# Frequency, pressure, and strain dependence of nonlinearelasticity in Berea Sandstone非线性的频率、压力和应变依赖性 贝里亚砂岩的弹性

Geophysical Research Letters Frequency, pressure, and strain dependence of nonlinear elasticity in Berea Sandstone Jacques Rivière1,2, Lucas Pimienta3, Marco Scuderi1,4, Thibault Candela1,5, Parisa Shokouhi6, Jér?me Fortin3, Alexandre Schubnel3, Chris Marone1, and Paul A. Johnson2 1Department of Geosciences, Pennsylvania State University, University Park, Pennsylvania, USA,2Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico, USA,3école Normale Supérieure (ENS), Laboratoire de Géologie, Paris, France,4Department of Earth Sciences, La Sapienza Universit di Rome, Rome, Italy,5TNO, Geological Survey of The Netherlands, TA Utrecht, Netherlands,6Department of Civil Engineering, Pennsylvania State University, University Park, Pennsylvania, USA AbstractAcoustoelasticity measurements in a sample of room dry Berea sandstone are conducted at various loading frequencies to explore the transition between the quasi-static (f → 0) and dynamic (few kilohertz) nonlinear elastic response. We carry out these measurements at multiple confi ning pressures and perform a multivariate regression analysis to quantify the dependence of the harmonic content on strain amplitude, frequency, and pressure. The modulus softening (equivalent to the harmonic at0f) increases by a factor 2–3 over 3 orders of magnitude increase in frequency. Harmonics at2f,4f, and6fexhibit similar behaviors. In contrast, the harmonic at1fappears frequency independent. This result corroborates previous studies showing that the nonlinear elasticity of rocks can be described with a minimum of two physical mechanisms. This study provides quantitative data that describes the rate dependency of nonlinear elasticity. These fi ndings can be used to improve theories relating the macroscopic elastic response to microstructural features. 1. Introduction The main motivation of this study is to improve our understanding of nonlinear elastic properties of rocks, in particular frequency (-time) and pressure dependences. In terms of modeling, one wants to improve theo- retical descriptions that relate nonlinear measures to microstructural properties of rocks [GuyerandJohnson, 2009]. Nonlinear elasticity is relevant for a broad range of applications in geosciences, including earthquake slipprocesses,stronggroundmotion[BeresnevandWen,1996;Fieldetal.,1997;Renaudetal.,2014;Trifunacand Todorovska, 1996], liquefaction phenomenon [Aguirre and Irikura, 1997], Earth tides [Hillers et al., 2015], and oil/gas exploration. Further, a better knowledge of frequency dependence facilitates comparisons between observations made at diff erent scales, e.g., from the laboratory to the fi eld scale. Anumberofstudies[WinklerandLiu,1996;WinklerandMcGowan,2004]havecharacterizednonlinearelastic- ity of rocks using acoustoelastic experiments. Such experiments consists in measuring speed of sound with high-frequency (HF) pulses across the sample while it is gradually stressed uniaxially and/or hydrostatically atincreasinglevels.Thesequasi-staticacoustoelasticexperimentsareequivalenttozero-frequencymeasure- ments. In dynamic acoustoelastic testing (DAET), the stepwise increases in pressure/stress are replaced by a low-frequency (LF) strain modulation [Renaudetal., 2011]. DAET is therefore a pump-probe scheme in which theHFelasticpulsesprobethestateofamechanicalsystemsetbytheLFwave,thepump.ForthepumpPwe often use the fundamental compressional mode of the sample driven to strain amplitudes10?7≤ ??P20 Hz. The ultrasonic source signal is three periods of 500 kHz broadcast. The ultrasonic detection is sampled at 50 MHz on a sep- arate data acquisition system (HF device). To synchronize