Lei Yu

Keqiang Wu

Li Liu

Na Liu

Xiaoran Ming

Eric H Oelkers

Dawsonite and ankerite formation in

Keywords: authigenic carbonate minerals, carbon and oxygen isotopes, dawsonite stability, carbon capture and storage (CCS), Yinggehai Basin

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Introduction

Carbon-dioxide emissions to the atmosphere by anthropogenic industrial activities are a key factor causing global climate change [START_REF] Ipcc | Underground geological storage, in Carbon dioxide capture and storage[END_REF](IPCC, , 2007;;[START_REF] Manabe | Century-scale effects of increased atmospheric CO 2 on the ocean-atmosphere system[END_REF]. The injection of anthropogenic CO 2 into deep geologic formations has been advocated to slow increasing atmospheric CO 2 concentrations and its effect on global climate [START_REF] Bachu | Aquifer disposal of CO 2 : hydrodynamic and mineral trapping[END_REF][START_REF] Bachu | Screening and ranking of sedimentary basins for sequestration of CO 2 in geological media in response to climate change[END_REF][START_REF] Ipcc | Underground geological storage, in Carbon dioxide capture and storage[END_REF]IPCC, , 2007;;[START_REF] Oelkers | Carbon dioxide sequestration a solution to a global problem[END_REF]. There are four mechanisms of carbon storage in the subsurface: structural and stratigraphic trapping, residual CO 2 trapping, solubility trapping and mineral trapping [START_REF] Ipcc | Underground geological storage, in Carbon dioxide capture and storage[END_REF][START_REF] Johnson | Reactive Transport Modeling of Geologic CO 2 Sequestration in Saline Aquifers: The Influence of Intra-Aquifer Shales and the Relative Effectiveness of Structural, Solubility, and Mineral Trapping During Prograde and Retrograde Sequestration[END_REF]. Of these, mineral trapping, and in particular in-situ carbon mineralization, is the most stable and safe carbon storage solution [START_REF] Lackner | Carbon dioxide disposal in carbonate minerals[END_REF][START_REF] Johnson | Reactive Transport Modeling of Geologic CO 2 Sequestration in Saline Aquifers: The Influence of Intra-Aquifer Shales and the Relative Effectiveness of Structural, Solubility, and Mineral Trapping During Prograde and Retrograde Sequestration[END_REF][START_REF] Matter | Rapid carbon mineralization for permanent disposal of anthropogenic carbon dioxide emissions[END_REF]Oelkers et al., 2008;[START_REF] Power | Carbon mineralization: from natural analogues to engineered systems[END_REF]).

Sandstone aquifers with impermeable cap-rocks are commonly considered for carbon storage as they are have adequate permeability and pore space, and are abundant in the subsurface [START_REF] Benson | CO 2 Sequestration in Deep Sedimentary Formations[END_REF][START_REF] Ipcc | Underground geological storage, in Carbon dioxide capture and storage[END_REF][START_REF] Xu | Mineral sequestration of carbon dioxide in a sandstone-shale system[END_REF]. A number of studies have shown that calcite, siderite, magnesite, dolomite, ankerite and dawsonite could form in various sandstones in response to CO 2 injection. [START_REF] Baker | Continental-scale magmatic carbon dioxide seepage recorded by dawsonite in the Bowen-Gunnedah-Sydney Basin system, eastern Australia[END_REF][START_REF] Fischer | CO 2 -brine-rock interaction -First results of long-term exposure experiments at in situ P-T conditions of the Ketzin CO 2 reservoir[END_REF][START_REF] Gao | Petrology and isotopic geochemistry of dawsonite-bearing sandstones in Hailaer basin, northeastern China[END_REF][START_REF] Gaus | Comparison of long-term geochemical interactions at two natural CO 2 -analogues: Montmiral (Southeast Basin, France) and Messokampos (Florina Basin, Greece) case studies[END_REF][START_REF] Hellevang | Kinetic modelling of CO 2 -water-rock interactions[END_REF][START_REF] Higgs | Changes in reservoir heterogeneity and quality as a response to high partial pressures of CO 2 in a gas reservoir, New Zealand[END_REF][START_REF] Higgs | The Pretty Hill Formation as a natural analogue for CO 2 storage: An investigation of mineralogical and isotopic changes associated with sandstones exposed to low, intermediate and high CO 2 concentrations over geological time[END_REF][START_REF] Hitchon | Aquifer disposal of carbon dioxide: hydrodynamic and mineral trapping-proof of concept[END_REF][START_REF] Klajmon | REPP-CO 2 : Equilibrium Modelling of CO 2 -Rock-Brine Systems[END_REF][START_REF] Liu | Genesis of authigene carbonate minerals in the Upper Cretaceous reservoir, Honggang Anticline, Songliao Basin: A natural analog for mineral trapping of natural CO 2 storage[END_REF][START_REF] Moore | Mineralogical and geochemical consequences of the long-term presence of CO 2 in natural reservoirs: An example from the Springerville-St. Johns Field, Arizona, and New Mexico[END_REF][START_REF] Pauwels | Chemistry of fluids from a natural analogue for a geological CO 2 storage site (Montmiral, France): lessons for CO 2 -water-rock interaction assessment and monitoring[END_REF][START_REF] Pearce | Reactivity of micas and cap-rock in wet supercritical CO 2 with SO 2 and O 2 at CO 2 storage conditions[END_REF][START_REF] Pham | On the potential of CO 2 -water-rock interactions for CO 2 storage using a modified kinetic model[END_REF][START_REF] Uysal | CO 2 degassing and trapping during hydrothermal cycles related to Gondwana rifting in eastern Australia[END_REF][START_REF] Van Pham | On the potential for CO 2 mineral storage in continental flood basalts-PHREEQC batch-and 1D diffusion-reaction simulations[END_REF][START_REF] Worden | Dawsonite cement in the Triassic Lam Formation, Shabwa Basin, Yemen: A natural analogue for a potential mineral product of subsurface CO 2 storage for greenhouse gas reduction[END_REF][START_REF] Xu | Numerical simulation of CO 2 disposal by mineral trapping in deep aquifers[END_REF][START_REF] Xu | Mineral sequestration of carbon dioxide in a sandstone-shale system[END_REF][START_REF] Xu | Numerical modeling of injection and mineral trapping of CO 2 with H 2 S and SO 2 in a sandstone formation[END_REF][START_REF] Xu | Reactive transport modeling to study changes in water chemistry induced by CO 2 injection at the Frio-I Brine Pilot[END_REF][START_REF] Yu | Petrological characterization and reactive transport simulation of a high-water-cut oil reservoir in the Southern Songliao Basin, Eastern China for CO 2 sequestration[END_REF][START_REF] Zhao | Petrographic and stable isotopic evidences of CO 2 -induced alterations in sandstones in the Lishui sag, East China Sea Basin, China[END_REF][START_REF] Zhou | Dawsonite formation in the Beier Sag, Hailaer Basin, NE China tuff: A natural analog for mineral carbon storage[END_REF]. Of these minerals, dawsonite (NaAl(OH) 2 CO 3 ) is particularly attractive as a carbon storage host as it contains sodium and aluminum, which can be abundant in sedimentary silicates. This mineral, however, is more reactive and less stable than many of the aforementioned divalent metal carbonate minerals [START_REF] Bénézeth | Dawsonite synthesis and reevaluation of its thermodynamic properties from solubility measurements: Implications for mineral trapping of CO 2[END_REF][START_REF] Hellevang | Can dawsonite permanently trap CO 2 ?[END_REF][START_REF] Hellevang | Why is Dawsonite Absent in CO 2 Charged Reservoirs? Oil Gas Sci[END_REF].

The LDX-1 structure in the Yinggehai basin of South China Sea provides an insightful analogy for the study of mineral carbonation in sandstone aquifers (Fig. 1).

The gas phase in the LDX-1 structure contains as much as 93 volume percent CO 2 and this CO 2 has a distinct inorganic carbon isotopic signature [START_REF] He | Origin, migration and accumulation of CO 2 in East China and offshore shelf basins[END_REF][START_REF] Huang | Geochemistry and origins of natural gases in the Yinggehai and Qiongdongnan basins, offshore South China Sea[END_REF]. In addition, past studies concluded that the CO 2 gas injection into this structure occurred after that of the hydrocarbon gases [START_REF] Huang | Geochemistry, origin, and accumulation of CO 2 in natural gases of the Yinggehai Basin, offshore South China Sea[END_REF][START_REF] Sun | Main gas sources and gas charge history in Yinggehai-Qiongdongnan region[END_REF], which is similar to scenerios that might be encountered by injecting into former oil field reserviors.

The purpose of this study is to characterize the mineralogy and paragenesis in two wells drilled into the LDX structure to investigate sandstone diagenesis and carbonate mineral formation in response to the influx of CO 2 . Results will be used to gain further insight into the potential for mineral carbonation in sedimentary sandstones as part of subsurface carbon storage efforts.

Geological setting

The Yinggehai basin, which developed during the Cenozoic, is a natural gas-rich rift basin. It is located between the Indian and South China plate, formed by the combination of lithosphere extension and the deep, strike-slip Red River fault. The direction of this basin is Northwest-Southeast, with a total area of approximately 11×10 4 km 2 [START_REF] Huang | Spatial distribution and geochemistry of the nearshore gas seepages and their implications to natural gas migration in the Yinggehai Basin, offshore South China Sea[END_REF][START_REF] Wan | The comparative study of hydrocarbon accumulation conditions in Yinggehai and Qiongdongnan basins, Northern South China Sea[END_REF]Fig. 1a, b). Two northwest-trending, deep-to-basement faults divide the Yinggehai basin into three sub-structural units: The Central Depression, the Ying Dong Slope and the Ying Xi Slope (see Fig 1). The central depression consists of two tectonic units: the Lin Gao uplift and the diapir structure belt (Fig 1b;[START_REF] Zhu | Palaeolimology and hydrocarbon potential in Beibu gulf basin of South China Sea[END_REF]. During the Miocene, the Yinggehai basin was filled with argillaceous sediments along the fault and formed a series of diapir structures in the central depression zone [START_REF] Huang | Spatial distribution and geochemistry of the nearshore gas seepages and their implications to natural gas migration in the Yinggehai Basin, offshore South China Sea[END_REF]. [START_REF] He | Origin, migration and accumulation of CO 2 in East China and offshore shelf basins[END_REF] reported that most of the natural gas reservoirs with high CO 2 contents are concentrated in these diapir structures.

