Determination of optimum relaxation coefficient using finite difference method for groundwater flow

Solution of Laplace’s equation can be done by iteration methods likes Jacobi, Gauss–Seidel, and successive over-relaxation (SOR). There is no new knowledge about the relaxation coefficient (ω) in SOR method. In this paper, we used SOR for solving Laplace’s differential equation with emphasis to obtaining the optimum (minimum) number of iterations with variations of the relaxation coefficient (ω). For this purpose, a code in FORTRAN language has been written to show the solution of a set of equations and its number of iterations. The results demonstrate that the optimum value of ω with minimum iterations is achieved between 1.7 and 1.9. Also, with increasing β?=??x/?y from 0.25 to 10, the number of iterations reduced and the optimum value is obtained for β?=?2.     

New mineral activity–composition relations for thermodynamic calculations in metapelitic systems

New activity–composition (a–x) relations for minerals commonly occurring in metapelites are presented for use with the internally consistent thermodynamic dataset of Holland & Powell ( 2011 , Journal of Metamorphic Geology, 29 , 333–383). The a–x relations include a broader consideration of Fe₂O₃ in minerals, changes to the formalism of several phases and order–disorder in all ferromagnesian minerals where Fe–Mg mixing occurs on multiple sites. The a–x relations for chlorite, biotite, garnet, chloritoid, staurolite, cordierite, orthopyroxene, muscovite, paragonite and margarite have been substantially reparameterized using the approach outlined in the companion paper in this issue. For the first time, the entire set of a–x relations for the common ferromagnesian minerals in metapelitic rocks is parameterized simultaneously, with attention paid to ensuring that they can be used together to calculate phase diagrams of geologically appropriate topology. The a–x relations developed are for use in the Na₂O–CaO–K₂O–FeO–MgO–Al₂O₃–SiO₂–H₂O–TiO₂–O₂ (NCKFMASHTO) system for both subsolidus and suprasolidus conditions. Petrogenetic grids in KFMASH and KFMASHTO are similar in topology to those produced with earlier end‐member datasets and a–x relations, but with some notable differences. In particular, in subsolidus equilibria, the FeO/(FeO + MgO) of garnet is now greater than in coexisting staurolite, bringing a number of key staurolite‐bearing equilibria into better agreement with inferences from field and petrographic observations. Furthermore, the addition of Fe³⁺ and Ti to a number of silicate phases allows more plausible equilibria to be calculated in relevant systems. Pseudosections calculated with the new a–x relations are also topologically similar to equivalent diagrams using earlier a–x relations, although with many low variance fields shifting in P–T space to somewhat lower pressure conditions.     

Consolidation analysis of saturated multi‐layered soils with anisotropic permeability caused by a point sink

This paper presents the analytical layer‐element method to analyze the consolidation of saturated multi‐layered soils caused by a point sink by considering the anisotropy of permeability. Starting from the governing equations of the problem, the solutions of displacements and stresses for a single soil layer are obtained in the Laplace–Hankel transformed domain. Then, the analytical layer‐element method is utilized to further derive the solutions for the saturated multi‐layered soils in the transformed domain by combining with the boundary conditions of the soil system and continuity conditions between adjacent layers. The actual solutions in the physical domain can be acquired by the inversion of Laplace–Hankel transform. Numerical results are carried out to show the accuracy and stability of the proposed method and evaluate the influence of sink depth and anisotropic permeability on excess pore pressure and surface settlement. Copyright © 2011 John Wiley & Sons, Ltd.     

