Prediction of mechanical property of sedimentary rock based on particle-to-particle parametric aggregation contact model

Abstract

To provide a method of, a computer system for, and a program for carrying out simulation of physical properties of a porous medium by a computer.SOLUTION: A method according to the present invention includes steps of receiving a micro CT3D image capturing a representative element volume of a porous medium, labelling the micro CT3D image as an individual voxel in accordance with a kind of mineral or fluid, and labelling the mineral voxel as one belonging to a separated and fixed individual particle. The method further includes steps of converting the labelled voxel to non-structuring conformal mesh representation for all particles, and applying the non-structuring conformal mesh representation to a parametric aggregation contact engine. The parametric aggregation contact engine executes a parametric aggregation contact model having critical separation δ0 conditioned in accordance with adjustable parameters and a compaction level.SELECTED DRAWING: Figure 5

Description

[Background technology]

[0001] Multi-component fluid flows through porous regions are a key feature of hydrocarbon reservoir rocks and represent a crucial input resource for the oil and gas industry, as well as other industries.

[0002] Numerical simulations of multi-component fluid flow in porous regions with complex solid structures are crucial for many industrial applications, such as enhanced oil recovery (EOR) and personal protective equipment (PPE). To achieve accurate simulation results, it is essential to capture relevant data from porous structures of all scales.

[0003] In the oil and gas industry, so-called digital rock simulations are being developed based on X-ray microtomography (micro-CT) imaging, which enables the capture of three-dimensional (3D) images of reservoir rock structures at the pore scale at the micrometer scale. Such images are used to perform computer simulations of simulated fluid (i.e., oil / water) flows under different production conditions.

[0004] However, digital rock simulations of rock mechanical properties have been considered challenging, given the inherent limitations of micro-CT images in capturing the degree of consolidation between particles within the rock, due to limitations in imaging resolution. Another drawback is that micro-CT images are typically acquired under ambient conditions that do not capture the actual net confinement pressure (NCS) on the pore structure.

[0005] Different levels of consolidation, compression, recrystallization, and diagenesis can generally produce a wide range of particle contact behaviors in sedimentary rocks, from loosely consolidated sandpacks to fully fused particles forming a single solid structure. Numerical methods in the literature typically use a single solid frame formed by mineral particles, which often overestimates the simulated rock stiffness compared to experimental measurements. Several ideas have been discussed that specify different properties for the contact region and the particle region. [Overview of the project]

[0006] A common feature of these conventional methods is that particle rearrangement is not permitted, which limits the particle-particle contact behavior of actual rocks and the fracture of rocks with large displacements.

[0007] According to one embodiment, a computer-based method for simulating the physical properties of a porous medium is to receive a micro-CT3D image capturing the volume of a representative element of the porous medium, the porous medium being defined as having mineral species and fluid species through individual particles and particle-to-particle contact; label the micro-CT3D image as individual voxels according to the mineral and fluid species; label the mineral species voxels as belonging to separated and fixed individual particles; convert the labeled voxels into an unstructured conformal mesh representation for all particles; and apply the unstructured conformal mesh representation to a parametric cohesive contact engine, the parametric cohesive contact engine being conditioned on a critical separation δ with adjustable parameters and a consolidation level. 0 This includes implementing and applying a parametric aggregate contact model having the following characteristics:

[0008] Embodiments of the method can include any one or more of the following features or other features, as disclosed herein.

[0009] The second differentiation is performed by the watershed method. Micro-CT 3D has a resolution sufficient to identify individual particles and the combined pore geometry. The particles are free particle contacts where only friction between the particles is modeled, and fixed particle contacts where the particles are completely fused. Converting the labeled voxelized 3D image further includes optimizing the elements of the same particles for finite element simulation, and the elements on each side of the contact between the particles fit together at the contact boundary without voids or overlaps.

