Detection and picking of shear wave arrival for stiffness evaluation of highly porous chalk

Elastic wave velocities of compressional and shear waves propagating through sedimentary rocks are often coupled with information of bulk density to derive the rock stiffness. Acquiring the transit time of compressional and shear waves often involves manual picking of wave arrival times from wave trains recorded in the laboratory or by well‐logging tools. Picking the compressional wave arrival time is commonly accepted as straightforward. Oppositely, detecting the shear wave arrival and picking its arrival time is often troublesome because the transmitted shear wave partly converts to compressional waves and back to a secondary shear wave, concealing the transmitted shear wave arrival in the wave train. In laboratory settings, we illustrate the difficulty of shear wave detection in wave trains recorded on highly porous chalk plug samples from the Danish North Sea Basin. Wave trains were recorded on plugs dry, Tap‐water or Isopar‐L saturated during uniaxial strain compaction. The recorded shear wave trains showed two distinct features, which could be interpreted as the transmitted shear wave first arrival; we denoted them as early and late arrivals. However, as only one feature can mark the arrival of the transmitted shear wave, we propose a semi‐empirical disclosure strategy combining a graphical representation of stacked wave trains with rock physical modelling. By stacking recorded wave trains in a graphical strain–time–amplitude domain, we demonstrate that an early shear wave feature marks a converted shear to compressional to shear wave and not the transmitted shear wave. We used physical modelling to identify early shear wave features and illustrate the consequences of adopting a falsely interpreted shear wave on stiffness properties.

. Quantifying elastic wave velocities, for example compressional and shear waves, inevitably involves detecting and picking wave arrivals to derive the wave transit time within a recorded wave train.In laboratory settings and older well-log datasets, the transit time is determined by picking the wave arrival time from a single wave train.However, using a single wave train for detecting and picking arrival times implies uncertainty because single wave trains may include misleading wave features.In modern sonic logging tools with multiple receivers, arrival times are robustly determined using slowness-time-coherence maps (Kimball & Marzetta, 1984).
The compressional wave travels with the highest velocity of the elastic wave types, and consequently, its arrival time is generally detected as the first amplitude deflection in a recorded wave train.Hence, velocity determination of the compressional wave seldom causes problems.In contrast, detecting and picking shear wave arrivals are often troublesome because recorded shear wave trains include features of shear-to-compressional wave conversion (Benavente et al., 2020;Wang et al., 2009).Wave conversion occurs at interfaces of two elastic solids from incident waves arriving inclined, inducing four wave types, two transmitted and two reflected, compressional and shear waves (Bourbie et al., 1992;Ewing et al., 1957).Therefore, compressional waves generated from the transmitted shear wave conversion may convert back to a secondary shear wave (Modiriasari et al., 2018;Nakagawa et al., 2000).Thus, the shear wave train composes features, amplitudes and wave periods from more than one wave type, sometimes rendering detection of the transmitted shear wave highly uncertain or even impossible (Benavente et al., 2020).The imminent consequence of an ambiguous detection of the shear wave and picking of its arrival time is an erroneously derived shear wave velocity and thereby erroneously derived stiffness parameters.In order to add confidence in detection and picking of the shear wave arrivals, several authors have attempted to develop automation processes for compressional and shear wave arrival picking on wave trains (Allen, 1982;Gentili & Michelini, 2006;Mousavi et al., 2016;Withers et al., 1998).Automated picking of compressional wave arrivals in single wave trains can remain effective, even under low signal-to-noise ratio (Zhang et al., 2003).Although picking-automation of shear wave arrival has shown relative improvements in overcoming high noise factors that distort the shear wave arrival (Benavente et al., 2020), their validity often relies on manual detection and picking (Benavente et al., 2020;Finas et al., 2016).
As part of a laboratory test programme with uniaxial strain compaction of highly porous chalk from the Danish North Sea Valdemar field, we continuously recorded ultrasonic wave trains on dry, water and oil (Isopar-L -isoparaffinic hydrocarbon fluid) saturated plug samples.Analysis of recorded wave trains illustrates the difficulties well-log interpreters and experimentalists often face in detecting shear wave arrivals in highly porous chalk.To identify the transmitted shear wave arrival in wave trains showing several prospective shear waves, we illustrate and propose a semi-empirical strategy of combining stacked wave trains in a graphical timestrain-amplitude domain with rock physical modelling.As stacking wave trains helps determine wave evolution during an experimental procedure (Modiriasari et al., 2017), we first graphically displayed stacked wave trains in a time-strainamplitude domain to add confidence in detection and picking of shear wave arrivals.Further, we combined bulk density with ultrasonic velocities derived from prospective wave arrivals and derived the corresponding elastic moduli.We successfully added additional confidence to the shear wave picking and exclusion of prospective wave arrivals by comparing derived elastic moduli with rock physical modelling results.