The basement of the Yinggehai basin is dominated by palaeozoic magmatic rocks [START_REF] Hao | Abnormal organic-matter maturation in the Yinggehai Basin, South China Sea: Implications for hydrocarbon expulsion and fluid migration from overpressured systems[END_REF]. According to the drilling strata, the Cenozoic sediments include the Eocene Lingtou Formation, the Oligocene Yacheng and Lingshui Formations, the Miocene Sanya, Meishan, and Huangliu Formations, the Pliocene Yinggehai Formation and the Quaternary Ledong Formation. The thickness of the Cenozoic sediments in this basin is up to 16-17 km [START_REF] Gong | The major oil and gas fields of China offshore[END_REF][START_REF] Huang | Geochemistry, origin, and accumulation of CO 2 in natural gases of the Yinggehai Basin, offshore South China Sea[END_REF][START_REF] Huang | Spatial distribution and geochemistry of the nearshore gas seepages and their implications to natural gas migration in the Yinggehai Basin, offshore South China Sea[END_REF][START_REF] Meng | Abnormal Transformation of the Clay Minerals in Yinggehai Basin and Its Significances[END_REF][START_REF] Zhu | North continental margin basin in the south China sea[END_REF] (Fig. 2). The Lingtou Formation mainly consists of sandstone and mudstone. The Yacheng and Lingshui Formations are composed of mudstone and silty mudstone interbeded with mudstone. Both the Sanya and Meishan Formations are dominated by calcareous mudstone. The Huangliu, Yinggehai and Ledong Formations are mainly siltstone, sandstone and interbedded mudstone. The Miocene mudstone is the main source rock of natural gas in this basin, while the interbedded mudstone and sandstone of the overlying Huangliu and Yinggehai Formations provide a good reservoir and cap rock system for the basin [START_REF] Wang | Dongfang 1-1 gas field in the mud diapir belt of the Yinggehai Basin, South China Sea[END_REF]. According to the logging data, the Yinggehai and Huangliu formations contain the main CO 2 gas reservoirs in the basin.

The LDX-1 Structure is a dome structure located in the southeast of the Ledong area in the central depression of the Yinggehai Basin (Fig. 1b). The gas reservoir in the LDX-1 structure is an anticline formed by a mud diapir. The rocks containing the CO 2 -rich gas reservoirs consist mainly of the marine sandstones of the Pliocene Yangehai Formation (Fig. 2) and are located at a depth of 392 m to 2300 m [START_REF] Huang | Geochemistry, origin, and accumulation of CO 2 in natural gases of the Yinggehai Basin, offshore South China Sea[END_REF][START_REF] Li | Geochemistry of Quaternary basaltic volcanic rocks of Weizhou island in Beihai City of Guangxi and a discussion on characteristics of their source[END_REF]. According to gas measurements, the LDX-1 reservoirs are dominated by hydrocarbon gas and CO 2 ; the CO 2 content ranges from 41.9% to 78.7% [START_REF] Huang | Geochemistry, origin, and accumulation of CO 2 in natural gases of the Yinggehai Basin, offshore South China Sea[END_REF].

There were two major episodes of gas emplacement into the Yinggehai basin, the first from an organic source and dominated by CH 4 and the second from an inorganic sourced and dominated by CO 2 . According to the [START_REF] Xie | Natural gas origin and accumulation model in major and excellent gas fields with high temperature and overpressure in Yinggehai basin:a case of DF13-2 gas field[END_REF], the arrival of the CH 4 dominated gas was at approximately 3.7 Ma. The CO 2 dominated gas arrived later. Estimates of its arrival time range from 0.3 to 1.9 Ma [START_REF] Fu | Diagenesis and reservoir quality of overpressured deep-water sandstone following inorganic carbon dioxide accumulation: Upper Miocene Huangliu Formation, Yinggehai Basin, South China Sea[END_REF][START_REF] Huang | Geochemical behaviors and reservoiring history of natural gas in Ledong gas field in Yinggehai basin[END_REF][START_REF] Wang | Heat flow action and its control on natural gas migration and accumulation in mud-fluid diapir areas in Yinggehai basin[END_REF][START_REF] Xie | Natural gas origin and accumulation model in major and excellent gas fields with high temperature and overpressure in Yinggehai basin:a case of DF13-2 gas field[END_REF]. As such this basin provides a close natural analogue to the filling of a former hydrocarbon reservoir with CO 2 as part of subsurface carbon storage efforts.

Sampling and research methods

A total 159 sandstone samples were collected from the gas reservoir layers of the LDX-1 structure of the Yinggehai Formation. The samples were collected from the depths of 1841.62 to 1869.02 m (from well LDX-1-1) and of 1958.07 to 1905.13 m (from well LDX-1-4). The compositions of the sandstone samples were determined by counting 300 points in each thin section following the approach of [START_REF] Dickson | A modified staining technique for carbonates in thin section[END_REF] using an OLYMPUS BX51 polarizing microscope. To identify the carbonate minerals, all the thin sections were stained with Alizarin Red S and K-ferricyanide solutions following the procedure of [START_REF] Dickson | A modified staining technique for carbonates in thin section[END_REF]. Authigenic mineral observations and energy spectrum analyses were performed at the Research Center of Paleontology and Stratigraphy of Jilin University, using a JEOL JSM-6700F scanning electron microscope (SEM), equipped with energy dispersive X-ray spectrometer (EDS) analyzer manufactured by Oxford, UK; mineral surfaces were gold coated prior to this analysis.

Based on a systematic petrographic study of the sandstone in the gas reservoirs, 10 dawsonite-bearing samples were selected for carbon and oxygen isotope analyses using MAT253 isotopic ratio mass spectrometers at the Analytical Laboratory of The China National Nuclear Corporation (CNNC) Beijing Research Institute of Uranium Geology.

The analytical method for obtaining stable isotopic compositions is based on past studies such as [START_REF] Liu | Genesis of authigene carbonate minerals in the Upper Cretaceous reservoir, Honggang Anticline, Songliao Basin: A natural analog for mineral trapping of natural CO 2 storage[END_REF]. Samples were grounded into a powder using an agate mortar and pestle. The ground powers were sieved and the > 74 μm fraction was collected, then dried for 2 h at 105 ºC under atmospheric pressure, to remove potential surface volatile phases, before analyzing. For carbon and oxygen isotope analyses, 99% phosphoric acid (produced by Merck KGaA, Darmstadt, Germany) was reacted with crushed whole rock powders at 25 ºC or 50 ºC [START_REF] Mccrea | On the isotopic chemistry of carbonates and a paleotemperature scale[END_REF]. For dawsonite analysis, CO 2 released from samples during the first 4 h reactions at 25 ºC was discarded, as this gas likely originated from the decomposition of calcite. After that, the reaction was continued for another 24h and the CO 2 was collected for the analysis of dawsonite isotopic compositions [START_REF] Becker | Carbon and Oxygen Isotope Ratios in Iron-formation and Associated Rock from the Hammersley Range of Western Australia and Their Implications[END_REF][START_REF] Rosenbaum | An isotopic study of siderites, dolomites and ankerites at high temperatures[END_REF][START_REF] Sharma | Measurement of O 18 /O 16 ratios of total oxygen of carbonates[END_REF]. For ankerite analysis, the CO 2 released during the first 24 h reaction at 25 ºC was discarded, as it likely originated from calcite and dawsonite decomposition, then the reaction temperature was raised to 50 ºC followed by the collection of the CO 2 produced by the decomposition reaction during subsequent 48 h [START_REF] Uysal | CO 2 degassing and trapping during hydrothermal cycles related to Gondwana rifting in eastern Australia[END_REF].

The precision of the analyses is ±0.1‰ for δ 13 C and ±0.2‰ for δ 18 O. All reported isotope values in this study are relative to the Belemnitella americana from the Peedee Formation (PDB) standard for carbon and Standard Mean Ocean Water (SMOW) for oxygen.

All the speciation and solubility calculations were conducted using PHREEQC v.

3 [START_REF] Parkhurst | Description of input and examples for PHREEQC version 3--A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF] with its Lawrence Livermore National Laboratory (LLNL) thermodynamic database after adding to it the stability constant for ankerite.

The stability of ankerite in this study is based on the formula (Ca(Fe 0.5 Mg 0.5 )(CO 3 ) 2 ) (see below).

Its thermodynamic properties were estimated from the stoichiometrically weighted sum of the calcite, siderite and magnesite stability constants [START_REF] Xu | Numerical simulation of CO 2 disposal by mineral trapping in deep aquifers[END_REF]. The thermodynamic properties required for this estimate were calculated using the SUPCRTBL database [START_REF] Zimmer | SUPCRTBL: A revised and extended thermodynamic dataset and software package of SUPCRT92[END_REF]. The resulting ankerite equilibrium constant and enthalpy of reaction are 0.642 and -69.326 kJ/mol at 70 °C and -0.087 and -77.803 kJ/mol 100 °C. Thermodynamic properties in the LLNL database are consistent with those of dawsonite reported by [START_REF] Bénézeth | Dawsonite synthesis and reevaluation of its thermodynamic properties from solubility measurements: Implications for mineral trapping of CO 2[END_REF].

The Truesdell-Jones equation was used to calculate aqueous activity coefficients and the fugacity for CO 2 gas was calculated using the ThermoSolver 1.0 software [START_REF] Barnes | ThermoSolver: an integrated educational thermodynamics software program[END_REF].

Results

Petrographic characteristics

The dawsonite in the LDX-1 structure is hosted in fine-grained subarkose and lithic quartz sandstones.

Authigenic mineralogy and paragenesis

The authigenic minerals observed in the rock samples are dawsonite, quartz, siderite, dolomite, ankerite, mixed-layer clays consisting of illite and smectite, and kaolinite.

Dawsonite

Dawsonite is principally found at depths of 1841.27 to 1867.23 m in well LDX-1-1 and 1863 to 2018 m in well LDX-1-4 with its content ranging from 0.2 to 6

Vol% as determined by point counting (see above). Dawsonite aggregates nucleate as microcrystalline grains then grow radially outward into the sandstone pores.

Dawsonite growth usually terminates at the edge of the detrital minerals (eg. quartz and feldspar). Dawsonite could also be observed growing on dolomite (Fig. 4a,b), potassium feldspar and plagioclase (Fig 4c,d). The EDS results show that the dawsonite mainly consists of carbon, oxygen, sodium and aluminum having weight percents of 7.25 to 22.23, 50.43 to 66.93, 3.58 to 14.69, and 7.27 to 30.26 respectively (see Table 1).

Authigenic quartz

Two kinds of authigenic quartz were observed: quartz overgrowths and microcrystalline quartz. The content of authigenic quartz is low, generally less than 0.5%. Quartz overgrowths are the among the earliest authigenic silicate minerals observed and are evident due to clay rims between detrital quartz and these overgrowths. The thickness of the quartz overgrowths ranges from 0.01 to 0.06 mm. Furthermore, we observed some radial dawsonite surrounded by quartz overgrowths (see Fig. 4e), suggesting some of the quartz precipitated concurrently with or after the dawsonite. In addition, SEM images show that the microcrystalline quartz was distributed in sandstone pores sporadically and is associated with kaolinite. This kaolinite is also associated with dawsonite in some places (Fig. 4f). Such observations further suggest that the microcrystalline quartz formed during same time period as the dawsonite.

Siderite

Most siderite was distributed in pores as subhedral intergranular crystals or aggregates along the boundaries of the detrital grains (Fig. 4g). The content of the siderite ranges from 0.5% to 13.5% in the bulk rock and the crystal sizes range from 0.01 mm to 0.03 mm. The color of the siderite is yellow brown to dark brown under single polarized light. The siderite crystals were distributed either in the inner side of quartz overgrowth or near the clay rims surrounding the original quartz grain boundaries, indicating that siderite formation was earlier than or concurrent with the quartz overgrowths.

