Results of examination of experimental data on non-linear elasticity of rocks
using experimentally determined pressure dependences of P- and S-wave
velocities from various literature sources are presented. Overall, over 90
rock samples are considered. Interpretation of the data is performed using an
effective-medium description in which cracks are considered as compliant
defects with explicitly introduced shear and normal compliances without
specifying a particular crack model with an a priori given ratio of the
compliances. Comparison with the experimental data indicated abundance
(

It is widely appreciated that most rocks exhibit strongly increased tensosensitivity, that is, giant elastic non-linearity as compared with atomic non-linearity of homogeneous solids and liquids. A bright manifestation of this non-linearity is a very pronounced dependence of rocks' elastic moduli on applied pressure. The main reason for this giant non-linearity is the presence of highly compliant cracks and contacts in the relatively hard matrix.

Important features of this “soft–hard paradigm” of giant non-linearity in
microstructured solids

Despite the usefulness of the above-mentioned 1-D models for understanding basic features of the influence of highly compliant inclusions on reduction of the elastic modulus and the origin of its giant stress sensitivity, closer comparison with seismo-acoustic properties of real rocks requires the effective-medium models that more adequately correspond to the 3-D character of real rocks. Even in the simplest isotropic approximation, rocks are characterized by a pair of independent elastic moduli. The most widely used are the bulk modulus, the shear modulus determining the velocity of shear S-waves, the Young modulus, as well as the modulus corresponding to the velocity of longitudinal P-waves. Among those moduli any two ones can be considered independent and the others are expressed via the chosen pair of independent ones.

Since cracks are the simplest and most important type of compliant defect in
consolidated rocks, considerable attention was paid to developing models that
describe crack-induced variations in elastic moduli. Although such
descriptions differ in the way of accounting for eventual interaction of
cracks (i.e. small perturbation or approximation of low crack concentrations,
without accounting for mutual crack interaction

Despite the differences in the methods accounting for interaction of cracks at larger concentrations, in the limiting case of small crack concentrations all such models predict identical complementary variations for the chosen independent elastic moduli. For example, the chosen crack geometry pre-determines a given very specific proportion between variations in the S- and P-wave velocities under hydrostatic pressure. Observations for real rocks, however, often demonstrate different proportions between crack-induced variations in the P- and S-wave velocity variations, such that playing with crack concentrations in the above-mentioned models in principle cannot help to reach better agreement between the predictions and observations.

The fact that variations of different moduli inferred from the measured wave velocities require different crack concentrations implies that real cracks could be characterized by significantly different proportions between their shear and normal compliances. Such variability of crack properties in principle cannot be accounted for in conventional effective-medium models based on cracks modelled as straight cuts of any geometry (e.g. penny-shaped) or thin ellipsoidal voids with a small aspect ratio. In such conventionally used models the ratio between those compliances is pre-determined and cannot exhibit significant variations.

This fact motivated the development of alternative effective-medium models in
which cracks are considered as highly compliant defects with independently
defined normal and shear compliance without a predetermined proportion
between them. Such an idea was realized in

Results from

In geophysics elastic non-linearity of rocks is well appreciated; however,
when considering non-linear propagation of elastic waves, the modelling is
often simplified by using 1-D approximation starting from a 1-D constituent
non-linear stress–strain relationship in which non-linearity is often
considered quadratic in strain. For the present consideration of non-linear
variations of elastic moduli under isotropic hydrostatic compression that
affects the state high compliant defects, 1-D description can also be used,
for example, in the form with small (quadratic in strain) non-linear
correction to the linear stress–stress relation:

In contrast to homogeneous materials with weak atomic non-linearity and the
non-linearity parameter of the order of unity

Important features of the microstructure-induced non-linearity can be
revealed in the framework of the above-mentioned 1-D description

Simultaneously with increase in the non-linearity parameter, the elastic
modulus

Non-linearity exhibited by dry Navajo, Nugget, and Weber sandstones

In contrast to the above-mentioned bubbly liquids, for which the bubbles have
the same contrast

This gradual increase in slope

For non-identical defects leading to a distribution in the compliance
parameter, Eqs. (3) and (4) should be modified to comprise the contributions
of defects with different compliance parameters

If one consider ranges of pressure

Now we recall that in the previous section we considered only 1-D
descriptions that can be fairly well applied to the reduction in the bulk
modulus under hydrostatic compression of real rock samples. However, in real
3-D rocks even under isotropic hydrostatic compression and fairly
isotropically oriented cracks, there are two independent elastic moduli, of
which usually the bulk modulus and shear modulus are considered. The
crack-like defects with isotropic orientations can also be characterized by
two independent compliances with respect to normal and shear loading. Using
such a representation of cracks like planar defects with two compliances that
are not a priori predetermined by a particular crack model, one can relate
the values of different elastic moduli to the crack-effective densities and
compliances by analogy with the above-considered 1-D case. Such expressions
were obtained in

Schematic of determining the

Similar equations were derived in

Note further that the total shear compliances

Assuming that both normal and shear compliances are localized at the same
defects (like at penny-shaped cracks in conventional models), the ratio

Since different effective elastic moduli are sensitive to normal and shear
compliances of the compliant defects differently, gradual variation of crack
density with pressure should correspond to different trajectories of the
point

This approach was discussed in detail in

In what follows we present results of examination of over 90 rock samples,
for which data on pressure dependences of P- and S-wave velocities were taken
in

Histograms for the Poisson ratios calculated using P- and S-wave
velocities for over 90 rocks

For those samples, the initial pressure dependences of the P- and S-wave
velocities were re-plotted in the plane of the normalized moduli

It was also found that for two tens of samples, the trajectories could be fairly
well fitted by a constant

Figure 4 shows the histogram for distribution over

Distribution over

Figure 5 shows histograms similar to Fig. 4, but separately for 34 samples
demonstrating negative Poisson ratios at low pressures and 37 samples with
positive Poisson ratios in the entire pressure range. As expected from the
above-presented arguments (see the discussion of Eq. 15), the

It can be mentioned that an increased

In the described examination of pressure-dependent (i.e. non-linear) elastic
rock properties we used approaches from

Distributions over

The performed examination has indicated that properties of compliant cracks
in many rocks agree reasonably well with the assumption of uniform
distribution

The usage of the theoretical description

Furthermore, the performed examination of pressure dependences for

Overall, the obtained results indicate the necessity for further development
of crack models to account for the revealed numerous examples of rocks with
defects demonstrating

The authors used experimental data from Coyner (1984), Freund (1992), and Mavko and Jizba (1994).

The authors declare that they have no conflict of interest.

This article is part of the special issue “Waves in media with pre-existing or emerging inhomogeneities and dissipation”. It is a result of the EGU General Assembly 2017, Vienna, Austria, 23–28 April 2017.

Vladimir Y. Zaitsev and Andrey V. Radostin acknowledge the support of the Russian Foundation for Basic Research (grant no. 15-05-05143). Arcady V. Dyskin and Elena Pasternak acknowledge the financial support from Australian Research Council linkage project LP 120200797, Australian Worldwide Exploration (AWE) limited, and Norwest Energy NL companies.Edited by: Sergey Turuntaev Reviewed by: two anonymous referees