空心圆柱饱和多孔介质热固结问题的解析解

基于考虑热渗效应和等温热流效应的热-水-力耦合的线性热弹性固结控制方程,建立无限长空心圆柱饱和多孔介质热固结问题的一种理论求解方法。该方法先给出Laplace变换域上的解,然后,利用Stehfest法求其数值逆变换。该理论解考虑了空心圆柱体内、外透水界面随时间变化的外力和温度荷载耦合作用过程。最后,通过一算例分析了饱和多孔介质的热固结特征,给出其温度、孔压、位移和应力的演化规律     

Numerical simulation of fully saturated porous materials

Fully coupled, porous solid–fluid formulation, implementation and related modeling and simulation issues are presented in this work. To this end, coupled dynamic field equations with u?p?U formulation are used to simulate pore fluid and soil skeleton (elastic–plastic porous solid) responses. Present formulation allows, among other features, for water accelerations to be taken into account. This proves to be useful in modeling dynamic interaction of media of different stiffnesses (as in soil–foundation–structure interaction). Fluid compressibility is also explicitly taken into account, thus allowing excursions into modeling of limited cases of non‐saturated porous media. In addition to these features, present formulation and implementation models in a realistic way the physical damping, which dissipates energy. In particular, the velocity proportional damping is appropriately modeled and simulated by taking into account the interaction of pore fluid and solid skeleton. Similarly, the displacement proportional damping is physically modeled through elastic–plastic processes in soil skeleton. An advanced material model for sand is used in present work and is discussed at some length. Also explored in this paper are the verification and validation issues related to fully coupled modeling and simulations of porous media. Illustrative examples describing the dynamical behavior of porous media (saturated soils) are presented. The verified and validated methods and material models are used to predict the behavior of level and sloping grounds subjected to seismic shaking. Copyright © 2008 John Wiley & Sons, Ltd.     

A single numerically efficient equation for approximating the Mohr–Coulomb and the Matsuoka–Nakai failure criteria with rounded edges and apex

This paper presents a novel formulation for defining soil failure. It plots in the principal stress space as a surface with the shape ranging between an approximation of the Matsuoka–Nakai and of the Mohr–Coulomb criteria depending on the value of a single parameter. The new function can be used as a replacement of the original equations of these well‐established criteria for implementing in a program for numerical analyses, and it is particularly effective for approximating the Matsuoka–Nakai criterion. Both the Mohr–Coulomb and the Matsuoka–Nakai failure criteria present numerical difficulties during implementation and also at run‐time. In the case of the Matsuoka–Nakai, the new formulation plots in the first octant only, whereas the original criterion plots in all octants, which causes severe convergence problems particularly for those Gauss points with low stress state, such as those on the side of a shallow footing. When the shape parameter is set to reproduce the Mohr–Coulomb failure criterion, on the other hand, the new formulation plots as a pyramid with rounded edges. Moreover, as the new function is at least of class C², the second derivatives are continuous, thus ensuring quadratic convergence of the Newton's method used within the integration scheme of the constitutive law. The proposed formulation can also provide both sharp and rounded apex of the surface at the origin of the stress space by setting accordingly one additional parameter. Copyright © 2013 John Wiley & Sons, Ltd.     

Using calculated chemical potential relationships to account for replacement of kyanite by symplectite in high pressure granulites

Calculated mineral equilibria are used to account for the formation of sapphirine–plagioclase, spinel–plagioclase and corundum–plagioclase symplectites replacing kyanite in quartz–plagioclase–garnet–kyanite granulite facies gneisses from the Southern Domain of the Athabasca granulite terrane, a segment of the Snowbird tectonic zone in northern Saskatchewan, Canada. Metamorphic conditions of >14 kbar and 800 °C are established for the high pressure, garnet–kyanite assemblage using constraints from P–T pseudosections and Zr‐in‐rutile thermometry. Replacement of kyanite by symplectites reflects the reaction of kyanite with the matrix following near‐isothermal decompression to <10 kbar. The chemical potential gradients developed between the kyanite and the matrix led to diffusion that attempted to flatten the gradients, kyanite persisting as a stable phase while it is consumed by symplectite from its edge. In this local equilibrium model, the mineral and mineral compositional spatial relationships are shown to correspond to paths in μ(Na₂O)–μ(CaO)–μ(K₂O)–μ(FeO)–μ(MgO) in the model chemical system, Na₂O–CaO–K₂O–FeO–MgO–Al₂O₃–SiO₂ (NCKFMAS), with SiO₂ and Al₂O₃ taken to be completely immobile. The values of μ(Na₂O) and μ(CaO) are constrained by fixing P–T conditions and choosing appropriate μ(Na₂O) and μ(CaO) values that correspond to the observed plagioclase compositions. μ(FeO)–μ(MgO) diagrams show the corresponding spatial relationships with kyanite and the symplectite phases. These results demonstrate that the replacement of kyanite by sapphirine–plagioclase and spinel–plagioclase appears to be metastable with respect to replacement by corundum–plagioclase. Replacement by corundum–plagioclase does also occur, apparently overprinting pre‐existing symplectite and also kyanite. Ignoring corundum, the resulting diagrams account for the spatial relationships and compositions observed in the spinel–plagioclase and sapphirine–plagioclase symplectites. They are predicted to occur over both a wide range of P–T conditions (6–11 kbar, 650–850 °C) and plagioclase compositions (X_An = 0.5–0.9). The wide range of P–T conditions that may result in identical spatial and compositional relationships suggests that such reaction textures may be of limited use in accurately quantifying the P–T conditions of retrograde metamorphism.     