Dolomite

The dolomite content (0.5 to 43 Vol%) varied substantially in the studied cores. As dolomite is not stained by Alizarin Red S and K-ferricyanide solutions, it is readily distinguished from other carbonates. Under the polarizing microscope, some dolomite was observed being replaced by dawsonite aggregates (Fig. 4h). SEM analysis revealed that the ends of some needle-like dawsonite crystals terminate at the faces of dolomite crystals exhibiting a "bridged phenomenon" (Figure 4a,b). Both sets of observations imply that the dolomite formed earlier than the dawsonite, yet the presence of some detrital dolomite cannot be discounted.

Ankerite

Ankerite (0.5 to 36 Vol%) mainly surrounds dolomite crystals as rim-like growths (Fig. 4i) or as euhedral to subhedral intergranular crystals (Fig. 4j,k). Under the Back Scattered Electron (BSE) mode, the ankerite is seen to form of nearly equal ring-like shapes along the dolomite surfaces as it replaced this mineral. Some ankerite is seen to have grown on dawsonite (Fig 4j,k). Such observations suggest that ankerite formation occurred later than both the dolomite and dawsonite. EDS analyses results for ankerite showed that its Ca, Mg, and Fe content ranged from 0.39 to 9.22, 2.4 to 6.52, and 1.47 to 7.82 atomic percent respectively (see Table 2), consistent with the formula Ca(Fe 0.5 Mg 0.5 )(CO 3 ) 2 .

Clay minerals

The XRD results indicate that the clay minerals in the CO 2 -containing reservoirs of the Yinggehai Basin mainly consist of mixed-layer illite and smectite, and kaolinite [START_REF] Huang | Geochemistry, origin, and accumulation of CO 2 in natural gases of the Yinggehai Basin, offshore South China Sea[END_REF][START_REF] Jiang | Geochemistry of pore water and associated diagenetic reactions in the diapiric area of Yinggehai basin, northwestern South China Sea[END_REF][START_REF] Tong | Pooling conditions of gas reservoirs in the Dongfang 1-1 Gas Field,Yinggehai Basin[END_REF]. The mixed layer illite and smectite commonly surround the detrital minerals as "clay rims", which could be identified by polarizing microscope. Mixed layer illite and smectite are not found in the sandstone pores, suggesting that it is the earliest formed secondary mineral in the diagenetic sequence. Honeycomb-like authigenic clay is seen to grow on K-feldspar (see Fig. 4l,m). According the EDS results shown in Table 3, most of the smectite and mixed layer illite are Ca depleted. This suggests that the Ca in the smectite and mixed layer of illite and smectite may have been removed over time through water-mineral reactions.

Authigenic kaolinite grows as vermicular assemblages in the sandstone pores.

Hexagonal kaolinite crystals are found attached to dawsonite crystals (Fig. 4n), at times adjacent to small "etch pits" on the dawsonite surface (Fig. 4o), suggesting that some kaolinite post-dates the dawsonite. However, vermicular kaolinite is commonly interspersed between acicular dawsonite crystals, suggesting that this kaolinite formed slightly earlier than or concurrently with dawsonite (Fig. 4p). In some places ankerite grew on the top of kaolinite aggregates (Fig. 4q). This suggests that the formation of kaolinite might have spanned from the period from when dawsonite grew through that of ankerite formation.

Chlorite was not observed in the samples considered in this study. Its presence in the rocks of the LDX system was reported by [START_REF] Fu | Diagenesis and reservoir quality of overpressured deep-water sandstone following inorganic carbon dioxide accumulation: Upper Miocene Huangliu Formation, Yinggehai Basin, South China Sea[END_REF]. Moreover, the presence of chlorite within and below the CO 2 gas reservoirs is inferred as a potential source of the iron leading to ankerite precipitation (see below).

Vertical characteristics of authigenic carbonates in the reservoirs

The dawsonite and ankerite contents across the study area, measured on 159 thin sections, are shown in Fig. 5. Over the depths of this study (from 1841.6 m to 1868.89 m), dawsonite (0.7 to 8.3%) and ankerite (1.7 to 26.0%) in the LDX-1-1 well mainly developed near the gas water contact (GWC) (Fig. 5a). Owing to lack of samples, few observations of these minerals were made above the GWC in this well. In well LDX-1-4, the limited measurements of mineral content in the gas reservoirs show the dawsonite and ankerite content ranging from 0.7 to 2.3% and 2.0 to 14.0% above the GWC, whereas the corresponding content results for dawsonite and ankerite under the GWC in this well are 1.0 to 6.3% and 1.0 to 24.0 % (Fig. 5b).

C and O isotope compositions

The stable isotopic compositions of dawsonite and ankerite are shown in Table 4 and Fig. 6. The carbon isotope (δ 13 C) values for dawsonite vary from -4.3‰ to -0.6‰

with an average value of -2.0‰. The oxygen isotope (δ 18 O) values for dawsonite range from +22.4‰ to +24.0‰. The δ 13 C values for ankerite range from -2.6‰ to -0.8‰ with an average value of -1.4‰ and the δ 18 O values range from +22.8‰ to +23.7‰.

The measured δ 13 C values of dawsonite and ankerite have been used to estimate the corresponding equilibrium δ 13 C of the CO 2 (δ 13 C CO2 ) from which they precipitated.

The δ 13 C CO2 for the CO 2 that formed the dawsonite, was estimated assuming dawsonite exhibits an identical CO 2 carbon isotope fractionation as that of calcite, using the equation proposed by [START_REF] Ohmoto | Isotopes of sulfur and carbon[END_REF] [START_REF] Baker | Continental-scale magmatic carbon dioxide seepage recorded by dawsonite in the Bowen-Gunnedah-Sydney Basin system, eastern Australia[END_REF]. The calculated δ 13 C for the CO 2 in equilibrium with ankerite was estimated assuming its fractionation was identical to the dolomite-CO 2 fractionation equation proposed by [START_REF] Horita | Oxygen and carbon isotope fractionation in the system dolomite-water-CO 2 to elevated temperatures[END_REF]. The δ 13 C of the CO 2 (δ 13 C CO2 ) calculated to be in isotopic equilibrium with dawsonite range from -9.66‰ to -6.69‰ with an average value of -7.90‰. The equilibrated δ 13 C CO2 for ankerite varies from -7.27‰ to -5.94‰ with a mean value of -6.4‰.

The formation temperature for dawsonite and ankerite were calculated from their oxygen isotope compositions. This was done by assuming oxygen fractionation between dawsonite and H 2 O is identical to the calcite-H 2 O fractionation equation proposed by [START_REF] O'neil | Oxygen isotope fractionation in divalent metal carbonates[END_REF] and that oxygen fractionation between ankerite and H 2 O is identical to the dolomite-H 2 O fractionation equation of [START_REF] Horita | Oxygen and carbon isotope fractionation in the system dolomite-water-CO 2 to elevated temperatures[END_REF]. The 18 O value for water used in these calculations was δ 18 O H2O = +2.7‰ [START_REF] Deng | A research on the formation water chemical characteristics in Yinggrhai basin of South China sea and its significance in petroleum geology[END_REF], which was measured in the formation waters of the DFY-1 structure in the Yinggehai-Huangliu Formation of the Yinggehai basin (Fig. 1). The calculated temperature for dawsonite formation varies from 67.8 ºC to 79.6 ºC and the calculated temperature of ankerite formation ranges from 89.6 ºC to 96.4 ºC (see Table 4). These temperatures are consistent with 1) the earlier formation of dawsonite and 2) the increasing temperature of the study site over time as the basin subsided. Note that the current temperature of the study site is 92 to 103 o C.

Discussion

The origin of the carbon in the dawsonite and ankerite

The δ 13 C CO2 of the gas forming dawsonite and ankerite reported above and in Table 4 are similar to the current δ 13 C values in the CO 2 -rich reservoirs of the LDX system; the current δ 13 C CO2 , are reported to range from -8.2‰ to -4.2‰ [START_REF] He | Origin, migration and accumulation of CO 2 in East China and offshore shelf basins[END_REF]. Moreover, the hydrogen and oxygen isotopic compositions of the formation water in the Yinggehai Basin reservoirs are significantly different from that of meteoric water, which implies the lack of exchange between meteoric water and that present in the LDX system [START_REF] Deng | A research on the formation water chemical characteristics in Yinggrhai basin of South China sea and its significance in petroleum geology[END_REF]. Such evidence suggests that the CO 2 in the gas reservoir is likely a major carbon source for the both dawsonite and ankerite formation.

Carbon dioxide in the LDX gas reservoirs at the present time is likely dominated by magmatic derived CO 2 . Firstly, the δ 13 C CO2 value in the gas reservoir ranges from -8.2‰ to -4.2‰, which is significantly different from that obtained from kerogen degradation; δ 13 C CO2 from kerogen degradation ranges from -25‰ to -20‰ [START_REF] Irwin | Isotopic evidence for source of diagenetic carbonates formed during burial of organic-rich sediments[END_REF][START_REF] Tissot | From kerogen to petroleum, Petroleum formation and occurrence[END_REF] and CO 2 from bacterial fermentation typically has a δ 13 C CO2 ~ +15‰ [START_REF] Irwin | Isotopic evidence for source of diagenetic carbonates formed during burial of organic-rich sediments[END_REF]. Secondly, the carbon isotope values of CO 2 in the LDX system exhibits a δ 13 C CO2 range similar to the -8‰ to -3‰ range observed for CO 2 sourced from mantle-derived magma [START_REF] Baker | Continental-scale magmatic carbon dioxide seepage recorded by dawsonite in the Bowen-Gunnedah-Sydney Basin system, eastern Australia[END_REF][START_REF] Clayton | Origin and migration of hydrocarbon gases and carbon dioxide, Bekes Basin, southeastern Hungary[END_REF][START_REF] Dai | CO 2 reservoir and its type in eastern China and continental shelf[END_REF][START_REF] Ohmoto | Isotopes of sulfur and carbon[END_REF][START_REF] Rollinson | Using geochemical data: evaluation, presentation, interpretation[END_REF]). Taking account of the observations summarized above, we suggest that origin of the CO 2 in the LDX-1 structures is mantle-derived magma degasing.