the pump and the probe broadcasts, the HF device simultaneously records the input signal of the piezoelectric actuator used for the pump oscillations. Themeasurementprotocolinvolvesthefollowingsteps.Attimet = t0= 0,theultrasonicsourceisturnedon, sending 500 kHz pulses at a sequence of timestj,j = 1,2,…. The time between successive pulses,ΔT, is cho- sensuchthatthecodareceivedinresponsetothejthpulsedecaystozerobeforesendingthe(j+1)thpulse. Due to the limited onboard memory of the acquisition card, we use diff erentΔTdepending on the pump frequency, ranging from 1 ms (atf =250 Hz) up to 50 ms forf = 0. 1Hz. A full acquisition always therefore consists of 3000 ultrasonic pulses, independent of pump frequency. The piezoelectric actuator, at frequency fand amplitudeA, is turnedonafterthebroadcastofaminimumof10ultrasonicpulses(Figure 1b),toprop- erly estimate the speed of sound across the sample in the absence of pump oscillations. The subsequent HF pulses are launched in the presence of pump oscillations. We record such acquisitions for pump frequencyfvarying from 0.2 to 250 Hz, pump strain amplitudes?? varying from9 × 10?7to2 × 10?5 and confi ning pressuresPcvarying from 0.1 (ambient) to 30 MPa. 2.1. Data Analysis TheLFmeasurementsrelyonthestress-strainmethod[e.g.,Batzleetal.,2006].Anaxialstressoscillation??axof agivenfrequencyisapplied,inducingaxial??axandradial??radstrainoscillationsoftherocksample.Examplesof typicalstress-straincurvesarereportedinFigureS1inthesupportinginformation.Thepumpstrainamplitude ??isdeterminedbycollectingthemaximumaxialstrainreachedduringtheoscillation.TheLFelasticproperties can be deduced from linear regressions (i) between??axand??axto obtain Young modulusE, and (ii) between ??radand??axto obtain Poisson ratio??. Young modulus is found to increase from about 25 GPa at ambient pressureto40 GPaatPc= 30MPa,whilePoissonratioincreasesfrom0.1to0.14.Becausetheprimarygoalof thepaperistofocusonacoustoelasticityanditsfrequency,amplitude,andpressuredependences,nofurther analysis on the elastic moduli is reported here. Eachpulses(t?tj)propagatingduringthepumposcillationiscomparedtoareferencepulsethatcrossesthe sample before the piezoelectric actuator is turned on, e.g.,s(t ? t0). The time between successive pulses,ΔT, ischosentobeincommensuratewithT = 1∕fsothatovertimetheultrasonicbroadcastsattimes{tj}sample all phases of the pump strain fi eld. RIVIèRE ET AL.FREQUENCY DEPENDENCE OF NONLINEAR ELASTICITY3227 Geophysical Research Letters10.1002/2016GL068061 Figure 1. Experimental setup and protocol. (a) The sample is placed inside a pressure vessel where the confi ning pressurePis increased from ambient up to 30 MPa. The piezoelectric actuator on top oscillates the sample around a constant deviatoric stress of 0.5 MPa at frequencies ranging from 0.2 up to 250 kHz and strain amplitudes roughly from 1 to 10 microstrains. Axial and radial strains (respectively stresses) are measured with strain gauges placed on the sample (respectively on the aluminum base). One pair of piezoceramics is glued on the perimeter of the sample to send/receive longitudinal ultrasonic waves. They launch high-frequency pulses centered at 500 kHz to probe the sample at a given strain level established by the oscillations. (b) Typical oscillations protocol applied to the sample (here at 5 Hz). No oscillations are applied during 0.5 s to serve as a reference for ultrasonic waveforms. (c) Typical received ultrasonic waveform. The inset (blue) shows that waveforms received during oscillations are slightly delayed (and of lower amplitude) than reference waveforms. The fi rst step in analysis ofs(t ? tj)from the steady state time domain is to compare it to the reference pulse s(t ? t0)by computing the cross correlation: