Supergiant porphyry Cu deposits are failed large eruptions

Porphyry copper deposits, the principal natural source of Cu and Mo, form at convergent margins. Copper is precipitated from uids associated with cooling magmas that have formed in the mantle and evolved at variably deep crustal levels, before raising close to the surface where they exsolve uids and copper. Despite signicant advances in the understanding of their formation, there are still underexplored aspects of the genesis of porphyry copper deposits. Here, we address the role played by magma injection rates into the shallow crust on the formation of porphyry copper deposits with different copper endowments. Using a mass balance approach, we show that supergiant porphyry Cu deposits (>10 Mt Cu) require magma volumes and magma injection rates typical of large volcanic eruptions. Because such volcanic events would destroy magmatic-hydrothermal systems or prevent their formation, the largest porphyry Cu deposits can be considered as failed large eruptions and this may be one of the causes of their rarity.


Introduction
The last decades have seen a continuous increase in the understanding of how porphyry Cu deposits form. Several studies have unveiled the different processes that control the formation of porphyry Cu deposits, including, among others, geodynamic setting, tectonics, magma genesis, magma evolution, and uid chemistry [1][2][3][4][5][6][7][8][9][10][11][12][13][14] . During the last years there has also been an effort towards a physical understanding of porphyry-related processes through thermodynamic and petrologic modelling [15][16][17][18] . From this point of view, simple mass balance constraints are very useful to set quantitative boundaries within which various processes interplay to form porphyry deposits. Arc magmas are characterized by a narrow range of Cu contents 19,20 and also uid-melt partition coe cients for Cu are limited to a relatively restricted range 21 . Thus, purely from mass balance considerations, the main parameter determining the size of a deposit (considering similar precipitation e ciencies from the hydrothermal uid) is the mass of uid, which, in turn, is closely associated with the mass of degassing magma 16,17 .
The mass of magma that can degas to fuel the formation of hydrothermal ore deposits depends on the thermal structure of the crust, as well as the depth and duration of magma accumulation. Thermal and petrological modelling shows that, for long-term average rates of vertical magma injection typical of arc settings (e.g., 5 mm/year; Ref. 22 ), the accumulation of the magma masses required for the formation of large ore deposits can only occur if magma is injected into the mid-to lower crust for periods of several Myr 17 . However, the accumulation of large magma volumes in mid-to lower crust systems does not guarantee per se the formation of a porphyry system. This is con rmed by the absence of a relationship between the duration of magma accumulation at deep crustal levels, which leads to the largest and thus most fertile transcrustal magmatic systems, and the Cu endowments of porphyry deposits ( Fig. 1; refs. [23][24][25][26] ).
Magmas in the mid-to lower crust are mostly H 2 O-undersaturated, because of the strong pressure dependency of H 2 O solubility in silicate melts 27 . Therefore, the exsolution of mineralising uids requires magma ascent to shallower crustal levels. The depth interval (8-17 km) at which deep fertile magmas saturate according to the calculations of ref. 17 overlaps with the estimated emplacement depths of magma reservoirs feeding the porphyry ngers around which mineralization develops (e.g., ref. 28 and references therein).
We propose that while large magma volumes in the mid-to lower crust are a pre-requisite for the potential formation of large porphyry-Cu deposits, it is the volume of magma transferred to shallower depths that releases the mineralising uids and, therefore, determines the ultimate Cu endowments of the deposits.
Correlation between Cu endowments and the overall duration of the mineralizing events 13,17 suggests that the longer lasting is the overall ore-forming process, the larger is the Cu endowment of the deposit ( Fig. 2; see also refs. 16,29 ). Therefore, ultimately the Cu endowments of porphyry deposits depend on the e ciency of transfer of the largest possible magma volumes from the mid-, lower crustal reservoir to the shallow parental magma chamber feeding the porphyry ngers around which ore deposition occurs. This transfer is accompanied by the release of the mineralizing uids ultimately forming the deposit.
Here, we show that Cu endowments of porphyry deposits increase with the volumes and rates of magma transfer from the mid-and lower-crustal reservoirs to shallower depths. The largest porphyry Cu deposits require magma volumes and injection rates into the upper crust that overlap those leading to large eruptions 15,38,39 . Because volcanic eruptions are detrimental for magmatic-hydrothermal systems, as they would either destroy them or prevent their formation [40][41][42] , the largest porphyry Cu deposits can be considered as failed eruptions.

