INTRODUCTION

The relationship between acidic plutonic and volcanic rocks (P : V) has long been debated by geologists, as a significant amount of research on petrology, geochemistry, geochronology, thermodynamic modeling, geophysical methods leads to conflicting interpretations (e.g., Winkler, 1962; Harris et al., 1972; Crisp, 1984; Lipman, 1984; Shaw, 1975; Glazner et al., 2015; Lundstrom and Glazner, 2016; and references in them). The main problem that arises in determining the ratios of intrusive and effusive rocks is that in most cases either the plutonic rock is similar to the associated volcanic deposits, or most of the volcanic rock has been destroyed, exposing the plutonic rock. Very few areas of the Earth’s crust have preserved or exposed major parts of both volcanic and associated plutonic rocks. A typical example of acidic rocks is Yellowstone Supervolcano. Geophysical studies indicate that there are approximately 32.800 ± 4.200 km3 of granite plutonic rock (Christiansen, 2001; et al.). The volume of volcanic rocks formed in Yellowstone field over the past 2.2 million years ranges from 3550 to 7250 km3. The comparison of plutonic and volcanic rock (P : V) volumes for the Yellowstone field gives a rather uncertain ratio between 4 : 1 and 10 : 1. The main factors that correlate with the observed in nature volume ratio of plutonic and volcanic rocks in the Earth’s crust (P : V) are: 1. the composition of magma, 2. its degree of crystallinity and fluid saturation, 3. oxygen potential f(O2), 4. crust thickness, 5. tectonic situation and regional stresses (e.g., Winkler, 1962; Harris et al., 1972; Lukanin, 1985; Di Genova et al., 2017). The ratios of intrusive and extrusive volumes are typically around 5 : 1 for the oceanic crust and 10 : 1 for the continental crust. This difference seems to reflect a different speed magma rise associated with different crust thickness and magma composition (e.g. Winkler, 1962; Harris et al., 1972; Marsh, 1981). On the basis of the analysis of plutonic and volcanic rock volumes in more than 170 provinces of the Earth (White et al., 2006), the authors state the same uncertainty on the one hand, and on the other hand, rightly note the systematic and significant correlation of the ratio (P : V) with the composition of magma, and hence with its viscosity. It is known that the viscosity of magmas is largely determined by the nature of the movement of magmatic melts in the Earth’s crust and upper mantle, texture features of intrusions and effusions and many other features of magmatic rocks (Zavaritsky and Sobolev, 1961; Sobolev, 1973; Persikov, 1984; Persikov, 1991; etc.). In the work (Sobolev, 1973), a general scheme of vertical movement of magmas in the Earth’s crust and upper mantle, formed at different values of Plit and \({{P}_{{{{{\text{H}}}_{{\text{2}}}}{\text{O}}}}}\) was proposed, and the impossibility of rise of magma to the surface with a water content in them – X(H2O) > 1–2 wt % at \({{P}_{{{{{\text{H}}}_{{\text{2}}}}{\text{O}}}}}\)/Plit ~ 0.1. At the same time, he suggested that the reason for the relative distribution of intrusive and effusive rock masses in the Earth’s crust lies in the patterns of viscosity of water-bearing magmas in a variable field of temperatures and pressures. Magma viscosity is a complex function of the chemical and phase composition, structure and degree of crystallinity of magmas, oxygen potential f(O2), temperature, pressure, concentration of volatile components, primarily water and its forms of dissolution in the melt. Therefore, in those years it was not possible to solve this problem on a quantitative physical-chemical basis, since experimental and theoretical studies of the viscosity of magmatic melts at high pressures were just beginning. It took more than three decades of such studies in IEM RAS and in many laboratories around the world. The main results of these studies were developed by the author of the physical-chemical model of reliable forecasts and calculations of the viscosity of heterogeneous magmas of almost any composition from granite to ultramafic rocks in a variable field of temperatures and pressures of the Earth’s crust and the upper mantle. The new model was published in detail (Pepsikov, 1991, 1998, 2007; Persikov and Bukhtiyarov, 2009; Persikov et al., 1990; and references therein). Model allows us to calculate and predict for the first time the viscosity of magmatic melts, as a function of the following parameters: 1. temperature; 2. lithostatic and fluid pressures; 3. the structure and chemical composition of the melt, including the volatile components (H2O, OH–, CO2, \({\text{CO}}_{3}^{{2 - }}\), F–, Cl–); 4. oxygen potential f(O2), determined using the ratio of Fe2+/(Fe2+ + Fe3+); 5. the ratio of cations: Al3+/(Al3+ + Si4+), Al3+/(Na+ + K+ + Ca2++ Mg2++ Fe2+); 6. volume content in the melts of crystals and bubbles (up to 45 vol %), using a developed computer program.