The timing of dawsonite and ankerite formation can be constrained by the timing of the basaltic magmatism related CO 2 charging of the LDX system. The Yinggehai Formation, where the dawsonite and ankerite developed, was formed between 5.5 Ma and 1.9Ma (Fig. 2). The Honghe fault zone, which controls the formation and distribution direction of Yinggehai basin is still active [START_REF] Zhu | The Red River Fault zone in the Yinggehai Basin, South China Sea[END_REF] (Fig. 1). The ages of alkaline basalt in this region range from 4 to 6 Ma and from 1 Ma up to present [START_REF] Zhu | Nd-Sr-Pb isotopic and chemical evidence for the volcanism with MORB-OIB source characteristics in the Leiqiong area[END_REF]. Several ages of the CO 2 charging into the Yinggehai Formation have been reported in the literature; these ages range from 0.3 to 1.9 Ma [START_REF] Fu | Diagenesis and reservoir quality of overpressured deep-water sandstone following inorganic carbon dioxide accumulation: Upper Miocene Huangliu Formation, Yinggehai Basin, South China Sea[END_REF][START_REF] Huang | Geochemical behaviors and reservoiring history of natural gas in Ledong gas field in Yinggehai basin[END_REF][START_REF] Wang | Heat flow action and its control on natural gas migration and accumulation in mud-fluid diapir areas in Yinggehai basin[END_REF][START_REF] Xie | Natural gas origin and accumulation model in major and excellent gas fields with high temperature and overpressure in Yinggehai basin:a case of DF13-2 gas field[END_REF]. Based on these ages, it seems reasonable to conclude the dawsonite and ankerite in the Yinggehai Formation is formed during the last 1.9 Ma.

Cation source of dawsonite and ankerite

The dissolution of feldspars in the LDX-1 structure provides the cations for dawsonite. The formation of dawsonite from feldspar has been observed in numerous systems. For example, feldspars were observed to be altered to dawsonite in the Lam Formation of Shabwa basin Yemen [START_REF] Worden | Dawsonite cement in the Triassic Lam Formation, Shabwa Basin, Yemen: A natural analogue for a potential mineral product of subsurface CO 2 storage for greenhouse gas reduction[END_REF], in the reservoirs of the Castillo structure of the Golfo San Jorge basin [START_REF] Comerio | The Role of Telogenetic Injection of Magmatically Derived CO 2 in the Formation of Dawsonite from the Castillo Formation, Chubut Group, Patagonia, Argentina[END_REF], and in the Songliao basin, the Hailaer basin, and the East Sea basin in China [START_REF] Gao | Petrology and isotopic geochemistry of dawsonite-bearing sandstones in Hailaer basin, northeastern China[END_REF][START_REF] Liu | Genesis of authigene carbonate minerals in the Upper Cretaceous reservoir, Honggang Anticline, Songliao Basin: A natural analog for mineral trapping of natural CO 2 storage[END_REF][START_REF] Zhao | Petrographic and stable isotopic evidences of CO 2 -induced alterations in sandstones in the Lishui sag, East China Sea Basin, China[END_REF][START_REF] Zhou | Dawsonite formation in the Beier Sag, Hailaer Basin, NE China tuff: A natural analog for mineral carbon storage[END_REF]. The reaction forming dawsonite from albite can be written [START_REF] Bénézeth | Dawsonite synthesis and reevaluation of its thermodynamic properties from solubility measurements: Implications for mineral trapping of CO 2[END_REF][START_REF] Hellevang | Can dawsonite permanently trap CO 2 ?[END_REF][START_REF] Ryzhenko | Genesis of dawsonite mineralization: Thermodynamic analysis and alternatives[END_REF][START_REF] Xi | Evidences of localized CO 2 -induced diagenesis in the Cretaceous Quantou Formation, southern Songliao Basin, China[END_REF]:

NaAlSi 3 O 8 + CO 2 + H 2 O ⇌ NaAlCO 3 (OH) 2 + 3SiO 2(aq) (Eq. 1)
Albite Dawsonite where some of the liberated SiO 2(aq) could provoke quartz precipitation. The equivalent reaction forming dawsonite from K-feldspar:

KAlSi 3 O 8 + Na + + CO 2 + H 2 O ⇌ NaAlCO 3 (OH) 2 + 3SiO 2(aq) + K + (Eq. 2)
K-feldspar Dawsonite has been discounted by [START_REF] Hellevang | Why is Dawsonite Absent in CO 2 Charged Reservoirs? Oil Gas Sci[END_REF] due to the likely build-up of K + in the aqueous phase arresting this reaction.

Evidence for the coupling of feldspar dissolution to dawsonite precipitation reactions could be found throughout the LDX system (Fig. 4c,d,r). If dawsonite originates from K-feldspar by Eq. 2, however, its growth also requires Na + from the fluid phase. This would require the system to be relatively open to fluid transport, which may be limited in this system. Note that according to mass conservation, for every 1 cm 3 of dawsonite formed from albite by reaction 1, there should be 1.16 cm 3 of SiO 2 precipitation [START_REF] Zhou | Dawsonite formation in the Beier Sag, Hailaer Basin, NE China tuff: A natural analog for mineral carbon storage[END_REF]. However, far less microcrystalline quartz is observed in our system than dawsonite (Fig. 4b), which suggests that some of the SiO 2 may have been consumed by other water-mineral reactions.

Geochemical calculations results suggest that chlorite dissolution provoked by the charging of the system by CO 2 could provide Fe 2+ for ankerite formation [START_REF] Xu | Mineral sequestration of carbon dioxide in a sandstone-shale system[END_REF][START_REF] Xu | Numerical modeling of injection and mineral trapping of CO 2 with H 2 S and SO 2 in a sandstone formation[END_REF][START_REF] Xu | On fluid-rock chemical interaction in CO 2 -based geothermal systems[END_REF][START_REF] Yu | Dawsonite fixation of mantle CO 2 in the cretaceous Songliao Basin, Northeast China: a natural analogue for CO 2 mineral trapping in oilfields[END_REF]. The dissolution of smectite also could provide some of the iron for ankerite growth [START_REF] Hellevang | Kinetic modelling of CO 2 -water-rock interactions[END_REF][START_REF] Pham | On the potential of CO 2 -water-rock interactions for CO 2 storage using a modified kinetic model[END_REF][START_REF] Worden | Clay minerals in sandstones: A review of the detrital and diagenetic sources and evolution during burial[END_REF][START_REF] Worden | Dawsonite cement in the Triassic Lam Formation, Shabwa Basin, Yemen: A natural analogue for a potential mineral product of subsurface CO 2 storage for greenhouse gas reduction[END_REF], as well as some aluminum promoting dawsonite formation. The dissolution of chlorite and smectite coupled to ankerite formation has been observed in the sandstone reservoirs of the Otway basin of Austrailia,, in the Songliao Basin of China, the Jurassic and late Cretaceous reservoirs of North Sea oil fields, and in the Eocene strata from Wilcox of Texas, [START_REF] Awwiller | Illite/smectite formation and potassium mass transfer during burial diagenesis of mudrocks: A study from the Texas Gulf Coast Paleocene-Eocene[END_REF][START_REF] Hendry | Ankerite cementation in deeply buried Jurassic sandstone reservoirs of the central North Sea[END_REF][START_REF] Liu | Genesis of authigene carbonate minerals in the Upper Cretaceous reservoir, Honggang Anticline, Songliao Basin: A natural analog for mineral trapping of natural CO 2 storage[END_REF][START_REF] Thyberg | Quartz cementation in Late Cretaceous mudstones, northern North Sea: Changes in rock properties due to dissolution of smectite and precipitation of micro-quartz crystals[END_REF][START_REF] Watson | The Ladbroke Grove-Katnook carbon dioxide natural laboratory: A recent CO 2 accumulation in a lithic sandstone reservoir[END_REF].

The chlorite content of the Yinggehai Basin CO 2 bearing reservoirs has been reported to be significantly lower than that in CO 2 -free reservoirs [START_REF] Tong | Pooling conditions of gas reservoirs in the Dongfang 1-1 Gas Field,Yinggehai Basin[END_REF]. Similar observations have been reported in wells of the Yinggehei Basin [START_REF] Huang | Geochemistry, origin, and accumulation of CO 2 in natural gases of the Yinggehai Basin, offshore South China Sea[END_REF][START_REF] Jiang | Geochemistry of pore water and associated diagenetic reactions in the diapiric area of Yinggehai basin, northwestern South China Sea[END_REF]. Such observations suggest that the dissolution of iron bearing clay minerals such as chlorite is the main Fe 2+ source for ankerite. The Mg 2+ and some Ca 2+ in the ankerite crystals could come from dolomite dissolution, as fresh ankerite is observed growing within a dolomite matrix (Fig. 4i). Additional Ca 2+ can be sourced from the transformation of clays in the system; a number of studies have reported Ca release as smectite transforms to mixed layered clays and illite (e.g. Boles and Frank, 1979;[START_REF] Mchargue | Dolomie from clay in argillaceous or shale-assocaited maring carbonates[END_REF]. A close relation between ankerite and smectite is evident in the studied core samples (see Fig. 7). Taking account of these observations, a representative reaction coupling chlorite dissolution to ankerite precipitation could be expressed as (modified from [START_REF] Kirste | Geochemical modelling of CO 2 -water-rock interaction in the Pretty Hill fomation, Otway Basin[END_REF]

Kaolinite

Note this reaction, balanced on Fe and Mg, requires substantial magmatic CO 2 for ankerite formation, consistent with its carbon isotope composition. This reaction also consumes SiO 2 , which may account, at least in part, for the low mass of authigeic quartz in the present day system.

CO 2 partial pressure and carbonate mineral stability

It is likely that both ankerite and dawsonite are formed after the arrival of mantle sourced CO 2 , providing a clear analogy for modern carbon capture and storage systems. The observations described in this study indicate that dawsonite is first formed at ~70 ºC, followed by ankerite at ~ 90 °C, in response to the arrival of CO 2 into the overlying reservoirs. The increase in temperature in the system likely stems from its increased burial depth over time. Ankerite is also reported to be the last forming mineral in the sandstone reservoirs in Songliao [START_REF] Liu | Genesis of authigene carbonate minerals in the Upper Cretaceous reservoir, Honggang Anticline, Songliao Basin: A natural analog for mineral trapping of natural CO 2 storage[END_REF][START_REF] Yu | Petrological characterization and reactive transport simulation of a high-water-cut oil reservoir in the Southern Songliao Basin, Eastern China for CO 2 sequestration[END_REF] and Hailaer basins [START_REF] Gao | Petrology and isotopic geochemistry of dawsonite-bearing sandstones in Hailaer basin, northeastern China[END_REF]. Some kaolinite is seen to precipitate coupled to the dissolution of dawsonite; some dissolution etch pits on dawsonite crystal were evident near the kaolinite (Fig. 4m). Kaolinite precipitation continues as ankerite precipitates, consistent with Eq. ( 3). The formation of ankerite also appears to be linked to dolomite dissolution.