Geological setting
The Valdemar field is located in the central part of the Danish Central Graben and comprises two structural anticlines named Bo and North Jens (Jakobsen et al., 2004).The reservoir is situated in the upper levels of the Lower Cretaceous Cromer Knoll Group at a depth of approx.2200 m (7200 ft) and with pre-production formation pressure and temperature in the order of 37 MPa (5400 psia) and 85˚C, respectively.In accordance with Jensen et al. (1986) and Kühnau and Michelsen (1994), the Lower Cretaceous Cromer Knoll Group is subdivided into five lithostratigraphic formations denoted as Åsgard, Vyl, Tuxen, Sola and Rødby.

Chalk core plugs
A series of induration H2 cores sections denoted with indices 15, 18 and 19 were recovered during horizontal drilling campaigns in the Upper Tuxen Formation of the Valdemar field.For laboratory testing, 12 core plugs with approximate dimensions of 1.5 in.diameter and 3 in.length were sampled from the drilled core sections using a rotary core drill with Isopar-L oil as coolant.Due to downhole core material sparsity, eight plugs had horizontal and four vertical orientations; some plugs had fractures, likely induced by unloading and equilibration to ambient conditions from core recovery.The vertical direction was established based on the geological interpreta-tion of fractures and trace fossils (Table 1).End surfaces were trimmed and paralleled within a precision of 0.1 mm.Using backscatter electron micrograph images of vertical side trims, the core sections are characterised as mudstones consisting of calcite fossils of mainly nanoconids, coccoliths and their fragments (Figure 1).Energy-dispersive X-ray spectroscopy shows traces of quartz, titanium oxide and clays, including kaolinite and illite/smectite.
In order to identify significant fractures prior to the mechanical testing, we scanned all plugs using a medical CT-scanner (Figure 2).Vertical plugs show fractures perpendicular to the vertical in situ stress direction and longitudinal plug direction.Similarly, fractures seen in horizontal plugs from core sections 18 or 19 are perpendicular to the vertical in situ stress direction but parallel to longitudinal plug direction.No fractures are seen in horizontal plugs from core section 15.The resolution of CT-scanning does not allow for a detailed analysis of mineralogy.
The 12 plugs were cleaned with the Soxhlet extraction method (Dean, 1998) using methanol and toluene as solvents before being oven-dried at 60˚C and equilibrated to ambient temperature in a desiccator.The dry density was calculated from measurements of dry mass and plug dimensions.Grain density and gas-porosity were derived from N 2 expansion measurement.Measured grain density and porosity are centred around 2.75 g/cm 3 and 43%, respectively, and show high similarity between individual plugs (Table 1).Each plug was placed in a core holder designed for flowthrough experiments and equipped with up-and downstream F I G U R E 2 Medical CT-scanning of the Upper Tuxen core plugs before testing.CT-scanning was performed on unconfined plugs prepared from full core sections recovered during horizontal drilling campaigns.Full core section orientation was estimated through geological interpretation of fractures and trace fossils.Thereby, plug orientation in pictures (horizontal and vertical) approximates in situ formation orientation.Lighter colours represent higher density.
discharge and pressure transducers.Using a confining pressure of 2.75 MPa and steady-state N 2 flow measurements at a minimum of three differential pressure steps, we derived the gas-permeability using Darcy's equation corrected to account for gas compressibility by the ideal gas low.Using Klinkenberg correction (Klinkenberg, 1941), we estimated the liquid permeability.The Klinkenberg corrected permeability is around 1 mD and appears independent of plug orientation (Table 1).Carbonate content, determined using HCl dissolution and NaOH titration on Soxhlet cleaned core plug after mechanical testing, is centred around 91% and shows high similarity between plugs (Table 1).

Mechanical experimental setup
As preparation for the mechanical testing, dry core plugs were jacketed with an impermeable membrane and placed between the upper and lower piston heads in a standard Hoek cell with independent control of axial (σ Α ) and radial stress (σ R ) (Figure 3).The piston heads enabled fluid flow through porous filter plates connected to a precision balance for expelled fluid mass recording and volumetric strain and porosity change computation.The setup controls the axial stress using a 250 kN load frame with virtual infinite stiffness correction for self-deflection.The radial stress was controlled using a pressure actuator.The axial strain (ε A,LVDT ) was recorded using two external linear variable differential transformers (LVDT).Internally, axial (ε A,SG ) and radial strain (ε R,SG ) were recorded locally using four strain gauges (SG) placed at the centre of the plug (Figure 3).As part of the plug installation process, axial and radial stresses were increased to 0.5 MPa, plugs evacuated by vacuum, and the saturating fluid introduced as part of a process attempting the highest possible saturation degree.While maintaining the applied axial and radial stresses, a flooding procedure was followed for 3-6 days at a flooding rate of 250 mm 3 /h, keeping the inlet pressure below 0.25 MPa.

