observations found for other injection tests conducted at different differential stresses (OB15-68 at 6.8 MPa and OB15-73 at 7.3 MPa) are shown in Figure 2a.

In order to check if the results of our injection tests were related to the differential stress applied in those tests, a water injection test under hydrostatic conditions was conducted as well. The term hydrostatic refers here to an injection test with no differential stress applied, at the same confining pressure of 1.5 MPa. Water was injected with the same pore pressure gradient as for the previously described injection tests and the water saturation reached at the end of the test was about 98%. Once again, the sample was instrumented with six PZTs in the same configuration as for the injection tests. It displays the same attributes as for the injection tests under deviatoric stress (Figure 2c). However, the peak-like shape of attenuation is not symmetrical as the amplitude is recovering very slowly after reaching a minimum value. This may be caused by the fact that, under deviatoric conditions, the effective mean stress (σ1 -2σ3)/3 is much higher than in hydrostatic condition, which may cause (1) signal enhancement and/or (2) the saturation to be close to 100%. For the drying test, data are displayed in reversed time order to be compared with the other data sets, i.e. the evaporation starts at relative time 1 and ends at 0 (the experiment proceeded from right to left).

Imbibition and drying tests at room conditions

An imbibition test was conducted with the same procedure as in David et al. (2015a). The rock sample is suspended under an electronic balance to register the sample mass increase during spontaneous capillary rise. A digital camera records the water rise at the sample surface during the test. The water is kept permanently in contact with the bottom end of the sample. The same PZTs configuration and processing scheme as for the injection test were used. The two stages observed in the injection tests are also recognizable in the imbibition test (Figure 2c); however, the VP in stage #2 does not experience a sharp increase like we observed in Figure 1, whereas the amplitude just slightly decreases. This different behavior is likely due to the air trapped in the pore space which prevents complete saturation. The imbibition test provides however the opportunity to relate the waterfront position with the change in VP. We used this criterion to obtain the water height as a function of time inside the pressure vessel so that we could compute the water saturation (Supporting Text S2).

Once the hydrostatic test described in the previous section was completed, the rock sample was quickly removed from the pressure vessel and immediately suspended under an electronic balance in order to measure the sample mass decrease with time during drying under room conditions. As for the mechanical tests, the waveforms were recorded with 6 PZTs. The rock sample took approximatively 3 days to return to the initial dry mass. Nevertheless, once more they display the same peak in attenuation correlated with the variation in P-wave velocity (Figure 2c). However it is important to note that the time scale for the drying test (several days) is significantly larger than the one for the mechanical tests (several hours).

Discussion

In our experimental data, it is possible to split the velocity-amplitude vs. time data sets in two parts (Figure 1); in stage #1 the P-wave velocity decreases and the first peak amplitude undergoes a fluctuation as a result of the upward motion of the water front, followed by stage #2 where the VP increases and AP experiences a peak in attenuation. The amplitude oscillation observed in the first stage can be induced by a superposition and interference of the direct wave and a reflected wave onto the moving waterfront (David et al., 2018;Kovalyshen, 2018).

Here we focus on stage #2, where the attenuation peak correlates with the VP inflection point. We believe that our experimental data set shows the onset of relaxation processes, marked by a shift from a relaxed regime towards an unrelaxed one with time, at a constant frequency. We postulate that such relaxation processes can be explained by variations in the fluid distribution inside the rock samples during injection, imbibition or drying. In the past, several models for frequency dispersion and attenuation have been developed based on theories like poroelasticity (Biot, 1956;Bourbié et al., 1987;Pride, 2005), on pressure relaxation effects between compliant and stiff pores (e.g. squirt flow, (Mavko & Nur, 1975)) or on viscoelastic relaxation processes. Such a behavior, involving the passage from a low velocity limit to a high velocity limit concomitant with a peak in attenuation is well described by classical viscoelastic models: using the outcome of the standard linear solid (or Zener) model for the evolution of velocity and attenuation in the vicinity of the relaxation frequency, we found that our data set is consistent with an exponential decrease of the relaxation frequency with time, thus with water saturation, intercepting the pulsing frequency fPLS = 0.2 MHz of our ultrasonic device (Figure 2b). Given this result, it is reasonable to link the observed behavior to a frequency-dependent mechanism induced by the progressive saturation and the evolving fluid distribution in the pore space.

Several models have been proposed to account for wave dispersion and attenuation in fluid bearing porous media, allowing for the calculation of the critical frequency at maximum attenuation from fluid-or matrix-related parameters. The outcome of some of these models for the prediction of the relaxation frequency is discussed in Supporting text S3 from which we concluded that among these, the patchy saturation model introduced by White (1975) represents the most suitable one. Indeed, for the Obourg chalk (i) the critical frequency for the drained to undrained transition is 2.3 Hz and (ii) the critical frequency for the transition from (undrained)

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relaxed to unrelaxed regime through the squirt flow model requires a crack aspect ratio far too small to be consistent with our ultrasonic measurements. Conversely, the patchy saturation model is based on relaxation phenomena triggered by fluid heterogeneities in rocks, which are likely to develop during water injection, and considers pockets of one fluid (in our case air) distributed across a medium saturated with a second fluid (in our case water). Kobayashi & Mavko (2016) explored the variation of patch size as a function of saturation. For a critical frequency equal to fPLS, the patchy saturation model requires an elementary patch size of nearly 50 µm (Supporting text S3), which is approximately two orders of magnitude higher than the pore size (≈ 0.3 µm) and three orders of magnitude lower than the wavelength (≈ 11 In White's model (White, 1975), compressional waves induce oscillatory fluid flow across the boundary between both fluids and, consequently, a loss of energy. The patchy saturation model was improved to make it consistent with the Biot poroelasticity theory (Dutta & Odé, 1979a, 1979b;Vogelaar et al., 2010). In the following we adopted the formalism proposed by Johnson (2001Johnson ( , 2007)), hereafter referred to as Johnson's model, who developed a patchy saturation model in the low-frequency limit of the Biot theory. Indeed, for Obourg chalk the Biot crossover frequency (Johnson, 2001) is fB = 76 MHz so that our data set at ultrasonic frequency belongs to the low-frequency regime of the Biot poroelasticity theory. In this regime viscous effects dominate, and the slow Biot wave is a diffusion wave with a wavelength of 54 µm much smaller than the fast P-wave wavelength (Müller at al., 2010). We checked that Johnson's model yields negligible errors with respect to the exact solution provided by Vogelaar et al. (2010). In the low and high frequency limit the bulk modulus tends to the Gassmann-Wood (𝐾𝐾 𝐺𝐺𝐺𝐺 ) and Gassmann-Hill (𝐾𝐾 𝐺𝐺𝐺𝐺 ) effective moduli respectively. In the intermediate frequencies, the complex bulk modulus (𝐾𝐾 * ) as a function of angular frequency 𝜔𝜔 is computed by:

𝐾𝐾 * (𝜔𝜔) = 𝐾𝐾 𝐺𝐺𝐺𝐺 - 𝐾𝐾 𝐺𝐺𝐺𝐺 -𝐾𝐾 𝐺𝐺𝐺𝐺 1 -𝜁𝜁 + 𝜁𝜁�1 - 𝑖𝑖𝜔𝜔𝑖𝑖 𝜁𝜁 2
(1)

where 𝜁𝜁 and 𝑖𝑖 are two parameters depending on the size and the surface to volume ratio of the patches (Johnson, 2001;Müller et al., 2010). Shear modulus variations with frequency and saturation are not considered.

In all the tests performed on chalk, due to the experimental configuration, heterogeneities in the air-water distribution are spread across the rock sample before full saturation. The elementary volume in the model is an outer water saturated sphere of radius b which surrounds an inner air-filled sphere of radius a. The characteristic length scale for interactions between patches is Lp=b-a. Changing the patch size at a fixed wave frequency will allow a transition from the relaxed towards the unrelaxed regime. It is important to note that in nature, the size and shape of the heterogeneities (a and b) should be described by a range of values rather than a unique one: for the sake of simplicity, we stick to the periodic spherical model initially introduced by White (1975). To determine the heterogeneity size, we computed the P-wave dispersion and attenuation as a function of frequency with Johnson's formulation and, by assigning a range of b values we compared the results to the experimental data using a trial-and-error method. This yielded a value b = 63 µm of the same order as the patch length derived from the theoretical relaxation frequency (Supporting text S3). This value for b was kept constant although in reality both a and b probably vary, but since we do not have any control over these parameters, we chose to vary only one of them. Moreover, in isotropic rocks, the pressure gradient generated by a passing wave on these fluid patches is controlled by the fluid-diffusion relaxation scale Ld defined as:

𝐿𝐿 𝑑𝑑 = � 𝑘𝑘𝐾𝐾 𝑓𝑓 𝜂𝜂𝜂𝜂𝜂𝜂 � 1/2
(2) which depends on porosity, permeability, fluid viscosity η, fluid bulk modulus Kf and frequency f. If the patch size Lp is larger than Ld, the pressure gradient between different patches has not enough time to equilibrate. For Obourg chalk, Ld is equal to 120 µm which is also in the same range as the previously mentioned value. Given these results, both the wavelength of the Biot slow wave and the patch size are much smaller than the wavelength of the fast P-wave and, for this reason, we proceeded by implementing the best choice value b = 63 µm as the radius of the outer sphere in Johnson's model. We computed the P wave dispersion and attenuation from the calculation of the complex bulk modulus (equation 1) as a function of water saturation by changing the radius a of the inner sphere (gas pocket) from a = b = 63 µm to a = 0, in a frequency range from 10 2 to 10 10 Hz. The water saturation is computed as Sw = 1 -(a/b) 3 . Doing so, we map out the real part (linked to velocity dispersion, Figure 3a) and imaginary part (linked to attenuation, Figure 3b) of the complex bulk modulus as a function of frequency and saturation. At fixed water saturation, the model reproduces well a peak in attenuation and the Sshape of the bulk modulus (from which the P wave velocity is inferred) as a function of frequency. At fixed frequency a peak in attenuation is observed when varying water saturation: when the frequency increases, the peak in attenuation and the velocity dispersion are smaller and occur at lower values of saturation. The path followed at fixed frequency f = fPLS has been drawn on Figures 3a andb. The computed results along the black line in Figure 3b allow us to calculate in Figure 3c the attenuation coefficient predicted by Johnson's model at f = fPLS as a function of saturation, which successfully matches the experimental data both qualitatively and quantitatively, where the attenuation coefficient α is computed as: with At/Ao the ratio between the amplitude at time t and the initial amplitude, and d the distance between the emitter and receiver. Note that, while Johnson's model is in water saturation domain, the experimental tests are in normalized time domain, revealing that water saturation increases from 0 to a nominal saturation close to 1 with time. The smaller peaks in attenuation occurring at early times (low saturation) are related to stage #1 (water front rise) and can obviously not be accounted for by Johnson's model. By increasing the water saturation, the critical frequency at the relaxed-to-unrelaxed transition decreases (Figure 3d) and the peak in attenuation moves toward lower frequencies, which is in good agreement with the prediction of the viscoelasticity model in Figure 2b (see also the movie in the Supporting Information).