All of these observations are consistent with the phase diagrams shown in Figs. 8 to 10. In Fig. 8, the relative stability of dawsonite in the Na 2 O-Al 2 O 3 -SiO 2 -CO 2 -H 2 O system balanced on aluminum and saturated with quartz is shown. Dawsonite and quartz become stable relative to albite at log (f CO2 ) > 0.15 and 0.9 at 70 and 100 °C, respectively. This diagram thus suggests that albite would be favored to transform to dawsonite as CO 2 fills the reservoirs in accord with reaction 1. As suggested by Fig. 9, some of the precipitated dawsonite and quartz will tend to react to form kaolinite due to an increase in temperature, which tends to shrink the dawsonite stability field. The value of log (f CO2 ) in equilibrium with the assemblage albite = dawsonite + quartz at 70 and 100 °C plot as vertical lines on the Fig. 10, which also shows the relative stability field of ankerite + kaolinite relative to that of chlorite, dolomite and quartz consistent with reaction 3. It can be seen that for all a Ca2+ /a 2 H+ > ~9.7 at 70 °C and > ~7.3 at 100 °C, dawsonite would first tend to form before ankerite as the CO 2 partial pressure in the reservoirs increase. The partial pressure of CO 2 in the LDX system would increase as the CO 2 fraction of the gas progressively increased in the reservoirs located just above the studied rock samples (Fig. 5). The persistence of dawsonite and ankerite in this system is assured by the continued presence of the CO 2 -rich gas reservoir, which maintains f CO2 at relatively high levels. Note the current temperature of the gas reservoirs are 92.3 and 102.9 ºC for well LDX-1-4 and well LDX-1-1 respectively, and the corresponding gas pressure in reservoirs, which is dominated by CO 2, are between 18.9 and 43.2 Mpa (China National Offshore Oil Company, personal communication). [START_REF] Hellevang | Can dawsonite permanently trap CO 2 ?[END_REF], noted, that dawsonite, would likely rapidly dissolve once it became thermodynamically unstable as the f CO2 decreased in the such systems.

The paragenetic sequence

Based on the observations described above, the proposed paragenetic sequence for the studied dawsonite-bearing sandstones is shown in Fig. 11. Secondary clays (mainly illite and smectite mixed layer clays) began forming before compaction during the early stage of diagenesis and coated detrital minerals as clay rims. Sideite is found to form next; siderite precipitation is readily observed in reducing, non-sulphidic pore waters in the suboxic and methanogenesis zones [START_REF] Morad | Carbonate cementation in sandstones: distribution patterns and geochemical evolution[END_REF]. Quartz overgrowths are then observed to form, perhaps due to the recrystallization of biogenic SiO 2 , but also accompanied with the formation of minor kaolinite in response to feldspar dissolution.

Subsequently, dolomite is observed to form, potentially promoted by Ca and Mg release from the dissolution or transformation of the earlier formed clays. After the influx of CO 2 into the reservoir. dawsonite and additional quartz forms from the dissolution of feldspar, and subsequently ankerite and kaolinite forms from the dissolution of dolomite, quartz, chlorite and potentially some clay minerals..

Implications for carbon storage in sandstone reservoirs

The formation and long-term stability of dawsonite and ankerite in the LDX system are likely good analogies for some modern carbon storage systems such as the Shenhua CCS demonstration project in China [START_REF] Yang | Reactive transport modeling of long-term CO 2 sequestration mechanisms at the Shenhua CCS demonstration project[END_REF]. The distribution of dawsonite and ankerite in the LDX reservoirs (Fig. 5) indicates that these minerals can form in the water layers beneath gas water contact (GWC). Some dawsonite and ankerite were also observed above the present day GWC, which may indicate that either 1) the position of the GWC has changed over time or 2) that these phases grew in the gas filled section of the reservoir, potentially on water-wet surfaces. This later, process, however, is likely limited as the dawsonite and ankerite forming reactions 1 and 3 consume H 2 O. The formation of such carbonate minerals adjacent to the CO 2 gas reservoir is also supported by the results of TOUGHREACT calculations reported by [START_REF] Pan | Evaluation of CO 2 -Fluid-Rock Interaction in Enhanced Geothermal Systems: Field-Scale Geochemical Simulations[END_REF] that suggest that when CO 2 is injected into sandstone reservoirs, carbonate minerals would form on both upper and lower layers of the reservoir.

Carbonate mineral precipitation in the water layers below the GWC has also been observed in New Zealand sandstone reserviors [START_REF] Higgs | The Pretty Hill Formation as a natural analogue for CO 2 storage: An investigation of mineralogical and isotopic changes associated with sandstones exposed to low, intermediate and high CO 2 concentrations over geological time[END_REF]. As such, we suggest that the CO 2 in the gas reservior maintains a sufficient CO 2 partial pressure in the formation waters below the GWC to promote the growth and long term stability of dawsonite and ankerite. However, according to the paragenetic sequence determined in this study, kaolinite precipitated from the fluid concurrently with the precipitation of dawsonite and ankerite. Numerous petrographic observations and experimental simulations suggest that kaolinite and dawsonite formation could significantly reduce sandstone permeability [START_REF] Luquot | CO 2 percolation experiment through chlorite/zeolite-rich sandstone (Pretty Hill Formation-Otway Basin-Australia)[END_REF][START_REF] Shiraki | Experimental study on water-rock interactions during CO 2 flooding in the Tensleep Formation, Wyoming, USA[END_REF][START_REF] Yu | An experimental study of CO 2 -brine-rock interaction at in situ pressure-temperature reservoir conditions[END_REF].

Kaolinite also commonly grows in response to CO 2 -feldspar or CO 2 -clay mineral interaction [START_REF] Luquot | CO 2 percolation experiment through chlorite/zeolite-rich sandstone (Pretty Hill Formation-Otway Basin-Australia)[END_REF][START_REF] Moore | Mineralogical and geochemical consequences of the long-term presence of CO 2 in natural reservoirs: An example from the Springerville-St. Johns Field, Arizona, and New Mexico[END_REF][START_REF] Shiraki | Experimental study on water-rock interactions during CO 2 flooding in the Tensleep Formation, Wyoming, USA[END_REF][START_REF] Yu | An experimental study of CO 2 -brine-rock interaction at in situ pressure-temperature reservoir conditions[END_REF]. [START_REF] Shiraki | Experimental study on water-rock interactions during CO 2 flooding in the Tensleep Formation, Wyoming, USA[END_REF] concluded that the precipitation and growth of kaolinite in the pores could block pore throats leading to the reduction of the permeability of reservoir. Similarly, [START_REF] Worden | Dawsonite cement in the Triassic Lam Formation, Shabwa Basin, Yemen: A natural analogue for a potential mineral product of subsurface CO 2 storage for greenhouse gas reduction[END_REF] suggested that the fiber-like crystal morphology of dawsonite would also reduce sandstone permeability. The reduced permeability in the sandstone could slow chemical transport, and including potentially the transport of dissolved CO 2 out of the storage system. Such processes could thereby promote the stability of carbonate minerals in this system. The decrease in permeability could also lead to fluid pressure build up in the gas reservoir as further minerals form in this system.

Conclusions

This study focused on the mineral reactions provoked by the influx of CO 2 into the subarkose and sublitharenite sandstones of the Yinggehai basin. This system is likely a good analogy for subsurface carbon storage in sandstone hosted hydrocarbon reservoirs as the influxed CO 2 replaces a methane-rich gas phase in an anticlinal structure. This leads to the presence of a CO 2 -rich gas phase stable over a ~1-2 million year time frame.

The influx of provoked the CO 2 provoked the formation of dawsonite from plagioclase and ankerite and kaolinite likely from chlorite, quartz, and dolomite, at temperatures of 68 to 80 ºC. and 90 to 96 o C, respectively, largely just below the CO 2 -gas water interface. The formation of such phases could provide for the mineral storage of some of the CO 2 injected during subsurface carbon storage efforts.

The key to maintaining dawsonite and ankerite stability in such systems appears to be preserving a high partial pressure of CO 2 . As such, although the formation of these minerals might lead to the storage of CO 2 injected into sandstones as part of engineered geologic storage efforts, such minerals may not provide for the 'permanent' mineral storage of this gas that one might anticipate from the formation of carbonate minerals such as calcite, which can be stable at far lower CO 2 partial pressure, unless the system tends to become impermeable as due to the formation of various secondary minerals.

Acknowledgement

  This manuscript is sponsored by the National Natural Science Foundation of China (No. 41572082), the China Scholarship Council and the project "Study on CO 2 accumulation in reservoirs and the risk assessment for key exploration regions of Chinese sea areas (No. YXKY-2015-ZY-01)" from China National Offshore Oil Corporation (CNOOC).

Figure 1 .

 1 Figure 1. Location map of LDX-1 structure in Yinggehai Basin, south China Sea area. a) The location of Yinggehai basin and basalt area modified after Shi and Yan (2011), basalt areas are modified from Hoang et al. (1996, 1998), Li et al. (2005), Wang et al. (2001); Zhou et al. (1997) and Zhu et al. (1989), b) Geotectonic units modified after Luo et al., (2003) and Meng et al. (2012).

Figure 2 .

 2 Figure 2. Simplified stratigraphic column of the Yinggehai basin modified after Gong et al. (1997) and Huang et al. (2003).

Figure 3

 3 Figure 3. a) Image of a thin section of a sandstone sample collected from a depth of 1860.72 m in the LDX-1-1 well showing its detrital minerals: Q=quartz, F=feldspar and R=rock fragment. b) Mineral composition diagram, after Folk (1968), of the dawsonite-bearing samples from well LDX-1-1 (blue rhombuses) and well LDX-1-4 (black triangles). All the data were obtained through point-counting -see text.

Figure 4 .

 4 Figure 4. Micrographs of diagenetic minerals. (a, b) Acicular dawsonite growing on top of dolomite (red crosses are EDS probe spots), SEM; (c, d) Dawsonite replacing feldspar and plagioclase with EDS analysis results (red crosses are EDS probe spots); (e) Dawsonite replacing quartz overgrowths, thin section micrograph, cross polar; (f) Microcrystalline quartz grows with kaolinite on the top of drusy dawsonite, SEM; (g) Siderite surrounding detrital

Figure 5 .

 5 Figure 5. Plot of volume percent versus depth for dawsonite (blue rhombuses) and ankerite (green triangles) in well LDX-1-1 (Fig 3a) and well LDX-1-4 (Fig 3b). The filled gray areas and dashed boxes indicate the location of gas layers, the water layers in the reservoir system are below the gas layers. The gas-water-contact (GWC), as reported by the China National Offshore Oil Company are located at 1844.1m and 1886.0m depth in wells LDX-1-1 and LDX-1-4, respectively.

Figure 6 .

 6 Figure 6. Plot of δ 13 C versus δ 18 O values for dawsonite and ankerite samples. Filled symbols represent the compositions of dawsonite, and open symbols for ankerite The filled red triangles and open blue rhombus correspond to compositions measured in this study from the LDX-1 structure, Yinggehai basin, China. Compositions of dawsonite from the Bowen-Gunnedah-Sydney basin system in Australia reported by Baker et al. (1995) are shown as filled blue rhombuses, those from the late Permian coal measures of the Upper Hunter Valley in Australia reported by Golab et al. (2006) are shown as filled dark red squares, those of the Cretaceous Hailaer basin in China reported by Gao et al. (2009) are shown as filled yellow crosses, those of the upper Cretaceous Songliao Basin in China reported by Liu et al. (2011) are shown as filled purple circles, and those of the Yaojie coalfield, Minhe basin in China reported by Ming et al. (2017) are shown as filled green triangles. Compositions of ankerite from the Cretaceous Hailaer basin in China reported by Gao et al. (2009) are shown as open red squares, and those of the upper Cretaceous Honggang anticline of the Songliao Basin in China reported by Liu et al. (2011) are shown as open green triangles.