[0010] The parametric aggregation contact engine relates the critical separation δ 0 to a consolidation level defined as C by the following equation,

Equation

[0011] = ∞, C = 1, and for δ 0 = 0, C = 0, and C represents both extremes of fixed and free particle-particle contacts. The porous medium is a porous rock sample, and the method includes applying the parametric aggregation contact engine to a finite element solver using specified strain / stress boundary conditions and particle-particle contacts by the parametric aggregation contact model, determining the contact behavior at different levels of particle coagulation to convert the parametric aggregation contact engine to a net confinement pressure model, and performing a flow simulation on the net confinement pressure model.

[0012] ​​Aspects also include a computer system and a computer program product.

[0013] One or more of the above aspects may provide one or more of the following advantages.

[0014] The methods described herein provide a workflow that more accurately models varying particle-particle cohesion to accurately simulate mechanical rock properties, while having the ability to recover the correct net confinement pressure (NCS) pore structure and enabling particle rearrangement capabilities, including fracture test simulations, to simulate petrophysical properties under correct conditions.

[0015] The described methods use a parameterized cohesive contact model that can capture the contact behavior of both extreme free and fixed particles, as well as intermediate degrees of particle compaction. The methods address problems including (1) modeling varying particle-particle cohesion to accurately simulate mechanical rock properties, (2) recovering the correct NCS pore structure to simulate petrophysical properties under field conditions, and (3) enabling particle rearrangement, including fracture test simulations.

[0016] The proposed method can be used to simulate petrophysical properties in NCS, such as fluid flow in porous rocks, and is applicable to the lattice Boltzmann method and other computational fluid dynamics methods including the finite volume method, the finite element method, etc.

[0017] Other features and advantages of the present invention will become apparent from the following description and the claims.

Brief Description of the Drawings

[0018] [Figure 1] A diagram showing a system for simulation of micromechanics modeling using a model of a porous structure. [Figure 2A]This figure shows the aggregate contact behavior, and this graph shows the typical tensile-separation response. [Figure 2B] This figure shows the aggregate contact behavior, and this graph shows the typical tensile-separation response. [Figure 3A-C] The tensile-separation response curves for stationary particles, intermediate particle contact, and free particle contact are shown, respectively. [Figure 4A] This figure shows a particle mesh with three-grain packing, and the particle behavior at fixed (C=1), free (C=0), and intermediate aggregation contact levels (C=0.5). [Figure 4B] This figure shows a particle mesh with three-grain packing, and the particle behavior at fixed (C=1), free (C=0), and intermediate aggregation contact levels (C=0.5). [Figure 5] This is a flowchart illustrating the parametric agglomeration contact engine process. [Figure 6A-C] This figure shows a binary microCT image of a sphere-packed structure, segmented particles indicated by shades of gray, and a particle mesh for the FE solver. [Figure 7A-C] This figure shows images of Grosmont Carbonate against a binary microCT image, segmented particles indicated by shades of gray, and a particle mesh for the FE solver (only the outer edges are shown for better visualization). [Figure 8A-C] This figure shows the Fontainebleau model for binary microCT images, segmented particles indicated by shades of gray, and a particle mesh for the FE solver. [Figure 9A] This figure shows graphs of the stress-strain response for various consolidation levels, and the normalized modulus of elasticity as a function of the consolidation level. [Figure 9B] This figure shows graphs of the stress-strain response for various consolidation levels, and the normalized modulus of elasticity as a function of the consolidation level. [Figure 10A-C]This figure shows binary microCT images of Fontainebleau sandstone, segmented particles, and images of the particle mesh. [Figure 11A] This figure shows the stress-strain response for various consolidation levels C, and the normalized modulus of elasticity as a function of the consolidation level. [Figure 11B] This figure shows the stress-strain response for various consolidation levels C, and the normalized modulus of elasticity as a function of the consolidation level. [Figure 12A-B] This figure shows a voxelized image of segmented particles (Figure 12A) and a particle mesh for FE simulation of a fracture test (Figure 12B). [Figure 13] This figure shows the evolution of the vertical displacement field as the Fontainebleau model is compressed vertically. [Figure 14] This figure shows the normal stress as a function of normal strain for various consolidation levels C. [Figure 15A-C] This figure shows the bulk modulus as a function of porosity for various samples. [Modes for carrying out the invention]

[0019] This application describes a parametric agglomeration contact model run in a parametric agglomeration engine to simulate a wide range of sedimentary rocks, from unconsolidated to well-consolidated, and a benchmark study on sandstone samples, where the degree of consolidation of the parametric agglomeration contact engine is compared with laboratory-measured moduli. In addition, numerical uniaxial compression tests are performed to demonstrate the effect of adequately capturing the degree of consolidation on rock strength and fracture patterns.

