Characterizing the nonlinear interaction of S- and P-waves in a rock sample

Abstract: The nonlinear elastic response of rocks is known to be caused by the rocks' microstructure, particularly cracks and fluids. This paper presents a method for characterizing the nonlinearity of rocks in a laboratory scale experiment with a unique configuration. This configuration has been designed to open up the possibility the nonlinear characterization of rocks as an imaging tool in a field scenario. The nonlinear interaction of two traveling waves: a low-amplitude 500 kHz P-wave probe and a high-amplitude 50 kHz S-wave pump has been studied on a room-dry 15 x 15x 3 cm slab of Berea sandstone. Changes in the arrival time of the P-wave probe as it passes through the perturbation created by the traveling S-wave pump were recorded. Waveforms were time gated to simulate a semi-infinite medium. The shear wave phase relative to the P-wave probe signal was varied with resultant changes in the P-wave probe arrival time of up to 100 ns, corresponding to a change in elastic properties of 0.2%. In order to estimate the strain in our sample, ae also measured the particle velocity at the sample surface to scale a finite difference linear elastic simulation to estimate the complex strain field in the sample, on the order of $10^{-6}$, induced by the S-wave pump. We derived a fourth order elastic model to relate the changes in elasticity to the pump strain components. We recover quadratic and cubic nonlinear parameters: $\tildeβ=-872$, $\tildeδ=-1.1\times10^{10}$, respectively, at room-temperature and when particle motions of the pump and probe waves are aligned. Temperature fluctuations are correlated to changes in the recovered values of $\tildeβ$ and $\tildeδ$ and we find that the nonlinear parameter changes when the particle motions are orthogonal. No evidence of slow dynamics was seen in our measurements.

• The paper demonstrates that pump-probe experiments reveal amplitude-dependent nonlinear elastic interactions between S- and P-waves in Berea sandstone, with travel-time modulations up to 100 ns.
• It employs rigorous laser vibrometer measurements and finite-difference simulations to directly relate elastic modulus changes to quadratic and cubic nonlinear parameters in rock.
• Results highlight significant anisotropy and environmental sensitivity, establishing a foundation for advancing field-scale nonlinear imaging techniques in subsurface characterization.

Nonlinear Interaction of Shear and Compressional Waves in Berea Sandstone

Introduction and Motivation

This study rigorously investigates the nonlinear elastic interactions between propagating shear (S-) and compressional (P-) waves in rocks, using a laboratory configuration devised for relevance to field-scale imaging scenarios. The work is motivated by the complex, amplitude-dependent nonlinear elastic properties observed in rocks resulting from their heterogeneous microstructure, notably the presence of cracks and fluids. Conventional imaging techniques largely recover linear elastic moduli; however, nonlinear elastic parameters provide orders-of-magnitude greater sensitivity to microscale properties and potentially enable new tomographic methodologies.

Earlier laboratory techniques, such as resonance-based measurements, inherently involve spatial and temporal averaging, eliding the dynamic localized nonlinear response. This paper circumvents those limitations by deploying a pump–probe experiment in a finite-sized Berea sandstone sample using a high-amplitude, low-frequency propagating S-wave pump and a low-amplitude, high-frequency P-wave probe. The experimental design emulates the local dynamic interaction expected in in-situ nonlinear elasticity imaging and expands on acoustoelastic frameworks typically limited to static or quasi-static strains.

Experimental Design and Measurement Protocol

A $15 \times 15 \times 3$ cm slab of Berea sandstone is interrogated using a $500$ kHz P-wave probe, traversing the long axis of the sample, while a $50$ kHz S-wave pump, polarized along the probe axis, is launched normal to the probe path. This configuration ensures that the probe path (approximately $30$ wavelengths) samples a sizable strain field, unlike conventional resonant setups. The arrival time of the P-wave probe is measured as a function of the relative phase between the probe and S-wave pump, enabling direct observation of phase-dependent elastic property shifts.

Precision measurement of the pump-induced strain field is accomplished by combining laser vibrometer surface velocity measurements with finite-difference elastic wavefield simulations, yielding reliable amplitude scaling and spatial strain distribution even in the presence of complicated boundary geometry. The probe strain is kept below $10^{-7}$, minimizing the probe’s contribution to nonlinear response and ensuring that the measured travel-time modulations reflect a pure pump-probe nonlinear interaction.