A comparison of observed and thermodynamically predicted phase equilibria and mineral compositions in mafic granulites

Recently published activity–composition (a–x) relations for minerals in upper amphibolite‐ and granulite facies intermediate and basic rocks have expanded our ability to interpret the petrological evolution of these important components of the lower continental crust. If such petrological modelling is to be reliable, the abundances and compositions of phases calculated at the interpreted conditions of metamorphic equilibration should resemble those in the sample under study. Here, petrological modelling was applied to six granulite facies rocks that formed in different tectonic environments and reached different peak metamorphic pressure–temperature (P–T) conditions. While phase assemblages matching those observed in each sample can generally be calculated at P–T conditions that approximate those of peak metamorphism, a consistent discrepancy was found between the calculated and observed compositions of amphibole and clinopyroxene. In amphibole, Si, Ca and A‐site K are underestimated by the model, while Al and A‐site Na are overestimated; comparatively, in clinopyroxene, Mg and Si are generally underestimated, while Fe²⁺ and Al are typically overestimated, compared to observed values. One consequence is a reversal in the Fe–Mg distribution coefficient (K_D) between amphibole and clinopyroxene compared to observations. Some of these mismatches are attributed to the incorrect partitioning of elements between the predicted amphibole and clinopyroxene compositions; however, other discrepancies are the result of the incorrect prediction of major substitution vectors in amphibole and clinopyroxene. These compositional irregularities affect mineral modal abundance estimates and in turn the position and size (in P–T space) of mineral assemblage fields, the effect becoming progressively more marked as the modal abundance of hornblende increases; hence, this study carries implications for estimating P–T conditions of high‐temperature metabasites using these new a–x relations.     

Approximate analytical and numerical solutions for radial non‐Darcian flow to a well in a leaky aquifer with wellbore storage and skin effect

This study investigated non‐Darcian flow to a well in a leaky aquifer considering wellbore storage and a finite‐thickness skin. The non‐Darcian flow is described by the Izbash equation. We have used a linearization procedure associated with the Laplace transform to solve such a non‐Darcian flow model. Besides, the Stehfest method has been used to invert the Laplace domain solutions for the drawdowns. We further analyzed the drawdowns inside the well for different cases. The results indicated that a smaller B_D results in a smaller drawdown at late times and the leakage has little effect on the drawdown inside the well at early times, where B_D is a dimensionless parameter reflecting the leakage. We have also found that the flow for the negative skin case approaches the steady‐state earlier than that for the positive skin. In addition, the drawdown inside the well with a positive skin is larger than that without skin effect at late times, and a larger thickness of the skin results in a greater drawdown inside the well at late times for the positive skin case. A reverse result has been found for the negative skin case. Finally, we have developed a finite‐difference solution for such a non‐Darcian flow model and compared the numerical solution with the approximate analytical solution. It has been shown that the linearization procedure works very well for such a non‐Darcian flow model at late times, and it underestimates the drawdowns at early times. Copyright © 2012 John Wiley & Sons, Ltd.     

