Fracture Toughness Evaluation Based on Tension-softening Model and its Application to Hydraulic Fracturing

This paper discusses the applicability of the tension-softening model in the determination of the fracture toughness of rocks, where the fracture toughness evaluated based on the tension-softening model is compared with the crack growth resistance deduced from laboratory-scale hydraulic fracturing tests. It is generally accepted that the fracture process is dominated by the growth of a fracture process zone for most types of rocks. In this study, the J-integral based technique is employed to determine the fracture toughness of Iidate granite on the basis of the tension-softening model, where compact tension specimens of different dimensions were tested in order to examine the specimen size effect on the measured fracture toughness. It was shown that the tension-softening relation deduced from the J-integral based technique allowed us to determine the specimen size independent fracture toughness K_c of Iidate granite. Laboratory-scale hydraulic fracturing tests were performed on cubic specimens (up to a 10 m sized specimen), where cyclic pressurization was conducted using a rubber-made straddle packer to observe the extent of the hydraulically induced crack. The experimental results of pressure and crack length were then used to construct the crack growth resistance curve based on the stress intensity factor K. The crack growth resistance obtained from the hydraulic fracturing tests was observed to initially increase and then level off, giving a constant K value for a long crack extension stage. The plateau K value in the crack growth resistance curve was found to be in reasonable agreement with the fracture toughness K_c deduced from the tension-softening relation. It was demonstrated that the tension-softening model provides a useful tool to determine the appropriate fracture toughness of rocks, which may be applicable for the analysis of the process of large-scale crack extension in rock masses.     

Seismic dispersion and attenuation in layered porous rocks with fractures of varying orientations

Wave‐induced fluid flow plays an important role in affecting the seismic dispersion and attenuation of fractured porous rocks. While numerous theoretical models have been proposed for the seismic dispersion and attenuation in fractured porous rocks, most of them neglect the wave‐induced fluid flow resulting from the background anisotropy (e.g. the interlayer fluid flow between different layers) that can be normal in real reservoirs. Here, according to the theories of poroelasticity, we present an approach to study the frequency‐dependent seismic properties of more realistic and complicated rocks, i.e. horizontally and periodically layered porous rock with horizontal and randomly orienting fractures, respectively, distributed in one of the two periodical layers. The approach accounts for the dual effects of the wave‐induced fluid flow between the fractures and the background pores and between different layers (the interlayer fluid flow). Because C₃₃ (i.e., the modulus of the normally incident P‐wave) is directly related to the P‐wave velocity widely measured in the seismic exploration, and its comprehensive dispersion and attenuation are found to be most significant, we study mainly the effects of fracture properties and the stiffness contrast between the different layers on the seismic dispersion and attenuation of C₃₃. The results show that the increasing stiffness contrast enhances the interlayer fluid flow of the layered porous rocks with both horizontal and randomly orienting fractures and weakens the wave‐induced fluid flow between the fractures and the background pores, especially for the layered porous rock with horizontal fractures. The modelling results also demonstrate that for the considered rock construction, the increasing fracture density reduces the interlayer fluid flow while improves the dispersion and attenuation in the fracture‐relevant frequency band. Increasing fracture aspect ratio is found to reduce the dispersion and attenuation in the fracture‐relevant frequency band only, especially for the layered porous rock with horizontal fractures.     

Topographic stress and rock fracture: a two‐dimensional numerical model for arbitrary topography and preliminary comparison with borehole observations

Theoretical calculations indicate that elastic stresses induced by surface topography may be large enough in some landscapes to fracture rocks, which in turn could influence slope stability, erosion rates, and bedrock hydrologic properties. These calculations typically have involved idealized topographic profiles, with few direct comparisons of predicted topographic stresses and observed fractures at specific field sites. We use a numerical model to calculate the stresses induced by measured topographic profiles and compare the calculated stress field with fractures observed in shallow boreholes. The model uses a boundary element method to calculate the stress distribution beneath an arbitrary topographic profile in the presence of ambient tectonic stress. When applied to a topographic profile across the Susquehanna Shale Hills Critical Zone Observatory in central Pennsylvania, the model predicts where shear fractures would occur based on a Mohr–Coulomb criterion, with considerable differences in profiles of stresses with depth beneath ridgetops and valley floors. We calculate the minimum cohesion required to prevent shear failure, C_min, as a proxy for the potential for fracturing or reactivation of existing fractures. We compare depth profiles of C_min with structural analyses of image logs from four boreholes located on the valley floor, and find that fracture abundance declines sharply with depth in the uppermost 15 m of the bedrock, consistent with the modeled profile of C_min. In contrast, C_min increases with depth at comparable depths below ridgetops, suggesting that ridgetop fracture abundance patterns may differ if topographic stresses are indeed important. Thus, the present results are consistent with the hypothesis that topography can influence subsurface rock fracture patterns and provide a basis for further observational tests. Copyright © 2014 John Wiley & Sons, Ltd.     