Model Rationale And Constraints
We used two simple constraints to obtain average magma injection rates from the mid-and lower crust accumulation zone into the upper crustal magma reservoir above which mineralization takes place: (i) the volume of magma required to precipitate a speci c Cu mass, with a conservative precipitation e ciency of 50% 43 ; (ii) the time interval within which such magma volume is transferred from the lower-and midcrustal reservoirs into the upper crust where Cu is precipitated, which is provided by geochronological data. The ratio between these two parameters gives the average magma injection rate corresponding to a speci c Cu endowment.

Magma volume and Cu endowments
The point (i) above is constrained by the relationship between potential Cu endowments of transcrustal magmatic systems and the mass (volume) of magma accumulated in the mid-, lower-crust (refs. 13,17 ; Methods and Table 1). Modelling of ref. 17 indicates that the largest potential Cu endowments are associated with magma accumulations occurring at pressures higher than 0.4 GPa ( > ~ 18 km) and for injection periods longer than 2.5-3.0 Myr. Table 1 Constraints on the parameters used in the Monte Carlo modelling of magma injection rates.

Parameter
Range of values or value Magma injection rate in the mid-to lower crust through a circular surface of 15 km diameter 0.0009 km 3  Parameter "a" in equation above 1 60-700 Parameter "b" in equation above 1 -0.187767*ln(a) + 1.507068 Number of magma pulses transferring a mid-to lower crust magma accumulation corresponding to a speci c Cu endowment to its H2O saturation depth in the shallower crust (N_pulses)

5-20
Duration of magma pulses transferring a mid-to lower crust magma accumulation corresponding to a speci c Cu endowment to its H2O saturation depth in the shallower crust (Pulse_duration)

10000-100000 years
Magma volume of each pulse transferring a mid-to lower crust magma accumulation corresponding to a speci c Cu endowment to its H2O saturation depth in the shallower crust (Pulse_volume)

MLC_magmaVol /N_pulses
Duration of magma transfer from the mid-to lower crust to the shallow crust

Pulse_duration *N_pulses
Magma injection rate from the mid-to lower crust accumulation reservoir into the shallow crust Pulse_volume/Pulse_duration 1 These ranges correspond to enveloping curves enclosing 95% of the data in the correlation between magma volumes accumulated in the mid-to lower crust and their potential Cu endowments exsolvable with uids at H 2 O saturation depth with a 50% precipitation e ciency for Cu (Fig. 3).  27 . These magmas can release uids and potentially lead to the formation of porphyry deposits only if they ascend to depths at which they become H 2 O-saturated (~ 8 and 17 km from ref. 17 ).
Fluid saturation would be achieved at higher depths considering CO 2 -bearing magma. However, early excess uids would be CO 2 -rich 44 and represent a small contribution to the total mass of uid released 27 .
For these reasons, in our calculations we simplistically consider that magmas contain exclusively H 2 O and that the volumes of magma generated in the lower crust (0.4 to 1.2 GPa), which determine the maximum potential Cu endowment of the porphyry deposits 17 (Fig. 3 (Table 1 and Methods). Figure 3 shows that the largest potential Cu endowments (up to a maximum of 150 Mt Cu in the model, for a 50% precipitation e ciency) are associated with volumes of magma accumulated in the lower crust in excess of 2500 km 3 .

Duration of ore deposition events
The second constraint of the model (ii) is provided by available geochronological data on the overall duration of ore deposition. Geochronological data indicate that the overall ore deposition durations of the largest porphyry deposit formation are < ~ 2 Myr (e.g., Chuquicamata 106  Fig. 2). This translates into average Cu deposition rates at these deposits of 74 ± 7 t Cu/yr. Taking into account all major porphyry Cu deposits with available geochronological estimation of ore deposition duration 13 , the average rate of Cu deposition for all these deposits is 65 ± 7 t Cu/yr (Fig. 2). We reasonably assume that the overall duration of the mineralizing events constrains the duration of the magma injection into the upper crust. In order to comply with the above gures, in our model we have taken as an upper temporal limit a duration of 2.0 Myr for the transfer into the upper crust of the magma volumes associated with deposition of up to 150 Mt Cu at 50% e ciency.

Magma injection rates
Models based on geological cross-cutting relationships 29,30 , geochronology [31][32][33][34] , geospeedometry based on element diffusion 29 , and thermodynamic 36 as well as numerical modelling 37 suggest that porphyry Cu deposits are formed by multi-step magmatic-hydrothermal pulses, re ecting a variable number of intrusions and uid release cycles. High-precision dating of both hydrothermal (e.g., molybdenite Re-Os dating) and magmatic activity (U-Pb dating of zircons) suggests that the duration of each pulse may be as short as a few tens of thousands of years [31][32][33][34] . These results are corroborated also by Ti diffusion in hydrothermal quartz, which indicates timescales of single pulses of magmatic-hydrothermal activity as short as a few thousands of years or even less 29 . Recent studies on Ti-in-quartz diffusivity 45,46 suggest slower Ti diffusion rates in quartz (up to two orders of magnitude less than those taken into account in ref. 29 ). These would result in longer durations of single hydrothermal pulses, which, nonetheless, remain on the order of tens to hundreds of thousand years. Thermodynamic and numerical modelling agrees with timescales determined for individual magmatic-hydrothermal pulses as well 36,37 .
We allow a random range of 5 to 20 pulses to inject into the upper crust the magma volume required for the deposition of a speci c amount of Cu (Fig. 3). We vary randomly the duration of the single magma pulse transfer events between 10000 and 100000 years to comply with the available geochronological data discussed above. In the model, any combination of random numbers of magma pulses (