α = -� ln � 𝐴𝐴 𝑡𝑡 𝐴𝐴 0 � 𝑑𝑑 � (3)
The P-wave velocity computed from Johnson's model is compared to the experimental data from test OB15-65 in the insert of Figure 3d (top right corner), for which the water saturation is calculated as explained in Supporting text S2. For comparison, the prediction of Gassmann-Wood and Gassmann-Hill limits (Müller et al., 2010) are also displayed. At lower saturation, the experimental results seem to match the Gassmann-Wood prediction, whereas, at higher saturation, a deviation from this trend leads toward the Gassmann-Hill model. The mismatch in the saturation value at the onset of velocity increase which occurs at Sw = 85% compared to 98% modelled with Gassmann-Wood has also been observed by Lebedev et al. (2009a) on Casino sandstone. They explained such behaviour by variations in the patch arrangement and size which lead the velocity to switch from the Gassmann-Wood to the Gassmann-Hill limits. A similar behavior has been described by Cadoret et al. (1995Cadoret et al. ( , 1998) ) where a departure of the P-wave velocity from the Gassmann-Wood limit is explained by heterogeneities in the fluid distribution. Unfortunately, our modelling does not provide any information about the formation of the patches and their possible merging. Probably a different combination of sample size, rock microstructure and experimental procedure will affect the shape and size of the air pockets which could be analyzed in a further work through imaging techniques. Figure 3d shows that our experimental data are qualitatively similar to the prediction of Johnson's model; in addition, we believe that the lower velocities measured in our experiment (by about 7% at 80% of saturation) and the higher peak in attenuation (Figure 3c) can result from the "water weakening effect" in Obourg chalk (Geremia et al., 2021). Indeed, chemical or surface effects on the grain contact area are known to affect the mechanical properties of rocks and the magnitude of the P-wave velocity passing from dry to water-saturated conditions (Murphy, 1984), which are not accounted for by Johnson's patchy saturation model.

Conclusions

In the present work we showed that fluid heterogeneities can be the cause of significant variation in P-wave velocity and attenuation during water saturation in the microporous Obourg chalk. An almost continuous recording of relaxation processes linked to variations in fluid saturation at fixed ultrasonic frequency was obtained. We believe that such a continuous recording of relaxation processes can feed new models or help to improve existing ones for a better understanding of wave induced fluid flow processes in reservoir rocks.

The observed behavior is similar to the relaxation processes in classical viscoelasticity models like the standard linear solid or Zener model. Indeed at saturation above 80% a transition from relaxed to unrelaxed regimes occurs, as the relaxation frequency, decreasing with saturation, matches at some point the frequency of the pulsing system used in the experimental setup. Application of effective medium models demonstrates that as saturation increases, the system undergoes a transition from the Gassmann-Wood (or isostress) to the Gassmann-Hill (or patchy) behaviour. By applying Johnson's (2001Johnson's ( , 2007) ) formulations for patchy saturation, we have been able to fit the experimental data. This result shows that, within the measurement errors, the spherical shell model originally proposed by White (1975) for patchy-saturated porous media is relevant and accurate enough for the microporous Obourg chalk.

The size of the patches, in Johnson's model, was of the order of 50 µm, which is larger than the pore scale (≈ 0.3 µm) but lower than the wavelength of the fast P wave (≈ 11 mm). It is very likely that (i) the very fine scale of the rock microstructure combining small pore size, low permeability and high porosity, and (ii) the way fluids are distributed in the experimental setup both permit this relaxation phenomenon to be observed at ultrasonic frequency. Nevertheless, the same kind of mechanisms might take place in wave-induced fluid flow occurrence at a larger scale and at seismic frequencies, even in the same type of rock, as long as the condition of patch scale lower than the wavelength is fulfilled.

The results gained from this work provide new insights into the mechanisms acting during water injection or fluid substitution in rocks and, for this reason, they can contribute to a better understanding of seismic monitoring of a reservoir during secondary or tertiary recovery.

Introduction

In the following, the supporting material is divided into three sections:

1. We describe the involved experimental setup and the protocol for the different tests. 2. We explain how we measured the water saturation as a function of time through the imbibition test and how this information is used in the injection tests. 3. We analyze different theories for wave dispersion and attenuation to identify the origin of the recorded relaxation processes.

  4.3.1 Continuous Recording of Viscoelastic Relaxation Processes at Constant Ultrasonic Frequency Due to

Figure 1 .

 1 Figure 1. Evolution of P wave velocity and amplitude normalized to their value in the dry state before the start of water injection. The sample is Obourg Chalk OB15-65 tested at 1.5 MPa confining pressure and 6.5 MPa differential stress. A sketch of the PZT sensors assembly on the rock sample is also shown. A very large drop in the waveform amplitude is observed at the peak of relaxation.

Figure 2 .

 2 Figure 2. a) Results for three injection tests at different deviatoric stresses for stage #2 only, showing similar features as in Fig. 1. b) Relaxation frequency shift with time as estimated through the Zener viscoelastic model. c) Normalized P-wave velocity and amplitude evolution vs. time for the injection test under hydrostatic loading, and for the drying and imbibition tests at room conditions. For the drying test, data are displayed in reversed time order to be compared with the other data sets, i.e. the evaporation starts at relative time 1 and ends at 0 (the experiment proceeded from right to left).

Figure 3 .

 3 Figure 3. Prediction of the patchy saturation model (Johnson, 2001) for the evolution of the complex bulk modulus vs. frequency and saturation: a) Real part and b) Imaginary part. c) Comparison of Johnson's model with experimental data for the attenuation coefficient. d) Variation of the relaxation frequency for the patchy saturation model vs. water saturation and velocity evolution vs. water saturation for effective medium models and experimental data.

  setup, injection test protocol and complementary data setA Sanchez Technologies TM pressure vessel mounted on a GDS Instruments Ltd TM VIS loading frame with servo-controlled axial displacement were used for all the mechanical tests. The core sample is placed inside a neoprene sleeve through which we connected six P-wave piezoelectric transducers (PZTs) with 5 mm in diameter emplaced in a brass holder. The sensors are either glued or simply pushed in contact with the sample using an ultrasonic couplant gel. The sensors are distributed in three planes at different heights, named bottom, mid and top, as shown in the sketch in Figure1, allowing for the measurement of P-wave velocity (called VP hereafter) and for the recording of waveforms in 15 different pathways. For the purpose of this work, only the horizontal/radial VP has been considered.An AMSY-5 Vallen Systeme GmbH (Germany) was used to operate with the sensors in active mode. An ultrasonic survey proceeds by switching each sensor between pulser and receiver. The pulsing frequency imposed during velocity surveys is fPLS=0.2 MHz and the voltage amplitude sent to the emitter is about 100 Volts. Four repeated pulsing cycles are conducted per sensor giving a total of 144 recorded waveforms (6 received signals, times 4 pulses, times 6 sensors). We applied a time interval between each survey of 45 to 60 seconds.The P-wave first arrival time is picked automatically through the Akaike Information Criterion (AIC)(Leonard & Kennett, 1999) which has proved very effective in discarding the possible noise located before the real signal. The VP is computed by applying a calibrated distance matrix knowing the sample dimensions and the sensors positions. No corrections were made for the change in travel path lengths due to strain during the experiments since we found this effect to be negligible. In addition to VP, the P-wave first peak amplitude (called hereafter AP) was also automatically measured in a time window after the P-wave first arrival.

  Recording of Viscoelastic Relaxation Processes at Constant Ultrasonic Frequency Due to Wave-Induced Fluid Flow in a Microporous Carbonate Rock[211] An injection test is carried out following the novel procedure of David et al.(2015b) and adopted inGeremia et al. (2021a). After applying a target differential stress (σaxial -σradial) below the yielding point of the rock and a period of stabilization from 30 to 60 minutes, water injection is started, as shown in the top right sketch in Figure1, with a bottom pore pressure of 0.12 MPa, while vacuuming from the top in order to avoid as much as possible air trapping in the pore space (giving a pore pressure difference of about 0.2 MPa from bottom to top). The described procedure permits to reach a nominal full saturation of the rock sample at the end of the experiment.Text S2: Estimation of water saturation through imbibition testThe imbibition tests give the possibility to link the waterfront position at the surface with the variation in VP. For the imbibition test, a drop in VP of about 10% induced by the rising water is observed in stage #1. This behaviour is observed in all travel paths and takes place systematically when the waterfront is about 5 mm ± 1 mm below the plane containing the PZTs. This distance is measured accurately on the pictures of the sample surface taken every second by the camera. This criterion can be used to infer the waterfront position as a function of time for the injection tests in which we record the VP decrease due to the rising water while we do not see the sample in the pressure vessel. Subsequently, we consider that the water-invaded area is not homogeneously saturated, and, for this reason, we modelled a water saturation profile from the bottom to the top of the area reached by the water. For instance, let us assume in FigureS1that the water reached approximatively the first centimeter of the rock sample, then the black line shows the fluid distribution. The known parameters are the height reached by the water and the total volume injected. With an iterative process, we match this volume with the area below the distribution profile. This is reproduced for each position step and the results give a rightward and upward movement of the distribution profile as shown on the bottom left corner of FigureS1.