Hydraulic Tomography Offers Improved Imaging of Heterogeneity in Fractured Rocks

Fractured rocks have presented formidable challenges for accurately predicting groundwater flow and contaminant transport. This is mainly due to our difficulty in mapping the fracture‐rock matrix system, their hydraulic properties and connectivity at resolutions that are meaningful for groundwater modeling. Over the last several decades, considerable effort has gone into creating maps of subsurface heterogeneity in hydraulic conductivity (K) and specific storage (S_s) of fractured rocks. Developed methods include kriging, stochastic simulation, stochastic inverse modeling, and hydraulic tomography. In this article, I review the evolution of various heterogeneity mapping approaches and contend that hydraulic tomography, a recently developed aquifer characterization technique for unconsolidated deposits, is also a promising approach in yielding robust maps (or tomograms) of K and S_s heterogeneity for fractured rocks. While hydraulic tomography has recently been shown to be a robust technique, the resolution of the K and S_s tomograms mainly depends on the density of pumping and monitoring locations and the quality of data. The resolution will be improved through the development of new devices for higher density monitoring of pressure responses at discrete intervals in boreholes and potentially through the integration of other data from single‐hole tests, borehole flowmeter profiling, and tracer tests. Other data from temperature and geophysical surveys as well as geological investigations may improve the accuracy of the maps, but more research is needed. Technological advances will undoubtedly lead to more accurate maps. However, more effort should go into evaluating these maps so that one can gain more confidence in their reliability.     

Cohesive Crack Analysis of Toughness Increase Due to Confining Pressure

Apparent fracture toughness in Mode I of microcracking materials such as rocks under confining pressure is analyzed based on a cohesive crack model. In rocks, the apparent fracture toughness for crack propagation varies with the confining pressure. This study provides analytical solutions for the apparent fracture toughness using a cohesive crack model, which is a model for the fracture process zone. The problem analyzed in this study is a fluid-driven fracture of a two-dimensional crack with a cohesive zone under confining pressure. The size of the cohesive zone is assumed to be negligibly small in comparison to the crack length. The analyses are performed for two types of cohesive stress distribution, namely the constant cohesive stress (Dugdale model) and the linearly decreasing cohesive stress. Furthermore, the problem for a more general cohesive stress distribution is analyzed based on the fracture energy concept. The analytical solutions are confirmed by comparing them with the results of numerical computations performed using the body force method. The analytical solution suggests a substantial increase in the apparent fracture toughness due to increased confining pressures, even if the size of the fracture process zone is small.     

Occurrence of Alaskan-type mafic-ultramafic intrusions in the North Qilian Mountains, northwest China: Evidence of Cambrian arc magmatism on the Qilian Block