C(??,tj)= ∫ ∞ 0 s(t ? t0)s(t + ?? ? tj)dt(1) todetermine??max(tj),thetime??atwhichthecorrelationfunctionismaximum[Renaudetal.,2009,2010].This time of fl ight shift can be converted into a relative velocity change using Δc c (t j ) = ? ??max(tj) t0 ,(2) wheret0 is the time of fl ight of the reference pulse. An example ofΔc∕ccurves (variations over the course of oneacquisition)isdepictedinFigure2.InFigure2a,onecanseethepumposcillationappliedtothesample.In thisexample,theoscillationsstartapproximatelyhalfasecondafterlaunchingtheultrasonicpulses.Asshown RIVIèRE ET AL.FREQUENCY DEPENDENCE OF NONLINEAR ELASTICITY3228 Geophysical Research Letters10.1002/2016GL068061 Figure 2. Change in ultrasonic velocity during pump oscillations. (a) Input oscillations of the piezoelectric actuator. Positive (negative) values corresponds to the dilation (compression) phase. In this example, the oscillation frequency is 5 Hz and the output axial strain amplitude is roughly 10 microstrains. (b) Relative change in ultrasonic velocity (see equation (2)) induced by pump oscillations and varying at frequenciesnfwithn = 0,1,2,…. The steady state regime is further decomposed applying Fourier analysis. The component at0fcorresponds to an elastic softening of the medium. (c) Instantaneous velocity change during steady state at two extreme frequencies (0.2 Hz and 200 Hz) and diff erent confi ning pressures. One can see a larger change in velocity (i.e., larger nonlinearity) at high frequencies and low confi ning pressures. inFigure2b,thevelocityremainsconstant(i.e., Δc c = 0)untilthepumpisturnedon.Assoonasthepumpison, thevelocitydropsandstartsoscillatingwiththepumpfrequencyf,aswellaswithmultiplesoff.Thevelocity drop, or off set, corresponds to an elastic softening of the medium and can be considered as a zero-frequency harmonic(0f) . The three terms (off set, elastic softening, and0fcomponent) will be used interchangeably throughout the document. We perform a dedicated Fourier analysis analogous to a lock-in amplifi er [Rivière etal.,2013,2015]toeachΔc∕csignaltoextracttheamplitudeoftheharmonics.Theseamplitudesarereferred toas Δc c |nfwithn = 0,1,2,….Itisworthnotingthatevencomponents(0f,2f,4f,and6f)forsuchpump-probe scheme are equivalent to odd harmonics (fundamental at frequencyf , third, fi fth, and seventh harmonics) in standard nonlinear acoustic techniques [Buck et al., 1978; Van Den Abeele et al., 2000], i.e., when the probing wavealsoactsasapump.Similarly,thecomponentat1fhereisequivalenttothesecondharmonicinstandard techniques. 3. Results Figure 2c shows the instantaneous change in velocity ( Δc c ) as a function of the LF input oscillations. These nonlinearsignaturescloselyresembletheonesinRenaudetal.[2013b]andRivièreetal.[2015]obtainedatfew kilohertz and ambient pressure. The signatures are complex at low pressure with the presence of hysteresis and elastic softening, roughly0. 5% drop in velocity, which corresponds to a1% drop in elastic modulusM (assumingaconstantdensity??, ΔM M = 2Δc c ? 1%withM = ??c2).Elasticsofteningaswellashysteresisbecomes smaller and smaller with increasing pressure. At 30 MPa, the nonlinearity (off set and hysteresis) approaches towardzero;thetwocurvesoverlappingalmostcompletelyonthelinearverticalplan ( Δc c = 0 ) .Finally,both the off set and hysteresis are larger at 200 Hz than at 0.2 Hz. We study the dependence of harmonic amplitudes ( Δc c | | |nf withn = 0,1,2,… ) as a function of fre- quency, pressure and strain amplitude (Figure 3) obtained through Fourier analysis of eachΔc∕csignal. Figure3ashowsthefrequencydependenceatthreeconfi ningpressuresandconstantpumpstrainamplitude RIVIèRE ET AL.FREQUENCY DEPENDENCE OF NONLINEAR ELASTICITY3229 Geophysical Research Letters10.1002/2016GL068061 Figure 