  4.3.1 Continuous Recording of Viscoelastic Relaxation Processes at Constant Ultrasonic Frequency Due to Wave-Induced Fluid Flow in a Microporous Carbonate Rock

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  4.3.1 Continuous Recording of Viscoelastic Relaxation Processes at Constant Ultrasonic Frequency Due to Wave-Induced Fluid Flow in a Microporous Carbonate Rock

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  4.3.1 Continuous Recording of Viscoelastic Relaxation Processes at Constant Ultrasonic Frequency Due to Wave-Induced Fluid Flow in a Microporous Carbonate Rock

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Acknowledgments

Many thanks to both reviewers and to the editor for their advices which helped us to significantly improve our work. This work has been funded by the GEO²FRI²SK project granted by the Initiative d'Excellence Paris Seine at CY Cergy Paris Université. DG is greatful to the Ecole Doctorale Sciences et Ingénierie who granted his PhD fellowship. This work has benefited from discussions with Beatriz Menéndez and Christophe Barnes, co-advisors of DG. Fruitful collaborations with our project partners at Université de Mons and CSIRO Perth are acknowledged. Many thanks to Laurent Louis for his careful reading of the final version.

Data Availability Statement

The data supporting this work are available at https://doi.org/10.6084/m9.figshare.14959608.v1.

Contents of this file

Hence, we can estimate the water saturation in any particular region of the sample like the grey area between 0.5 and 0.7 cm in Figure S1, from the volume given by the integral of the distribution profile, divided by the pore volume. With this criterion, the error in the estimated water saturation follows a power law with increasing the height of the water-invaded area. For instance, for any pore volume injected into 5 cm of the rock sample, the error in the water saturation is 1.53%.

Text S3: Identification of the relaxation process

Poroelasticity theory predicts a transition from a drained regime to an undrained one at a critical or cut-off frequency fc (poro) depending on the permeability k, the dry bulk modulus Kd, the fluid viscosity ηf and the sample length Ls (Cleary, 1978;Pimienta et al., 2015). For frequencies higher than fc (poro) the undrained regime is isobaric or relaxed, but another relaxation transition can occur between this relaxed regime and an unrelaxed one where pore pressure is heterogeneously distributed. Squirt flow can account for this relaxation process through local flow either from compliant cracks to stiffer pores (O'Connell & Budiansky, 1977) or from compliant grain contacts to pores (Murphy et al., 1986): the corresponding critical frequency fc (squirt) depends on the fluid viscosity, the dry bulk modulus and on the aspect ratio ξ of the compliant elements. White (1975) introduced a model for the relaxed-unrelaxed transition based on patchy saturation, which requires the porous media to be partially saturated, with discrete patches of one fluid (e.g. air) embedded in a medium saturated with a second fluid (e.g. water). The critical frequency fc (patchy) for this mechanism depends on the permeability, the water viscosity (for patches made of air embedded into water), the patch size Lp and an effective bulk modulus KE (White, 1975) depending on water, matrix and dry bulk moduli and porosity. A complete (but lengthy) definition of KE can be found in Carcione et al. (2003). Finally the Biot crossover frequency fc (Biot) depending on the porosity φ, the fluid density ρf and viscosity, the permeability and the tortuosity T features a transition between a regime dominated by viscous forces towards a regime where inertial forces decouple the fluid from the solid motion (Johnson, 2001). The critical frequencies for these relaxation processes are given in Equation S1.

Taking k = 6 10 -16 m², Kd = 6.1 10 9 Pa, Ls = 0.08 m for our Obourg Chalk samples and ηf = 0.001 Pa.s, we get fc (poro) = 2.3 Hz, which shows that the recorded relaxation processes are clearly not linked to the drained-undrained transition. Taking ρf=1000 kg/m 3 T~1.5 and φ=0.43 for Obourg chalk yields fc (Biot) =76 MHz, showing that our measurements are done in the low frequency limit for the Biot relaxation. For the squirt flow mechanism, a crack aspect ratio ξ = 3.2 10 -3 is needed to obtain a critical frequency equal to fPLS = 0.2 MHz, which according to our microstructure analysis (Geremia et al., 2021a) seems unrealistic.

For the patchy saturation mechanism, taking KE = 2.9 10 9 Pa for Obourg chalk, a patch length LP = 53 µm is required to get a critical frequency equal to fPLS = 0.2 MHz, which is approximately two orders of magnitude higher than the pore size (≈ 0.3 µm) and three orders of magnitude lower than the fast P wave wavelength (≈ 11 mm considering VP = 2250 m/s for our rock). This result makes the patchy saturation model the most realistic one.

Movie S1. The movie explains how the process of decreasing in critical frequency with water saturation can affect the attenuation and dispersion observed in the experimental data. The Critical frequency fc in between the relaxed and unrelaxed regimes, moves to lower frequencies intercepting the pulsing frequency of the PZTs used in the experiments, causing a peak in attenuation and an increase in bulk modulus and velocity.


Mohamed K Salah

Mohammad Alqudah

Christian David

Acoustics and Petrophysical Investigations on Upper Cretaceous Carbonate Rocks from Northern Lebanon

Keywords: Acoustics, Petrophysical Properties, Nannofossils, Chekka Formation, Northern Lebanon

Carbonate rocks represent significant aquifers and reservoir rocks in many parts of the world. In this study, twenty-one carbonate samples collected from the exposed rocks at Batroun and Chekka, northwestern Lebanon, are studied to investigate their lithological characteristics, nannofossil content, and their acoustic and petrophysical properties. Four microfacies are recognized in the studied rocks including packstone, wackstone, wackmudstone, and the wack-packstone. Nannofossils are detected in the top part of the studied section indicating marine-type depositional conditions for the upper Cretaceous carbonate rocks. The Petrographic analysis revealed also that the studied rocks have moderate to high porosity which is consistent with the measured porosity on core samples. The low porosity of some samples is induced by cementation, which also reduced their permeability. The bulk density varies widely because of the many diagenetic processes that impacted the studied rocks. Moreover, because the primary and secondary wave velocities (Vp, and Vs) are greatly influenced by porosity and density, the samples exhibit extreme variations in acoustic velocities from low values which characterize soft, highly-porous, rocks to higher velocities which are characteristic of compact, massive, and highly-cemented carbonates. Some relationships between the measured petrophysical and acoustic parameters are studied to investigate their mutual dependence, and discuss the impacts of rock composition and diagenetic alterations on the petrophysical properties of the collected rocks. The obtained results are also important in characterizing their mechanical strength, and in the analysis of seismic reflection profiles acquired for equivalent offshore limestone reservoirs in the Levant and the Eastern Mediterranean regions.

Introduction

Significant hydrocarbon reservoirs of enhanced porosity and permeability, resulting from prolonged diagenesis and karstification of carbonate rocks, are common in many sedimentary basins around the world [START_REF] Mazzullo | Hydrocarbon reservoirs in karsted carbonate rocks[END_REF][START_REF] Dürrast | Correlation between rock fabrics and physical properties of carbonate reservoir rocks[END_REF][START_REF] Jin | Rock-physics-model-based pore type characterization and its implication for porosity and permeability qualification in a deeply-buried carbonate reservoir, Changxing formation, Lower Permian, Sichuan Bain, China[END_REF], including the supergiant oil fields of the Middle East [START_REF] Nurmi | Carbonates, the inside story[END_REF]. In addition, carbonate rocks represent significant, water-bearing, karst aquifers formed by fracturing and dissolution of limestone, dolomite or magnesite rock [START_REF] Harmon | Perspectives on karst geomorphology, hydrology, and geochemistry -A tribute Volume to Derek C. Ford and[END_REF]. Carbonate rocks are also important building stones whose durability is determined mainly by their geological history, petrophysics, strength and elastic characteristics [START_REF] Cassar | The Historic and Archaeological Heritage: Pollution and Non-Urban Sites[END_REF]. In comparison to sandstone reservoirs, the exploration of carbonate rocks is challenging due to many intrinsic heterogeneities at various scales of observation and measurement (e.g., [START_REF] Soete | Acoustic properties in travertines and their relation to porosity and pore types[END_REF]. Such heterogeneities result mainly from variable lithology, sedimentary facies, chemistry/mineralogy, complex rock texture, pore geometry and connectivity, as well as subsequent diagenetic modifications [START_REF] Kerans | Karst-controlled reservoir heterogeneity in Ellenburger Group carbonates of west Texas: Reply[END_REF][START_REF] Loucks | Paleocave carbonate reservoirs: origins, burial-depth modifications, spatial complexity, and reservoir implications[END_REF][START_REF] Loucks | Modern analogs for paleocave-sediment fills and their importance in identifying paleocave reservoirs[END_REF][START_REF] Loucks | Origin and recognition of fractures, breccias, and sediment fills in paleocave reservoir networks[END_REF][START_REF] Loucks | Published, Paleocave facies classification and associated pore types[END_REF][START_REF] Jin | Rock-physics-model-based pore type characterization and its implication for porosity and permeability qualification in a deeply-buried carbonate reservoir, Changxing formation, Lower Permian, Sichuan Bain, China[END_REF][START_REF] Salah | Petrophysical and acoustic characteristics of Jurassic and Cretaceous rocks from Central Lebanon[END_REF]. These various variables affect also their acoustic properties and the relationships between various reservoir parameters [START_REF] Neto | Carbonate pore system evaluation using the velocity-porosity-pressure relationship, digital image analysis, and differential effective medium theory[END_REF].