Experimental procedures and interpretation of mechanical stress-strain curves
We performed compaction tests as uniaxial strain (K 0 -testing) on dry as well as Isopar-L and Tap-water saturated core plugs (Table 1).The plugs were compacted under the same applied stress geometry to strain levels three to four times beyond yield.We used Tap-water (calcite equilibrated) in order to minimize possible effects of water weakening.During testing, we increased axial stress by applying a constant strain rate of 0.1%/h while controlling the radial stress to inhibit radial strain until reaching the maximum radial stress capacity of 60 MPa.At maximum radial stress, we kept the axial stress constant in a creep phase over several days before initiating an unloading sequence to atmospheric pressure with an identical strain rate.To minimise initial bedding effects caused by, for example, tilting of plug and piston heads, we performed an unloading and loading cycle upon an initial loading to 10 MPa.We tested with major principal stress applied in the longitudinal plug direction.Consequently, we tested horizontal plugs with principal stress directions perpendicular to in situ stress state.By examining the axial stress versus axial strain compaction curves, we define distinct transition points at which a physical change in mechanical compaction behaviour occurs.We conceptually define physical change as an onset of change in deformation rate for the applied compaction energy (Figure 4).We name regions in the stress-strain domain with changes in deformation rate based on Kågeson-Loe et al. (1993), without necessarily reflecting physical phenomena (Figure 4).Denoted as pore collapse onset, the first transition point marks the onset of a deformation rate increase and the end of the elastic region (Figure 4).The pore collapse marks a point where the load exceeds the frame strength, leading to the onset of grain contact ruptures and yielding of the rock frame.The yield region ends with a transition point marking the onset of a region with a linear deformation rate, and both regions include rearrangement and compaction of grain clusters (Figure 4).Denoted as strain hardening onset, the third transition point marks the onset of a non-linear reduction in deformation rate.We understand this region as a compaction of grains with ruptured contacts during pore collapse.Other authors have suggested different regions notation (e.g.Jaeger et al., 2007), but we adhere to those of Kågeson-Loe et al. (1993) and emphasize that strain hardening only involves hardening by compaction or rearrangement of grains or both, and not re-cementation of ruptured grain contacts.

Ultrasonic wave velocities
Throughout mechanical testing, we continuously transmitted and recorded one ultrasonic compressional wave (P) and two orthogonal shear waves (S1 and S2) in the longitudinal plug direction (Figure 3).We used embedded piezoelectric crystals into the upper and lower piston heads as wave transmitters and receivers.We executed one-by-one in a P-S1-S2 sequence, exciting the transmitting transducer and reading signal through the receiving transducer.For both P and S, we transmitted 64 waves at a frequency of 10 Hz and calculated the average in order to establish the wave train used to interpret the wave arrival times.We applied a broadband voltage pulse, centred approximately at 20 MHz and recorded the received signal using an oscilloscope.The output frequency of the P-and S-waves was approximately 300 and 600 kHz, respectively.These frequencies correspond to the low-frequency regime for low-permeable chalk (Fabricius et al., 2010).For system calibration, we measured P-and S-wave transit times using polyether ether ketone plugs of different lengths and applied linear regression to deduct the delay time at zero-plug-length (axis intercept).We defined the P-wave arrival as the first deflection on the wave trains, and for the S-wave, we used the zero-amplitude crossings of the corresponding S-wave period.
Wave conversion at various interfaces during propagation through, for example, piston heads and core plugs complicates the detection of the transmitted S-waves in recorded wave trains.We stacked 64 wave trains to increase signal-tonoise ratio and assist with the detection and trace trajectory of amplitude changes from converted, reflected and transmitted waves.Using a graphical technique, we created maps from stacked wave trains within a 3D strain-time-amplitude domain and in a 2D strain-time domain.The 3D and 2D maps are the paramount element in our strategy and assist in detecting waves with S-wave wavelength and amplitude features.Further, the 3D and 2D maps are used to trace strain trajectories of unique wave features and compare trajectories to changes in deformation rate in the 2D stress-strain domain.
We derived ultrasonic P-and prospective S-wave velocities as a function of stress and strain from combined wave arrival (corrected for delay time) and deformation corrected plug length.Further, we derived dynamic elastic moduli by combining bulk density and wave velocities and employed the iso-frame model (see Appendix for details).The application of the iso-frame model assisted in distinguishing the presumably transmitted S-wave over prospective S-wave arrivals.