Field relationships, mineralogy and petrology, whole‐rock chemistry, and age of the Zhamashi mafic–ultramafic intrusion in the North Qilian Mountains, northwest China, have been studied in the present work. The Zhamashi intrusive body consists of ultramafic, gabbroic, and dioritic rocks in a crudely concentrically zoned structure. The ultramafic rocks are layered cumulates with rock types varying continuously from dunite through wehrlite and olivine clinopyroxenite to clinopyroxenite. The gabbroic and dioritic rocks are also layered or massive cumulates with rock types varying continuously from noritic gabbro through hornblende gabbro to diorite. The ultramafic and adjoining gabbroic rocks are discontinuous in lithology and discordant in structure across the interface. The interface is steep, sharp, and fractured. Contact metamorphic zones are well developed between the Zhamashi intrusive body and the country rock. The concentrically zoned structure of the intrusive body and the intrusion into the continental crust are the two main pieces of evidence for considering that the Zhamashi intrusion is Alaskan‐type. The mineral chemistry of the chromian spinels (Cr‐spinels) and clinopyroxenes, and the variation trend of the whole‐rock compositional plot in the (Na₂O + K₂O)–FeO–MgO (AFM) diagram are also supportive of this consideration. The age of the Zhamashi intrusive body, determined with sensitive high mass‐resolution ion microprobe on the zircon grains, is 513.0 ± 4.5 Ma. Parental magma of the Zhamashi intrusion is compositionally close to the primitive magma produced by partial melting of the mantle peridotite. It was differentiated by fractional crystallization at low total pressure and under H₂O‐rich conditions in an arc environment to form all the major rock types. The concentrically zoned structure of the Zhamashi intrusive body was constructed in two stages: formation of a stratiform‐type layered sequence, followed by diapiric re‐emplacement. The occurrence of the Alaskan‐type intrusion suggests an active continental margin and Cambrian arc magmatism for the northern margin of the Qilian Block.     

On the propagation of an in-plane shear fault with super-S-wave velocity

The speedv, especially the problem whether super S-wave velocity in the classical model (linear elasticity fracture mechanics) exists, of spontaneous propagation of a shear fault is investigated theoretically. An in-plane shear crack propagating in the crack plane is taken as the model of the shear fault. The results obtained firstly by Kostrov (1975) is extended from sub-Rayleigh wave velocity to super S-wave velocity, and the analytical expression for the stress intensity factorK ₂ in the case ofα>v>β is derived. It is proved that for Poisson mediumK ₂ is positive and real in the velocity range (β, 1.70β). This demonstrates that (β, 1.70β) is the velocity range which fulfils the conditions for spontaneous crack propagation. The existence, convergence and positiveness or negativeness ofK ₂ forv in individual sections are examined, and it is found that for an in-plane shear crack: 1. There are three sections forv, i.e., [0.v _R], (β, 1.70β), andα, respectively, and 2. There are two physically reasonable sections forv, the first is [v _R, β], and the second is [1.70β, α]. These two forbidden sections behave as barriers to fault propagation. The analytical expressions derived in this paper are not only suitable to classical model, but also to the other derivative models (e. g., the slip-weakening model and the renomalization model etc.). The model considered in this paper is more realistic than the static model employed by previous authors. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica, 15, 9–14, 1993.     

Radioactive heat production in rocks and its relation to other petrophysical parameters

Summary Radioactive heat productionA is a scalar and isotropic petrophysical property independent of in situ temperature and pressure. Its value is usually expressed in HGU units (1 HGU=10^–13 cal/cm³ sec) and depends on the amounts of uranium, thorium and potassium.A varies with rock type over several orders of magnitude and reflects the geochemical conditions during rock formation (magmatic differentiation, sedimentation or metamorphism).In order to assign realistic thermal parameters to deeper-seated rocks correlations with seismic velocity (which can be determined from the surface) have been looked for. In the range characteristic for crystalline rocks of the crust (5–8 km/sec)A is strongly correlated with density and compressional wave velocityv _p:A decreases with increasingv _p orp. From this relationship it is now possible to estimate heat production values for any particular layer of a crustal section from measured seismic velocities. Contrary to earlier belief there is, as shown by experimental determinations, no correlation between heat productionA and thermal conductivityK in igneous and metamorphic rocks. In sediments however, especially in sand/shale sequences, a correlation betweenK andA is most likely: increasing clay mineral content, characterized by increasingA, causes the decrease ofK in these rocks.Contribution No. 111, Institute of Geophysics, Swiss Federal Institute of Technology, Zurich, Switzerland.     

Common Evolution of Mechanical and Transport Properties in Thermally Cracked Westerly Granite at Elevated Hydrostatic Pressure

Increasing the damage and crack porosity in crustal rocks can result in significant changes to various key physical properties, including mechanical strength, elastic and mechanical anisotropy, and the enhancement of transport properties. Using a Non-Interactive Crack Effective Medium (NIC) theory as a fundamental tool, we show that elastic wave dispersion can be inverted to evaluate crack density as a function of temperature and is compared with optically determined crack density. Further, we show how the existence of embedded microcrack fabrics in rocks also significantly influences the fracture toughness (K_IC) of rocks as measured via a suite of tensile failure experiments (chevron cracked notch Brazilian disk). Finally, we include fluid flow in our analysis via the Guéguen and Dienes crack porosity-permeability model. Using the crack density and aspect ratio recovered from the elastic-wave velocity inversion, we successfully compare permeability evolution with pressure with the laboratory measurements of permeability.     