Analytical solutions for one-dimensional advection–dispersion equation of the pollutant concentration

We present simple analytical solutions for the unsteady advection–dispersion equations describing the pollutant concentration C(x, t) in one dimension. The solutions are obtained by using Laplace transformation technique. In this study we divided the river into two regions x ≤ 0 and x≥0 and the origin at x = 0. The variation of C(x, t) with the time t from t = 0 up to t → ∞ (the steady state case) is taken into account in our study. The special case for which the dispersion coefficient D = 0 is studied in detail. The parameters controlling the pollutant concentration along the river are determined.     

Interaction of deformation and metamorphism during subduction and exhumation of hydrated oceanic mantle: Insights from the Western Alps

Petrological investigations supported by multi‐scale structural analysis of eclogitized serpentinite in the Zermatt–Saas Zone of the Western Alps allows for the determination of mineral assemblages related to successive fabrics, upon which the P–T–d–t path of these hydrated mantle rocks can be inferred. Serpentinites of the upper Valtournanche, with lenses and dykes of metagabbro and meta‐rodingite, display an Alpine polyphase metamorphic evolution from eclogite to epidote‐amphibolite facies conditions associated with three successive foliations having different parageneses in these rocks. Serpentinite mainly consists of serpentine with minor magnetite; however, where S1 and S2 foliations are pervasive, metamorphic olivine, together with Ti‐clinohumite and clinopyroxene, are also found. The mineral assemblage associated with D1 includes serpentine1, clinopyroxene1, opaque minerals, titanite ± olivine1, Ti‐clinohumite1 and ilmenite; the D2 assemblage is the same (±chlorite) but minerals have different compositions. The assemblage associated with D3 comprises serpentine3, opaque minerals, ±chlorite3, ilmenite and amphibole3. Ti‐clinohumite is associated with veins that are older than D2 and pre‐date D3. Veins that post‐date D3 are characterized by amphibole + chlorite or by serpentine. P–T conditions for S2 parageneses evaluated using two pseudosections for different bulk compositions suggest that these rocks experienced pressures >2.5 ± 0.3 GPa at temperatures slightly higher than 600 °C. The late epidote–amphibolite facies re‐equilibration associated with D3 and D4 developed during late syn‐exhumation deformation related to folding and testifies to a small temperature decrease. These results, which were integrated in the regional framework, suggest that different portions of the Zermatt–Saas Zone registered different P–T peak conditions and underwent different exhumation paths. In addition, the inferred P–T–d–t path suggests that the Valtournanche serpentinites re‐equilibrated close to the UHP conditions registered by the Cignana meta‐cherts. These results imply that tectonic slices exhumed after UHP metamorphism might be wider than previously reported or that small‐size UHP units, tectonically sampled during the Alpine convergence, are more abundant than those that have been detected to date.     

Time‐scale of deformation and intertectonic phases revealed by P–T–D–t relationships in the orogenic middle crust of the Orlica‐Śnieżnik Dome,Polish/Czech Central Sudetes