Effect of plug orientation on the stress-strain relation during compaction
We examined the difference in the mechanical behaviour of vertical and horizontal plugs with Isopar-L or Tapwater saturation (Table 1).Horizontal plugs represent a single horizontal direction.Consequently, the perpendicular horizontal direction and 45˚orientation are unaddressed, leaving an incomplete picture of orientation effects on compaction behaviour.Nonetheless, because we consistently applied major principal stress along the longitudinal plug axis, it is feasible to evaluate effects on compaction behaviour in the principal stress direction and compare to the in situ state.The similar compaction behaviour of plugs with different orientations indicate near isotropy; however, anisotropy may have increased with increasing strain (Figure 5).In the axial stress-strain domain, excluding horizontal plug 19.C4(H) and 18.C2(H) with longitudinal fractures (Figure 2), vertical plugs show slightly lower strain (Figure 5a,c).Consequently, in terms of axial stress, the pore collapse onset is higher for the vertical plugs, which behave both stiffer and stronger.Strain increase beyond pore collapse increases the orientationrelated differences and reaches an order of 2% axial strain difference at maximum axial stress (Figure 5a,c).In the differential stress versus axial strain domain, horizontal and vertical plugs show similar stress-strain relation before the onset of strain hardening (Figure 5b,d).Strain increase beyond strain hardening shows an increasing strain difference for increasing stress, reaching an order of 2% at maximum stress (Figure 5b,d).In contrast, the diverging trend is noticed at the pore collapse onset in the axial stress domain.However, at maximum stress, the strain difference between plug orientations is similar in the two stress domains.The observed compaction behaviour shows that the observed strain reflects in situ orientation even after pore collapses with increased grain contact area (Alam et al., 2012;Gram et al., 2020).Longitudinal fractures in Tap-water saturated horizontal plugs 18.C2(H) and 19.C4(H) (Figure 2) parallel to the major principal stress direction (σ A ) induce an increased confining stress (σ R ) to counteract lateral buckling and maintaining zero lateral strain (Figure 5c,d).In contrast, fractures perpendicular to the major principal stress direction, that is, in vertical plugs, do not affect compaction (Figures 2 and 5).

Effect of pore fluid on stress-strain relation during compaction
Representing the full test matrix, we examined the Isopar-L and Tap-water saturated plug 15.A1(H), 15.A2(H), as well as the dry 18.C1(H) plug.The three plugs are practically identical with respect to initial petrophysical properties (Table 1).Nonetheless, the different saturation fluids induce contrasting compaction behaviour (Figure 6).Compared to dry conditions, the pore collapse onset and strain hardening onset of Isopar-L and Tap-water saturated plugs are lower in terms of stress (Figure 6).On the contrary, the pore collapse onset and strain hardening onset are indistinguishable in terms of strain, suggesting strain as yield and hardening controlling parameter (Figure 6).Previous experimental studies of chalks have shown similar contrasting mechanical behaviour related to the saturating fluid (e.g.Hellmann et al., 2002;Meireles et al., 2021;Nermoen et al., 2018;Newman, 1983;Sylte et al., 1999;Voake et al., 2019).

Single, stacked and mapped wave trains
P-wave trains recorded during compaction show a distinct deflection and arrival times are confidently picked at all stress levels (Figure 7).Further, a small amplitude P-wave arrival is  1).
detected in the S-wave train as the first zero-line deflection.Contrary, detecting and picking of S-wave arrivals from single wave trains are troublesome.In the elastic region, prior to pore collapse onset, the S-wave trains display only a single wave period with S-wave arrival features (Figure 7a).After the yield and linear region (Figure 4), two different wave periods show S-wave arrival features (Figure 7b).We denote these two wave periods/features and the corresponding arrival times as the early and late arrival time (Figure 7b).Further, we note that the two S-wave features have wavelengths and amplitudes with similar order of magnitude.In contrast, the polyether ether ketone calibration plugs S-wave trains showed a very distinct single S-wave feature (Figure 7c).
The Isopar-L saturated plug 15.A1(H) with intermediate compaction behaviour in the 2D axial stress-strain domain shows a distinct deflection as the first P-wave in the 3D strain-time-amplitude domain likewise identified in the single wave trains (Figures 7 and 8).The zero-amplitude contour lines projected in the strain-time plane display several later wave arrivals with trajectory parallel to the first arrival as well as trajectory changes when passing through the different mechanical regions (Figure 8).
Stacked S1-wave trains of the same plug 15.A1(H), as well as plug 19.B2(V) with initial fractures (Figure 2), show a sequence of low-energy first arrivals (Figures 9 and 10), which are presumably parasitic P-waves emitted by the Swave transducers (Yurikov et al., 2019).Moreover, we see that the early and late wave periods previously identified in single wave trains (Figure 7) are identified in the stacked S1-wave trains (Figures 9 and 10).Comparison of stacked wave trains from unfractured 15.A1(H) and fractured sample 19.B2(V) show minor variations in amplitude, shape and behaviour (Figured 9 and 10).The low pore collapse stress level probably limits fracture effect development on wave trains at strain levels above pore collapse.The early and late wave trajectories are close to parallel for strain levels below the onset of strain hardening (Figures 9 and 10).Beyond the pore collapse onset, the early wave trajectory is close to parallel with the P-waves, indicating that the early wave has, at least partly, propagated through the core plug as an S1 to P converted P-wave (Figures 9 and 10).In contrast, the late wave trajectory is distinctly different from P-waves beyond the strain hardening onset.The late wave shows a well-defined trajectory similar to the 2D stress-strain domain (Figures 4, 9 and 10).Thus, the compaction behaviour observed in the 2D stress-strain domain (e.g. Figure 4) is traced in the 3D strain-time-amplitude domain and the 2D strain-time plane.Figures 9 and 10 illustrate challenges in detecting the late wave at strain levels below strain hardening onset.
Except for a decrease around pore collapse, the early wave amplitude increases monotonously for increasing strain (Figures 9 and 10).Signal amplitudes often increase due to a better plug-to-piston head contact and increased grain contact area by strain increase.The late wave amplitude reaches a local maximum before the strain hardening onset and fluctuates whereafter (Figures 9 and 10).Fluctuation appears as wave interference, where converted waves with trajectory parallel to the P-wave cross the late wave.Out-of-phase wave F I G U R E 8 Stacked P-wave trains of Isopar-L saturated chalk plug 15.A1(H) tested with K 0 compaction.The plot shows wave trains as a function of linear variable differential transformers (LVDT) recorded axial strain ranging from 0.5% to 17.8%.The bottom plane shows zero-amplitude contour lines of the stacked wave trains in a strain-time plane projecting the highest and lowest amplitudes with black and whiter colour, respectively.Blue markings and notations represent distinct points and regions during compaction (Figure 4).With a consistent line in the plane, the first deflection on wave trains reflects the P-wave.

