The subducting oceanic lithosphere is typically composed of a section of igneous oceanic crust with its sedimentary cover (mostly < 1 km) above and an ultramafic lithospheric mantle section beneath. The geometry of the model is illustrated in Fig. 1. The target column represents a simplified section of the upper part of such oceanic lithosphere. It is composed of a layer of basaltic composition (Rock 1) overlaid by a layer of sediments (Rock 2, see below for details). Two different rock columns are considered: (1) a relatively water-rich system with altered basalts and terrigenous sediments and (2) a relatively water-poor system with unaltered basalts (of mid-ocean ridge basalt (MORB) composition) and carbonate sediments. The column has a fixed section of 1 m2, while the thickness of each rock unit can be set by the user. The model is conservative with respect to the mass, while the volume of each rock type changes according to fluid loss and density variation along the P–T path. The P–T structure of subduction zones depends on numerous variables, including the age of the incoming lithosphere and the amount of previously subducted lithosphere (e.g. Peacock, 1990). In this study, the calculation was performed following the subduction geotherm from Gerya et al. (2002) (Fig. 2) over a pressure range of 1.3–2.6 GPa, corresponding to a depth of ∼45 to ∼85 km to encompass the conditions of interest for the investigated processes. The modelled temperatures range from a minimum of 350 ∘C to a maximum of 700 ∘C. The lower limit was chosen to take into account the large uncertainties in the thermodynamic databases for many common low-T metamorphic minerals that lead to unsatisfactory models for phase equilibria and mineral parageneses for low-grade rocks (Frey et al., 1991; Vidal et al., 2016). The upper limit is fixed by the wet solidus of metasediments – the present model strictly applies to subsolidus conditions.

During burial and heating the different slab lithologies undergo dehydration reactions. The produced fluid escapes from the source rock and migrates upward, likely interacting with the surrounding units of different chemical and isotopic composition. The effect of an external fluid input on the δ18O value of growing minerals is strongly dependent on the isotopic composition of the infiltrating fluid (δ18Ofluid) and on the degree of fluid–rock interaction. The integrated F/R ratio is defined here as the total mass of aqueous fluid that has passed through and interacted with the rock normalized to the mass of the rock. To explore different scenarios, three models are discussed involving different associations of fresh or altered oceanic basalts with terrigenous or carbonate sediments (Fig. 1b): (1) the interaction between the fluid released from the metamorphosed mafic crust and the overlying metasediment is negligible and the two rocks evolve independently (no-interaction case, NI); (2) part of the fluid derived from the metabasalt (50 % when not specified differently) equilibrates with the metasediment, while the other part leaves the system (partial-interaction case, PI); and (3) all the fluid released by the metamorphosed mafic crust equilibrates with the metasediment before escaping the system (high-interaction case, HI). The fluid released by the entire system is a mixture of fluids derived from the progressive dehydration of the metabasalt and metasedimentary layers. In the NI case, both rock types behave like closed systems and the fluid is liberated from the metabasalt and from the metasediment separately. In the case of infiltration of fluid derived from the metabasalt in the metasediment, the amount of fluid released by this latter includes the fluid produced by dehydration reactions plus the excess fluid that enters the metasediment and cannot be incorporated into stable hydrous minerals.

The thickness and the degree of serpentinization of the lithospheric mantle subducting beneath the oceanic crust can be highly variable. The most important dehydration reactions in partly or fully serpentinized mantle are related to antigorite breakdown, which can release up to 12 wt % of water, playing an important role for water flows in subduction zones. Deserpentinization is assumed to result in two main subsequent fluid peaks (Padrón-Navarta et al., 2013; Scambelluri et al., 2004) related to the reactions 1antigorite+brucite→olivine+chlorite+water∼480∘C,1.7GPa;1to2wt%waterreleased,upto∼50kgm-3,2antigorite→olivine+orthopyroxene+chlorite+water∼660∘C,2.5GPa;≥6.5wt%waterreleased,≥170kgm-3.

The effect on the δ18O of the metabasalts and metasediments of an external fluid influx, i.e. caused by dehydration of the underlying serpentinites, was investigated by defining an amount of fluid with a specific δ18O value that infiltrates the basaltic layer (Rock 1) at two steps of the model (480 and 660 ∘C along the chosen P–T path) (Figs. 1c, 2).

The strategy behind PTLoop consists of forward modelling the evolution of the mineral assemblage and the oxygen isotope composition of a rock column composed of two lithologies (Fig. 1a) of assigned thickness and starting bulk chemical and oxygen isotope compositions along a defined P–T path using a stepwise procedure (Fig. 2). At each P–T step, (1) the equilibrium mineral assemblage, oxygen isotope composition of stable phases (δ18O ‰ vs. Vienna Standard Mean Ocean Water, VSMOW), mass (in kg) and isotopic composition of the excess fluid for the metabasalt are calculated; (2) any fraction of excess fluid deriving from dehydration reactions in the metabasalt can be transferred to the metasediment or directly escapes the system; (3) the equilibrium mineral assemblage, δ18O value of stable phases, and amount and δ18O of the excess fluid for the metasediment are evaluated by accounting for the changes caused by the fluid input from the metabasalt; (4) the mass and δ18O of the total fluid leaving the system are calculated. Furthermore, at each step a chosen amount of external fluid with a given δ18O can be input into the metabasalt and its contribution is accounted for in the subsequent steps. This model is based on the assumption of thermodynamic equilibrium applied to a partially reactive system, whereby phases are assumed to reach chemical and isotopic equilibrium at all steps within the reactive part of the system, i.e. removing from the reactive bulk the phases that are fractionated (Lanari and Engi, 2017). Such petrological models can account for element sequestration during prograde metamorphism. Mineral fractionation in relics and fluid input or loss are two processes that are allowed to modify the reactive bulk composition. No mineral resorption is permitted. Any fluid liberated during dehydration (excess fluid) does not interact further and leaves the rock. This process is termed Rayleigh volatilization (Rumble, 1982; Valley, 1986). In natural rocks, it often occurs combined with the opposite endmember, the batch volatilization,where the produced fluid stays within the system as the mineral reaction proceeds and remains in isotopic equilibrium with the rock until the reaction is completed. In most natural cases involving oxygen isotopes, the difference between the results calculated using the two processes is negligible (Baumgartner and Valley, 2001). The released fluid is regarded as pure H2O. Any other effect related to solute transport by the fluid is ignored in the calculation; the potential effect of a CO2 component in the fluid is discussed in Sect. 3.2.