Fracture toughness and subcritical crack growth during high-temperature tensile deformation of Westerly granite and Black gabbro

The double torsion testing method has been used to determine catastrophic and subcritical crack propagation parameters for pre-cracked specimens of Westerly granite and Black gabbro under a number of environmental conditions.The critical stress intensity factor for catastrophic crack propagation (fracture toughness) of granite and gabbro has been measured at temperatures from 20 to 400°C, in a vacuum. At 20°C, the fracture toughness of Westerly granite was $\text{1.79 ± 0.02}\text{MPa}\text{·}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, and for two blocks of Black gabbro it was $\text{3.03 ± 0.08}\text{MPa}\text{·}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and $\text{2.71 ± 0.15}\text{MPa}\text{·}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, respectively. These values are very close to those reported by other investigators for tests conducted in air of ambient humidity at room temperature. For both rocks, fracture toughness at first increased slightly, and then decreased steadily on raising the temperature above ambient conditions. This behaviour is explained in terms of the density and distribution of thermally induced microcracks, as determined by quantitative optical microscopy.Subcritical crack growth behaviour has been studied at temperatures up to 300°C, and under water vapour at pressures of 0.6 to 15 kPa. Both the load relaxation and incremental constant displacement rate forms of the double torsion testing method were utilised to generate stress intensity factor/crack velocity diagrams. Crack growth was measured over the velocity range 5 × 10^?3 to 10^?7 m · s^?1. Increasing both temperature and water vapour pressure resulted in substantially higher crack growth rates. The overall effect of raising the temperature over the range studied here (20–300°C) was to increase the crack growth rate in granite and gabbro by ～5 and 7 orders of magnitude, respectively, at constant stress intensity factor and vapour pressure of water. For both rocks, the slopes of stress intensity factor/crack velocity curves were sensitive to changes in both temperature and water vapour pressure at low values of the latter parameter. Slopes fell substantially on raising the water vapour pressure, but were relatively insensitive to changes in temperature at these higher pressures. No subcritical crack growth limit was encountered.Estimates of the uncertainty in our experimental data are given. From the results of multiple load relaxation experiments on Westerly granite specimens, we estimate the uncertainty in position of stress intensity factor/crack velocity curves along the stress intensity axis to be c. 10% of the fracture toughness, and the uncertainty in slope of such curves to be c. 12%.Problems associated with the extrapolation of our experimental data to regions of higher effective confining pressure in the Earth's crust are discussed.     

Velocities of elastic waves andv P/v S ratios in dry and water-saturated rock samples at high pressure

Elastic wave velocities and the ratiov _P/v _S were studied for dry and initially saturated samples of carbonate and crystalline rocks at pressures to 2 kbar. In initially saturated samplesv _P increases in crystalline rock, whereas in sedimentary rock it may either increase or decrease with increasing pressure. Under the same conditionsv _S remains approximately constant in crystalline rocks and decreases in sedimentary samples. The ratiov _P/v _S as a function of pressure increases in dry rocks and decreases in saturated ones. Saturated samples always have higherv _P/v _S ratios than dry ones. It appears that the geometry of the pore space influences the acoustic properties of saturated crystalline rocks.     

Spinifex-like textured metaperidotites from the Higo Metamorphic Rocks,Japan, a possible high-pressure dehydration product of antigorite serpentinite