[0020] Referring to Figure 1, a system 10 is shown for simulating the physical properties of a multiscale porous material using an aggregate contact model that captures both extreme contact behaviors, namely free and stationary particles, as well as intermediate degrees of particle consolidation. Finite element (FE) simulations are performed on the results of the parametric contact model to create a net confinement pressure (NCS) model as input to the flow simulation. The flow simulation can be used for various purposes, such as simulating the "wetness restoration" or "aging" process, i.e., "numerical aging," which is representative of subsurface reservoir conditions. Other simulations may include effects such as vapor flow on PPE. The flow simulation can use lattice Boltzmann or other computational fluid dynamics methods.

[0021] Generally, in this implementation, system 10 includes a client system 14, which includes a network 12, e.g., the Internet or another network, and a mesh supplied by the user, based on a client-server or cloud-based architecture, and a server system 16, which is functionally coupled to the client system 14 and implemented as a large-scale parallel computing system (standalone or cloud-based). The server system 16 includes memory 18, a bus system 22, interfaces 20 (e.g., user interface / network interface / display or monitor interface, etc.), and processing devices 24.

[0022] Memory 18 contains a parametric agglomeration contact engine 32 that operates on a digital display 32' of a physical rock sample. The digital display 32' of the physical rock sample represents the pore space, particles, and particle boundaries of the physical rock sample. The parametric agglomeration contact engine 32 can be used to simulate bonded interfaces with and without the possibility of bond damage and fracture. The parametric agglomeration contact engine 32 includes a linear elastic tensile-separation model 32a, a damage initiation criterion model 32b, and a damage propagation model 32c, all of which enable an accurate description of particle contact behavior.

[0023] The contact model can be used in conjunction with the FE analysis engine 33 to supply an NCS model as input to the flow simulation engine 34 for simulating multi-phase flow behavior occurring through reservoir rock adjacent to, for example, a gas well or oil well (e.g., a drilling rig 37). Determining the multi-phase flow behavior may include, for example, changes in the wettability of the physical rock sample.

[0024] The digital display 32' of the physical rock sample can be a third-party application running on the same or a different system as the server 16. System 10 simply requires the digital display 32' of the physical rock sample, and as a result the agglomeration contact engine 34 digitally provides the digital display 32' of the physical rock sample.

[0025] One method for obtaining a digital representation of rock samples is to acquire the representation from 3D images generated from micro-CT scans of rock samples, for example. Micro-CT 3D images are voxelized and used as input to identify individual particles and bound pore geometry of the rock. Limitations of micro-CT images for properly capturing micromechanical rock models concern insufficient contrast for identifying different mineral particles and particle-particle contacts between the same mineral particles. Particle mineralogy and contact identification can be improved by complementing 3D micro-CT imaging with higher 2D resolution, as well as mineralogical imaging such as scanning electron microscopy (SEM) and energy-dispersive X-ray spectroscopy (EDS).

[0026] To establish a finite element (FE) micromechanical rock model from micro-CT data, two preprocessing steps are required: (1) particle-to-particle separation and (2) conformal meshing of particle contacts. For a description of these steps, see Z. Sun, R. Salazar-Tio, L. Duranti, B. Crouse, A. Fager, and G. Balasubramanian, Prediction of Rock Elastic Moduli Based on a Micromechanical Finite Element Model, Comput. Geotech. 135, 104149 (2021). A brief explanation is given below.