A section of the orogenic middle crust (Orlica‐?nie?nik Dome, Polish/Czech Central Sudetes) was examined to constrain the duration and significance of deformation (D) and intertectonic (I) phases. In the studied metasedimentary synform, three deformation events produced an initial subhorizontal foliation S1 (D₁), a subsequent subvertical foliation S2 (D₂) and a late subhorizontal axial planar cleavage S3 (D₃). The synform was intruded by pre‐, syn‐ and post‐D₂ granitoid sheets. Crystallization–deformation relationships in mica schist samples document I_1–2 garnet–staurolite growth, syn‐D₂ staurolite breakdown to garnet–biotite–sillimanite/andalusite, I_2–3 cordierite blastesis and late‐D₃ chlorite growth. Garnet porphyroblasts show a linear Mn–Ca decrease from the core to the inner rim, a zone of alternating Ca–Y‐ and P‐rich annuli in the inner rim, and a Ca‐poor outer rim. The Ca–Y‐rich annuli probably reflect the occurrence of the allanite‐to‐monazite transition at conditions of the staurolite isograd, whereas the Ca‐poor outer rim is ascribed to staurolite demise. The reconstructed P–T path, obtained by modelling the stability of parageneses and garnet zoning, documents near‐isobaric heating from ~4 kbar/485 °C to ~4.75 kbar/575 °C during I_1–2. This was followed by a progression to 4–5 kbar/580–625 °C and a subsequent pressure decrease to 3–4 kbar during D₂. Pressure decrease below 3 kbar is ascribed to I_2–3, whereas cooling below ~500 °C occurred during D₃. In the dated mica schist sample, garnet rims show strong Lu enrichment, oscillatory Lu zoning and a slight Ca increase. These features are also related to allanite breakdown coeval with staurolite appearance. As Lu‐rich garnet rims dominate the Lu–Hf budget, the 344 ± 3 Ma isochron age is ascribed to garnet crystallization at staurolite grade, near the end of I_1–2. For the dated sample of amphibole–biotite granitoid sheet, a Pb–Pb single zircon evaporation age of 353 ± 1 Ma is related to the onset of plutonic activity. The results suggest a possible Devonian age for D₁, and a Carboniferous burial‐exhumation cycle in mid‐crustal rocks that is broadly coeval with the exhumation of neighbouring HP rocks during D₂. In the light of published ages, a succession of telescoping stages with time spans decreasing from c. 10 to 2–3 Ma is proposed. The initially long period of tectonic quiescence (I_1–2 phase, c. 10 Ma) inferred in the middle crust contrasts with contemporaneous deformation at deeper levels and points to decoupled P–T–D histories within the orogenic wedge. An elevated gradient of ~30 °C km^?1 and assumed high heating rates of c. 20 °C Ma^?1 are explained by the protracted intrusion of granitoid sheets, with or without deformation, whereas fast vertical movements (2–3 Ma, D₂ phase) in the crust require the activity of deformation phases.     

Dynamic analysis of a cylindrical casing–cement structure in a poroelastic stratum

The transient response of a cylindrical casing–cement structure in a poroelastic stratum under dynamic radial tractions is one of the significant issues during the analysis of downhole operations and the selection of safe material. Based on the Biot theory and general elastic mechanics, this paper gives a set of exact solutions for radial displacement, stresses for the casing–cement system and the pore pressure of the infinite surrounding poroelastic stratum in the Laplace transform space. Solutions are presented for three different types of transient radial loadings acting on the surface of casing, i.e., suddenly applied constant load, gradually applied step load and triangular pulse load. Time domain solutions are obtained using a reliable numerical method of inverse Laplace transforms. A detailed parametric study about the transient response is presented both at the casing–cement interface and the cement–stratum interface, and the distributions of the pore pressure and the effective stresses in the stratum are also examined. Copyright © 2017 John Wiley & Sons, Ltd.     

The problem of the spatial variation of a spectral band ratio: a first order approach based on probability theory

Numerous efforts have been made to study how the spatial distribution of ground surface objects controls the image semivariogram. The present paper is centered on how the histograms and semivariograms of the individual bands x and y influence the spatial variation of a simple spectral ratio u = arctan(x/y). The image histogram of each separate band is described by a proper distribution. The exponential model is used to describe the semivariograms of x and y. Taking the first derivatives of the spectral ratio u for x and y and taking into account the mathematical behavior of the histograms of bands x and y, an approximate expression for the semivariogram γ _u of the spectral ratio is derived. This mathematical expression shows how the spatial variation of the spectral ratio depends on the standard deviations of the histograms, as well as the ranges of the semivariograms of x and y. Experimentation with multispectral images is then carried out and it shows that theoretical predictions agree, in qualitative terms, with real data. The results and conclusions of this paper may be useful in assessing the efficiency of various spectral band ratios and vegetation indices, which are often used in geological and environmental research (mapping of hydrothermal zones and land cover types).     