F I G U R E 9
Stacked S1-wave trains of the 15.A1(H) Isopar-L saturated chalk plug tested with K 0 compaction.The plot shows wave trains as a function of linear variable differential transformers (LVDT) recorded axial strain ranging from 0.5% to 17.8%.The bottom plane shows zero-amplitude contour lines of the stacked wave trains in a strain-time plane projecting the highest and lowest amplitudes with black and whiter colour, respectively.Blue markings and notations represent distinct points and regions during compaction (Figure 4).With a consistent line in the plane, the first deflection on wave trains reflects a parasitic P-wave.Several waves later, the early and late wave periods are shown in both stacked wave trains and plane.The onset of strain hardening onset on the late wave period marks a shift in wave trajectory as a function of axial strain.
F I G U R E 1 0 Stacked S1-wave trains of the 19.B2(V) Isopar-L saturated chalk plug tested with K 0 compaction.The plot shows wave trains as a function of linear variable differential transformers (LVDT) recorded axial strain ranging from 0.3% to 16.9%.The bottom plane shows zero-amplitude contour lines of the stacked wave trains in a strain-time plane projecting the highest and lowest amplitudes with black and whiter colour, respectively.Blue markings and notations represent distinct points and regions during compaction (Figure 4).With a consistent line in the plane, the first deflection on wave trains reflects a parasitic P-wave.Several waves later, the early and late wave periods are shown in both stacked wave trains and plane.The onset of strain hardening onset on the late wave period marks a shift in wave trajectory as a function of axial strain.
interference diminishes potential amplitude increase, making picking a zero-amplitude crossing on the late wave arrival from a single wave train challenging even at strain levels above the strain hardening onset.
Stacking and mapping S1-wave trains in the strain-timeamplitude domain and in the strain-time plane, the Tap-water saturated 15.A2(H) plug and the dry 18.C1(H) plug show features identical to the Isopar-L saturated 15.A1(H) plug (Figures 11 and 12).However, trajectories are different between the saturating fluids, corresponding to observations from the 2D stress-strain domain (Figure 6).The late wave trajectory is short for the dry plug because experimental limitations prevented stress increase much beyond the strain hardening onset (Figures 6 and 12).