[0027] For particle-to-particle separation, a direct binary thresholding operation is applied to micro-CT to classify voxels as either pores or solids. After this thresholding operation, solid voxels are reclassified using an integer index that indicates which particle the voxel belongs to. For this purpose, we use a type called the watershed method, which is used to separate tangent objects in binary images. For further details, see D. Legland, I. Arganda-Carreras, and P. Andrey, Morpho Lib J: Integrated Library and Plugins for Mathematical Morphology with Image J, Bioinformatics 32, 3532 (2016). The watershed method first calculates a distance transformation of the 3D image, where the Euclidean distance value to the nearest pore / solid interface is recorded for each solid voxel. If this is considered a topographic depth map, the deeper parts of the image are the centers of the objects. Watershed classification considers each solid voxel by the center that rolls into this inverse distance-transformed topographic map. In the final particle-segmented 3D image, each particle voxel has a corresponding particle index value, while pore voxels are labeled as 0.

[0028] To obtain an initial static solution for the segmented particles, a conformal mesh is prepared at the particle-particle contact. The conformal mesh indicates that there is neither separation nor overclosure between any two contacting particles. The labeled voxelized 3D image is converted to an unstructured mesh display, where elements of the same particle are suitable for finite element (FE) simulation, and elements on each side of the contact between particles fit together perfectly without gaps or overlaps at the contact boundary. For a more thorough explanation, see Z. Sun, R. Salazar-Tio, L. Duranti, B. Crouse, A. Fager, and G. Balasubramanian, Prediction of Rock Elastic Moduli Based on a Micromechanical Finite Element Model, Comput. Geotech. 135, 104149 (2021). The mesh is generated together for all particles while maintaining indices representing the segmented particles, resulting in a conformal mesh at the particle-particle contact. The mesh can consist of either two separate surfaces or one combined interface, which corresponds to two case scenarios: free or stationary particle-particle contact.

[0029] The meshed model, along with the specified strain / stress boundary conditions and particle-particle contact model, is input into the FE solver 33. For a more complete explanation, see S. Qin, R. McLendon, V. Oancea, and AM Beese, Micromechanics of Multiaxial Plasticity of DP600: Experiments and Microstructural Deformation Modeling, Mater. Sci. Eng. A 721, 168 (2018); Abaqus, Abaqus 6.10 Online Documentation, Abaqus User Subroutines Ref. Man. (2010).

[0030] Parametric Cohesive Contact Engine The general parameterization of the basic contact model is used to capture different levels of contact behavior in particle solidification, including two extreme scenarios: free particles, where only friction between particles is modeled, and fixed particles, where particles are completely fused. From the perspective of FE simulation, free particle contact introduces replication nodes and elements that allow for particle rearrangement, while fixed particle contact models a solid framework as a whole.

[0031] After the results of the parameterized cohesive contact model are applied to the FE solver 33, the FE results are subsequently applied to, for example, a fluid simulation 34.

[0032] For details of the fluid simulation process 34, see U.S. Patent Application No. 15 / 880,867, filed January 26, 2018, entitled "Multi-Phase Flow Visualizations Based On Fluid Occupation Time," which has been assigned to the assignee of this application. For other exemplary cases, see U.S. Patent Application No. 16 / 243,285, filed January 9, 2019, entitled "Determining Fluid Flow Characteristics Of Porous Mediums," or U.S. Patent Application No. 16 / 545,387, filed August 20, 2019, entitled "Determination Of Oil Removed By Gas Via Miscible Displacement In Reservoir Rock," which has been assigned to the assignee of this application.

[0033] Memory 18 may store parameters used by engines 32, 33, and 34. Specifically, for one or more of the above applications of the subject matter of this disclosure, the parameters used may include particle surface properties obtained by specifying the mineral species to the particles to determine the surface properties for each of those mineral species, as well as surface texture and roughness properties. Memory 18 may also store parameters such as fluid properties, e.g., fluid density and viscosity of each expected fluid (e.g., two or more of water, gas, and oil), as well as fluid-fluid interface tension properties. Memory 18 also stores parameters such as chemical composition data of the fluid and affinity data of the fluid components for specific mineral species. Memory 18 also stores separation pressures for each mineral species in combination with the fluid used by the flow simulation engine 34. In addition, reservoir pressure and temperature data are also stored. The mineral species being evaluated may be those found or expected in the actual site of the reservoir.