The relationships between acoustic velocities, rock composition, and texture are fundamental to the understanding of sonic logs and seismic reflection profiles of sedimentary basins (e.g., [START_REF] Buckley | Carbonate Buildups in the Santos Basin, offshore Brazil. Programme and abstract volume: microbial carbonates in space and Time: implications for global exploration and production[END_REF][START_REF] Soete | Acoustic properties in travertines and their relation to porosity and pore types[END_REF]. Acoustic impedance, an important rock parameter in reflection seismology, is calculated from the primary wave velocity (Vp) and the bulk density (ρb). Moreover, knowledge of both rigidity and incompressibility of a dry rock is fundamental for the shear wave velocity (Vs) prediction and fluid substitution [START_REF] Neto | Carbonate pore system evaluation using the velocity-porosity-pressure relationship, digital image analysis, and differential effective medium theory[END_REF]. The Poisson's ratio (ν) is very sensitive to the presence of fluids, and is widely used in seismic exploration as a lithological gauge [START_REF] Kearey | An Introduction to Geophysical Exploration[END_REF]. In addition to porosity and lithology, these elastic parameters are controlled also by the geometry of pores as well as their connectivity [START_REF] Li | Elastic moduli of dry rocks containing spheroidal pores based on differential effective medium theory[END_REF].

Carbonate rocks exhibit very wide ranges in their petrophysical and elastic properties (e.g., Anselmetti andEberli, 1993, 2001). They are composed essentially of calcite with its well-known matrix density, which implies that other factors control the acoustic wave velocity variations in carbonate rocks. Porosity (φ), which is calculated as the ratio of the volume of the void part of a rock that is unoccupied by rock grains or mineral cement (Vpr) to the rock's total bulk volume (Vb), is the principal parameter affecting the acoustic properties of rocks (e.g., [START_REF] Zolotukhin | Calcareous nannofossil taxonomic list: Amphizygus Bukry 1969 A. brooksii Bukry 1969 Broinsonia Bukry 1969 B. parca[END_REF]. It can be divided into primary porosity, secondary porosity, and effective porosity (φe). However, results of extensive research done in this field during the last decades indicated that beside porosity, the whole fabric of carbonate rocks controls the variations of acoustic wave velocities (e.g., [START_REF] Wang | Seismic velocities in carbonate rocks[END_REF][START_REF] Salah | Effects of porosity and composition on seismic wave velocities and elastic moduli of Lower Cretaceous rocks, central Lebanon[END_REF].

Moreover, their high sensitivity to diagenetic changes may completely modify their porosity, crystal shape, and the rock rigidity [START_REF] Verwer | Acoustic properties of carbonates: effects of rock texture and implications for fluid substitution[END_REF], which will, in turn, affect their overall elastic properties.

Diagenesis can enhance or diminish significant reservoir properties such as porosity and permeability. The initial porosity of shallow carbonate deposits may be as high as 50-60%.

Unlike sandstone reservoirs, subsurface limestone reservoirs have much lower porosities, which decrease gradually with burial depth [START_REF] Schmoker | Empirical relation between carbonate porosity and thermal maturityan approach to regional porosity prediction[END_REF][START_REF] Brown | Porosity variation in carbonates as a function of depth: mississippian madison group, williston basin[END_REF][START_REF] Ehrenberg | Sandstone vs. carbonate petroleum reservoirs; a global perspective on porosity -depth and porosity -permeability relationships[END_REF]. Although mechanical compaction is a principal way by which sandstones lose porosity at depths less than about 2.5 km and temperatures of up to 90° C, it is more difficult to assess its effects on carbonates because of the particular grain shape and the high diagenetic potential [START_REF] Croizé | Petrophysical properties of bioclastic platform carbonates: implications for porosity controls during burial[END_REF]. Meanwhile, if carbonate sediments are not cemented, laboratory investigations revealed that mechanical compaction has the greatest effect on porosity reduction (e.g., [START_REF] Chuhan | Experimental compression of loose sands; relevance to porosity reduction during burial in sedimentary basins[END_REF].

Mineralogically, calcite, aragonite, and dolomite may co-exist in one carbonate rock with varying proportions. These minerals have different stabilities and respond differently to diagenetic processes. The early diagenetic processes may comprise dissolution of unstable components such as aragonite, dolomitization, and the precipitation of low Mg-calcite [START_REF] Scholle | Burial diagenesis; out of sight, out of mind![END_REF]. Dissolution, for example, may initially increase the pore volume, while late cementation can either reduce the pore size or result in complete pore closure and, therefore, decreased connectivity, and subsequently low permeability. Early diagenesis may add or remove significant rock constituents in a way that impacts mechanical compaction at shallow burial (e.g., [START_REF] Hamilton | Variations of density and porosity with depth in deep-sea sediments[END_REF]. Shallow platform carbonates, in particular, are very susceptible to early diagenesis [START_REF] Friedman | Early diagenesis and lithification in carbonate sediments[END_REF], which complicates the prediction of their petrophysical features (e.g., [START_REF] Croizé | Petrophysical properties of bioclastic platform carbonates: implications for porosity controls during burial[END_REF]. Many authors addressed the dominant problems in carbonate petrophysical evaluation resulting from these highly variable rock properties and complex diagenesis (e.g., [START_REF] Lucia | Characterization of petrophysical flow units in carbonate reservoirs: Discussion[END_REF][START_REF] Akbar | A snapshot of carbonate reservoir evaluation[END_REF][START_REF] Kennedy | Solutions to some problems in the analysis of well logs in carbonate rocks[END_REF][START_REF] Cerepi | High-resolution characterisation and integrated study of a reservoir formation: the danian carbonate platform in the Aquitaine Basin (France)[END_REF].

The Late Cretaceous carbonate rocks outcropping at Chekka and Batroun areas, northern Lebanon (Fig. 1), which are essentially shallow platform deposits, are selected to study their facies, nannofossil content, and the impact of diagenesis on their petrophysical and elastic characteristics. The obtained results are useful in predicting the rock porosity and elastic properties of the carbonate sediments formed elsewhere at similar conditions, and the respective role of the different diagenetic processes in modifying these parameters. Moreover, the measured parameters including porosity, density, permeability, and the elastic properties are of great importance in characterizing the rocks' mechanical strength and in the analysis of seismic reflection profiles acquired for equivalent offshore limestone reservoirs in the Levant and Eastern Mediterranean regions.

Geological Setting

The Lebanese Mesozoic stratigraphic succession is represented mainly by carbonate rocks [START_REF] Beydoun | The Middle East: Regional Geology and Petroleum Resources[END_REF][START_REF] Nader | The geology of Lebanon[END_REF]. The Late Cretaceous beds, in particular, are represented by shallow platform carbonate deposits [START_REF] Walley | A revision of the Lower Cretaceous stratigraphy of Lebanon[END_REF][START_REF] Walley | The lithostratigraphy of Lebanon: a review[END_REF]. During the Late Cretaceous, the Arabian Plate was subject to both compressional and extensional stresses. Compressional stresses resulted in a regional uplift of the Early Turonian up to the Early Campanian during which ophiolitic rocks were obducted at the northeastern and southeastern margins of the Arabian Plate. Extensional stresses in the Coniacian, primarily from the continued opening of the Mediterranean, have possibly affected the northern plate margin as illustrated by the formation of the Euphrates Graben and transgression of the sea in the northwestern side of the Arabian Plate [START_REF] De Ruiter | The Euphrates Graben of Eastern Syria: A New Petroleum Province in the Northern Middle East[END_REF][START_REF] Litak | Structure and Evolution of the Petroliferous Euphrates Graben System, Southeast Syria[END_REF]. During these extensional regimes, the carbonate platform was formed at the northwestern part of the Arabian Plate. More details on the nature of the depositional conditions and the Mesozoic sediment characteristics in Lebanon can be found in other published works (e.g., [START_REF] Beydoun | The Middle East: Regional Geology and Petroleum Resources[END_REF][START_REF] Walley | The lithostratigraphy of Lebanon: a review[END_REF][START_REF] Nader | The geology of Lebanon[END_REF].

3.Material and Methods

We collected a total of twenty-one representative rock samples from the exposed Late Cretaceous rocks at Chekka and Batroun (Fig. 2). These rocks have been studied petrographically and petrohpysically as explained in the following paragraphs.

Petrography

The exposed Late Cretaceous Chekka and Batroun rocks at northern Lebanon, and the location of the collected samples are shown in Fig. 3. The petrographic and Scanning Electron Microscopic (SEM) investigations have been carried out to determine the rock facies and the dominant pore types. A total of twenty-one thin sections were analyzed using the polarized microscope to characterize lithology and find out the paleo-environmental depositional conditions. Special attention was paid to evaluate the overall rock porosity, detect the primary and secondary porosities, and to investigate the different post-depositional diagenetic processes. Standard types of microfacies are assigned according to [START_REF] Wilson | Carbonate facies in geologic history[END_REF][START_REF] Flügel | Microfacies Analysis of Limestone[END_REF]. Porosity and other thin-section derived parameters are characterized in the same way explained in [START_REF] Salah | Effects of porosity and composition on seismic wave velocities and elastic moduli of Lower Cretaceous rocks, central Lebanon[END_REF]. In addition, the nannofossil content has been analyzed as described by [START_REF] Alqudah | Calcareous nannofossil, nummulite and ostracod assemblages from Paleocene to Miocene successions in the Bekaa Valley (Lebanon) and its paleogeographic implications[END_REF].

Petrophysical and acoustic measurements

The petrophysical measurements were conducted on twenty-one, one-inch-diameter, plugs. First, samples were prepared for measurements in the same way described by [START_REF] Freire-Lista | Thermal stress-induced microcracking in building granite[END_REF]. The effective porosity was measured by the saturation method (e.g., [START_REF] Freire-Lista | Historical quarries, decay and petrophysical properties of carbonate stones used in the historical center of Madrid (Spain)[END_REF][START_REF] Salah | Petrophysical and acoustic characteristics of Jurassic and Cretaceous rocks from Central Lebanon[END_REF], and calculated according to the following equation: φe = [(ws -wd)/(ws -wm)]

(1)

where ws, wd, and wm are the saturated, dry, and submerged sample weights. The numerator corresponds to the pore volume whereas the denominator is equivalent to the sample bulk volume.