Metamorphic history of a syn‐convergent orogen‐parallel detachment: The South Tibetan detachment system,Bhutan Himalaya

The South Tibetan detachment system (STDS) in the Himalayan orogen is an example of normal‐sense displacement on an orogen‐parallel shear zone during lithospheric contraction. Here, in situ monazite U(–Th)–Pb geochronology is combined with metamorphic pressure and temperature estimates to constrain pressure–temperature–time (P–T–t) paths for both the hangingwall and footwall rocks of a Miocene ductile component of the STDS (outer STDS) now exposed in the eastern Himalaya. The outer STDS is located south of a younger, ductile/brittle component of the STDS (inner STDS), and is characterized by structurally upward decreasing metamorphic grade corresponding to a transition from sillimanite‐bearing Greater Himalayan sequence rocks in the footwall with garnet that preserves diffusive chemical zoning to staurolite‐bearing Chekha Group rocks in the hangingwall, with garnet that records prograde chemical zoning. Monazite ages indicate that prograde garnet growth in the footwall occurred prior to partial melting at 22.6 ± 0.4 Ma, and that peak temperatures were reached following c. 20.5 Ma. In contrast, peak temperatures were reached in the Chekha Group hangingwall by c. 22 Ma. Normal‐sense (top‐to‐the‐north) shearing in both the hangingwall and footwall followed peak metamorphism from c. 23 Ma until at least c. 16 Ma. Retrograde P–T–t paths are compatible with modelled P–T–t paths for an outer STDS analogue that is isolated from the inner STDS by intervening extrusion of a dome of mid‐crustal material.     

Nonlinear radial consolidation of vertical drains under a general time‐variable loading

By incorporating the nonlinear variation of a soil's compressibility and permeability during the process of consolidation, an analytical solution for the radial consolidation of vertical drains has been developed for a general time‐variable loading. The general solution was verified for the cases of instantaneous loading and ramp loading. Detailed solutions were further derived for two special loading schemes: multistage loading and preloading–unloading–reloading. The nonlinear consolidation behavior of a vertical drain subjected to these two types of loading schemes was then investigated by a parametric study. The results show that the loading rate, the ratio of the compressibility index to the permeability index (C_c/C_k), and the initial stress state have a significant influence on the consolidation rate. A smaller value of C_c/C_k, a larger initial stress, or a fast loading rate always leads to a rapid consolidation rate. During the unloading period, a negative excess pore water pressure may occur, and a slower unloading rate may reduce this negative value. Copyright © 2014 John Wiley & Sons, Ltd.     

How continuous and precise is the record of P–T paths? Insights from combined thermobarometry and thermodynamic modelling into subduction dynamics (Schistes Lustrés,W. Alps)

Pressure–temperature (P–T) paths as complete as possible and with a precision on the km‐scale or less are needed to further improve the knowledge of deformation, re‐equilibration processes and element/fluid transfer, in particular along subduction zones. This contribution attempts to (i) critically evaluate the precision and continuity with which metamorphic P–T histories are retrieved today and (ii) discuss implications for regional‐scale accretionary processes in subduction zones, through application to the Schistes Lustrés complex (Haute Maurienne, W. Alps). P–T estimates are compared and combined using several independent approaches: (i) from minerals assumed to be in textural equilibrium; (ii) from electron microprobe compositional maps; and (iii) from pseudosection modelling predictions. Multi‐equilibrium calculations were performed with tweequ and thermocalc, and pseudosections were built with Perple_X and Theriak/Domino. These P–T estimates were also compared with maximum temperatures (T_max) deduced from the Raman spectroscopy of carbonaceous matter. The different methods used here yield the peak of pressure for the lower structural unit of the Schistes Lustrés at 480 °C and 23 kbar and document the retrograde path for both the Median and Lower Units. The results show that P–T conditions are recorded almost continuously and can be determined with a precision of ±1 kbar and ±30 °C at best. This study underlines the complementarity of the various thermobarometric methods and demonstrates that precision could be increased by improving solid solution models for chlorite. Observed tectonic patterns, major lithological boundaries, pressure–temperature and T_max data suggest that underplating processes and early structural development played a key role in the Schistes Lustrés accretionary complex.     