[0034] The simulation engine 34 may include a module for setting up a rock sample simulation environment, a module for performing drainage / swelling simulations, and a module for calculating the local curvature of the surface within the pore space. The simulation engine 34 may also include a module 50 for processing multiscale porous materials that have regions with a low degree of decomposition.

[0035] System 10 accesses a data repository 38 that stores 3D meshes, coordinate systems, and libraries that can be used to apply meshes to 3D images generated from micro-CT scans of rock samples, and for fluid simulations such as drainage / swelling simulations using computational fluid dynamics or any well-known computational technique such as the so-called lattice Boltzmann method.

[0036] The concept of "bonded particles" is widely adopted in the discrete element method (DEM), which simulates each individual particle as a discrete rigid body, as discussed in M. Obermayr, K. Dressler, C. Vrettos, and P. Eberhard, A Bonded-Particle Model for Cemented Sand, Comput. Geotech. 49, 299 (2013), DO Potyondy and PA Cundall, A Bonded-Particle Model for Rock, Int. J. Rock Mech. Min. Sci. 41, 1329 (2004), or Z. Sun, DN Espinoza, and MT Balhoff, Discrete Element Modeling of Indentation Tests to Investigate Mechanisms of CO2 -Related Chemo-Mechanical Rock Alteration, J. Geophys. Res. Solid Earth 121, 7867 (2016).

[0037] Similarly, FE simulations can simulate bonded particle interfaces based on the cohesive contact model, both when there is potential for bond damage and failure and when there is not (Figure 2A). The parametric cohesive contact engine 32 includes a linear elastic traction-separation model 32a, a damage initiation criterion 32b, and damage evolution 32c, which enable an accurate description of the contact behavior. What is discussed below is an overview of the contact model. Further details regarding the contact model can be found in the FE solver accompanying documentation (see Abaqus 6.10 Online Documentation, Abaqus User Subroutines Ref. Man. (2010), H. Harkness, G. Ang, P. Vijalapura, and D. Cojocaru, Contact Stress Accuracy with Robust and Broadly Applicable Implicit Contact Algorithm, in NWC (Nafems World Congress) (2011)).

[0038] The parametric cohesive contact engine 32 assumes a traction-separation behavior that is initially elastic and continues with damage initiation and evolution. The elastic matrix K relates the normal and shear stresses to the normal and shear separations.

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[0039] Next, referring to Figure 2A, an overview of the cohesive contact behavior is shown, and Figure 2B shows a typical tensile-separation response plot for the parametric cohesive contact engine 32. The parameters of the parametric cohesive contact engine 32 are the stiffness K in the normal and shear directions relative to the elastic behavior. RR and K SS , critical separation δ R 0 and δ S 0, as well as effective separation δ in total destruction relative to damage behavior m f The parametric cohesive contact engine 32 includes stiffness normal and shear direction K in order to reduce the number of parameters. RR =K SS , δ R 0 =δ S 0 =δ 0 , and δ m f = 2.5 × δ m 0 Assume the stiffness K in the normal and shear directions. RR and K SS It is predetermined to reproduce the mechanical behavior of stationary particle-particle contact when no damage occurs. That is, to ensure that the results are consistent with stationary particle-particle contact, K RR and K SS It is possible to determine the value of [this].

[0040] Therefore, the parametric agglomeration contact engine 32 is critically separated δ 0 It has only one adjustable parameter, which is conditioned by the consolidation level "C". The parametric agglomeration contact engine 32 is critically separated δ 0 This is related to the consolidation level C as follows.

number

[0041] Next, referring to Figures 3A to 3C, typical tensile-separation response curves for various consolidation levels are shown.