The bulk density (in g/cm 3 ), is calculated from the dry sample weight and its bulk volume as follows:

ρb = wd/(ws -wm) (2)
The matrix density (ρma, g/cm 3 ), is calculated from the porosity and the ρb using the following simple equation:

φ = 1-(ρb /ρma), and thus ρma = ρb/(1 -φ) (3)
Permeability is a measure of the fluid amount transported by a porous medium within the network of its connected pores (e.g., [START_REF] Amyx | Petroleum reservoir engineering -physical properties[END_REF]. For an incompressible fluid and a horizontal flow in the direction of decreasing pressure, permeability (K) is defined, according to Darcy's law, as follows:

K = ηQL/AΔp (4)
where η is viscosity of fluid, Q is the flow rate, L is the sample length, A is the crosssectional area of the flow, and Δp is the pressure drop across the flow. Permeability was measured for the dry rock samples using nitrogen gas as the flowing fluid. The measured gas permeability is corrected to the equivalent liquid permeability (KL; [START_REF] Klinkenberg | The permeability of porous media to liquids and gases[END_REF] as described in [START_REF] Salah | Petrophysical and acoustic characteristics of Jurassic and Cretaceous rocks from Central Lebanon[END_REF]. A relationship between the values of five gas permeability measurements and the reciprocal of the applied mean pressure across a rock sample is constructed, with the resulting straight line extrapolated to the vertical axis to give the correct KL of the rock sample (Fig. 4).

Acoustic wave velocities are measured using the pulse transmission tool (e.g., [START_REF] Birch | The velocity of compressional waves in rocks to 10 kilobars, part 1[END_REF] first on the dried plugs. After the core samples were fully saturated with water for porosity determination, the acoustic velocities were measured again. Both Vp and Vs measurements were accomplished using Panametrics Pulser-Receiver (Model 5058PR) and an

Agilent DSO-X-2014A Digital Storage Oscilloscope (100 MHz) with a travel time accuracy of 0.01 µs. The P-and S-wave transducers are adhered to the core ends with a thin rubber membrane in between. The waveform is displayed on a screen and the first arrival time is picked on a monitor. The measured travel time for the two rubber membranes are subtracted to get the travel time only across the sample; then the acoustic velocity in the core sample (Vp or Vs) is computed as: V = hs/t, where hs is the sample length as digitally determined by a micrometer.

The elastic moduli can fully describe the mechanical behavior of the rock. The rock rigidity (µ) quantifies the rock resistance to change in shape as a response to the applied shear without any change in volume, whereas the bulk modulus (κ) measures the incompressibility of rocks. On the other hand, the Young's modulus (E) quantifies the rock response to variations in its length under uniaxial pressure (e.g., [START_REF] Yu | Effects of porosity on seismic velocities, elastic moduli and Poisson's ratios of solid materials and rocks[END_REF]. These elastic constants are calculated from the measured seismic wave velocity and bulk density as explained in [START_REF] Mavko | The Rock Physics Handbook[END_REF]. The ν ratio is calculated from the Vp and Vs as follows:

ν = [(Vp 2 -2Vs 2 )/2(Vp 2 -Vs 2 )] (5)
The elastic constants are also calculated using the following relationships (e.g., [START_REF] Russell | The relationship between dry rock bulk modulus and porosity-an empirical study[END_REF]:

µ = ρbVs 2 (6) κ = ρb[Vp 2 -(4/3)Vs 2 ] (7) E = ρbVs 2 [(3Vp 2 -4Vs 2 )/(Vp 2 -Vs 2 )] (8)

Results

Petrography and SEM imaging

Four limestone microfacies are recognized in the investigated samples. These are packstone, wackstone, mud-wackstone, and wack-packstone facies (Table 1, Fig. 5). Table 1 includes also the bioclast and non-bioclast ratios, grain types and percentages, etc. The microfacies are assigned based on the [START_REF] Dunham | Classification of carbonate rocks according to their depositional texture[END_REF] classification.

A) Packstone

This microfacies is detected in samples C1-C4 (Table 1). The samples are characterized generally by the absence of bioclasts, dominance of calcitic interlocking crystal grains (Fig. 5A), and rhombic dolomite (Fig. 5B). Matrix does not exist; instead, a cement represented by the secondary precipitation of CaCO3 is common. The cement occurs in the form of geodes as a free crystal growth (Fig. 5A). Iron oxides are also observed within the cement. The samples are grain-supported with voids that resulted from high dissolution during postsedimentological processes (Fig. 5A andB). These processes have affected the porosity partially first by reducing it through the secondary precipitation, and second by enhancing it with new fenestral pores that were created effectively by dolomitization. In accordance with these observations, [START_REF] Cerepi | Effects of carbonate rock-fabrics on the petrophysical properties[END_REF], described the packstone microfaciesas as having a monomodal pore network with low porosity and low permeability. The notable absence of matrix and signs of intensive diagenesis observed in the samples indicate that the sea level was at minimum at the bottom of the section.

B) Wackstone

This microfacies, with a matrix of up to 80%, is the most common as it was detected in the majority of the samples: C5 -C10, and C13 -C18 (Table 1). The matrix is mainly calcium carbonate, although a dolomitic matrix was also observed in samples C5 (Fig. 5C) and C6. Non-bioclasts consist mainly of calcite grains (Fig. 5D), whereas bioclastic skeletons of bone fragments (Fig. 5E), spines (Fig. 5F), gastropods (Fig. 5G), ostracods (Fig. 5H), echinoids (Fig. 5I), benthonic and planktonic foraminifera (Fig. 5J) are also detected in the studied samples. Calcareous nannofossils are found in samples C16-C18. These fossil assemblages have wide ranges of preservation from well-preserved (Fig. 5E, G and H), to those that are poorly-preserved as they were partially destroyed during the reworking and dissolution of calcium carbonates during post-sedimentary processes (Fig. 5F andJ).

Different kinds of porosity are observed such as the intragranular pores (Fig. 5K) in which the calcium carbonate skeletons were dissolved totally leaving some bird's eye voids. Fractures are also observed in some samples which might be induced by the stresses affecting the sediments. It is clear from the studied samples that most of the fractures are filled with a secondary precipitation of calcium carbonate (Fig. 5L). [START_REF] Cerepi | Effects of carbonate rock-fabrics on the petrophysical properties[END_REF] described the wackestone microfacies as having a monomodal pore network, with low porosity and low permeability.

C) Wack-Mudstone

The wack-mudstone microfacies, with dominant calcite matrix (~93%), and scarcity of grains, is found only in two samples (C11 and C12; Table 1). Few calcitic non-bioclastic grains (Fig. 5M), and bioclasts such as ostaracods exist in the samples which are dominantly matrix-supported. Porosity is observed in two forms: first as intergranular pores within the matrix and, second, as fractures (Fig. 5N), which are sometimes partially filled with calcium carbonate. [START_REF] Cerepi | Effects of carbonate rock-fabrics on the petrophysical properties[END_REF], characterized the wackestone-mudstone microfacies as having a monomodal pore network, low porosity, and low permeability; features that are consistent with our observations.

D) Wack-Packstone

This microfacies is detected in the upper part of the section; mainly in samples C19, C20 and C21. Grain content is again high with predominant bioclastic skeletons such as ostracods (Fig. 5O), planktonic and benthonic foraminifera (Fig. 5P) and gastropods (Fig. 5Q). Fossils are mostly well-preserved, although some with moderate to poor preservation were also found (Fig. 5R). The percentage of bioclasts is about 30%, whereas the non-bioclasts are rare. This microfacies is enriched in calcareous nannofossil assemblages. Porosity is almost intragranular where voids exist inside the chambers of the organisms (Fig. 5S). However, some pores are filled with the secondary calcium carbonate precipitation; hence reducing the porosity of the rock (Fig. 5T). Existing microfossils and planktonic foraminifera indicate that the sea level reached its maximum at the time of deposition.

Calcareous nannofossils and porosity

Many statigraphic units are geographically undifferentiated in Lebanon. Therefore, we here consider the calcareous nannofossil content of the studied strata, which provides information on the pore types and their effects on the acoustic and petrophysical characteristics. In carbonate sediments, the grain shape, the intragrain porosity, and sorting have large impacts on the rock porosity [START_REF] Lucia | Carbonate reservoir characterization[END_REF].

The lowermost fifteen samples of the section (C1 -C15) are barren whereas various types of nannofossils are detected in samples C16 -C21 (Fig. 6). Intragranular pores are clearly seen in many samples (Fig. 6; Table 1). The tiny shields of coccolithophores are composed of two parts: the rim, which is an arranged set of microplates of calcium carbonate, and the central area that has many shapes such as closed plates, sieved-shape plates, and open central area. The sieved-shape central area like

Broinsonia parca includes micro-pores. The same is happening in open central area nannofossils where the micro-porosity is higher and more effective. For instance, cross-shape central area in Predicosphaera cretacea created relatively large voids. The existence of clays in some samples reduces the porosity by filling the space in the central area as in Amphizygus brooksii (Fig. 6).

Petrophysical and acoustic properties

The measured porosity, bulk and matrix densities, the Klinkenberg permeability, Vp, and Vs (of both the dry and saturated plugs) as well as the calculated elastic moduli are given in Table 2. Porosity varies widely from 0.02 to 0.34, the bulk density from 1.70 to 2.77 g/cm 3 , and the matrix density from 2.58 to 2.83 g/cm 3 . In contrast, permeability is generally very low and ranges only between 0.004 and 0.075 mD. Although the studied rocks have generally moderate to high porosity (average 0.18), their permeability is very low due to the small size of the majority of the pores and the high percent of non-connected microporosity (Fig. 6).