Equilibrium assemblage calculation for a given bulk rock composition at any P and T is performed with the program Theriak (de Capitani and Brown, 1987; de Capitani and Petrakakis, 2010) and is based on Gibbs energy minimization. A complete description of the minimization algorithm is given by de Capitani and Brown (1987). The reacting bulk composition may evolve in the course of the metamorphic history of a rock because of mineral fractionation, fluid loss or input of external fluids. The effective bulk composition is recalculated by PTLoop at each subsequent stage following the strategy of Lanari et al. (2017).

As for phase assemblage determination, equilibrium is a common assumption of stable isotope transport (Baumgartner and Rumble, 1988; Baumgartner and Valley, 2001; Bowman et al., 1994; Gerdes et al., 1995a, b). Thus, a molar equilibrium constant (K) can be defined to describe the thermodynamic stable isotope equilibrium between two substances i and j (Sharp, 2017): 3K=18Oi/16Oi18Oj/16Oj.

The fractionation factor (α) can be related to the equilibrium constant K as 4α=K1/n, where n is the number of exchanged atoms, normally 1 for simplicity. In isotope geochemistry, the isotopic composition is commonly expressed in terms of δ values: 5δi=RiRSt-1⋅103(‰), where Ri and RSt are the isotope ratio measurements for the compound i and the defined isotope ratio of a standard sample respectively. For differences in δ values or for δ values of less than ∼10 ‰, 61000ln⁡αi-j=δi-δj is a valid approximation that is used in most cases (Hoefs, 2015; Sharp, 2017). For oxygen isotope fractionation, the equation that can reproduce most of the available calibrations describing the stable isotope fractionation function between two phases is a second-order polynomial of 103/T. Hence the stable isotope fractionation between two phases i (with k endmembers) and j (pure) as a function of T is described by Eq. (7): 7δ18Oi-δ18Oj=∑1kAk,j⋅106T2+Bk,j⋅103T+Ck,j⋅Xk,i⋅Nk,iNi. The conservation of the bulk δ18O in the system is described by Eq. (8): 8δ18Osys⋅Nsys=∑k=1pMk⋅Nk⋅δ18Ok, where δ18Oi, δ18Oj and δ18Osys are the isotopic compositions of phase i, phase j and the system (bulk δ18O) respectively; Ak,j, Bk,j and Ck,j are the fractionation parameters for endmember k of mineral i vs. phase j; Xk,i is the fraction of endmember k in the phase i; Nk,i, Ni and Nsys are the total number of moles of oxygen in endmember k, in mineral i and in the system respectively; p is the number of phases; Mk is the number of moles of phase k; Nk is the its number of oxygen; and δ18Ok is its oxygen isotope composition. Given a stable mineral assemblage at any P–T condition, the oxygen isotope partitioning among the stable phases is calculated by solving the linear system described by the sets of p-1 equatiomns of type (7) and Eq. (8); Kohn, 1993; Vho et al., 2020). In closed systems, the first term in Eq. (8) is constant. Open-system behaviour can either modify the δ18Osys or the number of moles of the phases (Nsys). The parameters A, B and C between phases were taken from the internally consistent database for oxygen isotope fractionation DBOxygen version 2.0.3 (Vho et al., 2020).

In order to represent the variability in the basaltic portion of the oceanic crust two different bulk compositions were used (Table 1): (1) a representative average MORB basalt (Gale et al., 2013) that has been hydrated but without any other addition or removal of element (metabasalt(h)) and (2) a basalt that underwent extensive sea-floor alteration during hydration (Baxter and Caddick, 2013 after Staudigel et al., 1996) that will be referred to as metabasalt(a) in the following. Those compositions are in good agreement with other compilations reported in the literature (e.g. Sun and McDonough, 1989; Albarède, 2005; Staudigel, 2014; White and Klein, 2014). Oceanic sediments were modelled with two distinct bulk compositions (Table 1): (1) terrigenous sediment referred to as metasediment(t) (clay from the Mariana Trench; Hacker, 2008, after Plank and Langmuir, 1998) and (2) nanno ooze carbonate referred to as metasediment(c) (Plank, 2014). Nanno oozes are widespread carbonate sea-floor sediments (Plank, 2014), and they are close in composition to carbonate-rich sediments observed in HP terrains (e.g. Bebout et al., 2013; Kuhn et al., 2005). Thicknesses of 1000 m for the basaltic layer, of 175 m for the clay sediment and 75 m for the carbonate sediment were chosen in order to maintain proportions between oceanic crust and sediments comparable with the values reported in various compilations (e.g. Hacker, 2008; Plank, 2014). This results in a total thickness of the rock column 2 to 3 times smaller than the real thickness to encompass the assumption of homogeneous temperature over the whole column within ∼20 ∘C. To overcome the effects of possible temperature variations within the column, a discretization step size of ∼20 ∘C along the P–T path was applied.