Theoretical Framework: Nonlinear Hooke’s Law and Parameter Estimation

The experimental observations require a constitutive model incorporating both quadratic and cubic nonlinearity terms. The authors derive a fourth-order elastic energy expansion for isotropic media that includes all relevant invariants, following the formalism by Landau and Lifshitz, and Destrade et al. The resultant stress–strain relationship for this wave mixing scenario is dominated by cubic interaction terms, as quadratic terms vanish for pure shear strain contributions in isotropic theory.

Key effective nonlinear coefficients, denoted $\tilde{\beta}$ (quadratic) and $\tilde{\delta}$ (cubic), are defined through a spatially averaged nonlinear modulus perturbation. Analytical expressions relate the arrival time modulation (as measured experimentally) to these coefficients, by integrating the strain and squared-strain fields sampled by the probe along its propagation path. The frequency content of the observed time modulation—comprising components at the pump frequency and its harmonics—permits separation and individual estimation of $\tilde{\beta}$ and $\tilde{\delta}$ by targeted filtering and model fitting.

Experimental Results and Analysis

The principal findings can be summarized as follows:

• Time Modulation and Nonlinearity: The arrival time of the probe P-wave is modulated by up to $100$ ns, equivalent to a $0.2\%$ change in the elastic modulus, under a peak pump-induced strain on the order of $10^{-6}$. This is directly attributable to nonlinear elastic interaction, with negligible slow-dynamics effects observed at these strain levels.
• Contrast in Materials: Analogous experiments in aluminum and lucite revealed negligible nonlinear time modulation, confirming that the effect in Berea sandstone is intrinsic and not an artifact of the apparatus.
• Parameter Estimation: The model fit yields $\tilde{\beta} = -872$ and $\tilde{\delta} = -1.1 \times 10^{10}$ (in consistent units), with the cubic term dominating the observed modulus reduction under pump action. The cubic coefficient is notably higher than previously reported values, and this difference is traced to experimental and definitional variations rather than fundamental contradiction.
• Amplitude Dependence and Anisotropy: Both quadratic and cubic nonlinear parameters were found to be amplitude dependent—$\tilde{\delta}$ increases linearly with strain, while $\tilde{\beta}$ decreases above approximately $0.6$ microstrain, confirming known amplitude-dependent nonlinear softening in rocks. Systematic variation of the pump polarization demonstrated strong anisotropy: cubic nonlinearity increased and quadratic nonlinearity diminished as S-wave pump orientation transitioned from parallel to orthogonal relative to the probe.
• Environmental Effects: Precise monitoring over a period of days under artificially induced temperature swings established a strong correlation of both nonlinear coefficients with temperature (correlation coefficients up to $0.91$ for $\tilde{\delta}$), emphasizing that field and laboratory studies must rigorously control or compensate for environmental parameters.

Implications and Future Directions

The rigorous experimental demonstration of localized, amplitude- and orientation-dependent nonlinear elastic signatures paves the way for the development of field-applicable imaging modalities exploiting nonlinear acoustic wave mixing. Such approaches have strong potential for subsurface characterization, including detection of microcracks, fracture density, and fluid content with sensitivity unattainable via linear measurements. The theoretical advancements, notably the adaptation of higher-order elasticity theory for specific wave interaction geometries, provide a foundational model for interpreting field data and guiding experiment design.

Moreover, the pronounced environmental sensitivity and anisotropy of the nonlinear coefficients advocate for the integration of environmental monitoring and adaptive measurement protocols in any practical nonlinear imaging methodology.

Conclusion

This work establishes both the feasibility and interpretative framework for characterizing nonlinear elastic interactions between propagating S- and P-waves in rocks. The study demonstrates that propagating pump–probe experiments can reveal rich, localized nonlinear signatures attributable to microstructure—robustly distinguished from instrumental or environmental artifacts. The observed amplitude dependence, anisotropy, and environmental sensitivity of the nonlinear parameters must inform future translation to field practice. Further research is necessitated to address scalability across rock types, sample geometries, and to develop robust inversion algorithms for imaging spatial spatial variations in nonlinear properties for subsurface characterization.