[0042] Figure 3A shows the tensile-separation response curves for fixed particle contact, where the curves are C=1 and δ 0 The parameter has a value of =∞. Figure 3B shows the tensile-separation response curve for intermediate particle contact, where C∈(0,1). Figure 3C shows the tensile-separation response curve for free particle contact, where C=0. These tensile-separation curves for various consolidation levels are quantified by the parameter C. Figures 3A-3C represent the parametric agglomeration contact engine 32 and show a general parameterization that captures the contact behavior at different levels of particle agglomeration. This parametric agglomeration contact engine 32 enables particle rearrangement capabilities instead of modeling the particle space as a single solid, whole framework of particles.

[0043] The agglomeration contact model 32 can simulate linear moduli comparable to experimental values ​​and can simulate significant compression found in situ, which is difficult to capture using micro-CT imaging under ambient conditions. The parametric agglomeration contact engine 32 can simulate a wide range of sedimentary rocks, from unconsolidated to well-consolidated rocks.

[0044] Next, referring to Figures 4A and 4B, a particle mesh packed with three particles is shown (Figure 4A), where the particles are distinguished by shades of gray. Figure 4B shows that the parametric agglomeration contact engine 32 can reproduce the behavior of fixed particle contact 42 and free particle contact 44 by changing the consolidation level C.

[0045] To validate this proposed simplification, we first test the parametric agglomeration contact engine through a simple scenario of three-particle packing. The particles are extracted from micro-CT images of a Fontainebleau sandstone model (see FDE Latief, B. Biswal, U. Fauzi, and R. Hilfer, Continuum Reconstruction of the Pore Scale Microstructure for Fontainebleau Sandstone, Phys. A Stat. Mech. Its Appl. 389, 1607 (2010)). The lower part of the packing is fixed, and a tensile stress is applied acting on the upper part of the packing.

[0046] Figures 4A and 4B show that the parametric agglomeration contact engine 32 can reproduce the fixed and free particle contact behavior when C=1 and C=0, respectively. Figure 4A shows the particle mesh used in the FE simulation. Figure 4B shows three simulation results for the agglomeration contact model with different consolidation levels C, as well as two simulation results for fixed and free particle contact (agglomeration contact is not applied). When C=0, the particles become easily separated by tensile stress. External work increases as a result of particle displacement due to the applied force. When C=1, the three particles behave as a solid framework without particle separation, and external work causes tension and overall displacement. When C=0.5, the result exhibits a transition from fixed particle contact to free particle contact due to bond damage and fracture. The transition is not very smooth due to the fact that there are only two bonded surfaces in the packing.

[0047] Referring to Figure 5, process 50 is shown to accurately model the changing particle-particle solidification in order to accurately simulate mechanical rock properties. Process 50, followed by process 33, restores the correct NCS pore geometry to simulate petrophysical properties (petroleum containing rock samples) under correct conditions. Process 50 also enables particle rearrangement capabilities, including fracture test simulations.

[0048] Process 50 receives a micro-CT3D image as input in 52, which is voxelized in 54, meaning that this is fitted using a computer-generated mesh to yield a 3D voxelized image. The voxelized image has sufficient resolution to identify individual particles and pore geometry within the rock and captures the representative elemental volume of the rock.

[0049] Process 50 labels the first segment of the micro-CT image in 56, labeling individual voxels as belonging to different mineral and fluid species. Process 50 labels the second segment of the mineral species voxels in 58, labeling them as belonging to another individual particle. Such processes can be carried out by the so-called watershed method and derived methods (see the discussion above).

[0050] The labeled voxelized 3D images are converted to an unstructured mesh representation, e.g., a particle mesh model, where elements of the same particles are optimized for FE simulation, and the elements on each side of the contact between particles fit together without gaps or overlaps at the contact boundary (see discussion above). The particle mesh model is input into the Abacus simulation engine, and strain / stress boundary conditions are set using the micromechanics plugin (see Qin et al., 2018 above).