The bulk compositions were simplified to the Na2O–CaO–K2O–MgO–FeO–Al2O3–TiO2–SiO2–H2O system; C is present in the initial bulk composition of the metabasalt(a) and the metasediment(c) (Table 1). MnO was excluded because it overemphasizes the stability of garnet at low metamorphic conditions (T≤350 ∘C). The conditions of the garnet-in reaction in Mn-absent systems match the results obtained for garnet nucleation in natural rocks (e.g. Laurent et al., 2018). Thermodynamic modelling was performed using the internally consistent dataset of Holland and Powell (1998) and subsequent updates (tc55, distributed with Theriak-Domino 4 February 2017; see Supplement S1). The following activity models where used for the solid solutions: Holland and Powell (2003) for calcite–dolomite–magnesite; Holland and Powell (1998) for garnet, white mica and talc; Holland and Powell (1996) for omphacite; Holland et al. (1998) for chlorite; Diener et al. (2007) for amphibole. In the presented model garnet undergoes fractional crystallization both in Rock 1 and Rock 2 fractionating from the reactive bulk for the subsequent steps. The amount of initial H2O in each rock was set at saturation and is reported in Table 1. No pore fluid expulsion, diagenetic and low-grade (T < 350 ∘C) devolatilization reactions are considered in this study (see above).

A starting bulk δ18O for the metabasalt(h) of 5.7 ‰ was chosen and represents the reference value for an unaltered MORB (e.g. Cartwright and Barnicoat, 1999; Eiler, 2001; Staudigel, 2014; White and Klein, 2014), while a starting bulk δ18O of 9.0 ‰ is used for metabasalt(a), representative of basaltic material that underwent sea-floor alteration at T≤400 ∘C (e.g. Alt et al., 1986; Cartwright and Barnicoat, 1999; Eiler, 2001; Gregory and Taylor Jr., 1981; Miller and Cartwright, 2000; Staudigel, 2014; White and Klein, 2014). The starting bulk of the terrigenous sediment of 15 ‰ represents the average for the δ18O of clastic sediments reported by Eiler (2001). The chosen δ18O starting bulk of the carbonate sediment is 25 ‰, which represents a conservative estimate of marine carbonate δ18O (typically 25 ‰–35 ‰; Eiler, 2001). It is ∼5 ‰ higher than the values for metasedimentary carbonates in the Italian Alps (e.g. Cook-Kollars et al., 2014) that are likely to have interacted with lower- δ18O fluids during subduction.

In order to define the contribution of an external fluid originating in the lithospheric mantle by serpentine breakdown, a layer of 150 m of pure serpentine containing 12 wt % bulk H2O was considered and the mass of water released at each reaction was calculated by mass balance, resulting in an input of 7800 kg of water at 480 ∘C and of 25 350 kg at 660 ∘C to satisfy the reactions in Eqs. (1) and (2) respectively. In order to fit the thicknesses chosen for the oceanic crust and the sedimentary layer (2 to 3 times thinner than an average lithospheric section), the 150 m of pure serpentine correspond to a conservative estimate of 3000 m of serpentinized peridotite with an average serpentine content of 5 % in volume. This is in agreement with the values used by Barnes and Straub (2010) and John et al. (2011) based on the estimate by Sharp and Barnes (2004). Serpentine oxygen isotope compositions reported in the literature are highly variable (Cartwright and Barnicoat, 1999, 2003; Früh-Green et al., 2001; Mével, 2003; Miller et al., 2001), typically ranging from 1 to 10 ‰. In mid-oceanic ridge environments, the distribution has a peak between 2 ‰ and 5 ‰ (Mével, 2003). A value of 2.5 ‰ was chosen from the lower-δ18O side of this peak. This results in the liberation of a fluid with a characteristic low-δ18O value, but one which is still feasible for natural serpentinites; this value is clearly distinct with respect to the overlying lithologies. The effect of infiltration into the metabasalts and metasediments of serpentinite-derived fluids will become smaller as the fluid δ18O becomes higher, approaching the equilibrium with the overlying lithologies. The δ18O value of the released fluid is ∼4.5 ‰ at T>550 ∘C (serpentine–water oxygen isotope fractionation factors from Vho et al., 2020). Further details on the modelling input data are given in Supplement S2.

The evolving stable mineral assemblages and bulk water contents of each lithology, without external fluid input, were calculated for each rock composition along the prograde P–T path. Results are provided as mode-box diagrams in Fig. 3. The H2O field represents the volume fraction of excess water in each rock type. The fluid is progressively extracted becoming isolated from the reactive part of the system. Garnet is the only phase prevented from re-equilibrating in the model, thus fractionating from the reactive bulk composition. Below 450 ∘C and 1.80 GPa, in the metabasalts glaucophane, actinolite and lawsonite comprise ∼80 vol. % of the paragenesis, with minor phengite (Siapfu=3.67–3.63, XMg=0.62–0.56 in metabasalt(h); Siapfu=3.68–3.65, XMg=0.67–0.58 in metabasalt(a)), omphacite (XNa=0.45–0.42, XMg=0.81–0.72), chlorite and titanite. Metabasalt(h) is richer in SiO2, FeO and MgO with respect to the metabasalt(a), and chlorite is stable up to 480 ∘C. Metabasalt(a) contains ∼5 vol. % of Ca carbonate that remains stable over the entire P–T path. For either composition, the volume of glaucophane, actinolite and lawsonite gradually decreases from 480 ∘C and ∼1.90 GPa until complete consumption at 600–620 ∘C and 2.30–2.36 GPa. Those represent the major hydrous phases contributing to the dehydration, while a secondary role is played by talc and zoisite at higher P–T conditions (T≥580 ∘C, P≥2.24 GPa). Most of the water still retained in the rocks is stored in phengite, the abundance of which is primarily controlled by bulk K2O content, higher in the metabasalt(a), and that remains stable beyond the model conditions. Garnet production starts at ∼500 ∘C and ∼2.00 GPa (Xalm=0.60, Xgrs=0.35), and garnet grows continuously until constituting ∼20 vol. % of the metabasalt(a) and ∼35 % of the metabasalt(h).