The ratio of the predicted by model and experimental data for water – containing granitic melts was considered in detail earlier (Persikov, 1984; Persikov, 1991, 1998; Persikov et al., 1990). Figures 1 and 2 show a comparison of new experimental and calculated data on the temperature and pressure dependences of the viscosity of ultrabasic (model dunite) and basalt melts in a wide temperature range (1300–1950°C) and pressures (0.1–7.5 GPA) in the Earth’s crust and upper mantle.

Fig. 1.
figure 1

Temperature dependences of viscosity of ultramafic (model dunite) and basalt melts at fluid pressures (PAr = 100 MPa, basalt melt; \({{P}_{{{\text{C}}{{{\text{O}}}_{2}}}}}\) = 100 MPa, ultramafic melt). The error of experimental and calculated data is ±30 rel. % (Persikov, 1998; Persikov and Bukhtiyarov, 2009; Persikov et al., 2018).

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Fig. 2.
figure 2

Isothermal (1800°C) dependences of viscosity of basalt and ultramafic (model dunite) melts on pressure (error of experimental and calculated data ±30 rel. % (Persikov, 1998; Persikov and Bukhtiyarov, 2009; Persikov et al., 2018).

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It is obvious from the given data that even for such extreme conditions (T, P, compositions) the data on the viscosity of such melts obtained from the model very well correspond to the experimental results within the specified errors.

It is necessary to emphasize the following: 1. It is now reliably established that from a wide range of fluid composition of the acidic magma (H2O, CO2, HCl, NaCl, HF, NaF, H2S), water dissolving in magmas in two forms, has a decisive influence on the viscosity of such magmas. Chemically dissolved water in the form of hydroxyl OH– strongly reduces the viscosity, significantly increases the degree of depolymerization (basicity) of the melt, and the limit capacity of the granitic melt for OH– is 6.4 wt % (Persikov, 1984; Persikov, 1998). Physically dissolved water in the form of molecular H2O weakly reduces the viscosity of the acid melt, without changing its basicity, and pseudo-binary system granitic magma–water with a good degree of approximation simulates the rheological behavior of such magmas in the entire range of depths of the Earth’s crust. Note, that the problem of reliable determination of the numerical ratio of the two forms of OH–/H2O water dissolved in granitoid melts at different temperatures and pressures is still debatable (Stolper, 1982; Persikov, 1984; Burnham, 1983; Kadik et al., 1971; Khitarov et al., 1963; etc.). Our experimental and theoretical results, as well as data obtained in a number of other works (Burnham, 1983; Nowak and Behrens, 1995), definitely confirm the mentioned limit capacity of granitic melts by the amount of chemically dissolved water in such melts at high temperatures. However, it is known that in quenched melts (glasses) obtained by isobaric quenching of water-bearing acidic melts (granite, albite), especially after experiments at \({{P}_{{{{{\text{H}}}_{{\text{2}}}}{\text{O}}}}}\) ≥ 200 MPa, molecular H2O prevails (Stolper, 1982; Persikov, 1984; Persikov, 1991, 1998; Persikov et al., 1990; Nowak and Behrens, 1995; and the references in them). 2. The anomalous pressure dependence of the viscosity of granitic magmas has been experimentally established. Their viscosity decreases significantly with the growth of lithostatic and water pressures (Kushiro, 1981; Brearley et al., 1986; Giordano et al., 2004; Mysen, 1991; Scarfe et al., 1987; Persikov, 1991, 1998, 2007 and references therein). 3. There is no geological evidence of significant overheating of acidic magmas in different facies conditions of the Earth’s crust. Therefore, the work performed calculations of the viscosity of such magmas in our model in relation to the structure of melts with thermodynamic parameters of their subliquidus, established in a number of works (Perchuk, 1973; Kadik et al., 1971; etc.).