Figure 7 .

 7 Figure 7. Images of clay minerals observed in this study. (a) detrital smectite in the sandstone and (b) image showing the relationship between smectite and ankerite. Rhombic ankerite crystals grew and adhered to the mixed layer of illite/smectite, suggesting the transformation of smectite could provide some of the materials for ankerite. I/S: mixed

Figure 8

 8 Figure 8 Log fugacity of CO 2 (log(f CO2 )) versus temperature (°C) diagram depicting mineral stability fields in the system Na 2 O-Al 2 O 3 -SiO 2 -CO 2 -H 2 O balanced on Al and at quartz equilibrium. The grey area indicates the stability field of dawsonite, and the white area indicates the stability field for albite. The orange and blue lines highlight the log(f CO2 ) values in equilibrium with dawsonite and albite at 70 and 100 o C, respectively. This figure was generated using PHREEQC together with its llnl databasesee text.

Figure 9 .

 9 Figure 9. Log fugacity of CO 2 (log(f CO2 )) versus log a Na+ /a H+ diagrams depicting mineral stability fields in the system Na 2 O-Al 2 O 3 -SiO 2 -CO 2 -H 2 O balanced on Al and at quartz equilibrium at 70 °C and 100 °C. The dashed area represents the current log (f CO2 ) range in the reservoirs considered in this study. This figure was generated using PHREEQC together with its llnl databasesee text.

Figure 10 .

 10 Figure 10. Phase diagram of log CO 2 fugacity (log (f CO2 )) versus log a Ca2+ /a 2 H+ . The dashed vertical lines correspond to equilibrium compositions of reaction 1, whereas the sloped lines correspond to equilibrium compositions of reaction 3, The grey area is the likely aqueous activity ratio of a Ca2+ /a 2 H+ in the studied system and the arrow illustrates the direction of the CO 2 fugacity evolution in the system as the reservoir fills with CO 2 . This figure was generated using PHREEQC together with its llnl database after adding estimated values for the thermodynamics properties of ankerite-see text.

Figure 11 .

 11 Figure 11. Paragenetic sequence for dawsonite-bearing sandstones in this study. The black bars are indicated by petrologic observations, the grey dashed bars are estimated based on the mass balance considerations and reactions (1) and (3).

Table 1 . EDS analysis data for dawsonite.

 1

			Weight Percentage, %			Atomic Percentage, %
	Well
		C	Na	Al	O	Si	C	Na	Al	O	Si
	LDX-1-1	18.76	4.98	12.99 63.27	n.d.	25.14	3.48	7.75	63.63	n.d.
	LDX-1-1	15.88 11.13 14.19	58.8	n.d.	22.01	8.06	8.76	61.18	n.d.
	LDX-1-1	11.81 12.44 12.96 55.55	7.23	17.14	9.44	8.38	60.55	4.49
	LDX-1-4	14.88	13.8	14.24 57.09	n.d.	20.87 10.11	8.89	60.13	n.d.
	LDX-1-4	15.61 12.53	13.7	58.15	n.d.	21.71	9.11	8.48	60.7	n.d.
	LDX-1-4	12.62 14.69 17.98 54.72	n.d.	18.19 11.06 11.54 59.21	n.d.
	LDX-1-4	12.64	8.37	14.29	57.5	7.2	18.15	6.28	9.14	62	4.43
	LDX-1-4	7.83	6.83	8.83	55.28 21.23 11.88	5.41	5.96	62.97 13.78
	LDX-1-4	12.58 12.01 19.97 55.44	n.d.	18.13	9.04	12.82 60.01	n.d.
	LDX-1-4	9.14	14.68 24.74 51.45	n.d.	13.75 11.54 16.57 58.13	n.d.
	LDX-1-1	10.16 13.06 23.91 52.87	n.d.	15.09 10.13 15.81 58.96	n.d.
	LDX-1-1	11.59	9.4	23.75 55.26	n.d.	16.9	7.16	15.42 60.52	n.d.
	LDX-1-1	18.76	4.98	12.99 63.27	n.d.	25.14	3.48	7.75	63.63	n.d.
	LDX-1-1	11.97 10.19 18.04 55.91	3.88	17.36	7.72	11.64 60.86	2.41
	LDX-1-1	7.25	12.06 30.26 50.43	n.d.	11.18	9.71	20.76 58.35	n.d.
	LDX-1-1	13.42 13.99 16.92 55.67	n.d.	19.16 10.43 10.75 59.66	n.d.
	LDX-1-1	13.68 13.29 16.93 56.11	n.d.	19.46	9.88	10.72 59.94	n.d.
	LDX-1-1	18.6	7.9	11.21 62.28	n.d.	24.98	5.54	6.7	62.78	n.d.
	LDX-1-1	12.75 11.24 20.18 55.83	n.d.	18.34	8.45	12.92 60.29	n.d.
	LDX-1-1	10.17 12.49	24.3	53.04	n.d.	15.1	9.69	16.07 59.14	n.d.
	LDX-1-4	13.71	7.2	12.99 58.84	7.26	19.43	5.33	8.2	62.63	4.4
	LDX-1-4	11.98 13.11 16.51 55.02	3.38	17.39	9.93	10.66 59.92	2.1
	LDX-1-4	10.86 13.85 21.98 53.31	n.d.	16	10.65 14.41 58.94	n.d.
	LDX-1-4	22.23	3.58	7.27	66.93	n.d.	28.65	2.41	4.17	64.77	n.d.
	LDX-1-4	10.18 12.73	24.1	52.99	n.d.	15.12	9.88	15.93 59.07	n.d.
	LDX-1-4	12.26 13.58 19.45	54.7	n.d.	17.75 10.27 12.53 59.44	n.d.

n.d.: not be detected.

Table 2 . EDS analysis data for ankerite

 2

	Weight Percent
	Well

n.d.: not be detected.

Table 3 . EDS analysis data for smectite and the mixed layer of illite and smectite

 3

	938
	Well	Weight Percentage, %

/ 39


Shalom Eliahou
email: eliahou@univ-littoral.fr

Jean Fromentin
email: fromentin@univ-littoral.fr

Rénald Simonetto
email: renalds@microsoft.com

Is the

Keywords: Collatz conjecture, 3x + 1 problem, stopping time, glide record, jump function

published or not. The documents may come

Introduction

We denote by N the set of positive integers. Let T : N → N be the notorious 3x + 1 function, defined by T (n) = n/2 if n is even, T (n) = (3n + 1)/2 if n is odd. For k ≥ 0, denote by T (k) the kth iterate of T . The orbit of n under T is the sequence O T (n) = (n, T (n), T (2) (n), . . . ).

The famous Collatz conjecture states that for all n ≥ 1, there exists r ≥ 1 such that T (r) (n) = 1. The least such r is denoted σ ∞ (n) and called the total stopping time of n. An equivalent version of the Collatz conjecture states that for all n ≥ 2, there exists s ≥ 1 such that T (s) (n) < n. The least such s is denoted by σ(n) and called the stopping time of n. For instance, we have

σ(n) = 1 if n is even, 2 if n ≡ 1 mod 4, (1)
as is well known and easy to check. A stopping time record is an integer n ≥ 2 such that σ(m) < σ(n) for all 2 ≤ m ≤ n -1.

For the original slower version C : N → N, where C(n) = n/2 or 3n + 1 according as n is even or odd, the analog of the stopping time is called the glide in [START_REF] Lagarias | [END_REF]. The list of all currently known glide records, complete up to at least 2 60 , is maintained in [START_REF][END_REF]. It is quite likely that glide records and stopping time records coincide; we have verified it by computer up to 2 32 .

It is well known that σ(n) is unbounded as n grows. For instance, since

T ( ) (2 -1) = 3 -1, (2)
as follows from the formula T (2 a 3 b -1) = 2 a-1 3 b+1 -1 for a ≥ 1, we have σ(2 -1) ≥ for all ≥ 2. In this paper, we propose an accelerated version of the function T . The idea, somewhat as in [START_REF] Tao | Almost all orbits of the Collatz map attain almost bounded values[END_REF], is to apply an iterate of T to n depending on the number of digits of n in base 2. Accordingly, we introduce the following function.

Definition 1.1 The jump function jp : N → N is defined for n ∈ N by jp(n) = T ( ) (n), where = log 2 (n) + 1 is the number of digits of n in base 2.

Example 1.2 We have jp(1) = T (1) (1) = 2, and jp(2) = T (2) (2) = 2 since 2 is of length = 2 in base 2. For n = 27, written 11011 in base 2, hence of length = 5, we have jp(27) = T (5) (27) = 71. In turn, 71 is of length = 7 in base 2 since 2 6 ≤ 71 < 2 7 , whence jp(71) = T (7) (71) = 137. The orbit of 27 under jumps is displayed below in (4).

Example 1.3 A single jump at n = 2 -1 with ≥ 1 yields jp(2 -1) = 3 -1.
(

) 3
This follows from the equalities = log 2 (2 -1) + 1 and (2).

Example 1. [START_REF] Eliahou | The 3x + 1 problem: new lower bounds on nontrivial cycle lengths[END_REF] We have jp(2n) = jp(n) for all n ≥ 1. Indeed, 2n is of length one more than n in base 2.

In analogy with the stopping time relative to T , we now introduce the falling time relative to jumps. As jp(1) = jp(2) = 2, we only consider n ≥ 3.

Definition 1.5 Let n ≥ 3. The falling time of n, denoted ft(n), is the least k ≥ 1 such that jp (k) (n) < n, or ∞ if there is no such k.
Note that, for a presumed cyclic orbit under T with minimum m ≥ 3, we would have jp(m) = ∞.

There is no tight comparison between stopping time and falling time. It may happen that σ(a) < σ(b) whereas ft(a) > ft(b). For instance, for a = 41 and b = 43, we have

σ(41) = 2 < σ(43) = 5, ft(41) = 8 > ft(43) = 2.
It may also happen that ft(n) > σ(n), as shown by the case n = 41.

Of course, the Collatz conjecture is equivalent to ft(n) < ∞ for all n ≥ 3. In Section 2, we provide computational evidence leading us to a stronger conjecture, namely that ft(n) is in fact bounded for all n ≥ 3. Specifically, all integers n we have tested so far satisfy ft(n) ≤ 16. See Conjecture 2.4. In Section 3, in analogy with the falling time, we introduce the Syracuse falling time sft(n), and corresponding conjectures, by only considering the odd terms in the orbits O T (n). In Section 4, we report surprising computational results on ft(2 -1) and sft(2 -1) for ≤ 500 000, and we formulate corresponding conjectures. In the last Section 5, inspired by the case n = 2 -1, we formulate still stronger conjectures on ft(n) and sft(n) for very large integers n. We conclude the paper with some supporting heuristics.