In the metasediment(c) calcium carbonate, quartz, phengite (Siapfu=3.42, XMg=0.53) and jadeite (jadeite content Xjd=0.95, XNa=0.96, XMg=0.39) compose ∼80 vol. % of the solids at 350 ∘C and 1.30 GPa. Glaucophane and lawsonite are stable up to 460 and 560 ∘C respectively. Jadeite abundance increases from ∼10 vol. % at 440 ∘C to ∼16 vol. % at 460 ∘C; ankerite is stable in minor amounts (≤3 vol. %) at 440–560 ∘C. Garnet (Xalm=0.57, Xgrs=0.41) is stable only at 540–580 ∘C and 2.12–2.24 GPa, reaching ∼5 vol. %, and is then preserved because of the assumption of fractionation from the bulk in the model. The metasediment(t) at T<500 ∘C shows a paragenesis of phengite (Siapfu=3.68–3.55, XMg=0.53–0.44), glaucophane, lawsonite, quartz and omphacite (XNa=0.50–0.48, XMg=0.87–0.70), with minor titanite. At 520 ∘C and 2.06 GPa, lawsonite is consumed and the amphibole proportion reduces from ∼30 vol. % to < 20 vol. %, while garnet (Xalm=0.63, Xgrs=0.34) is produced and reaches ∼10 vol. %. At these conditions, the omphacite content decreases and a clinopyroxene of more jadeitic composition (Xjd=0.72, XNa=0.76, XMg=0.45) becomes stable. These models are in line with the first-order mineralogical changes observed in subducted (and exhumed) crustal material. Thermodynamic calculations predict the coexistence of a calcic amphibole and a sodic amphibole in the metabasalts and of jadeite and omphacite in the metasediment(t). From an oxygen isotope partitioning perspective, the interpretation of the modelled coexistence of a sodic and a calcic amphibole either as two endmembers of a solid solution or as coexisting minerals is equivalent. The same applies to pyroxenes, for which the modelled coexistence of two pyroxenes can be interpreted as a continuous solid solution. Therefore, this does not affect the oxygen isotope partition model final results for the other phases and the bulk.

At the initial conditions, all the lithologies are saturated in H2O (Table 1). Up to 500 ∘C, lawsonite, actinolite and glaucophane are the main repositories of H2O in the metabasalts, followed by chlorite and minor phengite. A significant pulse of water is modelled at 500–520 ∘C and 2.00–2.06 GPa for both the metabasalts (Fig. 3a, b). This pulse is caused by a decreasing abundance of lawsonite and amphibole and a breakdown of chlorite followed by growth of garnet and omphacite. This first dehydration stage releases ∼25 % of the total water loss from the metabasalt(a) (∼4.0 vol. % H2O liberated) and ∼45 % from the metabasalt(h) (∼6.5 vol. % H2O liberated). The second significant pulse in the metabasalts occurs at 620–640 ∘C and 2.36–2.42 GPa, releasing ∼40 % of the total water loss from the metabasalt(a) and ∼15 % from the metabasalt(h). This pulse is caused by the final breakdown of lawsonite and of amphibole in the metabasalt(a) (Fig. 3b). In the metabasalt(h), glaucophane and actinolite breakdown takes place at 600 ∘C and 2.30 GPa. Growth of talc only incorporates half of the water released by amphibole breakdown, causing an intermediate fluid pulse of minor magnitude at these P–T conditions (Fig. 3a). If an initial undersaturated basaltic composition was considered, the amount of released fluid due to the breakdown of hydrous phases would be smaller.

The metasediment(c) is the rock type that dehydrates the least: the two main pulses of fluid production are at 480 ∘C and 1.92 GPa (∼0.4 vol. % H2O liberated) and from 540 ∘C and 2.12 GPa to 560 ∘C and 2.18 GPa (∼1.7 vol. % H2O liberated), caused by a breakdown of glaucophane and lawsonite respectively (Fig. 3c). The water produced from these two dehydration stages represents < 0.02 wt % of the total water released by the system composed of metabasalt(h) and metasediment(c). In the metasediment(t), the main fluid pulse occurs at 520 ∘C and 2.06 GPa (∼3.0 vol. % of H2O liberated), caused by the breakdown of lawsonite and a decrease in glaucophane abundance (Fig. 3d). The water produced from this dehydration stage represents ∼0.07 wt % of the total water released by the system composed of metabasalt(a) and metasediment(t).

As specified in Sect. 2.2, the aqueous fluid used for the calculation considers only the water component (aH2O=1). The release of CO2 occurs during progressive metamorphism of metabasalt(a) and metasediment(c). CO2 is absent or present in negligible amounts (< 1 mol % of the total fluid) up to 440 ∘C and 1.76 GPa. At higher temperatures, the X(CO2) content increases significantly up to ∼10 mol % in the fluid released by metabasalt(a) and ∼30 mol % in the fluid released by metasediment(c) at 700 ∘C and 2.60 GPa. However, the total amount of fluid produced at these conditions is negligible (< 0.01 vol. %). The oxygen isotope fractionation database used for this study (Vho et al., 2020) does not include fractionation data for CO2. Limited calibrations are available for oxygen isotope fractionation between H2O and CO2 (Friedman and O'Neil, 1977; O'Neil and Adami, 1969; Zheng, 1994). The fractionation values diverge significantly, for example between -4.9 ‰ and -8.3 ‰ at 450 ∘C and between 1.5 ‰ and -4.4 ‰ at 700 ∘C. At 520 ∘C, the fluid released by metabasalt(a) contains 2 mol % of CO2 and 6 mol %–7 mol % at 620–640 ∘C. The consideration of CO2 would produce a negligible shift on the fluid δ18O (0.1 ‰–0.2 ‰ at 520 ∘C and fractionation used). The amount of CO2 in metasediment(c) derived-fluid is 2 mol % at 480 ∘C and 5 mol % at 540 ∘C. The consideration of CO2 would also produce negligible to minor shifts in the fluid δ18O (0.1 ‰–0.3 ‰ at 480 ∘C and 0.2 ‰–1.1 ‰ at 620 ∘C, depending on the fractionation).