Results And Discussion
The average rate of magma injection into the upper crust controls whether magma will freeze and form plutonic intrusions or whether it will accumulate and form variably large reservoirs that eventually may erupt catastrophically 15,48,49 . Recent studies have started to include magma injection rates in the formation models of porphyry deposits 15,16,42 . Intuitively, high magma injection rates favor explosive volcanic eruptions 15,48,49 and are detrimental for the formation of porphyry deposits [40][41][42] . This is supported by the occurrence of porphyry-type deposits at the end of variably long periods of precursor volcanic activity and coinciding with periods characterized by the lack of or by very low volcanic activity (e.g., Yanacocha 50 ). In contrast, magma injection rates that allow the accumulation of magma at shallow levels without its eruption may eventually result in a magmatic system exsolving uids and generating a mineralized magmatic-hydrothermal system 42 . However, there are no studies that have addressed quantitatively and on a global scale how and if different rates of magma injection, encompassing the broad range below the threshold of those leading to eruption, control the formation and size of porphyry deposits.
Zircon age distribution modelling 15 and zircon thermometry 51 as well as inversion of hydrothermal and magmatic activity ages 16 suggest a broad range of > 2 orders of magnitude of average magma injection rates (~ 0.0001-~0.04 km 3 /yr) potentially associated with the formation of porphyry deposits. Ref. 42 suggests that magma injection rates higher than 1.3x10 − 3 km 3 /yr are necessary to form porphyry deposits. Our results allow us to narrow down the ranges of average upper crust magma injection rates associated with porphyry deposits towards the high value side, and to unveil a relationship between magma injection rates and Cu endowments of porphyry deposits. Figure 4 shows simulations in magma volume versus magma injection rate space. Also shown are the elds corresponding to large porphyry Cu deposits based on zircon age distribution modelling 15 , those corresponding to various individual porphyry Cu deposits based on inversion of hydrothermal and magmatic activity ages 16 , and the eld for Bingham inferred from geochemical and thermal modelling of zircons 51 . Our constrained magma injection rates overlap with the variably broad ranges de ned by these previous studies, but further constrain the magma injection rates associated with speci c porphyry Cu deposits to narrower ranges, especially trimming out the low magma injection rate side (Fig. 4). Figure 4 further shows that increasing Cu endowments require increasing minimum magma injection rates to transfer increasingly larger amounts of magmas and Cu from the deep accumulation zone to shallower levels within the timescales constrained by geochronology. All simulations for the largest possible Cu endowments (> 100 Mt Cu) require magma injection rates > ~ 0.001 km 3 /yr (Fig. 4). Additionally, the broadly normal density distributions of the simulations for potential Cu endowment intervals (< 10, 10- It should be further emphasized that, if the 50% precipitation e ciency of Cu is a realistic one 43 , the rates of magma transfer to the upper crust obtained here are minimum values, because the overall ore deposit durations considered are maximum values, bracketing the beginning and end of the mineralizing process. If, within these temporal intervals, most of the metals are precipitated within shorter timescales, the transfer rates would be higher than those obtained here. Conversely, for deposits in which there is a higher Cu precipitation e ciency (> 50%), the rates of magma transfer would decrease at equal overall duration of the mineralization. Nonetheless, the broadly linear correlation between Cu endowments and duration of ore deposition for porphyry Cu deposits (Fig. 2) suggests that precipitation e ciencies are probably similar for most porphyry Cu deposits.
Our results show that all deposits fall within magma injection rates between 10 − 2.65 and 10 − 3.0 km 3 /yr (i.e., ~ 0.0022 − 0.001 km 3 /yr; Fig. 6). Interestingly, the higher limit of the magma injection rate interval (~ 10 − 2.65 km 3 /yr) appropriate for the formation of most porphyry Cu deposits, broadly coincides with magma injection rates into the upper crust that may lead to large eruptions 15 . The lower limit (~ 10 − 3 km 3 /yr) is the same estimated for the highest possible magma injection rates that may result in the build-up of non-eruptible large magma bodies at shallow crustal levels 15 . However, an important aspect to consider is that there is no xed threshold of magma injection rate for large eruptions. Protracted magmatic activity results in the long-term modi cation of the physical properties of the crust and the magma within the plumbing system. For instance, the viscosity of the crust decreases with increasing temperature, and the magma within the plumbing system becomes progressively richer in uids. Both these phenomena contribute to dampen the pressure developed by magma injection into the shallow portion of the plumbing system 49,51,53 thus decreasing the probability of volcanic eruptions to occur and generating conditions that are suitable for the formation of porphyry Cu deposits. Such a scenario is consistent with the long-lived precursor magmatic activity recorded for porphyry Cu deposits for which geochronological data are available (Fig. 1).
We suggest that the e cient transfer to upper crustal depths of large volumes of uid-rich magma that fractionated in the middle-middle-crust is key for the formation of the largest porphyry Cu deposits.
Changing stress conditions in the crust 1,54 could be a likely cause of the modulation of magma volume transfer and injection rates into the upper crust, and control the size of porphyry Cu deposits (Figs. 4-6).
The data here presented and discussed imply that all supergiant porphyry Cu deposits (> 10 Mt Cu) are formed by magma volumes 31 and average magma injection rates (i.e, > 0.001 km 3 /yr) 15,30 into the upper crust that largely overlap those typically leading to large eruptions, supporting similar conclusions on the formation of the Bingham deposit, for which magma injection rates of ≥ 0.0065 km 3 /yr, based on thermal and geochemical modelling of zircons 51 , have been proposed (Fig. 4). Since eruption is obviously detrimental to the formation of porphyry deposits, our results suggest that supergiant porphyry Cu deposits can be considered as failed large eruptions. Our conclusion that high magma injection rates into the upper crust are associated with all supergiant porphyry Cu deposits nds its ultimate explanation in the prolonged magma accumulation occurring at deep crustal levels during long-lived compression, which is an essential condition for the formation of porphyry Cu deposits 17 . Such accumulation not only builds up enormous amounts of magma, volatile and metal in the deep crust, but is also responsible for the thermal pre-conditioning of the upper crust that prevents eruption of magmas even when magma injection rates at shallow depth are high 49 .