Consequences of adopting an implausible S-wave on Poisson's ratio and elastic moduli
In order to illustrate the consequences of adopting an implausible S-wave, we adopted both the early and late wave as the actual transmitted S-wave.As both the early and late wave features have wavelength and amplitudes with similar order of magnitude (Figure 7), it indicates that the S-wave crys-tals receiving the early and late wave are both excited by an S-wave.Thus, because of the higher velocity in P-waves, the early feature is presumably an S-P-S converted S-wave (Modiriasari et al., 2018;Nakagawa et al., 2000).
We derived the corresponding early and late S-wave velocities (V S,early and V S,late ) and combined V S,early and V S,late with the P-wave velocity (V P ) to derive Poisson's ratio from Equation (A.4).Denoted as ν early and ν late , we examined Poisson's ratio of the three saturating conditions (Isopar-L: 15.A1(H), Tap-water: 15.A2(H), dry: 18.C1(H)) as function of axial strain.Poisson's ratio derived from the early wave period (ν early ) follows a distinct trajectory reflecting the trajectory of the P-wave velocity (Figure 13), suggesting that ν early is computed from an S-P-S converted S-wave falsely interpreted as the actual V S , V S,early in the present case.Further, adopting V S,early , we compute negative values of ν early for the dry plug (Figure 13).Envisaging Equation (A.4), negative Poisson's ratio suggests a falsely interpreted V S (V S,early ) as confidence in the P-wave velocity is high.Further, radial shrinkage of the rock frame from axial compaction goes against current experience.In contrast, adopting the late wave period (V S,late ), the derived ν late shows only minor deviations across the various mechanical regions for the three saturating conditions (Figure 13).
Combining bulk density with ultrasonic wave velocities, we derived the compressional, shear and bulk modulus (M, F I G U R E 1 1 Stacked S1-wave trains of the 15.A2(H) Tap-water saturated chalk plug tested with K 0 compaction.The plot shows wave trains as a function of linear variable differential transformers (LVDT) recorded axial strain ranging from 0.6% to 20.1%.The bottom plane shows zero-amplitude contour lines of the stacked wave trains in a strain-time plane projecting the highest and lowest amplitudes with black and whiter colour, respectively.Blue markings and notations represent distinct points and regions during compaction (Figure 4).With a consistent line in the plane, the first deflection on wave trains reflects a parasitic P-wave.Several waves later, the early and late wave periods are shown in both stacked wave trains and plane.The onset of strain hardening onset on the late wave period marks a shift in wave trajectory as a function of axial strain.
F I G U R E 1 2 Stacked S1-wave trains of the 18.C1(H) dry chalk plug tested with K 0 compaction.The plot shows wave trains as a function of linear variable differential transformers (LVDT) recorded axial strain ranging from 0.2% to 11.7%.The bottom plane shows zero-amplitude contour lines of the stacked wave trains in a strain-time plane projecting the highest and lowest amplitudes with black and whiter colour, respectively.Blue markings and notations represent distinct points and regions during compaction (Figure 4).With a consistent line in the plane, the first deflection on wave trains reflects a parasitic P-wave.Several waves later, the early and late wave periods are shown in both stacked wave trains and plane.The onset of strain hardening onset on the late wave period marks a shift in wave trajectory as a function of axial strain.G and K) from Equations (A.1)-(A.3).The compressional modulus is independent of adopting the early or late wave periods.Again, we adopted V S,early and V S,late to derive G early , K early and G late , K late for the representative plugs 15.A1(H), 15.A2(H) and 18.C1(H).Corresponding to a negative Poisson's ratio and Equation (A.4), derived bulk modulus (K early ) for the dry plug (K dry,early ), is lower than the shear modulus (G dry,early ) (Figures 13 and 14).Confidence in derived den-sity and P-wave velocity again suggest V S,early as a falsely interpreted V S .In the elastic and yield regions, the derived bulk modulus from Isopar-L and Tap-water saturated plugs (K sat,early ) has similar numerical value as the shear modulus (G sat,early ) (Figure 14).Even as a similar range of K sat,early and G sat,early is not theoretically impossible, it is improbable for the studied chalk to the best of our knowledge, further suggesting that V S,early is an S-P-S converted S-wave.In contrast, adopting the late wave shows a bulk modulus of larger numerical magnitude than the shear modulus (Figure 14), thus linking results to previously published studies by, for example, Japsen et al. (2004), Gommesen et al. (2007), Katika et al. (2015) and Meireles et al. (2021).
Wave conversion occurs when an incident wave arrives inclined at interfaces between two elastic solids (Ewing et al., 1957).In the present experimental setup (Figure 3), potential wave conversion exists at horizontal interfaces between the piston head and each plug end as well as at the lateral interface between the plug wall and jacketing membrane.Due to interface conversion, the transmitted wave presumably travels with an indirect path and arrives at the receiver after reflecting on the plug wall approximately mid-sample.The indirect travel path is seen as the early wave in recorded wave trains (Figure 7).Combining the recorded arrival time of the early wave with travel length in plug and piston as well as P-and S-wave velocity in saturated chalk and piston (steel), back calculating using trigonometry shows that the early wave propagates about half the plug length as a Pwave.The higher P-wave velocity allowed the early wave to arrive at the receiver before the late wave arrival (Figure 7).Additional wave conversion may occur due to rock fractures and inhomogeneities; however, it is unlikely to have induced the early wave, as a significant amplitude was recorded in both intact and initially fractured plugs identified from CT scans (Figures 2, 9 and 10).Rock anisotropy can invoke shear wave splitting (Crampin, 1985).However, it is unlikely that the early and late waves are a fast and slow shear wave pair in the present case because the faster wave of a shear wave pair has in-phase shear motions and determines the shear wave velocity (Mavko et al., 2009).However, the early wave derives an unreasonably high shear wave velocity and unrealistic moduli (Figure 14), rendering it unlikely that the early and late waves are shear wave pair products.