The largest initial bulk δ18O difference occurs between metabasalt(h) and metasediment(c) (14.3 ‰, the relatively water-poor system), while the smallest initial bulk δ18O difference is observed between metabasalt(a) and metasediment(t) (6.0 ‰, the relatively water-rich system). In the following, the results are presented in detail for a selection of two endmember scenarios (Figs. 4, 5): (1) metasediment(c) associated with metabasalt(h) and (2) metasediment(t) associated with metabasalt(a) when not specified differently. Other scenarios (i.e. metabasalt(h) associated with metasediment(t) and metabasalt(a) associated with metasediment(c)) give intermediate results in terms of oxygen isotope composition variations as a consequence of fluid–rock interaction. Further details and the results for the intermediate scenarios are given in Supplement S3.

The changes in mineral assemblage, modes and compositions along a prograde P–T path control (1) the oxygen isotope partitioning between the stable phases and (2) the amount and δ18O of the fluid released by the system. At the same time, oxygen isotope fractionation between the stable phases is controlled by temperature. Thus, the effects of evolving paragenesis and increasing temperature are systematically overlapping in nature. In the case of a closed system, the bulk concentrations of 18O and 16O remain constant and a change in one phase is compensated for exactly by adjustments in other phases (Baumgartner and Valley, 2001; Kohn, 1993). In this situation, major changes in mineral assemblage do not play a significant role in shifting the δ18O of stable phases: this is demonstrated by the limited (< 0.5 ‰) shift in δ18O values of quartz, garnet, phengite, omphacite and rutile in the metabasalt(h) after (1) the breakdown of amphibole and lawsonite and (2) the crystallization of talc and kyanite over a narrow temperature range between 500 and 580 ∘C (Fig. 4a, b). Similar effects can also be anticipated for rocks with different chemical composition that undergo major changes in the mineral assemblage (see Supplement S4).

In a closed system evolving at equilibrium, the initial chemical bulk composition and bulk δ18O do not change along the P–T evolution. However, in metamorphic rocks, compositional zoning and metamorphic overgrowths are often preserved in refractory minerals (Lanari and Engi, 2017) indicating that parts of the minerals have become isolated from the reactive volume of the rock. This scenario is commonly referred to as “partially re-equilibrated open systems”, because the chemical and the isotopic compositions vary as a consequence of the fractionation of solid and fluid phases (i.e. garnet fractionation and excess fluid removal) even in the absence of external fluid input. Phase fractionation is expected to affect the bulk δ18O as function of both the amount of fractionated or expelled phases and their isotopic composition. Fractionation of a phase lighter than the bulk in δ18O leads to an increase in the reactive bulk δ18O value, while fractionation of an isotopic heavier phase leads to a decrease in the reactive bulk δ18O value.

The most common example of fractionating metamorphic mineral is garnet, which systematically records compositional zonation at low- to medium-grade (Evans, 2004; Giuntoli et al., 2018; Konrad-Schmolke et al., 2008; Spear, 1988; Tracy, 1982). Therefore, garnet fractionation was incorporated into the model in order to better approximate the behaviour of natural systems. Note that this effect is reduced at higher grade where intra-crystalline diffusion becomes efficient to partially re-equilibrate garnet (Caddick et al., 2010; Lanari and Duesterhoeft, 2019). As already documented by Konrad-Schmolke et al. (2008), garnet fractionation controls the extent of the garnet stability field. Garnet crystallization is not systematically expected to occur near the peak conditions, if the matrix was strongly depleted due to garnet fractionation and the volume of garnet remains constant (i.e. for the metabasalt(a), Fig. 3b). While garnet fractionation is recognized to significantly affect isopleth thermobarometry and volume fractions (Lanari and Engi, 2017), its effect on oxygen isotope bulk composition and partitioning is negligible (< 0.5 ‰) in all the studied lithologies. In the model, the garnet fraction varies from ∼5 vol. % in the metasediment(c) to ∼35 vol % in the metabasalt(h) (Fig. 3), and its δ18O is 0.8 ‰ to 1.7 ‰ lower than the bulk (Fig. 4a, b, e, f).

Beside garnet fractionation, dehydration due to hydrous mineral breakdown and expulsion of excess water may contribute to changing the starting chemical and isotopic bulk compositions. Baumgartner and Valley (2001) postulated that the liberation of metamorphic fluids might have a profound effect on the stable isotope composition of the residual rock. In the present study, the maximum fluid loss is from the metabasalts that release ∼15 vol. % (∼5 wt %) of H2O with δ18O values 0.9 ‰ to 1.5 ‰ higher than the bulk rock (Fig. 4a, b) at T≥500 ∘C. This significant fluid flux produces a decrease in the bulk δ18O of less than 0.2 ‰ (Fig. 4c, d). Even if more extensive dehydration occurs, the effect on the bulk δ18O value will be typically lower than 1.0 ‰. No significant differences in the effect of stable assemblage evolution and phase fractionation are observed between the four lithologies. Therefore, the bulk δ18O of a rock that experienced a succession of dehydration reactions, without rehydration by external fluids or major compositional changes through decarbonation or mineral dissolution, is likely to be representative of its protolith composition. In this regard, integrated thermodynamics and oxygen isotope modelling represent a key tool for quantifying the potential effects of different processes and for assessing closed- or open-system behaviours.