Methods
The Monte Carlo modelling (500000 simulations) was carried out using the conceptual framework developed in refs. 13,17 .
The starting point of the model is the covariation derived from refs. 13,17 between potential Cu endowments of magmatic systems (at 50% precipitation e ciency) and magma volumes for the potentially most productive magmatic systems, i.e., those accumulated at pressures of 0.4-1.2 GPa (Fig. 3). Compared to refs. 13,17 , where the maximum pressure of magma accumulation was 0.9 GPa, we extended the upper pressure limit to 1.2 GPa. Figure 3 shows > 10000 simulations and their density distribution. More than 95% of the simulations fall within the two curves of Fig. 3, which are reproduced mathematically by exponential equations of the type Magma Volume = a*MtCu b (1) Where parameter "a" ranges randomly between 60 and 700, parameter "b" = -0.187767*ln(a) + 1.507068, and "MtCu" are the Cu endowments (at 50% precipitation e ciency) that are allowed to vary randomly between 0 and 150 Mt Cu (Table 1) Additionally, because geological and geochronological evidences indicate that porphyry Cu deposits are formed by repeated magmatic-hydrothermal pulses with durations of few to several tens of kyr, we have allowed in the model that the magma volume transfer from the deep to the shallow crust is accomplished within any random combination of 5 to 20 magmatic pulses, each one varying randomly between 10000 to 100000 years. This means that in our model the transfer of any magma volume from the deep to the shallow crust can occur from a minimum of 50000 years (5 pulses with a 10000 year duration of each pulse) to a maximum of 2 Myr (20 pulses with a 100000 year duration of each pulse). This random temporal variability for the transfer of any magma volume de ned by Eq. (1) above translates into broadly different average magma injection rates that are plotted in Figs. 4-6.
The volume of each magma pulse is calculated as the ratio between the accumulated magma volume in the lower crust, that corresponds to a speci c Cu endowment and must be transferred to the shallow crust to exsolve uids and Cu, and the random number of pulses (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) through which the deep accumulated volume is transferred. Average magma injection rates from the deep to the shallow crust are then calculated as the ratios between the volume of the magma pulses and the random durations of each pulse in the interval 10000-100000 years that, incrementally (through the random number of pulses), results in the transfer of the overall magma volume associated with a certain Cu endowment (considering a 50% precipitation e ciency).
The full RStudio script of the Monte Carlo modelling is provided in the Supplementary Material.