A new acoustoelastic model for velocity-hydrostatic-pressure in rocks based on general acoustoelastic theory for elastic solids

Acoustoelastic model based on classic acoustoelastic theory of solids fails to describe rock acoustoelasticity, where wave velocities in rocks has complex non-linear relations with the applied hydrostatic pressure. In order to extend acoustoelastic model into better prediction for rock acoustoelasticity, a new acoustoelastic model is presented based on general acoustoelasticity of elastic solids having complicated constitutive law of rocks, which combines third-order elastic constants of compact rocks with the exponential terms for the influence of pore/cracks in rocks. The data for P- and S-wave velocities of diabase and greywacke samples subjected to the pressure up to 400 MPa are taken as examples to validate the presented model. The results show that the presented model could not only predict rock acoustoelasticity but also be linked with frequently-used empirical model and meso-mechanical model. The intrinsic connection among mechanical parameters, acoustical parameters, and meso-mechanical parameters could be established clearly, which will benefit the determination of high-resolution constitutive law and the compliant porosity by the measured ultrasonic-wave velocities of rock samples. General acoustoelasticity of elastic solids having complicated constitutive law yields a new acoustoelastic model of rock acoustoelasticity. New model predicts better non-linear characteristics of rock acoustoelasticity. New model benefits for high-resolution evaluation of constitutive law and meso-mechanical parameters for rocks. General acoustoelasticity of elastic solids having complicated constitutive law yields a new acoustoelastic model of rock acoustoelasticity. New model predicts better non-linear characteristics of rock acoustoelasticity. New model benefits for high-resolution evaluation of constitutive law and meso-mechanical parameters for rocks.


Introduction
Rock acoustoelasticity (Hydrostatic-pressure-induced wave-velocity changes of rocks) are of great importance for investigating the inner structure, the composition, and the stress state of the earth [1][2][3][4][5][6][7][8][9][10][11], which could be traced back to high-temperature and high-pressure experiments for metals and minerals by Bridgman [12][13][14]. He gave linear or quadratic relationships between ultrasonic-wave velocity,c , and hydrostatic pressure,P , by the ultrasonic measurements of metals and minerals as c=c 0 + AP or c=c 0 + AP + BP 2 ( c 0 denotes wave velocity at zero pressure, A and B are the first and second derivatives of c with respect to P , respectively), which could be deduced from classic acoustoelastic theory of elastic solids [15][16][17][18][19].
However, many rock experiments show a rapid, nonlinear increase in wave velocities with the increase of hydrostatic confining pressure at low pressure and then a slow increase linearly in wave velocities at high pressure, which is attributed to stress-induced change of cracks or/ and pores in rocks [1,3,5,7,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. At present, there are three kinds of theoretical models to describe rock acoustoelasticity, including empirical models, meso-mechanical models, and acoustoelastic models. Table 1 gives frequently-used empirical models describing rock acoustoelasticity, where non-linear exponential terms are usually introduced [5-7, 20-23, 27]. These empirical models could provide good approximation for rock acoustoelasticity in both dry and saturated rocks, whose unknown coefficients are fitting parameters for a given set of measurements. The combination of poro-elasticity and models of stress-induced crack/pore-space geometry change could give meso-mechanical models for rock acoustoelasticity [29,[32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Cheng and Toksöz [34], David and Zimmerman [33] related the pore aspect ratio distribution from the pressure dependence of dry velocities, assuming that the rock contains a distribution of cracks with different aspect ratios. Shapiro [29], Shapiro and Kaselow [32] considered stress dependencies of the stiff and compliant porosities based on the theory of poro-elasticity to give the pressure dependence of rock velocities. Although these internal variables in meso-mechanical models (pore aspect ratio distribution, the stiff and compliant porosities, etc.) are not easy to measure directly in the laboratory, these mesomechanical models could benefit us to understanding the physics of the empirical models [32]. Assuming that rocks behave like non-linear elastic bodies, acoustoelastic models are derived from classic acoustoelastic theory of elastic solids [17], where the third-order or fourth-order elastic constants are introduced into the constitutive law [15,[46][47][48]. However, the present acoustoelastic models only predict the linear or quadratic relationships for rock acoustoelasticity. Higher-order elastic constants need to be introduced into the constitutive law in order to achieve a better prediction of the non-linear characteristics of rock acoustoelasticity, but it is impossible because of the great increase of the complexity.
In order to extend acoustoelastic models into better prediction for the non-linear characteristics for rock acoustoelasticity, general acoustoelastic theory of elastic solids having complicated constitutive law is introduced to present a new acoustoelastic model for rock acoustoelasticity. Firstly, the assumption of small-amplitude wave motion superimposed on the pre-deformation state of solids and introduction of second-order elastic constants at the predeformation state of solids yields general acoustoelastic equation of elastic solids of any form of constitutive law. Secondly, constitutive law of rocks combining third-order elastic constants of compact rocks with exponential terms for the influence of pore/cracks in rocks is considered to give quantitative relation of rock acoustoelasticity. Thirdly, Greenfield and Graham [21] 5 c = a(ln P) 2 + b ln P + c 1 (P ≤ P c ) c = c 0 + DP (P ≥ P c ) The critical pressure P c is the pressure where the pores or cracks in rocks are completely closed, a and b denote the closure coefficient of pores or cracks, c 1 is the wave velocity for hydrostatic pressure of 1 MPa, D is the pressure derivative of wave velocity over critical confining pressure the data for P-wave and S-wave ultrasonic velocities of diabase and greywacke samples subjected to the pressure up to 400 MPa are taken as examples to validate this new acoustoelastic model and discuss its intrinsic connection with empirical models and meso-mechanical models.