[0051] The general parameterization of the parametric agglomeration contact engine 32 is used to capture different levels of contact behavior of particle solidification, including two extreme scenarios: free particles where only friction between particles is modeled (see Figure 3C) and fixed particles where particles are completely fused (see Figure 3A). Free particle contact introduces replication nodes and elements that allow particle rearrangement, while fixed particle contact models a solid framework as a whole. Strain / stress boundary conditions from micromechanics plug-ins lead to the parameterization of the parametric agglomeration contact engine 32. The general parameterization of the parametric agglomeration contact engine 32 is used to capture different levels of contact behavior of particle solidification between these two poles (see Figure 3B).

[0052] Below, we discuss the results of tests performed on sandstone samples to calibrate the degree of consolidation, compared with the elastic modulus measured in the laboratory. Conventional simulations that do not include particle contact modeling tend to overestimate the elastic modulus. The parametric agglomeration contact engine 32 discussed herein accurately captures particle contact behavior through parameter C.

[0053] Test case results Next, referring to Figures 6A-6C, the simulation test results for rock samples containing sphere packing are shown. Monodisperse sphere packing is a very good example of free-particle-particle contact behavior. Numerical simulations that are directly based on image voxels and allow only fixed-particle-particle contact tend to overestimate the modulus and stiffness. Micromechanical finite element models allow for particle rearrangement and, as a result, produce moduli that are very close to "laboratory" data (solutions from fine-grained dynamic simulations).

[0054] Figure 6A is a binary microCT image of the original sphere packing, Figure 6B shows the segmented particles indicated by shades of gray, and Figure 6C is the particle mesh for the FE solver. Using this sphere packing model, we demonstrate that a parametric agglomeration contact engine can mimic free particle contact. According to Z. Sun, R. Salazar-Tio, L. Duranti, B. Crouse, A. Fager, and G. Balasubramanian, Prediction of Rock Elastic Moduli Based on a Micromechanical Finite Element Model, Comput. Geotech. 135, 104149 (2021), the material properties of quartz for sphere packing are as follows, with a density of 2.65 g / cm³. 3 The bulk modulus is 37.0 GPa, and the shear modulus is 44.0 GPa. Table 1 summarizes the specifications for the sphere-packing model. Table 1. Specifications for sphere filling

[0055] [Table 1]

[0056] A parametric agglomeration contact engine has two parameters: a consolidation level C and a stiffness K. The consolidation level C is equal to 0 for free particle contact. The stiffness K is predetermined to replicate the mechanical behavior of stationary particle contact.

[0057] A hydrostatic compression test may be performed to measure the bulk modulus, and a shear test may be performed to measure the shear modulus. Table 2 summarizes the results. Table 2. Volume and shear modulus of sphere-filled structures [Table 2] A parametric aggregate contact engine at C=0 yields an elastic modulus very similar to that of free-particle contact. See Z. Sun, R. Salazar-Tio, L. Duranti, B. Crouse, A. Fager, and G. Balasubramanian, Prediction of Rock Elastic Moduli Based on a Micromechanical Finite Element Model, Comput. Geotech. 135, 104149 (2021). The results are not expected to be identical, considering that numerical simulations are based on different contact mechanisms.

[0058] Grossmont carbonates are a good example of stationary particle-particle contact behavior, given that carbonates are typically recrystallized through diagenetic processes. Therefore, all particles behave as a solid framework. In this section, we perform particle partitioning to demonstrate that aggregate contact can mimic the behavior of stationary particle contact when C=1.

[0059] Figure 7A is the original binary microCT image of Grossmont carbonate, Figure 7B shows the segmented particles indicated by different shades of gray, and Figure 7C is the particle mesh for the FE solver (only the outer edges are shown for better visualization).

[0060] Table 3 summarizes the specifications for Grossmont carbonate. The material properties for Grossmont carbonate with 50% calcite and 50% dolomite are shown, with a density of 2.79 g / cm³. 3 The bulk modulus is 81.6 GPa, and the shear modulus is 36.7 GPa. The consolidation level C is equal to 1 for contact with fixed particles, and the stiffness K is a calibrated parameter. Table 3. Specifications of Grosmont carbonate [Table 3]

[0061] Table 4 shows a comparison of this model with our previous simulations, demonstrating that the parametric aggregate contact engine at C=1 yields an elastic modulus very similar to that of a stationary particle contact engine. Table 4. Volume and shear modulus of Grosmont carbonates [Table 4]

[0062] Many sedimentary rock samples can exhibit different levels of consolidation because the sediments are compressed and joined under various geotechnical and geochemical conditions. In this section, we focus on an intermediate-consolidated rock, Fontainebleau Sandstone.