Spinifex-like textured metaperidotites from the Higo Metamorphic Rocks (HMR), west-central Kyushu, Japan, may be formed by high-pressure dehydration of antigorite, and may indicate deep subduction of serpentinite reaching a pressure–temperature condition of 1.6 GPa and 740–750 °C. Three rock types have been identified based on mineral assemblage and rock texture: Type I (L) consisting of medium-grained (1–5 cm long) olivine + enstatite + chromite ±tremolite with secondary talc and anthophyllite that occurs in low-grade metamorphic rocks of the biotite zone, Type I (H) of coarse-grained (up to 10 cm long) olivine + enstatite (with clinoenstatite lamella) + chromite ±tremolite with secondary talc that occurs in high-grade metamorphic rocks of the garnet-cordierite zone, and Type II composed of Al-spinel + chlorite + olivine + apatite + ilmenite with minor sodic gedrite in the garnet-cordierite zone together with Type I (H). Olivines in all rock types are mostly serpentinized during exhumation. The chromite-olivine thermometer gives 560–690 °C for Type I (L) rocks, and the spinel-olivine thermometer gives 610–740 °C for Type II rocks. The peak metamorphic pressure will be higher than 1.6 GPa based on the location of the experimentally determined invariant point (P = 1.6 GPa and T = 670 °C) of antigorite + forsterite + enstatite + talc + H₂O. This estimate is consistent with the occurrence of chlorite in Type II rocks, which is stable up to 890 °C at 2.0 GPa. The spinifex-like textured metaperidotites occur as small bodies in the low P/T type gneisses, implying tectonic juxtaposition of them probably during exhumation of the HMR. Recent findings of medium pressure (0.9–1.2 GPa) granulites and gneisses from the HMR may indicate that the HMR has a deep root into the wedge mantle from which the spinifex-like textured metaperidotites have derived.     

Flow and fracture maps for basaltic rock deformation at high temperatures

Understanding how the strength of basaltic rock varies with the extrinsic conditions of stress state, pressure and temperature, and the intrinsic rock physical properties is fundamental to understanding the dynamics of volcanic systems. In particular it is essential to understand how rock strength at high temperatures is limited by fracture. We have collated and analysed laboratory data for basaltic rocks from over 500 rock deformation experiments and plotted these on principal stress failure maps. We have fitted an empirical flow law (Norton’s law) and a theoretical fracture criterion to these data. The principal stress failure map is a graphical representation of ductile and brittle experimental data together with flow and fracture envelopes under varying strain rate, temperature and pressure. We have used these maps to re-interpret the ductile–brittle transition in basaltic rocks at high temperatures and show, conceptually, how these failure maps can be applied to volcanic systems, using lava flows as an example.     

A method of estimating the stress exponent in the flow law for rocks using fold shape

On the basis of experiment and theory, we expect rocks to deform in a linear fashion when diffusive processes control deformation, and nonlinearly in most other situations. The geometric characteristics of buckle folds in layered materials are dependent on rheological parameters, and in particular depend strongly on the stress exponent,n _L , of the stiff layers involved. Thus, information about the deformation rocks have undergone and their rheological state during deformation can be obtained by studying fold shapes and strain distributions. This is important because there is uncertainty in extrapolating laboratory-derived flow laws to the very slow natural strain rates and large strains found in nature.We have studied the development of buckle folds in linear and nonlinear materials using finite-element modeling, and interpolated the numerical results to construct plots relating several geometric parameters to variations in power-law exponent,n _L , and viscosity ratio,m, of layer to matrix. Such plots allow for a comparison of the results of numerical models with data for many natural and experimentally-produced folds, and there is consistency among the data for folds produced in physical models, using both linear and nonlinear materials and the numerical simulations. data for folds from the Appalachian Mountains, the Alps and elsewhere, however, suggest high values ofn _L in the flow laws for a number of rock types. The unexpectedly high estimates ofn _L suggest that other factors, such as strain softening or anisotropy, may influence fold shape, and thus complicate the estimation of the rheological properties of rocks.     

SEISMIC REFRACTION METHOD TO STUDY THE FOUNDATION ROCK OF A DAM*

Dynamic elastic moduli like E, μ, K and μ of the foundation rock of a dam have been determined by finding V_p- and V_s-velocities by seismic refraction with a hammer as source. Some parameters such as “fracture frequency” and “rock quality designation” (RQD) of the foundation rock have been derived using “average regression curves” and V_p-velocities. By comparing K/μ with V_p/V_s, a few locations showing weathered conditions have been demarcated. This compares well with RQD values of those locations.     