In the last decades, the significant advances of oxygen stable isotope analyses by SIMS (secondary ion mass spectrometer) have allowed zoned metamorphic minerals to be analysed in situ with a precision down to 0.2 ‰–0.3 ‰ (2σ) (e.g. Ferry et al., 2014; Martin et al., 2014; Page et al., 2010). The magnitude of the intra-crystalline δ18O variation in key metamorphic minerals has been widely used to establish whether a metasomatic stage is related to an internal fluid deriving from the breakdown of hydrous phases or if it reflects equilibration with an external fluid of different isotopic composition (e.g. Putlitz et al., 2000; Errico et al., 2013; Page et al., 2013; Russell et al., 2013; Martin et al., 2014; Rubatto and Angiboust, 2015; Engi et al., 2018). Understanding the scale of fluid migration at depth and the magnitude of the interaction between fluids and minerals is of special interest and can be enhanced by modelling of such fluid flow and isotopic exchange (Baumgartner and Valley, 2001). The definition of different interaction cases (NI, PI, HI) is useful to represent various degrees of isotopic exchange between the fluid and the rock. If the flow is channelled, all the interaction cases can possibly occur in close proximity and the modelled scenarios can be linked to the evolution of different domains around the vein (Fig. 8). The flow of a pervasive fluid leads to the homogenization of the chemical potential of all components, including stable isotopes (Baumgartner and Valley, 2001), and it is represented by the HI case, as long as integrated F/R ratios are high. In contrast, the flow of a channelled fluid results in local chemical heterogeneities, allowing some portions of the rock and of the fluid to remain unaffected (NI case) and some others to be only partially affected (PI case).

The first step for a meaningful interpretation of an observed intra-grain variation in δ18O value is the quantification of the possible effects of changes in T and mineral assemblage. Such effects are characteristic of each phase (Fig. 4a, b, e, f). Quartz, calcite and rutile are the minerals most sensitive to temperature changes. Their composition is expected to vary up to 1 ‰ per 100 ∘C, and they are stable over a wide range of temperature. For such phases, care is required in interpreting significant intra-grain δ18O variations (i.e. up to 3.0 ‰) since it does not necessarily reflect interaction with an external fluid having a different isotopic composition. However, quartz and calcite are not necessarily robust minerals that can preserve chemical and isotopic zoning during metamorphism. On the other hand, the variation in δ18O value over 150 ∘C in a mineral such as garnet that commonly retains growth zoning is typically within 0.5 ‰ when no phase fractionation is involved and still less than 1.0 ‰ when considering the effect of previous garnet and/or excess fluid fractionation (Fig. 4a, b, e, f). Thus, any larger variation has to be linked to a significant change in bulk δ18O. Similar behaviour is observed for other key metamorphic minerals such as phengite, amphibole and clinopyroxene. These minerals have been widely used in metamorphic petrology as thermometers and geobarometers (Dubacq et al., 2010; Ferry and Spear, 1978; Parra et al., 2002) and are expected to be robust targets to link the fluid evolution along the P–T path, especially when mineral relics are preserved. Due to its large capacity to preserve growth chemistry, garnet has been a primary target for microscale measurement of oxygen isotopes. Protocols and reference materials for SIMS measurements for a range of garnet compositions are well established (e.g. Martin et al., 2014; Page et al., 2010; Vielzeuf et al., 2005b) and its retentivity to high T resetting by diffusion has been investigated (Higashino et al., 2018; Vielzeuf et al., 2005a). In HP rocks, various degrees of intra-grain variations in garnet δ18O associated with external fluid infiltration have been reported in the literature. Where field constraints on the fluid source (i.e. oxygen isotope measurements on the feasible source lithologies) are available, the intra-grain variation can assist in the calculation of F/R ratios. Martin et al. (2014) describe a shift of -2.5 ‰ associated with infiltration of serpentinite- or altered gabbro-derived fluids in metasediments from the continental basement in Corsica. Rubatto and Angiboust (2015) observed a shift of 3.5 ‰–4.0 ‰ in garnet from an eclogite from Monviso that they attributed to sediment-derived fluid infiltration. Vielzeuf et al. (2005b) measured a decrease of 2.5 ‰–3.0 ‰ between garnet core and rim in the Dora Maira gneiss probably caused by the interaction with a fluid derived from the whiteschists (Gauthiez-Putallaz et al., 2016). Studies that used in situ measurement of micas are limited (Bulle et al., 2019; Siron et al., 2017), and thus the potential of mica to trace fluid–rock interaction is still underexplored. The matrix complexity of pyroxenes and amphiboles remains a challenge for SIMS measurements.

The effect of pervasive fluid flow deriving from the breakdown of serpentine (δ18O = 4.5 ‰) in a serpentine layer of different thicknesses (150, 300, 600 m, see above) on the bulk δ18O of the two metabasalts is negligible (< 0.5 ‰, Fig. 7, Supplement S3). This is mainly due to (1) the minor difference in δ18O between the serpentinite-derived fluid and the metabasalts (1.2 ‰ for the metabasalt(h) and 4.5 ‰ for the metabasalt(a)) and (2) the very low integrated F/R ratio (0.01, 0.02 and 0.04 for the three cases). In this case, the integrated F/R ratio seems to play a bigger role, since the limited change in δ18O is similar for both the metabasalts even if the initial Δ18O between fluid and rock is larger for the metabasalt(a). With the same total volume of fluid and rock, a channelled fluid flow would imply larger volumes of fluid interacting with smaller volumes of rock (higher F/R ratios) and would thus be expected to drive larger variations in isotopic composition. For instance, by increasing the F/R ratio by a factor of 10 (from 0.01 to 0.1) the bulk δ18O decreases by 0.6 ‰ (metabasalt(h)) and 1.1 ‰ (metabasalt(a)) upon infiltration of the serpentine-derived fluids.