For a wealth of information, developments and commented references related to the 3x + 1 problem, see the webpage and book of J. C. Lagarias [START_REF] Lagarias | 3x + 1 problem and related problems[END_REF][START_REF]The Ultimate Challenge: The 3x + 1 Problem[END_REF]. To date, the Collatz conjecture has been verified by computer up to 2 68 by D. Barina [START_REF] Barina | Convergence verification of the Collatz problem[END_REF]. Using this bound, it follows from [START_REF] Eliahou | The 3x + 1 problem: new lower bounds on nontrivial cycle lengths[END_REF] that any non-trivial cycle of T must have length at least 114 208 327 604.

Falling time records

In this section, we only consider those positive integers n satisfying σ(n) ≥ 3, i.e. such that n ≡ 3 mod 4 by [START_REF] Barina | Convergence verification of the Collatz problem[END_REF]. Let us denote by 4N + 3 the set of those integers. Here is our first computational evidence that the falling time remains small. Proposition 2.1 We have ft(n) ≤ 14 for all n ∈ [1, 2 44 -1] such that n ≡ 3 mod 4.

Proof. In a few days of computing time with CALCULCO [START_REF] Calculco | a computing platform at Université du Littoral Côte d'Opale[END_REF]. [START_REF][END_REF].

As shown in

Table 1 shows that the number 12 and a few smaller ones fail to occur as falling time records. One may then wonder about the smallest n ∈ 4N + 3 reaching ft(n) = 12.

The answer is to be found in Table 2. Let us define a new falling time as an integer n ∈ 4N + 3 such that ft(n) is distinct from ft(m) for all smaller m ∈ 4N + 3. Of course, every falling time record is a new falling time. The list of new falling times we know so far, which are not already falling time records, is given in Table 2. for all m ≥ 3, a single integer n satisfying ft(n) > 14 yields infinitely many integers N satisfying ft(N ) > 14, namely N = 2 r n for all r ≥ 1. However, the latter numbers have stopping time equal to 1, and hence are not particularly interesting.

Only those integers n satisfying ft(n) > 14 and having a reasonably large stopping time are really interesting, for their apparent rarity and their relevance to the Collatz conjecture. Hence, we shall restrict our search to 24-persistent integers, i.e. to those n satisfying

σ(n) ≥ 24.
The property for n of being 24-persistent only depends on its class mod 2 24 . See [START_REF] Terras | A stopping time problem on the positive integers[END_REF] for more details on the description of the condition σ(n) ≥ k by classes mod 2 k . See also [START_REF] Everett | Iteration of the Number-Theoretic Function f (2n) = n, f (2n + 1) = 3n + 2[END_REF]. For k = 24, the number of 24-persistent classes mod 2 24 is exactly 286 581.

As shown below, the occurrence of ft(n) > 14 among 24-persistent numbers seems to be very rare. Here is our first computational result. Proof. In a few days of computing time with CALCULCO [START_REF] Calculco | a computing platform at Université du Littoral Côte d'Opale[END_REF].

The neighborhood of g 30

It turns out that n 0 is a glide record, and as such is listed under the name

n 0 = g 30
in Table 5 of Section 2.2. We have verified by computer that g 30 is the smallest 24-persistent integer n satisfying ft(n) > 14. However, because of the restriction σ(n) ≥ 24, we do not know whether g 30 is an actual falling time record. In a small neighborhood of g 30 in the Collatz tree, we found 11 more 24-persistent integers n satisfying ft(n) > 14. By small neighborhood of g 30 , we mean here integers m such that

T (i) (m) = T (j) (g 30 )
for some i ∈ [0, 40], j ∈ [0, 30]. It turns out that these 11 integers all satisfy ft(n) = 15, as g 30 itself. They are displayed in Table 3.

The neighborhood of g 32

There is another glide record in Table 5 with falling time 15, namely g 32 = 180 352 746 940 718 527.

In this case, looking at a somewhat larger neighborhood of g 32 in the Collatz tree, namely at all m such that

T (i) (m) = T (j) (g 32 )
for some i ∈ [0, 50], j ∈ [0, 30], we found four 24-persistent integers n reaching ft(n) = 16. These four integers are displayed in Table 4. However, these four integers n have a small stopping time. They all satisfy σ(n) ∈ [35, 48], as compared to σ(g 32 ) = 966. Hence again, they are not particularly interesting. This leads us to the following conjecture.

Conjecture 2.4 There exists B ≥ 16 such that ft(n) ≤ B for all n ≥ 3.

An even bolder conjecture, based on the data we currently have, would be to take B = 16 above. Anyway, with whatever value of B, Conjecture 2.4 constitutes a strong form of the Collatz conjecture.

Glide records

Eric Roosendaal maintains the list of all currently known glide records [START_REF][END_REF], complete up to at least 2 60 . At the time of writing, there are 34 of them, denoted g 1 , . . . , g 34 below. As noted in [START_REF][END_REF], only the first 32 ones have been independently checked. The ten biggest are displayed in descending order in Table 5. It turns out that ft(g 1 ), . . . , ft(g 34 ) ≤ 15.

Moreover, among them, the highest value ft(n) = 15 is only reached by g 30 and g 32 . Table 4 displays four 24-persistent integers n satisfying ft(n) = 16 in the neighborhood of g 32 . We do not know whether ft(n) ≥ 17 is at all reachable.

Falling time distribution

In three distinct graphics, we display the distribution of the values taken by the falling time function in large integer intervals. These graphics show that the proportion of the case ft(n) ≥ 3 in the integer intervals [2 , 2 +1 -1] tends to 0 as grows.

• Figure 1 displays the proportion of the occurrence of ft(n) = 1, ft(n) = 2 and ft(n) ≥ 3, respectively, among all odd integers in the integer intervals [2 , 2 +1 -1] for 2 ≤ ≤ 40.

• Figure 2 does the same but separates the cases n ≡ 1 mod 4 and n ≡ 3 mod 4. The purpose is to show that the former case, with stopping time 2, behaves like the more interesting latter case, and so may be safely ignored.

• Finally, Figure 3 is restricted to 24-persistent integers in the integer intervals [2 , 2 +1 -1] for 24 ≤ ≤ 50.

A variant of jumps

Let h ∈ N. For all n ∈ N, we define

jp h (n) = T (h ) (n)
where, as before, is the number of digits of n in base 2. This is not the same, of course, as the h-iterate jp (h) (n). Note also that for h = 1, we recover i.e. jp 1 (n) = jp(n). For n ≥ 3, the h-falling time ft h (n) is defined correspondingly, as the smallest k ≥ 1, if any, such that ft

(k) h (n) < n.
It turns out that for h = 18, and for the glide records g 1 , . . . , g 34 , we have ft 18 (g i ) = 1 for all 1 ≤ i ≤ 34. In view of that fact, is it true that ft 18 (n) = 1 for all n ≥ 3? We do not know. But we have verified it up to n ≤ 2 30 , and it cannot be outright dismissed, given the conjectural behavior of ft(n) for very large n as discussed in Section 5. Of course, a positive answer would imply the Collatz conjecture. On the other hand, uncovering any counterexample would be quite a feat. where ν ≥ 1 is the largest integer such that 2 ν divides 3x + 1. That is, syr(x) is the largest odd factor of 3x + 1. This specific version is called the Syracuse function in [START_REF] Tao | Almost all orbits of the Collatz map attain almost bounded values[END_REF]. It has been amply investigated in the past, though under different notation or names. For instance in [START_REF] Crandall | On the " 3x + 1" Problem[END_REF], where lower bounds on the length of presumed nontrivial cycles of syr(x) are given in terms of the convergents p n /q n to log 2 (3); or in [START_REF] Kontorovich | Benfords law, values of L-functions, and the 3x + 1 problem[END_REF][START_REF] Kontorovich | Structure Theorem for (d, g, h)maps[END_REF][START_REF] Ya | Statistical (3x + 1)-Problem[END_REF], where statistical properties of syr(x) and related maps are studied using the Structure theorem of Sinai, which we briefly recall in Section 3.2 below.

In analogy with the functions jp(n) and ft(n) related to the 3x+1 function T (x), we now introduce the corresponding functions sjp(n) and sft(n) related to the Syracuse version syr(x). Here is the corresponding Syracuse falling time. As one may expect, the inequality sft(n) ≤ ft(n) holds very often, but not always. For instance, for n = 199, we have ft(199) = 1 but sft(199) = 5. The former equality follows from the orbit O T (199) = (199, 299, 449, 674, 337, 506, 253, 380, 190, . . . ) and the value log 2 (199) + 1 = 8, yielding jp(199) = 190, while the latter one follows from the orbit O syr (199) = (199, 323, 395, 479, 577, 1, . . . ). Definition 3.5 A Syracuse falling time record is an integer n ∈ 4N + 3 such that n ≥ 7 and sft(m) < sft(n) for all m ∈ 4N + 3 with m < n.

Definition 3.3 Let n ∈ O \{1}. The Syracuse falling time of n, denoted sft(n), is the least k ≥ 1 such that sjp (k) (n) < n, or ∞ if there is no such k.
The complete list of Syracuse falling time records up to 2 35 is displayed in Table 6. Compared with Table 1, it turns out that all current Syracuse falling time records are also falling time records. The converse does not hold, as shown by the falling time records 60 975 and 1 394 431 in Table 1.

Current maximum

The Collatz conjecture is equivalent to the statement sft(n) < ∞ for all n ∈ O \{1}. Again, it is likely that a stronger form holds, namely that sft(n) is bounded on O \{1}. Besides the computational evidence above and below, some heuristics point to that possibility in Section 5. Similarly to Proposition 2.1, here is a computational result in that direction.

Proposition 3.6 We have sft(n) ≤ 9 for all n ∈ [3, 2 35 -1] such that n ≡ 3 mod 4.
Proof. By computer with CALCULCO [START_REF] Calculco | a computing platform at Université du Littoral Côte d'Opale[END_REF].

As yet another hint pointing to the same direction, it turns out that sft(g 1 ), . . . , sft(g 34 ) ≤ 10

for the 34 currently known glide records. For definiteness, Table 7 displays the Syracuse falling times of the top ten glide records as listed in Table 5. Among the g i , and as in Section 2.2 for the falling time, only g 30 and g 32 reach the current maximum of the Syracuse falling time, namely sft(n) = 10. Interestingly, the biggest currently known glide record, namely n = g 34 , only satisfies sft(n) = 8. With Proposition 3.6 and (5) in the background, here is our formal conjecture. Again, the truth of this conjecture would yield a strong positive solution of the Collatz conjecture. At the time of writing, no single positive integer n ≡ 3 mod 4 is known to satisfy sft(n) ≥ 11. Thus, a still bolder conjecture would be to take C = 10 in Conjecture 3.7, or C = 12 to be on a safer side. Whence the title of this paper.