Hydrostatic-stress-induced velocity change in rocks based on general acoustoelasticity
In general acoustoelasticity of elastic solids having complicated constitutive law, a dynamic motion is superimposed on the static deformation i of an elastic solid, which yields three states including the natural state (free of stress and strain), the initial state (the deformed state subjected to the static loading), and the final state (the superimposed state of the dynamic motion and the initial state) [17]. Physical variables at the three states including displacements, strains, stresses, and elastic constants are denoted by a superscript label 0, i, and f, respectively. The natural Cartesian coordinate is set, where the position of a particle in the solid at the natural state is described by . If | | << | | i | | , equations of motion for the dynamic motion ( , t) in the natural coordinate can be given as, which are the same as those for classic acoustoelasticity. Here i ( , t) are the Kirchhoff stress tensors at the initial state [17]. 0 is mass density at the natural state. ( , t) are the Kirchhoff stress tensors induced by dynamic motion in the natural coordinate. In general acoustoelasticity, ( , t) are expressed by the second-order elastic constants at the initial state, c i , as (2) is different from that in classic acoustoelasticity [17]. In classic acoustoelasticity, the introduction of the second-order, c 0 , and third-order elastic constants,c 0 , at the natural state into ( , t) limits the classic acoustoelasticity to the small pre-deformation of elastic solids having simple constitutive law [19,31,46]. The introduction of second-order elastic constants at the initial state, c i ,in Eq. (2) could extend the classic acoustoelasticity to the large pre-deformed elastic solids having complicated constitutive law.
Substitution of Eq. (2) into Eq. (1) and consideration of the homogeneous pre-deformation yields where.
If the dynamic motion travels as a plane sinusoidal wave form in the direction of the unit vector, , the following eigenvalue equation will give natural wave velocity,c , in the natural coordinate If an isotropic solid subjected to hydrostatic stress,T i = T i , and the corresponding Lagrangian strain, E i = E i , is considered, P-wave velocity, c L , and S-wave velocity,c S , can be derived from Eq. (4) as, Equation (5) shows that the constitutive equation of elastic solids must be provided for the determination of wave velocities because they are influenced by hydrostatic strain, E i , hydrostatic stress,T i , and the second-order elastic constants at initial state, c i 3333 or c i 1313 . For compact rocks free of any pores/cracks, the thirdorder elastic constants at the natural state are introduced into the constitutive equation [17], where c 0 ijkl and c 0 ijkl are the second-and third-order elastic constants at the natural state. For isotropic compact rocks, c 0 ijkl and c 0 ijkl in Voigt notation have two independent second-order Lame constants ( 0 = c 0 12 and 0 = c 0 66 ) and three independent third-order Lame constants ( l 0 = c 0 112 2, m 0 = (c 0 111 − c 0 112 ) 4 , and n 0 = (c 0 111 − 3c 0 112 + 2c 0 123 ) 2 ) [46]. However, the applied hydrostatic pressure on noncompact rocks is smaller than that on compact rocks because of the existence of pores/cracks for a given hydrostatic strain. Their difference increases exponentially from zero to a constant with the increase of the strain, which can be expressed as [29], Here b ij are the difference of the applied stresses between compact and non-compact rocks for the infinite strain. d ij are the constant representing relative sensitivity of the compressibility to porosity of rocks. The Research Article SN Applied Sciences (2022) 4:135 | https://doi.org/10.1007/s42452-022-05011-1 second-order elastic constants at the initial state, c i , are given from Eq. (7), Equation (8) shows that c i are the combination of the second-order elastic constants of compact rocks at the initial state, c 0 + c 0 E i , and elastic constant drop induced by pores/cracks at the initial state,−b d e d mn E i mn . Second-order elastic constants of compact rocks at the initial state, c 0 + c 0 E i , depend linearly on the initial Lagrangian strain. Elastic constant drop induced by pores/ cracks at the initial state,−b d e d mn E i mn , has a minimum value of −b d , and then approaches exponentially to zero with the increase of the applied pressure.
For isotropic non-compact rocks subjected to hydrostatic stress, T i , Eqs. (7) and (8) can be reduced into . Substitution of Eqs. (9)-(11) into Eq. (5) gives the dependence of P-wave and S-wave velocities on E i and T i , respectively, where c 0 L = √ 0 + 2 0 0 and c 0 S = √ 0 0 are P-wave and S-wave velocities of compact rocks at the natural state, respectively. It is shown in Eq. (12) that there are eight independent mechanical parameters including 0 , 0 ,b 0 ,c 0 ,d 0 ,b 33 ,d 33 ,and b 13 d 13 . According to non-linear characteristics for rock acoustoelasticity, mechanical parameters should satisfy the following conditions:

Experimental results of rock samples
Because wave velocities shown in Eqs. (5) and (12) belong to the natural wave velocities described in natural coordinate, no corrections for dimensional changes of samples under hydrostatic pressure must be introduced for the wave-velocity measurement to validate the new acoustoelastic model in Eq. (12). Here, the data of ultrasonic P-and S-wave natural velocities of three rock samples are shown in Table 2 [49]. These samples include two diabase (13) Fig. 3. The c L and c S increase approximate linearly with the increase of the hydrostatic pressure. The c S increases about 4.5% from 0 to 400 MPa, which is two times of that for c L .

Non-linear least square fits by the new model and high-resolution determination of constitutive law
The mechanical parameters of three samples from nonlinear least square fits of the measured wave velocities in Table 2 by Eqs. (12) and (13) are shown in Table 3, whose corresponding fitted results are shown in parts (a) and (b) of Figs.1, 2, 3, respectively. The fitted results of three samples are well consistent with the data shown in Table 2 having goodness of the fit coefficients R 2 P > 0.99 and R 2 S > 0.98 , respectively. For two diabase samples, the standard deviations of the fitted second-order elastic con-  Although there is the very large value of b 0 for sample II compared with sample I, the influence of b 0 below the stain of 2000 is very little. The smaller value of b 33 for sample II yields the little difference between constitutive law in Eq. (7) and linear constitutive law as shown in part c of Fig. 2. Part c of Fig. 3 also shows little difference between non-linear constitutive law in Eq. (6) and linear constitutive law because of the very small values of b 0 and b 33 for sample III. Although the difference of three constitutive laws is commonly small, the difference of their derivatives is very large. This means that the proposed model for velocityhydrostatic-pressure could yield high-resolution constitutive laws of rock samples compared with the common compressional test.