RESULTS AND DISCUSSION

A modified theoretical Arrhenian–Frenkel–Eyring equation (Pepsikov, 1998, 2007; Persikov et al., 2018) is used to calculate the concentration, temperature and pressure dependences of the viscosity of the granitic magma–water system:

$$\eta _{T}^{P} = {{\eta }_{0}}\exp ({{E_{{\text{X}}}^{P}} \mathord{\left/ {\vphantom {{E_{{\text{X}}}^{P}} {RT}}} \right. \kern-0em} {RT}}),$$
((1))

where \(E_{{\text{X}}}^{P}\) is the activation energy of viscous flow, which is a function of the composition of the magmatic melt, concentration of two forms of dissolved water in it (ОН–, Н2О) and pressure (J/mol); \(\eta _{T}^{P}\) is the viscosity of magma at a given temperature and pressure (Pa s); η0 – pre-exponential constant, characterizing the viscosity of the melt at T → ∞, log η0 = –4.5 ± 0.14 (Pa s); T – temperature (K); R = 8.3192 (J/mol K) is the universal gas constant.

To characterize the features of the chemical composition and structure of magmas, the structural-chemical parameter (100NBO/T) is used, which is determined by a simple method from the chemical composition of the melt, expressed in wt % of oxides, including volatile components (ОН–, Н2О, СО2, \({\text{CO}}_{3}^{{2 - }}\), etc.), using the following equation:

$${{100{\text{NBO}}} \mathord{\left/ {\vphantom {{100{\text{NBO}}} T}} \right. \kern-0em} T} = {{200\left( {O - 2T} \right)} \mathord{\left/ {\vphantom {{200\left( {O - 2T} \right)} T}} \right. \kern-0em} T},$$
((2))

where O is the total amount of gram-ions of oxygen in the melt, T = (Al3+ + Si4+ + Fe3+ + Ti4+ + Р5+ + B3+) is the sum of network-forming cations located in the tetrahedral coordination of oxygen and included in the anionic part of the melt structure. An example of calculation of this parameter for water-bearing granitic melt is given in a number of works (e.g., Persikov, 1984; Persikov, 1991, 1998; Persikov et al., 1990). It has been proved that this parameter, called as the degree of depolymerization or the magma basicity coefficient, adequately and with the greatest sensitivity reflects the structure and composition of magmas (Persikov, 1984; Persikov, 1991, 1998, 2007). For granitic magmas this parameter varies from 0 to 17 (see Fig. 3), while often used in western literature, a similar parameter NBO/T (Mysen, 1991) is close to 0 for polymerized granitic melts and therefore cannot reflect the features of the chemistry and structure of polymerized granitic magmas (Persikov, 1984; Persikov, 1991; Di Genova et al., 2017). While setting 100NBO/T has no such limitation as to the water-bearing granitic melts, varies significantly (see Table 2, below). Previously, and generalized structural-chemical dependence of the activation energy of the viscous flow of magmatic melts was obtained, according to which the entire range of compositions of natural magmas is divided into 4 sections, with different degrees of change in the activation energy, and accordingly the viscosity, depending on the composition and structure of magma (Persikov, 1998, 2007):

Fig. 3.
figure 3

Generalized structural-chemical dependence of activation energies of a viscous flow of model and magmatic melts (bracketed are the basic terms of the melt anion structure (Persikov, 1998). (1) Qz100; (2) Ab100 (model granite); (3) Jd100; (4) Nepf100; (5) Ab93(H2O)7; (6) Ab85(H2O)15; (7) Ab75(H2O)25; (8) Ab80Di20; (9) Ab57Di43; (10) Ab30Di70; (11) Di92(H2O)8; (12) Di96(H2O)4; (13) Di100 (melts composition in mol %).

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(1) significantly polymerized acidic magmas, 0 ≤ 100NBO/T ≤ 17;

(2) partially depolymerized medium-mafic magmas, 17 ≤ 100NBO/T ≤ 100;

(3) depolymerized mafic-ultramafic magmas, 100 ≤ 100NBO/T ≤ 200;

(4) significantly depolymerized ultramafic magmas, 200 ≤ 100NBO/T ≤ 400.

It is obvious that the activation energy of the viscous flow, and, consequently, the melt viscosity (see Eq. (1)), decreases most significantly with the increase in the basicity of polymerized acidic magmas (see Fig. 3). It should also be noted that for such melts their activation energies, and, consequently, viscosity, significantly decrease with the increase in the ratio of Al3+/ (Al3+ + Si4+), i.e. in the transition from pure silicate (quartz melt) to aluminosilicate melts (Fig. 3), that for the first time was taken into account when calculating the viscosity of our model.

The influence of the crystal phase on the viscosity of the heterogeneous melt is calculated in the model using the following empirical equation:

$${{\eta }_{{{\text{ef}}}}} = {{\eta }_{0}}{{(1 - {{V}_{{{\text{cr}}}}})}^{{ - 3.35}}},$$
((3))

where η0 is the viscosity of the liquid phase, Vcr is the volume fraction of the crystalline phase.