A variant of Syracuse jumps

As in Section 2.4 for jumps, we propose here an accelerated variant of Syracuse jumps. Let h ∈ N. For all n ∈ O, we define

sjp h (n) = syr (h ) (n),
where is the number of digits of n in base 2. Of course, sjp 1 (n) = sjp(n). For n ≥ 3, the Syracuse h-falling time sft h (n) is defined correspondingly, as the smallest k ≥ 1, if any, such that sft (k) h (n) < n. It turns out that for h = 12, and for the glide records g 1 , . . . , g 34 , we have sft 12 (g i ) = 1 for all 1 ≤ i ≤ 34. Again, we may ask whether sft 12 (n) = 1 holds for all odd n ≥ 3. A positive answer would imply the Collatz conjecture. We have verified it up to n ≤ 2 30 , and our semi-random search did not yield any counterexample.

Anyway, the occurrence of sft 12 (n) = 1 as n grows to infinity is probably overwhelming; and, just possibly, tools such as Sinai's structure theorem and its applications [START_REF] Kontorovich | Benfords law, values of L-functions, and the 3x + 1 problem[END_REF][START_REF] Kontorovich | Structure Theorem for (d, g, h)maps[END_REF][START_REF] Ya | Statistical (3x + 1)-Problem[END_REF] might help prove that this is indeed the case. But we shall not pursue here this line of investigation.

For convenience, let us briefly recall the statement of that theorem. Given x ∈ 6N + {1, 5}, let x i = syr (i) (x) for all i ≥ 0, and let k i ≥ 1 be such that

x i = (3x i-1 + 1)/2 k i for all i ≥ 1. Moreover, for m ≥ 1, set γ m (x) = (k 1 , . . . , k m ).
Sinai's structure theorem states that given any (k 1 , . . . , k m ) ∈ N m , the set of all x ∈ 6N + {1, 5} such that γ m (x) = (k 1 , . . . , k m ) consists of a unique and full congruence class mod 6

• 2 k 1 +•••+km in N.

The case 2 -1

In sharp contrast with the stopping time of 2 -1, for which σ(2 -1) ≥ for all ≥ 2, the falling time and the Syracuse falling time of 2 -1 seem to remain very small as grows. Here is some strong computational evidence. Proof. In a few days of computing time with CALCULCO [START_REF] Calculco | a computing platform at Université du Littoral Côte d'Opale[END_REF].

Moreover, the value ft(2 -1) = 5 seems to occur finitely many times only, the last one being presumably at = 132. In turn, the value ft(2 -1) = 4 seems to occur infinitely often. Whence the following conjecture, verified by computer up to = 500 000. Proof. In a few days of computing time with CALCULCO [START_REF] Calculco | a computing platform at Université du Littoral Côte d'Opale[END_REF].

This leads us to the following conjecture, true up to ≤ 500 000.

For very large n

As hinted by the computational evidence and conjectures of Section 4 on the case n = 2 -1, by intensive semi-random search, and by the heuristics below, it appears to be increasingly difficult for integers n to satisfy ft(n) ≥ 5 or sft(n) ≥ 3 as they grow very large. Here then are still bolder conjectures. However, this integer is congruent to 3 mod 16 and hence has stopping time equal to 4 only.

Here is the analogous conjecture for the Syracuse falling time. Its threshold of 2 5000 is similarly inspired by Proposition 4.

Heuristics

Besides the computational evidence leading to Conjectures 2.4, 3.7, 4.2, 4.4, 5.1 and 5.2, a heuristic argument would run as follows. It is well known that the Collatz conjecture is equivalent to the statement that, starting with any integer n ≥ 1, the probability for T (k) (n) to be even or odd tends to 1/2 as k grows to infinity. Thus, even if n written in base 2 is a highly structured binary string, as e.g. for n = 2 -1, one may expect that for = the length of that string, then T ( ) (n) in base 2 will already look more random. That is, a single jump or Syracuse jump at n ≥ 3 should already introduce a good dosis of randomness, all the more so as n grows very large. And therefore, a bounded number of jumps or Syracuse jumps at n might well suffice to fall below n.

A challenge

We hope that the experts in highly efficient computation of the 3x+1 function will tackle the challenge of probing these conjectures to much higher levels than the ones reported here. For instance, as both a challenge and a request to the reader, and in view of Conjecture 5.1, if you do find any n ≥ 2 500 satisfying ft(n) ≥ 5, please e-mail it to the authors. Your solution will be duly recorded on a dedicated webpage.

For

  a, b ∈ Z, we denote by [a, b] = {n ∈ Z | a ≤ n ≤ b} the integer interval they span. Recalling Example 1.4, namely that ft(2m) = ft(m)

Proposition 2 . 3

 23 The smallest 24-persistent integer n such that ft(n) > 14 is n 0 = 1 008 932 249 296 231.It satisfies ft(n 0 ) = 15, σ(n 0 ) = 886 and log 2 (n 0 ) + 1 = 50.

Figure 1 :

 1 Figure 1: Proportion of odd integers in [2 , 2 +1 -1] with falling time equal to 1, 2 and greater than 2, respectively. The integer goes from 2 to 40

Figure 2 :

 2 Figure 2: Same plot as for Figure 1 except that integers are separated with respect to their class 1 or 3 modulo 4. Gray curves are for integers congruent to 1 modulo 4 while black ones are for those congruent to 3 modulo 4.

3 3

 3 The Syracuse version Let O = 2N + 1 denote the set of odd positive integers. Another well-studied version of the 3x + 1 function is syr : O → O, defined on any x ∈ O by syr(x) = (3x + 1)/2 ν ,

Definition 3 . 1 Example 3 . 2

 3132 We define the Syracuse jump function sjp : O → O by sjp(n) = syr ( ) (n), where = log 2 (n) + 1 . We have sjp(1) = 1, sjp(3) = 1 and sjp(27) = syr (5) (27) = 107.

Figure 3 :

 3 Figure 3: Proportion of 24-persistent integers in [2 , 2 +1 -1] with falling time equal to 1, 2 and greater than 2, respectively. The integer goes from 24 to 50

Example 3 . 4

 34 We have sft(27) = 6, as witnessed by the orbit of 27 under Syracuse jumps, namely O sjp (27) = (27, 107, 233, 377, 911, 53, 1, 1, . . . ).

Conjecture 3 . 7

 37 There exists C ≥ 10 such that sft(n) ≤ C for all n ≡ 3 mod 4.

Proposition 4 . 1

 41 Besides ft(2 5 -1) = ft(2 6 -1) = 8, we have ft(2 -1) ≤ 5 for all 2 ≤ ≤ 500 000 with / ∈ {5, 6}.

Conjecture 4 . 2 Proposition 4 . 3

 4243 We have ft(2 -1) ≤ 4 for all ≥ 133. Here are the analogous statement and conjecture for the Syracuse falling time. Besides sft(2 5 -1) = sft(2 6 -1) = 5, and sft(2 24 -1) = 4, we have sft(2 -1) ∈ {2, 3} for all ∈ [2, 4 624] \ {5, 6, 24}, sft(2 -1) = 2 for all ∈ [4 625, 500 000].

Conjecture 4 . 4

 44 We have sft(2 -1) = 2 for all ≥ 4 625.

Conjecture 5 . 1

 51 We have ft(n) ≤ 4 for all n ≥ 2 500 . This threshold of 2 500 is inspired by Conjecture 4.2, of course with a margin for safety. It cannot be significantly lowered, since ft(2 132 -1) = 5 as noted before Proposition 4.1. Moreover, intensive random search revealed one integer n ∈ [2 70 , 2 71 -1] satisfying ft(n) = 5, namely n = 1 884 032 044 420 885 877 201 579 449 071 924 925 072 300 117 065 411.

  3 and Conjecture 4.4. Conjecture 5.2 We have sft(n) ≤ 2 for all odd n ≥ 2 5000 .

Table 1 ,

 1 the smallest n ∈ 4N+3 such that ft(n) ≥ 14, namely n = 12 235 060 455, actually satisfies ft(n) = 14 and n > 2 33 .

	Definition 2.2 A falling time record is an integer n ∈ 4N + 3 such that
	ft(m) < ft(n) for all m ∈ 4N + 3 with m < n.

Table 1 :

 1 Falling time records up to 235

	n ≡ 3 mod 4 log 2 (n) + 1 ft(n)
	3	2	2
	7	3	3
	27	5	8
	60 975	16	9
	1 394 431	21	10
	6 649 279	23	11
	63 728 127	26	13
	12 235 060 455	34	14
	The list of falling time records up to 2 35 is given in Table 1. It was built
	while establishing Proposition 2.1. For instance, we have ft(3) = 2, ft(7) = 3
	and ft(n) ≤ 3 for all 3 ≤ n < 27 such that n ≡ 3 mod 4. The value ft(27) = 8
	follows from the fact that 8 jumps are needed from 27 to fall below it, as
	shown by the orbit of 27 under jumps:
	O jp (27) = (27, 71, 137, 395, 566, 3 644, 650, 53, 8, 2, 2, . . . ).	(4)

Table 2 :

 2 Some new falling times

	n	111 103 71 55 217 740 015
	ft(n)	4	5	6 7	12

2.1 Integers satisfying ft(n) > 14

Table 3 :

 3 24-persistent integers satisfying ft(n) = 15.

	1 513 398 373 944 347, 1 702 573 170 687 391, 2 017 864 498 592 463,
	2 553 859 756 031 087, 3 405 146 341 374 783, 3 830 789 634 046 631,
	5107719512062175, 5 746 184 451 069 947, 6 464 457 507 453 691,
	7 272 514 695 885 403, 22 370 169 558 105 279.

Table 4 :

 4 24-persistent integers satisfying ft(n) = 16.

	49 312 2600 554 790 303, 739 683 900 832 185 455,
	986 245 201 109 580 607, 1 479 367 801 664 370 911.

Table 5 :

 5 Top ten known glide records

	n		log 2 (n) + 1 glide of n σ(n) ft(n)
	g 34 2 602 714 556 700 227 743	62	1 639	1005	13
	g 33 1 236 472 189 813 512 351	61	1 614	990	14
	g 32	180 352 746 940 718 527	58	1 575	966 15
	g 31	118 303 688 851 791 519	57	1 471	902	12
	g 30	1 008 932 249 296 231	50	1 445	886 15
	g 29	739 448 869 367 967	50	1 187	728	12
	g 28	70 665 924 117 439	47	1 177	722	13
	g 27	31 835 572 457 967	45	1 161	712	13
	g 26	13 179 928 405 231	44	1 122	688	14
	g 25	2 081 751 768 559	41	988	606	12

Table 6 :

 6 Syracuse falling time records up to 2 35

	n ≡ 3 mod 4 log 2 (n) + 1 sft(n)
	7	3	2
	27	5	6
	6 649 279	23	7
	63 728 127	26	9

Table 7 :

 7 Syracuse falling times of top ten glide records n g 25 g 26 g 27 g 28 g 29 g 30 g 31 g 32 g 33 g 34

	sft(n)	9	8	8	8	8 10 8 10 9	8

Acknowledgements

The computations performed for this paper were carried out on the CAL-CULCO high performance computing platform provided by SCoSI/ULCO (Service COmmun du Systme dInformation de lUniversit du Littoral Cte dOpale).

Authors' addresses: Shalom Eliahou , Jean Fromentin