Formation of conical fractures in sedimentary basins: Experiments involving pore fluids and implications for sandstone intrusion mechanisms

Large sand intrusions often exhibit conical morphologies analogous to magmatic intrusions such as saucer-shaped or cup-shaped sills. Whereas some physical processes may be similar, we show with scaled experiments that the formation of conical sand intrusions may be favoured by the pore-pressure gradients prevailing in the host rock before sand injection. Our experiments involve injecting air into a permeable and cohesive analogue material to produce hydraulic fractures while controlling the pore pressure field. We control the state of overpressure in the overburden by applying homogeneous basal pore pressure, and then adding a second local pore pressure field by injecting air via a central injector to initiate hydraulic fractures near the injection point. In experiments involving small vertical effective stresses (small overburden, or high pore fluid overpressure), the fracturing pressure (λ_fract) is supralithostatic and two dipping fractures are initiated at the injection point forming a conical structure. From theoretical considerations, we predict that high values of λ_fract are due to strong cohesion or high pore fluid overpressure distributed in the overburden. Such conditions are favoured by the pore pressure/stress coupling induced by both pore pressure fields. The dips of cones can be accounted for elastic-stress rotation occurring around the source. Contrary to magmatic chamber models, the aqueous fluid overpressure developed in a parent sandbody (and prevailing before the formation of injectites) may diffuse into the surrounding overburden, thus favouring stress rotation and the formation of inclined sheets far from the parent source. For experiments involving higher vertical effective stresses (thick overburden or low pore fluid overpressure), the fracturing pressure is lower than the lithostatic stress, and a single fracture is opened in mode I which then grows vertically. At a critical depth, the fracture separates into two dilatant branches forming a flat cone. We make use of a P.I.V. (Particle Imaging Velocimetry) technique to analyse plastic deformation, showing that these inclined fractures are opened in mixed modes. Close to the surface, they change into steep shear bands where fluids can infiltrate. The final morphology of the fracture network is very similar to the common tripartite architecture of various injection complexes, indicating that different mechanisms may be involved in the formation of dykes. Feeder dykes under the sill zones may open as tensile fractures, while overlying dykes may be guided by the deformation induced by the growth of sills. These deformation conditions may also favour the formation of fluid escape structures and pockmarks.     

非均匀性对岩石介质中裂纹扩展模式的影响

运用岩石破裂过程分析ＲＦＰＡ２Ｄ系统,通过对岩石试样中预置的倾斜裂纹扩展过程的数值模拟,研究了材料非均匀性对岩石介质中裂纹扩展模式的影响．数值模拟再现了受压试样在裂纹扩展过程中逐步演变的应力场和应变场,以及与岩石非均匀性有关的声发射和断裂扩展模式的“岩桥”现象．模拟结果说明,岩石的非均匀性对含裂纹试样的变形、破裂过程及其破坏模式有很大的影响．     

Modeling of mineralogical composition,density and elastic wave velocities in anhydrous magmatic rocks

We use the technique of direct minimization of the Gibbs free energy of the 8-component (K₂O-Na₂O-Fe₂O₃-FeO-CaO-MgO-Al₂O₃-SiO₂) multiphase system in order to determine the equilibrium mineral assemblages of rocks of different bulk chemical compositions equilibrated at various P-T conditions. The calculated modal compositions of rocks and experimental data on elastic moduli of single crystals are then used to calculate densities and isotropic elastic wave velocities of rocks together with their pressure and temperature derivatives. Sufficient accuracy of the calculations is confirmed by comparison with experimental data on the gabbro-eclogite transformation and precise ultrasonic measurements of elastic wave velocities in a number of magmatic and metamorphic rocks.We present calculated phase diagrams with isolines of density, elastic wave velocities, and their pressure and temperature derivatives for several anhydrous magmatic rocks, from granite to lherzolite. Density and elastic properties of rocks are controlled by their chemical compositions, especially the SiO₂ content, and by P-T of equilibration, and they increase with pressure due to mineral reactions changing mineral assemblages from plagioclase-bearing and garnet-free to garnetbearing and plagioclase-free. TheV _p -density correlation is high, and shows two clear trends: one for iron-poor ultramafic rocks and another for all the other rocks considered. Mineral reactions, which occur at high pressures, changeV _p and density of anhydrous magmatic rocks following the well-known Birch (or a similar) law.Felsic, intermediate and mafic rocks can be well distinguished in theV _p -V _p /V _s - diagram, although their values ofV _p can be close to one another. TheV _p -V _p /V _s -density diagrams together with calculated phase diagrams can serve as efficient instruments for petrologic interpretation of seismic velocities.