Wide variations in Vp and Vs are encountered even though the averages of Vp/Vs and the ν ratios are 1.73 and 0.24, respectively. The Vp varies from 2834 to 6135 m/s, while the Vs varies from 1605 to 3429 m/s (Table 2). For a given sample, the value of Vs is, on average, equal to 0.58 of its Vp value. Samples having the lowest porosity (C2 and C3) have the highest density, the highest Vp, the highest Vs, and high ν (Table 2), and in turn, large elastic moduli. The ν is generally low, where only six, out of the studied twenty-one samples, have ν > 0.25. The acoustic impedance varies from 4.8 to 17.0 * 10 6 kg/m 2 .s, with an average of 9.3 * 10 6 kg/m 2 .s. In the following paragraphs, we construct some relationships between these different measurements and discuss the effect of the lithological composition and diagenetic processes on the measured parameters.

A) Petrophysical interrelationships

Porosity is inversely proportional to bulk density with a high correlation coefficient (Fig. 7A) indicating the reliability of our measurements. On the other hand, porosity is directly proportional to permeability although some scatter is clearly observed (Fig. 7B).

Similar to the relationship between porosity and bulk density, the matrix density and porosity are inversely correlated which reflects the effect of lithological composition on porosity (Fig. 7C). The bulk and matrix densities are directly proportional to each other with a good correlation (R 2 = 0.83; Fig. 7D). These relationships indicate that dissolution and leaching of some heavier components in the studied rocks are accompanied by an increase in porosity and a corresponding decrease in density.

B) Porosity versus seismic velocities

The relationships between porosity and both Vp and Vs are strongly inverse with high correlation coefficients (Fig. 8A, andB). In addition, at higher porosities (> ~0. The relationship between φ and ν seems interesting as it displays different behaviors for the dry and water-saturated samples. In general, the ν increases with saturation and the rate of that increase is larger at higher porosities (Fig. 8C). Higher Vp/Vs ratios, at saturation, have also been observed by [START_REF] Croizé | Petrophysical properties of bioclastic platform carbonates: implications for porosity controls during burial[END_REF]. For the dry carbonate samples, an inverse relationship is obtained although with a small correlation coefficient. Meanwhile, a weak direct relationship is observed between φ and the ν of the water-saturated samples. The relationship between porosity and acoustic impedance (A.I.) is also inverse with a high correlation coefficient (Fig. 8D). The high correlation coefficient (R 2 =0.94) between the two parameters is highly significant for the interpretation of subsurface seismic sections and well log analysis of carbonate reservoirs.

C) Porosity versus elastic moduli

All the elastic moduli of the dry and water-saturated samples decrease clearly with the increase of porosity (Table 2). This is also clear from the inverse relationships with the moderate/high correlation coefficients (Fig. 9A-C). The bulk moduli of the dry limestone samples are generally lower than those of the water-saturated carbonate rocks (Fig. 9A), which is attributed to the slight increase in the Vp of most samples with saturation (Table 2).

D) Density versus acoustic velocities

Acoustic velocities are strongly dependent on bulk density, which is inversely related to porosity. Thus, the bulk density is directly proportional to the acoustic velocities with high correlation coefficients (Fig. 10A). The ρb is a function of both porosity and composition and has a strong impact on the elastic properties of rocks, which explains the strong dependence of the seismic velocities on density. In contrast, the relationship between ρma and acoustic velocities (Fig. 10B) exhibits weaker correlation. These two relationships together indicate the strong impact of porosity on the acoustic characteristics of the investigated samples.

In Table 3, we provide the correlation coefficients ('r' values) which relate the different petrophysical and elastic parameters. Porosity is positively correlated with permeability, but negatively correlated with density, acoustic wave velocities, and the elastic moduli of the dry and water-saturated rocks. Except for permeability and the Poisson's ratio of the watersaturated rocks (νsat), the bulk density is positively correlated with the elastic properties of the studied samples. In the same way, the matrix density is also positively correlated with the elastic parameters. Permeability is negatively correlated with the elastic parameters even though with low/fair correlation coefficients. The νsat is negatively weakly correlated with other elastic parameters. Finally, the acoustic velocities and the elastic moduli are all strongly positively correlated with each other (Table 3).

Discussion

Petrophysical characteristics

The petrographical investigations revealed that the studied carbonates are classified to wackstones, packstones, wackstone-packstone, and mud-wackstones (Table 1). The grain sizes in these microfacies is very small, usually less than 100 µm [START_REF] Lucia | Rock-Fabric/Petrophysical Classification of Carbonate Pore Space for Reservoir Characterisation[END_REF][START_REF] Lucia | Characterization of petrophysical flow units in carbonate reservoirs: Discussion[END_REF], pointing to the existence of a large proportion of micropores which contribute to the very low permeability (Table 2). Both bulk density and porosity of the studied plugs exhibit large variations (with averages of 2.21 g/cm 3 and 0.18, respectively) implying variable impacts of sediment nature and the subsequent diagenesis on the measured parameters (e.g., [START_REF] Ehrenberg | Porosity-permeability relationships in interlayered limestone-dolostone reservoirs[END_REF].

The poro-perm (the porosity-permeability) relationships are very important in reservoir zonation and characterization. In carbonate rocks, these relations are generally poorly-defined especially for continental and shallow platform carbonate deposits formed in thin water films and under constantly-changing temperature and humidity [START_REF] Soete | Acoustic properties in travertines and their relation to porosity and pore types[END_REF]. Permeability is dependent not only on porosity, but also on grain size, its distribution, and the connectivity of individual pore spaces. Previous estimates revealed that at a certain reservoir porosity of, for example, 0.25, permeability in carbonate rocks varies over four orders of magnitude because of pore complexities [START_REF] Neto | Carbonate pore system evaluation using the velocity-porosity-pressure relationship, digital image analysis, and differential effective medium theory[END_REF]. Meanwhile, some authors (e.g., [START_REF] Lucia | Rock-Fabric/Petrophysical Classification of Carbonate Pore Space for Reservoir Characterisation[END_REF][START_REF] Lucia | Dolomitisation: A Porosity-Destructive Process[END_REF] derived correlations for specific pore type and grain size relationships. The observed scatter in our poro-perm relationship (Fig. 7B), and the small φ-K correlation coefficient (Table 3) confirm the fact that in addition to porosity, other factors including the grain size distribution, pore size, and mud/clay proportion control permeability as well (e.g., El Sayed et al., 2015).

Carbonate rocks with their very complex pore types, initial fabrics, and higher susceptibility to diagenesis (e.g., dissolution, cementation, recrystallization, dolomitization, etc.), possess various pore networks which complicate the poro-perm relationship [START_REF] Nurmi | Carbonates, the inside story[END_REF][START_REF] Lapponi | From outcrop to 3D modelling: a case study of a dolomitized carbonate reservoir, Zagros Mountains[END_REF][START_REF] Dewit | Reservoir characteristics of fault-controlled hydrothermal dolomite bodies: Ramales Platform case study[END_REF]. This specific nature of carbonate rocks makes it difficult also to understand the relationship between permeability and seismic velocity.

Pore types and elastic properties of carbonates

Diversity of pore types in carbonate rocks can produce great changes in seismic velocity at a given porosity [START_REF] Sun | Effects of pore structure on 4D seismic signals in carbonate reservoirs[END_REF]. [START_REF] Soete | Acoustic properties in travertines and their relation to porosity and pore types[END_REF] found that carbonate rocks with mouldic pores plot above the regression line of the φ-Vp relationship compared to those with micro-moldic porosity, which plot well below the regression line. In addition, samples with vugs appear close to, and around, the best-fit line, whereas those of framework porosity have lower velocities and plot below the average trend.

In our study, some samples on the porosity-velocity show different porosities at close velocities and different velocities at similar porosities (Fig. 8A andB). According to previous studies, these discrepancies arise from the complex pore system, where an increase in the percentage of micropores results in lower acoustic velocities for a given sample porosity [START_REF] Eberli | Factors controlling elastic properties in carbonate sediments and rocks[END_REF][START_REF] Ali | Depositional constraints and diagenetic pathways controlling petrophysics of Middle Miocene shallow-water carbonate reservoirs (Leitha limestones), Central Paratethys, Austria-Hungary[END_REF]. Samples C15-C18, have the highest porosities and lowest acoustic wave velocities. The high secondary porosity is induced mainly by fracturing and dissolution of some intragranular components (bioclasts) which are common in these matrix-supported wackstones ( 1, and Fig. 5A). They have also very low permeability (Table 2) that is induced by the small grain size and the presence of micropores. There is no major difference in the acoustic velocities between pure calcite and dolomite (Anselmetti andEberli, 1993, 2001). Therefore, we speculate that the observed variations in the acoustic velocities arise mainly from the rock texture and the diagenetic processes [START_REF] Verwer | Acoustic properties of carbonates: effects of rock texture and implications for fluid substitution[END_REF]. In the case of different velocities at the same porosity (Fig. 8B), the formation of a rigid structure at the time of early diagenesis gives a greater strength to the rocks than what is usual for their porosity. The cement type and the pore geometry may also contribute to the scatter in the φ-Vp plots [START_REF] Dürrast | Correlation between rock fabrics and physical properties of carbonate reservoir rocks[END_REF][START_REF] Eberli | Factors controlling elastic properties in carbonate sediments and rocks[END_REF]. In general, the dense and less-porous marine carbonates have high Vp of 6260-6640 m/s, which resemble that of pure calcite [START_REF] Mavko | The Rock Physics Handbook[END_REF]. Similar high Vp of 6060-6135 m/s and Vs (>3300 m/s) have also been observed in our samples for very low-porosity (φ ~ 0.02) carbonate rocks (samples C2 and C3; Table 2), which are mainly grain-supported packstones (Table 1).

Results of µCT imaging on travertine samples examined by [START_REF] Soete | Acoustic properties in travertines and their relation to porosity and pore types[END_REF] revealed that samples having similar porosities but varying velocities are significantly different.