In contrast to the metabasaltic layers, the overlying metasediments have (1) larger compositional differences with the serpentinite-derived fluid and (2) a smaller mass. The effect of the serpentinite-derived fluid input at the base of the column on the metasediment bulk δ18O compositions can be up to 10 times larger than the effect that the same amount of fluid has on basaltic compositions. This is the case even when the serpentinite-derived fluid completely equilibrates with the metabasalt before interacting with the overlying metasediment. In our model, the fluid infiltrating the metasediment is always a mixture of metabasalt-derived and serpentinite-derived fluid. Considering instead an input of serpentinite-derived fluid directly in the metasediment and applying an F/R ratio of 0.1, the final bulk δ18O of the metasediment(c) decreases by 4.6 ‰ and that of the metasediment(t) by 2.6 ‰. These values are significant and are comparable with variations observed in natural samples. For instance, Martin et al. (2014) describe a shift in δ18O of -2.5 ‰ among different generations of HP garnet in a sample from the Corsica continental basement (garnet mantle δ18O = 7.2±0.4 ‰, garnet rim δ18O = 4.7±0.5 ‰). The authors associate this shift with an infiltration of serpentinite-derived fluids and, to a lesser extent, altered gabbro-derived fluid. Williams (2019) describe an extreme δ18O shift of -15 ‰ between garnet core and rim in a metasediment from the Lago di Cignana Unit. Such an oxygen isotope composition variation has been related to a strongly channelled fluid influx originating from the dehydration of serpentinites. These results demonstrate that metasediments can be a good target to detect fluids from an ultramafic source migrating upward through the subducting slab or along the subduction interface, even though the two lithologies may not be in direct contact in the field.

It is important to note that the relatively water-poor system composed of the metabasalt(h) and the metasediment(c) is more sensitive to external fluid infiltration and thus affected by the highest changes in δ18O, according to observations in natural systems (e.g. Page et al., 2019).

As discussed in detail by previous studies (Baxter and Caddick, 2013; Hacker, 2008; Hernández-Uribe and Palin, 2019; Syracuse et al., 2010), the subduction geotherm has an important effect on hydrous phase stability and P–T conditions of fluid release into the mantle wedge. Along the average D80 geotherm by Syracuse et al. (2010) (Fig. 2), the top of the slab crust releases ∼95 % of the water at ∼80 km, during the transition from partial to full coupling. Along the cold geotherm by Penniston-Dorland et al. (2015), the first significant fluid pulse (20 %–40 % of water released) occurs as a consequence of the breakdown of glaucophane and actinolite at greater depths than predicted by our model (∼500 ∘C and 2.7–2.8 GPa, 90–100 km depth), and the remaining water is released at depth > 100 km. Both those models imply a relatively dry mantle forearc region, contradicting what has been described by Bostock et al. (2002). By contrast, along the warm geotherm by Penniston-Dorland et al. (2015), the breakdown of chlorite, epidote and actinolite releases 40 %–50 % of the water at 460–470 ∘C and ∼0.6 GPa (∼20 km depth); after this stage, ∼95 % of the water is released at depth < 70 km. This implies that little to no water is available at subarc depths. The average geotherm by Penniston-Dorland et al. (2015) is hotter than the one by Gerya et al. (2002) used in our model for T<650 ∘C and P<2.5 GPa (Fig. 2). Along this P–T gradient, chlorite is the main water carrier in low-grade conditions and the first two main dehydration pulses occur at depths of 30–40 km (significant decrease in chlorite, 20 %–25 % of water released) and of ∼50 km (complete breakdown of chlorite, 20 %–25 % of water released). Along this geotherm, the most water is released at shallower depths than in our model. Differences can be investigated in detail by modelling each case with PTLoop. Nevertheless, the effects of fluid–rock interaction on the bulk and mineral δ18O compositions follow the general trends described above. Different parageneses are expected to form during hydration, but the shift in bulk δ18O remains constrained by the F/R ratio and the isotopic composition of the incoming fluid.

Infiltration of the slab-derived fluid into the mantle wedge is important for subduction zone settings because mantle minerals are strongly depleted in volatiles. At equilibrium, a free aqueous fluid is not stable in the mantle wedge at T<650 ∘C until a fully hydrated mineral assemblage has formed (i.e. serpentine, chlorite, talc, and amphibole) (Manning, 2004). As shown above, the P–T conditions of H2O release from the subducting slab, as well as the volume and the δ18O of the liberated H2O, can vary according to the geometry of the subduction zone and the composition of the subducting lithosphere (e.g. Hacker, 2008; Hernández-Uribe and Palin, 2019; Poli and Schmidt, 2002). The program PTLoop allows the P–T conditions, amount and oxygen isotope composition of the released fluid to be calculated for the system of interest. The presented model uses an intermediate P–T gradient that stabilizes lawsonite and results in a first significant fluid pulse at 65–70 km depth and a second pulse at ∼80 km depth. Below that depth, phengite is the main carrier of H2O. This implies that the majority of fluid is released in the forearc region, in agreement with previous studies investigating the dehydration of basaltic and sedimentary components of the slab (e.g. Baxter and Caddick, 2013; Kerrick and Connolly, 2001; Schmidt and Poli, 1994).