As for the separate effect of fluid bubbles on the viscosity of heterogeneous melt, the data obtained by us are satisfactorily described by the following empirical equation:

$${{\eta }_{{{\text{ef}}}}} = {{\eta }_{0}}{{\left( {1 - 1.5{{V}_{{\text{b}}}}} \right)}^{{ - 0.55}}},$$
((4))

where Vb is the volume fraction of bubbles in the liquid (Persikov et al., 1990; Persikov, 1998, 2007; Persikov and Bukhtiyarov, 2009; Persikov et al., 2018).

The average composition of granite (rhyolite) obtained in the work (Le Meitre, 1976) on the basis of more than 2800 analyses of granite compositions from almost all regions of the world was chosen as representative compositions of acidic rocks (Table 1).

Table 1.   Chemical composition (wt %) and structural-chemical parameter (100NBO/T) of the starting granite melt
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Table 2.   Viscosity (\(\eta _{T}^{P}\) ) and structural-chemical parameters (100NBO/T) of the subliquidus hydrous granitic magmas in thermodynamic parameters of the Earth’s crust
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Figure 4 shows a diagram of a viscosity of the system of granite melt–water at the thermodynamic parameters of granite subliquidus (T = Tliq – 50°C), in a wide range of depths of the Earth’s crust (1–30 km).

Fig. 4.
figure 4

Patterns of viscosity of subliquidus (T = Tliq – 50°C) water-bearing granitic magmas in the earth’s crust, (P(H2O) = (0–300) MPa, Plit—up to 1000 MPa, T = 620–925°С, the contents of crystals and bubbles in the magma up to 25 vol %, the degree of depolymerization 100NBO/T = 3.5–54).

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In its construction, experimental and theoretical results of studies of thermodynamic parameters of granite melting in the presence of water vapor (Lebedev and Khitarov, 1964; Khitarov et al., 1963; Kadik, 1971; Persikov, 1984; Perchuk, 1973; Burnham, 1983) were used, and the melt viscosity under these conditions was calculated by our model (see Table 2).

Analysis of the presented results (Table 2, Fig. 4) shows that granitic magmas with water content from 2 to 9 wt % are sufficiently mobile melts in a wide range of thermodynamic parameters and depths of the Earth’s crust (1–30 km). You can see a relatively low viscosity of water-saturated granitic magmas (~105.9 Pa s), and most importantly, almost complete independence of magma viscosity in these conditions on the water content and the degree of their depolymerization (100NBO/T = 19–54) in them (see the highlighted area in Fig. 4). These results definitely indicate the real possibility of homogenization in the Earth’s crust at different depths of large masses of plutonic granitic batoliths by convection, fluid-magmatic mass transfer and their subsequent crystallization. The results obtained (Table 2, Fig. 4) prove also fundamentally different patterns of viscosity of granitic magmas with low contents of magmatic water ≤ 2 wt % (see the highlighted area in Fig. 4). They reflect many features of polymerized acidic magmatism in effusive and subvolcanic facies. With almost complete dehydration of granitic magmas, their viscosity increases by 3 orders of magnitude and reaches ~108–109 Pa s, and the degree of depolymerization decreases significantly (100NBO/T from 19 to 3.5, see Table 2). At this viscosity, polymerized granitic magmas are not able to move freely in the form of lava flows, and erupt relatively rarely in the form of extrusions or explosive catastrophic eruptions. The main mass of such granitic magmas will not reach the Earth’s surface, and the formation of huge arrays of granite intrusions, and their subsequent crystallization will occur in the plutonic facies.

CONCLUSIONS

(1) Reliable patterns of viscosity and structure of near-liquidus water-bearing granitic magmas were obtained for the first time at thermodynamic parameters corresponding to a wide range of crustal depths (1–30 km). Thus, the possible ranges and limits of movement of such magmas at different hypsometric levels in the Earth’s crust are justified on a quantitative physical-chemical basis.

(2) The established patterns of the viscosity of sub-liquidus water-bearing granitic magmas quantitatively confirm the assumption (Sobolev, 1973) about the relationship of the patterns of viscosity of water- bearing magmas with the relative distribution in the Earth’s crust of the masses of intrusive and effusive acidic rocks. Thus, the observed wide distribution of intrusive granite masses in the Earth’s crust in comparison with the masses of effusive rhyolites is quantitatively confirmed on the physical-chemical basis.