3. Frequency, pressure, and strain dependences of the harmonic amplitudes. Note that the harmonic amplitudes Δc c | | |nf, n≥1 are positive by defi nition (result of Fourier analysis). On the other hand, because the component at0f corresponds to an elastic softening of the medium (therefore negative),? Δc c | | |0f is actually represented in the log plot. (a) Pump frequency dependence of the harmonic amplitudes at three confi ning pressures and constant strain amplitude (?? ?14 microstrains). One sees an increase in nonlinearity by a factor 2–3 over three orders of magnitude in frequency for all harmonics but the component at1fand at all pressures. Result of the multivariate regression is also shown at ambient pressure. (B) Pressure dependence at 0.2 and 200 Hz and constant strain amplitude(?? ?14 microstrains). Result of the multivariate regression for data at 200 Hz shows the approximate exponential decrease of nonlinearity with confi ning pressure. (c) Strain dependence at 0.2 Hz, 200 Hz (ENS data), and 4500 Hz (LANL data) at ambient pressure. Result of the multivariate regression for data at 200 Hz show the power law dependence of nonlinearity with strain. The arrows show changes in dependence occurring at strains where higher harmonics at4fand6femerge from noise (vertical dashed line). Strain dependence at larger pressures is presented in Figure S3 in the supporting information. RIVIèRE ET AL.FREQUENCY DEPENDENCE OF NONLINEAR ELASTICITY3230 Geophysical Research Letters10.1002/2016GL068061 (?? ? 1. 4×10?5).Theharmoniccontentat0f,2f,4f,and6f(equivalenttooddharmonicsinstandardnonlinear techniques[Bucketal.,1978;VanDenAbeeleetal.,2000])increasesbyafactor2or3over3ordersofmagnitude in frequency, whereas Δc c | | |1f (equivalent to the second harmonic in standard nonlinear techniques) seems to be rather invariant. Components at3fand5fare below the noise level. All harmonics are decreasing with increasing confi ning pressure. In particular, components at4fand6fbecome progressively unmeasurable with increasing pressure. In Figure 3b, we focus on the pressure dependence of the fi rst three harmonics (n = 0,1,2) at two extreme frequencies(f = 0. 2and200Hz).Thenonlinearitydecreasesbymorethananorderofmagnitudewhenpres- sure increases from 1 to 30 MPa, following an approximate exponential decrease. At 30 MPa, the harmonic amplitudesstillpersist,suggestingthatsmallnonlinearitymightstillbepresentatsuchpressuresandpoten- tiallymeasurableifonefurtherreducesnoiseleveloftheexperimentalsetup.Inaddition,asseeninFigure3a, the nonlinearity is larger at 200 Hz than at 0.2 Hz for components at0fand2f, whereas it is similar for the component at1f. Figure 3c shows the strain dependence of the harmonic amplitudes at two extreme frequencies (0.2 and 200 Hz) and ambient pressure. Data from Rivière et al. [2015] obtained atf = 4500Hz and ambient pressure arealsopresentedforcomparison.Thestraindependencefoundat0.2and200Hzissimilartothatpreviously found at 4500 Hz [Rivièreetal., 2013, 2015]. The component at0f(elastic softening) exhibits a power law behavior ranging between 1 (linear) and 2 (quadratic).Thecomponent Δc c | | |1f evolveslinearlywithstrain.Asfor Δc c | | |2f,thepowerlawbehaviorisnotcon- stant over the strain range considered. It evolves progressively from a slope larger than 1 at lower strains to a slope less than 1 at higher strains. Finally, one sees higher harmonics at4fand6femerging from noise at largestrains.Thelargerthefrequency,thelowerthestrainatwhichtheyemerge:3microstrainsat4500Hz,6 microstrains at 200 Hz, and 11 microstrains at 0.2 Hz. We perform a multivariate regression analysis to quantify the dependence of the harmonic amplitudes on strain, frequency, and confi ning pressure and highlight main diff erence