Wide-blocky veins explained by dependency of crystal growth rate on fracture surface type: Insights from phase-field modeling

Vein microstructures contain a wealth of information on coupled chemical and mechanical processes of fracturing, fluid transport, and crystal growth. Numerical simulations have been used for exploring the factors controlling the development of vein microstructures; however, they have not been quantitatively validated against natural veins. Here we combined phase-field modeling with microtextural analysis of previously unexplained wide-blocky calcite veins in natural limestone and of the fresh fracture surface in this limestone. Results show that the wide-blocky vein textures can only be reproduced if ∼ 10%–20% of crystals grow faster than the rest. This fraction corresponds to the amount of transgranularly broken grains that were observed on the experimental fracture surfaces, which are dominantly intergranular. We hy-pothesize that transgranular fractures allow faster growth of vein minerals due to the lack of clay coatings and other nucleation discontinuities that are common along intergranular cracks. Our simulation results show remarkable similarity to the natural veins and reproduce the nonlinear relationship between vein crystal width and vein aperture. This allows accurate simulations of crystal growth processes and related permeability evolution in fractured rocks.

A number of different approaches have been presented for simulating crystal growth kinematics in veins. Early models used geometric projections for exploring kinematics of polycrystals growing in a confined space (Urai et al., 1991;Dickson, 1993) to explain fibrous and elongateblocky crystal growth. Subsequent numerical methods including front-tracking algorithms of the Elle numerical simulation platform (www. http://elle.ws/; Bons, 2001;Hilgers et al., 2001;Nollet et al., 2005) and cellular automation-type models (Lander et al., 2008;Lander and Laubach, 2015) explored anisotropic crystal growth in antitaxial veins, where crystals can develop facets during epitaxial growth from a rough surface toward the inert vein wall. However, the vast majority of veins in rocks at depths of 1-10 km and at 100-350 °C are syntaxial: crystals grow on both fracture sides (Durney and Ramsay, 1973;Bons et al., 2012). More recently, phase-field models were introduced to model both syntaxial and antitaxial veins. Due to the different approach to moving boundaries, these models are more efficient, allowing simulations against dynamic, moving interfaces as crystals from both sides seal a syntaxial vein, and enabling three-dimensional (3-D) modeling (e.g., Ankit et al., 2015;Wendler et al., 2016;Kling et al., 2017;Prajapati et al., 2020). However, up to now, the phase-field models have not been quantitatively compared with natural vein microstructures.
This study fills this gap by applying the phasefield method to simulate enigmatic wide-blocky microstructures in natural calcite veins from Somerset, UK. We were able to replicate the natural microstructures quantitatively and provide new insights on vein formation mechanisms, showing how variations in transgranular and intergranular segments on fracture surfaces lead to heterogenous crystal growth, producing microstructures that are not predicted by previous models.

SAMPLES AND METHODS
We analyzed arrays of calcite microveins in Liassic (Lower Jurassic) limestones near Blue Anchor, Kilve, and Lilstock beaches in Somerset, UK (Figs. 1A and 1B). The veins are normal to bedding in the damage zone of regional normal faults (Caputo and Hancock, 1999;Nixon et al., 2019). Double-polished thin sections, cut perpendicular to the veins, were imaged and studied in the PetroScan Virtual Microscope (see Item S1 in the Supplemental Material 1 ) as well as by scanning electron microscope (SEM)-based cathodoluminescence (CL), secondary electron (SE), and backscattered electron (BSE) imaging (Item S1).
Analysis of fresh fracture surface was performed on 4.5 × 2 × 1 cm blocks of the same rocks broken by three-point loading to obtain mode I fractures parallel to existing veins. Fracture surfaces were imaged in SEM-SE, and transgranular fractures in these images were quantified using Fiji software (Schindelin et al., 2012).
Phase-field vein growth simulations in two and three dimensions (2-D and 3-D) were performed using a multiphase-field approach (e.g., Nestler et al., 2005) (Items S2-S4). Computational fluid dynamics analysis of 3-D microstructure was used to solve fluid flow and permeability changes during the sealing process of the fracture (e.g., Kling et al., 2017).

MICROSTRUCTURAL RESULTS
The host rock is a micritic limestone with ∼95% calcite grains, carbonate fossils, and ∼5% accessory minerals (quartz, dolomite, albite, clays, and pyrite) (Figs. 1C, 1D, and 2A). It contains sets of subparallel calcite veins with apertures from a few micrometers to several centimeters. The veins that are wider than 1 mm show elongate-blocky microstructures (Fisher and Brantley, 1992;Bons, 2001) with solid inclusion bands suggesting crack-seal processes and growth competition. We focused on microveins (aperture <1 mm) filled by laterally wide, blocky crystals whose formation mechanism remains unexplained (Figs. 1C and 1D). These microveins lack host-rock inclusion bands, suggesting a single crack-seal cycle (Ramsay, 1980 Experimental fracture surfaces consist mostly of intergranular, rather than transgranular, segments that follow nano-porous fossil, micritic grain, and accessory mineral boundaries, as recognized in SE images by their surface morphologies (Figs. 2B and 2C). In most cases, such grain boundaries show clay-mineral coatings on calcite. Only ∼10% of the fracture surface is transgranular, exposing clean calcite cleavage planes (Fig. 2D).