Dual-thermodynamic estimation of stoichiometry and stability of solid solution end members in aqueous–solid solution systems

The dual-thermodynamic (DualTh) approach is shown to provide a useful alternative to other methods in: (i) forward modelling of equilibrium speciation, activities, and element partitioning in a heterogeneous system involving several variable-composition phases, such as the aqueous–solid solution system; (ii) estimation of interaction parameters of a non-ideal mixing model from known bulk compositions of coexisting aqueous and solid-solution phases; and (iii) retrieval of unknown stoichiometries and apparent standard chemical potentials of trace solid-solution end-members. Inverse-modelling tasks (ii) and (iii) can be performed when the solid solution of interest is shown experimentally to co-exist with the aqueous phase either in the equilibrium or at the minimum stoichiometric saturation state.DualTh calculations exploit the ability of Gibbs energy minimisation (GEM) algorithms to find simultaneously two numerical solutions of the isobaric–isothermal chemical equilibrium speciation problem: (1) primal solution x — a vector of amounts of components (species) in phases; and (2) dual solution u — a vector of chemical potentials of stoichiometry units (usually chemical elements and charge). Conversely, the chemical potential of a phase component can be found in two complementary ways: (i) primal via its standard-state potential, concentration and activity coefficient (the latter two are functions of the x vector); and (ii) dual through its formula stoichiometry multiplied by the u vector. The DualTh methods compare primal and dual values of the chemical potential in simple and straightforward equations that can be easily computed in a spreadsheet, or implemented in GEM geochemical modelling codes.     

Integrated garnet and zircon–titanite geochronology constrains the evolution of ultra‐high–pressure terranes: An example from the Sulu orogen

Dating ultra‐high–pressure (UHP) metamorphic rocks provides important timing constraints on deep subduction zone processes. Eclogites, deeply subducted rocks now exposed at the surface, undergo a wide range of metamorphic conditions (i.e. deep subduction and exhumation) and their mineralogy can preserve a detailed record of chronologic information of these dynamic processes. Here, we present an approach that integrates multiple radiogenic isotope systems in the same sample to provide a more complete timeline for the subduction–collision–exhumation processes, based on eclogites from the Dabie–Sulu orogenic belt in eastern China, one of the largest UHP terranes on Earth. In this study, we integrate garnet Lu–Hf and Sm–Nd ages with zircon and titanite U–Pb ages for three eclogite samples from the Sulu UHP terrane. We combine this age information with Zr‐in‐rutile temperature estimates, and relate these multiple chronometers to different P–T conditions. Two types of rutile, one present as inclusions in garnet and the other in the matrix, record the temperatures of UHP conditions and a hotter stage, subsequent to the peak pressure (‘hot exhumation') respectively. Garnet Lu–Hf ages (c. 238–235 Ma) record the initial prograde growth of garnet, while coupled Sm–Nd ages (c. 219–213 Ma) reflect cooling following hot exhumation. The maximum duration of UHP conditions is constrained by the age difference of these two systems in garnet (c. 235–220 Ma). Complementary zircon and titanite U–Pb ages of c. 235–230 Ma and c. 216–206 Ma provide further constraints on the timing of prograde metamorphism and the ‘cold exhumation' respectively. We demonstrate that timing of various metamorphic stages can thus be determined by employing complementary chronometers from the same samples. These age results, combined with published data from adjacent areas, show lateral diachroneity in the Dabie–Sulu orogeny. Three sub‐blocks are thus defined by progressively younger garnet ages: western Dabie (243–238 Ma), eastern Dabie–northern Sulu (238–235 Ma) and southern Sulu terranes (225–220 Ma), which possibly correlate to different crustal slices in the recently proposed subduction channel model. These observed lateral chronologic variations in a large UHP terrane can possibly be extended to other suture zones.