Samples of uniformly distributed pores give rise to a scattered wavefront and reduced velocities. On the other hand, samples with predominantly moldic pores have a stiff framework which envelops the sparite-depleted molds and give a limited wavefront scatter, and, therefore, higher velocities [START_REF] Anselmetti | Controls on sonic velocity in carbonates[END_REF]. Results of [START_REF] Saleh | Revisiting theWyllie time average equation in the case of near spherical pores[END_REF] revealed also that pores possessing high aspect ratios (e.g., vugs and molds) give a stronger support than intercrystalline and interparticle pores, hence making the rock much stiffer by reducing the pore compressibility at the same porosity. Therefore, at a given porosity, the seismic velocity decreases with the increase of the amount of micropores and increases with the increase of the stiff moldic pores (e.g., [START_REF] Xu | modeling elastic properties in carbonate rocks[END_REF][START_REF] Fournier | Pore space evolution and elastic properties of platform carbonates (Urgonian limestone, BarremianeAptian, SE France)[END_REF]. According to [START_REF] Brigaud | Acoustic properties of ancient shallow-marine carbonates: effects of depositional environments and diagenetic processes (Middle Jurassic, Paris Basin, France)[END_REF], the widely-variable pore sizes and their shapes result in the encountered velocity fluctuations.

Samples C1-C4, C6-C10, and C13-C14 possess the highest incompressibility (Table 2).

These samples are either packstone or wackstone and all have low/moderate porosity. Greater cementation may also contribute to the high incompressibility of some rocks in the present study. On the other hand, the wack-packstone and mud-wackstone microfacies as well as few wackstone rocks have lower incompressibilities and higher porosity. For example, sample C11 is a fractured, matrix-supported, mud-wackstone whose porosity and bulk modulus are 0.18 and 1.85*10 10 Pa, respectively. The lowest incompressible samples are C15 to C21, which are highly porous, matrix-supported, wackstones or wack-packstones. In addition, dry limestone samples have generally higher Young's and shear moduli compared to those of the saturated samples in accordance with many previous studies (e.g., [START_REF] Vanorio | The effect of chemical and physical processes on the acoustic properties of carbonate rocks[END_REF].

Upon saturation, incompressibility increases at lower and intermediate porosities (φ ˂ ~0.25). On the other hand, at porosities of larger than 0.30, there is a drop in the Vp, and accordingly the incompressibility of the rock decreases (Fig. 9A). In addition, the φ-Vp and φ-κ correlations are strong for the dry samples (Figs. 8A and9A), but weak under saturation conditions in accordance with the results of [START_REF] Croizé | Petrophysical properties of bioclastic platform carbonates: implications for porosity controls during burial[END_REF]. The better correlation under the dry conditions reflects basic similarities in pore geometry and matrix nature, which may revert after saturation. The observed high correlations between porosity and incompressibility and elasticity imply also that porosity is the main factor controlling such rock properties (Table 3). Indurated carbonates have high Vp/Vs ratios between 1.8 and 2.0 [START_REF] Anselmetti | Controls on sonic velocity in carbonates[END_REF]. However, our samples exhibit generally lower ratios, which vary from 1.6 to 1.83, and with an average of 1.73. Therefore, we speculate that cementation, rather than mechanical compaction, played a major role in the evolution of the physical properties of the studied rocks. Previous researchers found weak correlations between the elastic moduli and porosity implying that other parameters related mainly to the rock pore system control the stiffness of carbonate rocks [START_REF] Croizé | Petrophysical properties of bioclastic platform carbonates: implications for porosity controls during burial[END_REF].

A possible diagenetic sequence

The diagenetic potential of carbonates is controlled largely by the rock texture and the amount of metastable components [START_REF] Schlanger | The pelagic ooze-chalk-limestone transition and its implications for marine stratigraphy[END_REF]. Fine-grained, mudsupported, carbonate rocks have very low permeability; thus diagenetic alterations are very limited, resulting in low acoustic velocities [START_REF] Awais | Relating petrophysical parameters to petrographic interpretations in carbonates of the Chorgali Formation, Potwar Plateau, Pakistan. Carbonates and Evaporites[END_REF].

Through the petrographic analysis and the results obtained from the petrophysical measurements, we propose a complete paragenetic sequence for the studied rocks (e.g., [START_REF] Awais | Relating petrophysical parameters to petrographic interpretations in carbonates of the Chorgali Formation, Potwar Plateau, Pakistan. Carbonates and Evaporites[END_REF]). The initial environment was marine. Subsequent diagenesis represented by the mixed marine-meteoric, meteoric and burial diagenesis followed and affected the sediments (Fig. 11). Samples C1-C4 showed the mixed marine-meteoric to meteoric diagenesis which is evidenced by the existence of cementation, dissolution and dolomitization, while early marine diagenesis has been observed in samples C5-C10 and C13-C18 through the clear signals of micritization. Samples C11 and C12 exhibit different diagenesis during burial of the sediments where they were highly fractured.

Conclusions

In this study, we examined the effects of porosity, pore types, and sediment characteristics on the petrophysical and acoustic properties of twenty-one carbonate rock samples collected from northern Lebanon. The petrographical and SEM analyses revealed that the studied rocks are either grain-supported packstone or matrix-supported wackstone, mudstone/wackstone, and wackstone/packstone with variable grain and porosity types.

Porosity and bulk density vary generally widely. The primary rock framework and diagenetic alterations contribute to the observed wide ranges of porosity (up to 32%). The rock textures and the high percentage of microporosity in matrix-supported rocks explain the observed low permeability. The acoustic wave velocities vary also widely and are inversely correlated with porosity. Some samples exhibit similar velocities at different porosities. The acoustic velocities in porous samples are reduced by more than 50% of the velocity in the compact, less-porous, rocks. The inverse correlation between velocity and porosity points to the strong impact of the later on acoustic velocity. The soft, porous, limestones are more compressible and have lower elastic moduli, and thus lower velocities. On the other hand, stiff, less-porous, rocks have higher elastic moduli and higher acoustic velocities. The Vp/Vs ratio varies from 1.60 to 1.83 (average 1.73), which is low compared to values characteristic for marine carbonates and can be interpreted by the cementation of less mechanicallycompacted rocks. To sum up, the acoustic properties of carbonate rocks are controlled by their depositional conditions and the subsequent diagenetic processes which modify their pore types and sizes. φ = porosity (%);ρb = bulk density (g/cm 3 ); ρma = matrix density (g/cm 3 ); K = permeability (mD); Vp = primary wave velocity (m/s); Vs = secondary wave velocity (m/s); A.I. = Acoustic impedance (*10 7 kg/m 2 .s); ν = Poisson's ratio; E = Young's modulus; κ = bulk modulus; µ = shear modulus (the three elastic moduli are the given values times 10 10 N/m 2 ). The suffix 'sat' denotes the acoustic properties of the water-saturated samples. Two samples (C2 and C3) are almost impermeable.

  20), Vp decreases with saturation but increases at low porosity values. There is, however, very little change in Vp with saturation at moderate porosity ranges (0.08 -0.20). This behavior of Vp after saturation is different from the results of Adam et al. (2006); and Croizé et al. (2010), who detected an overall increase of Vp with saturation. In contrast, there is a marked decrease in Vs with saturation where almost all the Vs values of the water-saturated samples are lower than those of the dry rocks. Only one packstone sample, with a porosity of 0.073%, show a higher Vs after saturation.

Fig. 1 .

 1 Fig. 1. Geological map showing the ages and major structures of the Lebanese rocks. the inset shows the locations of the sampled sections in Batroun and Chekka areas in north central Lebanon.

Fig. 2 .

 2 Fig. 2. Panoramic view of Batroun (1), and Chekka (2) outcrops along with the locations of the drilled core samples.

Fig. 3 .

 3 Fig. 3. Two columnar sections representing the Cenomanian-Turonian succession in Batroun and the Campanian succession in Chekka.

Fig. 4 .

 4 Fig. 4. The Klinkenberg correction plot for the accurate estimation of rock permeability. Extrapolation of the straight lines intersects the vertical axis in a value which corresponds to the liquid permeability. Three samples: C8, C15, and C21 are shown on the plot with corresponding permeabilities of 0.075, 0.06, and 0.067 mD, respectively.

Fig. 5 .

 5 Fig. 5. The different types of microfacies detected in the studied samples. Note the heterogeneities in the pore sizes and shapes (see text for more details). Interlocking crystal grains are clear in A; while rhombic dolomite is clear in B. Calcitic grains within dolomitic matrix are observed in C and D; bioclasts are represented in the samples by bone fragments (E), spines (F), gastropods (G and Q), ostracods (H and O), echinoids (I), and both benthonic and planktonic foraminifera (J and P). The moderate to poor preservation of fossils is shown in R; different porosity types are observed in K, L, M, N, S, and T (refer to the text for more details).

Fig. 6 .

 6 Fig. 6. Different calcareous nannofossil assemblages detected in the studied samples (refer to the text for more details).

Fig. 7 .

 7 Fig. 7. Cross plots between different petrophysical parameters: porosity vs. bulk density (A); porosity vs. permeability (B); grain density vs. porosity (C); and grain density vs. bulk density (D).

Fig. 8 .

 8 Fig. 8. Cross plots between different petrophysical and acoustic parameters: porosity vs. Vp (A); porosity vs. Vs (B); porosity vs. Poisson's ratio (C); and porosity vs. acoustic impedance (D), for the dry (blue rhombs) and the water-saturated (red squares) cores.

Fig. 9 .

 9 Fig. 9. Cross plots between porosity and different elastic moduli: porosity vs. bulk modulus (A); porosity vs. shear modulus (B); and porosity vs. Young's modulus (C), for the dry (blue rhombs) and the water-saturated (red squares) cores.

Fig. 10 .

 10 Fig. 10. Cross plots between density and acoustic velocities: bulk density vs. acoustic velocities (A); and grain density vs. acoustic velocities (B). The blue rhombs and the red squares denote Vp, and Vs, respectively.

Fig. 11 .