SETUP OF THE PHASE-FIELD MODEL
Based on our microstructural observations, we hypothesized that precipitation of vein calcite is faster on clean, transgranular microcracks formed along cleavage planes in the host-rock calcite, and slower on intergranular cracks that are coated by clay minerals (Lander et al., 2008;Ajdukiewicz and Larese, 2012;Williams et al., 2015). The host rock in the 2-D models was generated with a Voronoi algorithm ( Fig. 2E; Item S2), where grains have random crystallographic orientations to mimic micrite. The starting fracture shape was based on the geometry of one of the natural veins, with normalized apertures (D m ) varied from 1 to 16 in different simulations. Vein crystals were set to grow epitaxially on host-rock grains into the open fractures, as inferred from microstructures in the natural veins (Figs. 1C and 1E). Crystals were assigned anisotropic surface energy so they could develop facets and vertices corresponding with calcite crystal symmetry. The relative growth rate differences for different fracture surfaces were incorporated as a dimensionless factor ξ. Three types of surfaces were distinguished based on the analysis of natural samples and experimental fracture surfaces (Figs. 2A-2E; Table S3): (1) inert accessory minerals with no epitaxial calcite growth (ξ = 0), (2) clay-covered intergranular microcracks in calcite with slow growth rates (ξ = 1), and (3) clean transgranular microcracks along calcite cleavage planes with fast growth rates (ξ = 5-20). The proportions of these surfaces were based on measurements along profile lines on an experimental fracture surface (Fig. 2D). Only grains that underwent transgranular fracture are mirrored on both sides of the fracture wall in the model.

SIMULATION RESULTS
Curves of the W/D m relationship produced by numerical models show good correspondence with our measurements performed on natural veins (Fig. 3A). Models show that larger initial fracture apertures result in wider crystals, but there is a maximum width that vein crystals can reach (plateau) that is dependent on the proportion of "fast-growing" grains in comparison to "slow-growing" and "inert" grains on fracture surfaces. Furthermore, the growth-rate difference (ξ) determines the slope of W/D m , so that a larger ξ produces wider crystals. Figure 3B shows that, regardless of fracture aperture, wide-blocky microstructures form only in cases where ξ >1. If ξ = 1, the veins consist of blocky crystals with typical growth competition microstructures and a prominent median line. Increasing the initial fracture aperture results in fewer and wider grains at the median line (as in Prajapati et al. [2018]).
In simulations where ξ >1, fast-growing grains on transgranular cleavage planes outgrow slow-growing grains at early stages. However, for low ξ values and small fracture apertures (D m <2), most slow-growing grains can still reach the other side of the fracture wall, blocking the lateral expansion of the fast-growing grains (Item S3). If the growth rate difference is large (ξ ∼20), slow-growing grains are outcompeted almost immediately, and the width of the wideblocky grains remains constant regardless of the initial fracture aperture and/or further changes in ξ (Item S3).
Most of the simulations were performed in 2-D to allow reasonable computational time for setups with a large number of grains. However, 3-D runs are also possible, and we show the results of some such runs in Figure 4.  show the evolution of porosity and fluid flow at three stages of vein filling. At early stages (step 1), the fracture is highly permeable and fluid-flow rate is high. As filling progresses (step 2), the two parts of the broken, fast-growing crystals on transgranular fracture surfaces touch near the median plane of the fracture and form bridges, then continue to grow laterally. Fluid pathways and pores remain mostly interconnected until >50% of the grains have crossed the median line. After that, the porosity structure changes and permeability decreases as pores become isolated (step 3).

DISCUSSION
Our models show that incorporation of differential crystal growth rates for transgranular and intergranular fracture segments is a key for simulating the evolution of wide-blocky veins. We recognize that distinction of only two growth classes (fast and slow) is a simplification. In natural veins, intergranular fracture segments might have different amounts of clay coatings, or surface defects and transgranular segments might have cleavage steps, both of which affect the precipitation rate of vein minerals. Despite that, our results show a very good correspondence between modeled and natural microstructures, including the quantified trend of increasing crystal width with increasing vein aperture (Fig. 3A) as well as the microstructural characteristics, including distributions of bridge crystals, equidimensional crystals, and rims of small grains along the vein wall (Figs. 1C,1D,3B,and 3C).
In addition to the studied location in the UK, wide-blocky vein textures have been documented in micritic limestones from Sestri Levante, Italy (Bons et al., 2012), and the Oman Mountains (Holland and Urai, 2010) but never explained in detail. We infer that wide-blocky textures might be characteristic in clay-rich, micritic host rocks with medium grain-boundary cohesion so that fractures are mostly intergranular, with 25%-5% transgranular segments. However, the heterogeneity of the fracture surface, not the lithology or grain size of the host rock, is the controlling factor in these veins, therefore the observed crystal  The 3-D models in Figure 4 show how permeability and fluid-flow regime in the wideblocky veins are strongly affected by the development of crystal bridges early in the filling process. Although these bridges locally block fluid pathways and increase the tortuosity of the flow paths, high pore connectivity is maintained until late stages in the sealing process (Figs. 4B-4D). This is very different from veins where all crystals grow at the same rate and the flow paths are more direct due to the lack of crystal bridges, maintaining higher permeability in the early stages of crystal growth, as was reported by Kling et al. (2017) for quartz veins. However, the early bridge formation in wide-blocky veins might prevent fracture collapse, keeping veins open between fluid pulses.
We infer that once the veins become fully sealed, they are mechanically stronger than the micritic host rock due to the large, bridging crystals that connect both vein walls. Higher mechanical strength implies that new fractures form more easily in the host rock than by a failure of an existing vein. The observed abundance of subparallel arrays of microveins rather than larger veins with crack-seal textures suggests that once filled, the studied veins are rarely reactivated.
This study demonstrates the power and versatility of the phase-field method to model the evolution of veins. Its ability to incorporate crystallography, host-rock composition, realistic fracture geometries, differential growth rates, and thermodynamics allows the fine tuning of models to natural vein microstructures.