Using rock physical modelling to detect falsely interpreted S-waves
Given the high calcium carbonate content of the samples (Table 1), we assumed that calcite was the load-bearing and frame constituting mineral.We combined mineral and saturating fluid stiffness (Table 2) with derived frame moduli to construct the iso-frame model (Equations A.5-A.9).We employed the iso-frame model for the compressional modulus as well as G early , K early and G late , K late .Because we regard the P-wave velocities as accurate, we use the IF value of the compressional modulus as a benchmark to determine whether early (G early , K early ) or late (G late , K late ) is the correct shear and bulk modulus.Hence, we determine if the IF values for F I G U R E 1 5 Iso-frame model of high porosity chalk.The iso-frame is constructed as function of porosity for dynamic moduli of M, G and K from left to right (Figure 14).Top panel: Isopar-L saturated plug 15.A1(H), middle panel: Tap-water saturated plug 15.A2(H), bottom panel: dry plug 18.C1(H).Adopting the early or late wave period (Figure 13) as the actual S-wave arrival is plotted with grey and orange colour for the shear and bulk modulus.Initial porosity is given in Table 1, whereas porosity decrease was calculated from expelled liquid mass and recorded axial deformation for the dry plug.The mechanical regions each plug undergoes during compaction are indicated with background greyscale colour.Error bars are approximately equal to the symbol size.
the derived elastic moduli (M, G and K) are congruous, that is, that identical fractions of the solid constituent in the rock frame govern resistance to shear and compression.Independent of saturation fluid, the compressional modulus shifts to higher IF values as the fraction of stress resisting solids increases with axial strain and porosity decrease (Figure 15, first column).We see the shear and bulk modulus derived from the late wave period (i.e.V S,late ) and across the different saturating conditions follow congruent IF trajectory as the compressional modulus (Figure 15).Congruent IF trajectories  15.A2(H)) bulk modulus compared to dry plug bulk modulus.Grey lines show iso-frame values.Adopting the early or late wave period (Figure 13) as the actual S-wave arrival is plotted with grey and orange colour.Initial porosity is given in Table 1, whereas porosity decrease was calculated from expelled liquid mass and recorded axial deformation for the dry plug.Error bars are approximately equal to the symbol size.
across the three moduli add argument and confidence to the late wave being the actual S-wave (Figure 15).In contrast, if we adopt the early wave, different IF trajectories suggest that the fractions of the solid constituent resisting deformation differ for the different elastic moduli (Figure 15).To our knowledge, no argument of why different factions of the solid constituent should control the different moduli exist, further suggesting that the early wave is an S-P-S converted S-wave and not the transmitted S-wave.Moreover, in the elastic region, the Tap-water K early derived using V S,early is below the Reuss bound (IF = 0) (Figure 15, second row).
As the Reuss bound defines a solid-fluid suspension mixture and because the plug at this stage is intact, it adds to previous observations with respect to the origin of the early wave.
Keeping in mind that Gassmann's equation (Equation A.10) does not take electrostatic effects on the rock frame stiffness into account, we compared fluid substituted bulk modulus of the Isopar-L and Tap-water saturated plugs to that of the dry plug (K dry ) (Figure 16).In the iso-frame domain, we find that the order of 1% lower porosity of the dry as compared to the Isopar-L and Tap-water saturated plugs (Table 1) causes a slight left shift in the dry plug trajectory (Figure 16).Counting in the initial porosity difference, we find that adopting V S,early or V S,late , the Isopar-L fluid substituted bulk modulus possesses trajectories and numerical values closely similar to the dry plug bulk modulus (Figure 16a).
Comparing dry plug bulk modulus to fluid substituted bulk modulus of the Isopar-L saturated plug, we find an order of 1 GPa overestimated fluid substituted bulk modu-lus (Figure 16a).The observed overestimation is independent of adopting V S,early or V S,late and was noticed even after pore collapse and grain rearrangement, where the effect of fractures of the dry sample (Figure 2) in shear wave propagation vanishes.Thus, we cannot argue that the adoption of either V S,early or V S,late causes poor or unrealistic performance by Isopar-L substitution by Gassmann's equation.Correspondingly, we cannot use the performance of Gassmann's fluid substitution to detect falsely interpreted S-waves in Isopar-L saturated plugs.
Comparing the dry plug bulk modulus to the fluid substituted bulk modulus of the Tap-water saturated plug, we find an underestimation of the fluid substituted bulk modulus in the order of 2 GPa.Adopting V S,early , the underestimation decrease with increasing compaction and decreasing porosity.Further, we find unphysical and negative values of the fluid substituted bulk modulus for low compaction and high porosities when adopting V S,early (Figure 16b).Violation of preconditions (i.e.electrostatic effects) in Gassmann's equation presumably accounts for the underestimation.The electrostatic effect is often referred to as water weakening and observed softening correspond with results from, for example, Rutter (1972), Carter and Mallard (1974), Sylte et al. (1999), Nermoen et al. (2018), Meireles et al. (2020) and Meireles et al. (2021).Even as fluid substituted using V S,early results in negative bulk modulus, the numerical overestimation is approximately independent of adopting V S,early or V S,late .Thus, the detection of falsely interpreted S-waves from the performance of Gassmann's fluid substitution of Tap-water is unfeasible.