Comparison with empirical and meso-mechanical models
As mentioned above, empirical models have been applied to the statistical analysis of the measured ultrasonic-wave velocities subjected to hydrostatic pressure. Here, we build their intrinsic connection between the new model and empirical models. Because the relative change of wave velocities for common rocks subjected to confining pressure are within 10% , Eq. (12) can be simplified into the following form within 0.5% error If omitting the higher orders of strain E i and taking the linear constitutive law E i = T i 3k 0 approximately, Eq. (14) can be simplified into empirical model of P-wave and S-wave velocity-confining pressure further, k 0 is decay constant. The pressure derivatives of P-wave and S-wave velocities D L and D S for compact rocks can be expressed as, respectively, Table 3 Fitted mechanical parameters of rocks samples from measured wave velocities by Eq. (12) and the corresponding calculated acoustical and meso-mechanical parameters by Eqs. (15) and (18)  The differences in P-wave and S-wave velocities between compact and non-compact rocks B L and B S at zero confining pressure are denoted as, respectively, The difference between the simplified model in Eq. (15) and the empirical model provided by Ji et al. [5][6][7] is that the decay constant k in the simplified model in Eq. (15) is the same for P-wave and S-wave but not for the empirical model provided by Ji et al. [5][6][7]. Therefore, the simplified model in Eq. (15) can be called the coupled empirical model. This simplification involves two-fold errors. The first-type error is from the simplification of Eq. (12) into Eq. (14). But this error is only within 0.5% . The secondtype error is from the simplification of non-linear constitutive law into linear constitutive law. Because the relative change of wave velocities for most rocks subjected to the confining pressure are within 10% , the large deviations of the coupled empirical model in Eq. (15) from the new model in Eq. (12) for the sample I as shown in parts a and b of Fig. 1 are mainly contributed by the simplification of the constitutive law. Calculated results of coupled empirical model for samples II and III are consistent with those of the new model because of the small second-type error.
Shapiro also gave the coupled empirical model in Eq. (15) from meso-mechanical model [29], which means that the mechanical parameters in the new model could be linked with the meso-mechanical parameters c , c0 , cu , s C drs − C gr , and su C drs − C gr as following, are the relative sensitivity of the compressibility,C dr , of the dry (drained) rock skeleton to compliant porosity, c ,and the stiff porosity, s ,in case of a closed compliant porosity ( c = 0 ) and the stiff porosity, s , equal to s0 , respectively.
are the relative sensitivity of the shear modulus, dr , of the dry (drained) rock skeleton to compliant porosity, c , and the stiff porosity, respectively. C drs and drs are the compressibility and the shear modulus of the dry (drained) rock skeleton in case of c = 0 and s = s0 , respectively. C gr is the compressibility of the grain material. c0 is compliant porosity supported by thin cracks and grain contacts vicinities when P = 0. s C drs − C gr and su C drs − C gr represent approximately the sensitivity of the compressibility and shear modulus of the dry (drained) rock skeleton to the stiff porosity, respectively. Table 3 gives the calculated acoustical parameters of the coupled empirical model in Eq. (15) and the calculated meso-mechanical parameters in Eq. (18) based on the fitted mechanical parameters by the new model in Eq. (12). Ji et al. define the critical pressure P c to represent the initial state of full closure of the microcracks and pores in rocks, which depends on the k value as P c ≈ 6.215∕k [5][6][7]. The P c of three samples are 545 MPa, 299 MPa, and 10 MPa, respectively. High P c values commonly correspond to the low B L but B S not. The higher D L values mean that the P-wave velocities of samples II and III are more sensitive to the hydrostatic pressure for the pressure beyond P c than that of sample I. For the S-wave velocity, sample III has the largest D S . The compliant porosities c0 of three rock samples are 0.181, 0.069, and 0.00008, respectively. Because the non-linear characteristics of rock acoustoelasticity is mainly determined by the compliant porosity from c0 , the smaller values of compliant porosity c0 mean the stronger linear characteristics of rock acoustoelasticity, the larger values of c , cu , and k , and the smaller values of P c . However, the meso-mechanical parameters s C drs − C gr and su C drs − C gr have no above-mentioned rules because they are determined by the stiff porosity of rock samples.

Conclusions
In conclusion, general acoustoelastic theory of elastic solids having complicated constitutive law is introduced to present a new acoustoelastic model for rock acoustoelasticity. The case study shows that the presented model can not only predict better non-linear characteristics of rock acoustoelasticity but also be linked with frequently-used empirical models and meso-mechanical models. The intrinsic relation among the mechanical parameters, acoustical parameters, and meso-mechanical parameters could be established clearly, which will benefit the determination of high-resolution constitutive law and the compliant porosity by the measured ultrasonic-wave velocities of rock samples. Although this new acoustoelastic model is only suitable for isotropic solids and hydrostatical pressure, general acoustoelastic theory of elastic solids could be extended to the description of anisotropic acoustoelastic phenomenon and acoustical hysteresis in rocks.