 11 Fig. 11. The paragenetic sequence of the different diagenetic processes observed in the studied samples.

  Figure 1

Figure 11

 11 Figure 11

Table 1

 1 of 0.072, 0.16, and 0.17. These samples have a variety of pore types such as intergranular, intragranular, fracture, and dissolution pores (Table

	). Moreover, samples C15 to C21 possess similarly low
	acoustic velocities (ranging from 2833 to 3236 m/s for Vp, and from 1605 to 1928 m/s for Vs)
	at moderate-high porosity ranges. Samples C19 and C21 have very similar Vp of 3093 and

3150 m/s but porosities of 0.27 and 0.23, respectively. In addition, samples C1, C6, and C13 have very similar velocities of 2557, 2580, and 2512 m/s, respectively. Yet, they have very different porosities

  Different pore shapes have different compressibilities and accordingly, different acoustic velocities. Upon saturation, the Young's modulus and rigidity decreased compared to their equivalent values of the dry rocks. Porosity value is critical in the behavior of Vp under saturation; it is reduced if the porosity is greater than about 0.15. Cementation by fine components plays a major role on velocity at higher porosities causing a positive departure from the best-fit line. On the other hand, stiffening by cementation of the rock frame at lower porosities is limited because of the already compact nature and the high packing of the lowporosity rocks.Acoustic velocities and the elastic moduli are positively correlated to bulk densities. The acoustic impedance of the studied samples varies widely due to the wide variations in seismic velocity and bulk density. It exhibits a strong inverse correlation with porosity, therefore, not all reflectors encountered in carbonate rocks are the result of different mineralogy or even lithology. Rather, they may be induced by complex pore geometry and different types of pores encountered. The obtained results of porosity, permeability, density, acoustic velocities, and the calculated elastic moduli of the studied rocks are important in seismic exploration and well log studies in equivalent subsurface marine carbonate reservoirs.

	Prediscosphaera Vekshina 1959
	P. cretacea (Arkhangelsky 1912) Gartner 1968
	Rotelapillus Noël, 1973
	Rotelapillus laffittei, syn. Rotelapillus crenulatus (Stover, 1966) Perch-Nielsen, 1984
	Tranolithus Stover 1966
	T. minimus (Bukry 1969) Perch-Nielsen 1984
	T. orionatus (Reinhardt 1966) Perch-Nielsen 1968 = T. phacelosus
	Watznaueria Reinhardt 1964
	W. barnesiae (Black 1959) Perch-Nielsen 1968
	Zeugrhabdotus Reinhardt 1965
	Figure Captions

Table 2 .

 2 Measured and calculated petrophysical and elastic parameters of the studied core samples.

	S. No.	φ	ρb	ρma	K, mD	Vp	Vs	A.I	ν	E	κ	µ	Vp sat	Vs sat	ν sat	E sat	κ sat	µ sat
	C1	0.072 2.50 2.70 0.004 4681 2557 1.17 0.29 4.21 3.30 1.63 5950 2806	0.36	5.34	6.22	1.97
	C2	0.021 2.77 2.83	-	6135 3429 1.70 0.27 8.28 6.08 3.25 6353 3241	0.32	7.70	7.29	2.91
	C3	0.020 2.75 2.81	-	6061 3313 1.67 0.29 7.78 6.09 3.02 6090 3237	0.30	7.52	6.37	2.89
	C4	0.104 2.43 2.71 0.004 4707 2723 1.14 0.25 4.50 2.98 1.80 4741 2426	0.32	3.78	3.55	1.43
	C5	0.189 2.20 2.72 0.005 3986 2395 0.88 0.22 3.08 1.82 1.26 3884 1833	0.36	2.01	2.34	0.74
	C6	0.159 2.28 2.71 0.030 4605 2580 1.05 0.27 3.86 2.81 1.52 4538 2338	0.32	3.28	3.03	1.24
	C7	0.094 2.48 2.73 0.010 4866 2695 1.21 0.28 4.60 3.47 1.80 4820 2357	0.34	3.70	3.92	1.38
	C8	0.177 2.25 2.74 0.075 4108 2423 0.92 0.23 3.26 2.04 1.32 3940 1974	0.33	2.34	2.32	0.88
	C9	0.180 2.23 2.71 0.041 4033 2322 0.90 0.25 3.01 2.02 1.20 3675 2031	0.28	2.35	1.78	0.92
	C10	0.114 2.43 2.75 0.015 4763 2713 1.16 0.26 4.52 3.13 1.79 4680 2491	0.30	3.94	3.32	1.51
	C11	0.179 2.22 2.71 0.020 3908 2286 0.87 0.24 2.88 1.85 1.16 3615 1902	0.31	2.10	1.83	0.80
	C12	0.178 2.24 2.72 0.029 3918 2318 0.88 0.23 2.96 1.83 1.20 3644 1952	0.30	2.21	1.83	0.85
	C13	0.142 2.32 2.70 0.010 4569 2634 1.06 0.25 4.02 2.70 1.61 4423 2303	0.31	3.23	2.89	1.23
	C14	0.168 2.30 2.77 0.072 4310 2512 0.99 0.24 3.61 2.34 1.45 4128 2051	0.34	2.59	2.63	0.97
	C15	0.341 1.70 2.58 0.060 2834 1605 0.48 0.26 1.11 0.78 0.44 2810 1332	0.36	0.82	0.94	0.30
	C16	0.323 1.78 2.63 0.023 2864 1748 0.51 0.20 1.31 0.73 0.54 2774 1396	0.33	0.92	0.91	0.35
	C17	0.318 1.80 2.64 0.030 2897 1727 0.52 0.22 1.32 0.80 0.54 2712 1326	0.34	0.85	0.90	0.32
	C18	0.306 1.82 2.62 0.028 2947 1734 0.54 0.24 1.35 0.85 0.55 2788 1372	0.34	0.92	0.96	0.34
	C19	0.267 1.96 2.68 0.051 3093 1928 0.61 0.18 1.72 0.90 0.73 2741 1274	0.36	0.87	1.05	0.32
	C20	0.263 1.98 2.69 0.023 3236 1868 0.64 0.25 1.73 1.16 0.69 2914 1320	0.37	0.95	1.22	0.35
	C21	0.234 2.06 2.68 0.067 3150 1915 0.65 0.21 1.82 1.03 0.75 2759 1415	0.32	1.09	1.02	0.41
	Min	0.02	1.70 2.58 0.004 2834 1605 0.48 0.18 1.11 0.73 0.44 2712 1274	0.28	0.82	0.90	0.30
	Max	0.34	2.77 2.83 0.075 6135 3429 1.70 0.29 8.28 6.09 3.25 6353 3241	0.37	7.70	7.29	2.91
	Avg	0.18	2.21 2.71 0.031 4080 2354 0.93 0.24 3.38 2.32 1.35 3999 2018	0.33	2.79	2.68	1.05

Acknowledgments

SEM images were taken in the Central Research Science Laboratory/American University of Beirut and the Digital photographs have been taken in the Laboratory Center at the Yarmouk University.


Julien Gaudric

Thibault Couture

Jeremy Arzoine

Laurent Chiche

Response to "Comment on 'Short-and Mid-term Outcomes of Cryopreserved Abdominal Aortic Allografts Used as a Substitute for Infected Prosthetic Grafts in 200 Patients"'

Response to Dr. Tresson's comment

Julien Gaudric a* , Thibault Couture a , Jeremy Arzoine b , Laurent Chiche a Although polymerase chain reaction (PCR) is an interesting tool in the diagnostic armamentarium, its contribution appears limited since it allows identifying a microorganism in only 20% of negative cultures [START_REF] Ajdler-Schaeffler | Increased Pathogen Identification in Vascular Graft Infections by the Combined Use of Tissue Cultures and 16S rRNA Gene Polymerase Chain Reaction[END_REF] .

Genus-specific PCR may be more accurate, but discordant results in the identification of the pathogen responsible for vascular graft infection between culture and PCR have been shown in 60% of patients [START_REF] Puges | Comparison of Genus Specific PCR and Culture with or without Sonication for Microbiological Diagnosis of Vascular Graft Infection[END_REF] . In addition, PCR is not yet widely used to detect antibiotic resistance genes, is not available for all bacteria, and does not provide information on the minimum inhibitory concentrations.

18F-fluoro-D-deoxyglucose positron emission tomography-computed tomography (18F-FDG-PET/CT) seems to be a promising technique to confirm vascular graft infection.

Although its sensibility is good, its specificity remains low [START_REF] Folmer | Diagnostic Imaging in Vascular Graft Infection: A Systematic Review and Meta-Analysis[END_REF] .

In our experience, 18F-FDG-PET/CT had a limited place because all patients had a major infection criterion before surgery. In case of uncertain diagnosis, patients were monitored and the diagnosis was confirmed or not depending on the evolution of the inflammatory syndrome and CT-scan findings.

In our opinion, patients not suitable for total explantation and treated medically with antibiotics for 3 months could benefit from 18F-FDG-PET/CT by continuing antibiotic therapy throughout patients' life in case of persistent signs of vascular graft infection.

a

  Sorbonne Université, Department of Vascular Surgery, Pitié-Salpêtrière Hospital, Assistance Publique Hôpitaux de Paris, France, b Sorbonne Université, Department of Anesthesiology, Pitié-Salpêtrière Hospital, Assistance Publique Hôpitaux de Paris, France, * : corresponding author We thank Dr. Tresson for highlighting an important issue regarding the use of new strategies currently available for the management of arterial graft infection.

The temporary use of endograft in case of aorto-enteric fistula is theoretically a good option.

Actually, in most cases, the occurrence of gastrointestinal hemorrhage is only the consequence of the erosion in the gut mucosa. True communications between the aorta and the intestinal lumen are rare and result in severe hemorrhagic shock. In addition, the incidence of bifurcated grafts decreases the likelihood to be suitable for an endograft because of the short length between renal arteries and graft bifurcation. For example, in our series, among the 56 patients with prostheto-digestive fistula, only 9 patients underwent emergency surgery and 7 of them already had a bifurcated graft.

We agree with Dr. Tresson that the outcomes still need to be improved in such a devastating condition but recent advances in the diagnosis or treatment currently available are moderately helpful in the real life. Nevertheless, they must be used whenever possible.