The influence of the slab-derived fluid on (1) the degree of hydration and (2) the δ18O modification of the overlying mantle rocks was estimated on the basis of the results presented above. A slab composed by the metabasalt(a) and the metasediment(t) (left column in Fig. 1a, assuming the real thickness of the slab to be 3 times the modelled one) subducting at 1 cm yr-1 was considered (Fig. 9a). This subduction rate represents a conservative estimate, considering that previous averages have proposed 4–5 cm yr-1 (Stern, 2002). Any faster subducting rate may shorten the timescale of the processes discussed below, allowing larger fluid fluxes in the mantle wedge over the same interval. Mechanical decoupling between the slab and the wedge and a steady-state cold mantle wedge are assumed (e.g. Abers et al., 2006; Hirauchi and Katayama, 2013; Wada et al., 2008). The fluid released by the slab at 500–520 ∘C with a characteristic δ18O = 15.0 ‰ (Fig. 6a) was allowed to infiltrate into an initially dry peridotite (composition KLB-1 from Walter, 1998, simplified to the Fe–Mg–Al–Si system; Table S2.4 in Supplement S2) with T=570 ∘C and an initial δ18O = 5.5 ‰ (Eiler et al., 1997; Mattey et al., 1994). This simplified model ignores the dynamics of fluid infiltration and assumes pervasive flow. Given the column length of 1 m, a subduction rate of 1 cm yr-1 implies that, in 100 years, any fixed point (i.e. fixed P–T conditions) at the slab–mantle interface receives the total amount of fluid that a single column can liberate in those conditions. Hence, in this example 4892.6 kg of water per 100 years (i.e. the amount released by the considered column at the chosen conditions, see above) infiltrate the mantle wedge. The released fluid first interacts with a small volume at the slab–mantle interface (V1 1000×1000×1 m; Fig. 9b) and once it has equilibrated and saturated V1, it infiltrates volume V2 (3000×2000×1 m; Fig. 9b). Both the volumes were scaled to 1:3 for the calculation in order to maintain the volume proportion with the modelled slab. The slab-derived fluid, which is in continuous supply during subduction of new material, infiltrates V1 and causes a progressive change in mineralogy from olivine + orthopyroxene +  garnet to serpentine + chlorite + minor olivine until the rock reaches saturation after 0.35 Myr of subduction. At this stage, V1 has a bulk δ18O of ∼8 ‰ that is significantly higher with respect to the initial mantle value (Fig. 9c). With ongoing subduction, the continuing release of water from new slab material under a static mantle drives the δ18O of the volume of mantle wedge toward higher values. The water that will infiltrate V2 has a δ18O that depends on (1) the δ18O of the slab-derived water and (2) the buffering capacity of V1. The same changes in mineral assemblage described for V1 occur also in V2, while the bulk δ18O of V2 increases more moderately than the one of V1 and reaches a δ18O of ∼6 ‰ after 0.75 Myr of subduction (Supplement S4).

In the proposed model, most of the fluid is released by the slab at forearc depths. However, in most subduction zones no melting appears to occur in the forearc region and the serpentinite acts as the effective H2O absorber (Iwamori, 1998), recording the possible variation in δ18O induced by the slab-derived fluid. Progressive oceanward migration of the slab (“slab rollback”) has been regarded as an important mechanism acting in most active subduction zones (e.g. Heuret and Lallemand, 2005; Nakakuki and Mura, 2013). The rollback of the slab results in a lateral extension of the serpentinized wedge. As a consequence, the melt ascending below the arc can interact with serpentinized, high-δ18O mantle portions that were originally part of the forearc mantle and modify its original isotopic composition. High-δ18O arc lavas and melt in arc-setting peridotites have been described (e.g. phenocrysts in lavas from central Kamchatka: olivine δ18O = 5.8 ‰–7.1 ‰ and clinopyroxenes δ18O = 6.2 ‰–7.5 ‰, Dorendorf et al., 2000; New Guinea: silicate glass inclusions in olivine δ18O = 8.8 ‰–12.2 ‰, clinopyroxenes in metasomatized lehrzolite δ18O = 6.2 ‰–10.3 ‰, Eiler et al., 1998), but the mechanism of crustal contamination is still debated. Our results support the model proposed by Auer et al. (2009) that relates such high-δ18O lavas to the interaction between primitive basaltic melts with an uppermost mantle that was hydrated and enriched as part of the forearc mantle prior to trench migration.

The presented approach has a broad range of applications for modelling fluid–rock interaction in different tectonic settings. We have presented here an example of subducted crust but the same principles apply also for regional metamorphism or hydrothermal systems. The model also provides new ways to quantify the degree of interaction of an external fluid within the same rock unit. We have shown that the observed effect on the δ18O of a rock of channelled vs. pervasive hydration is strictly coupled with the composition of the fluid source. Nevertheless, important insights can be given by linking observations of ideal cases with modelling even if the composition of the infiltrating fluid is not known a priori. For instance, the oxygen isotope composition of a fluid source can be retrieved when a variation in δ18O is observed within the same rock type from the more hydrated to the less hydrated portions, even in the absence of a clear presence of a vein or vein system. Alternatively, the degree of equilibration of the host rock around a vein can be calculated when the isotopic composition of the fluid is known. Therefore, the combination of mineral-scale in situ oxygen isotope analyses with major and trace element mapping will provide much more detailed information on quantitative element mobility during fluid–rock interaction.

Fluids play a central role as a catalyst for chemical reactions in rocks. Generally re-equilibration reactions occur only in the presence of fluids that either derive from breakdown of hydrous phases or from external sources (e.g. Airaghi et al., 2017; Cartwright and Barnicoat, 2003; Engi et al., 2018; Konrad-Schmolke et al., 2011; Rubatto and Angiboust, 2015). The program PTLoop calculates at which P–T conditions the breakdown of hydrous phases occurs, and, consequently, metamorphic reactions and free fluid are expected. If fluid-driven reactions occurred in a rock at P–T conditions where no release of internal fluid is expected, the role of an external fluid should be considered, and its amount and isotopic composition can be retrieved using the approach outlined in this paper.

Fluid–rock reactions in subducted lithosphere are likely to involve more complex open-system processes than the dehydration and rehydration considered for this model, driving silicate and carbonate dissolution, transport and re-precipitation (e.g. Ague and Nicolescu, 2013; Piccoli et al., 2016). Carbon release via the dissolution of calcium carbonate has been recognized to have important implications for CO2 release from subduction zones, and it is controlled by H2O-rich fluid infiltration (e.g. Ague and Nicolescu, 2013; Frezzotti et al., 2011; Gorman et al., 2006; Kerrick and Connolly, 2001). Future models may account for such variations in reactive major element bulk composition of the rocks along the P–T evolution as a consequence of mineral net transfer reactions occurring simultaneously with fluid–rock exchange (Baumgartner and Valley, 2001), in addition to water liberation and mineral fractionation (see above). This would require additional considerations on the changes in the fluid isotopic composition during transfer of solute species through the fluid. The implementation of other fluid species beside H2O, such as CO2, could be assessed, provided that (1) reliable constraints on the oxygen isotope fractionation between these species and water or minerals are determined and (2)  their consistency with other available data is established. Other species such as CH4 and H2 do not contain any oxygen, and thus they are likely to be less relevant to this model.