T A B L E 1
Initial properties and saturation condition of chalk plugs tested under K 0 stress symmetry.

Note:
Plugs are denoted with ID of their corresponding core section.(H) and (V) refer to horizontal and vertical orientation, respectively.The plugs correspond in true vertical depths between 2306 and 2309 mbsl.a Measured with N 2 porosimetry.b Klinkenberg corrected N 2 permeability.c Calculated as calcium carbonate content; measured by HCl dissolution and NaOH titration on tested plugs.F I G U R E 1 Backscattered electron micrography images of polished thin sections: (a) core section 15, (b) core section 18 and (c) core section 19.All sections are characterised as mudstone consisting primarily of calcite fossils of mainly nanoconids, coccoliths and their fragments.

F
Illustration of the experimental setup, including Hoek-cell and piston heads.Source: Courtesy of Geo.F I G U R E 4 Compaction curve showing the relationship between applied axial stress and the resulting axial strain of plug 15.A1(H) tested under uniaxial strain compaction and Isopar-L as saturating fluid.Initial unload and load cycle with red marking is attempted to eliminate bedding effects from, for example, plug alignment to piston heads.Circular markings on the compaction curve show the onset of a new mechanical region.Mechanical regions are indicated with greyscale background colour, and the conceptual notation is based on Kågeson-Loe et al. (1993).

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I G U R E 5 Stress-strain relation during compaction of plugs with identical saturation fluid and perpendicular orientation: (a) and (b) Isopar-L saturated; (c) and (d) Tap-water saturated.Left column shows axial stress versus axial strain and right column differential stress versus axial strain.Labelling shows plug ID where V is for vertical and H is for horizontal orientation.Vertical plugs are plotted with red colour.F I G U R E 6 Stress-strain relation of horizontal plugs with different saturation fluids (Isopar-L: 15.A1(H), Tap-water: 15.A2(H) and dry: 18.C1(H)).(a) Axial stress versus axial strain and (b) differential stress versus axial strain.Blue markings show the onset of pore collapse and strain hardening.For experimental reasons, the radial stress increased only to 46 MPa for the dry plug compared to 60 MPa for the additional plugs.F I G U R E 7 Example of single wave trains from P-, S1-and S2-transducers.(a) Wave trains recorded on the Isopar-L saturated plug 15.A1(H) in the elastic region at σ A = 8.1 MPa, σ R = 2.3 MPa, and axial strain from linear variable differential transformers (LVDT) measurements of 0.7%.(b) Wave trains recorded on the Isopar-L saturated plug 15.A1(H) in the strain hardening region at σ A = 108.3MPa, σ R = 60 MPa and axial strain from LVDT measurements of 18.2%.The late S-wave in (b) is detected as a potential but not clear arrival in (a).(c) Wave trains recorded on a calibrations plug at σ A = 4 MPa using the same load frame and P-, S1and S2-transducers as in (a) and (b).The figure is representative of the test matrix (Table

F
Development of axial stress (solid line), axial strain (dashed line), porosity (dotted line), V P , V S and Poisson's ratio throughout K 0 compaction.Top panel: Isopar-L saturated plug 15.A1(H), middle panel: Tap-water saturated plug 15.A2(H), bottom panel: dry plug 18.C1(H).Development of V S and Poisson's ratio if adopting the early or late wave period (Figures 9, 11 and 12) as the actual S-wave arrival is plotted with grey and orange colour.Porosity in plugs 15.A1(H) and 15.A2(H) is derived from initial porosity using expelled fluid measurements, while in dry plug 18.C1(H) from initial porosity and strain measurements.The mechanical regions during compaction are indicated with background greyscale colour.Error bars in wave velocities and Poisson's ratio are approximately equal to the symbol size.

F
Development of axial stress (solid line), compressional (circles), shear (triangles) and bulk modulus (squares) as function of axial strain throughout K 0 compaction.Top panel: Isopar-L saturated plug 15.A1(H), middle panel: Tap-water saturated plug 15.A2(H), bottom panel: dry plug 18.C1(H).Moduli are derived from deformation corrected bulk density and ultrasonic velocities adopting the early or late wave period as the actual S-wave arrival for the shear and bulk modulus (Figures 9, 11 and 12).The mechanical regions during compaction are indicated with background greyscale colour.Error bars in different moduli are approximately equal to the symbol size.

a
Mavko et al. (2009); based on formulas inBatzle and Wang (1992).b Derived from measurements of density as function of four pressure steps from 1 to 15 bar.c Based on citations inMavko et al. (2009).

F I G U R E 1 6
Gassmann fluid substitution of bulk modulus in the iso-frame model domain.(a) Fluid substitution of Isopar-L saturated (15.A1(H)) bulk modulus compared to dry plug bulk modulus.(b) Fluid substitution of Tap-water saturated (