[0063] For illustrative purposes, a parametric agglomeration contact engine 32 is applied to the Fontainebleau model, using voxels with a relatively small volume of 100 × 100 × 100 and a voxel size of 7.3 μm. In Figures 8A-8C, Figure 8A shows a binary voxelized image, Figure 8B shows the segmented particles, and Figure 8C shows the conformal particle mesh used in the FE solver. The specified material properties are for quartz, as in the case of sphere packing.

[0064] Uniaxial strain tests are performed on this Fontainebleau model, which requires incremental strain in the vertical direction and zero strain in the horizontal direction. Figure 9A shows the stress-strain behavior at various consolidation levels C. At C=0 and C=1, the agglomeration contact model 32 can reproduce the modulus of free and stationary particle contact, respectively. The modulus shows an increase as the consolidation level C increases.

[0065] Figure 9B shows the normalized modulus, [M(C)-M(0)] / [M(1)-M(0)], as a function of the consolidation level. The modulus tends to asymptotically approach the two ends. The data can be well fitted by the logistic function.

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[0066] Using knowledge gained from simulations on a small Fontainebleau model, the parametric agglomeration contact engine 32 is applied to a larger Fontainebleau sandstone micro-CT model with 288 × 288 × 300 voxels. The voxel size is 7.5 μm. The sample porosity is 14.7%. The material properties are specified as quartz. Figure 10A shows the voxelized image, Figure 10B shows the segmented particles, and Figure 10C shows the conformal particle mesh.

[0067] Static compression tests are performed to measure the bulk modulus. Figure 11A shows the stress-strain response for various consolidation levels C. Figure 11B shows the normalized modulus as a function of consolidation level. The results can be fitted by a logistic function (Equation 8) that describes the stress-strain behavior for various consolidation levels C. The parametric agglomeration contact engine can reproduce the moduli of fixed and free particle contact when C=1 and C=0, respectively. Two intermediate consolidation levels result in a modulus between them. The parameters k and x0 of the above logistic fitting function (Equation 8) are 15.9 and 0.5, respectively.

[0068] The experimentally measured bulk modulus is approximately 22.2 GPa, which corresponds to a consolidation level C of 0.56, based on the simulation results in Figure 11B. A shear test can be performed to measure the shear modulus when C=0.56, which yields a shear modulus of 24.9 GPa, which is also close to the shear modulus measured in the laboratory.

[0069] The parameterized agglomeration contact model 32 allows users to select the consolidation level as an input parameter based on their knowledge of the diagenetic history of the sedimentary rock sample.

[0070] Rock fracture behavior clearly demonstrates the importance of accurately capturing the consolidation level. Uniaxial compression tests are widely used to determine the uniaxial compressive strength (UCS) and deformability of rock specimens. The test specimen is loaded axially without radial confinement. The height / diameter ratio of the specimen is typically 2-3. A ratio smaller than 2 results in a higher uniaxial compressive strength.

[0071] In this section, uniaxial compression tests are performed on cylindrical volumes from the Fontainebleau model used above at various consolidation levels C. All other model parameters remain constant. The sample has a height of 500 voxels and a diameter of 250 voxels. The voxel size is 7.3 μm. Figure 12A shows the voxelized image of the segmented particles, and Figure 12B shows the corresponding particle mesh used in the FE simulation of the uniaxial compression test.

[0072] Figure 13 shows the progression of the vertical displacement field from the initial state to the intermediate state as the sample is compressed vertically (C=0.2). The Fontainebleau sample is initially undamaged and eventually ruptures due to the shear zone.

[0073] Figure 14 shows the normal stress as a function of normal strain for various consolidation levels C. When C=1, the particles are fused together, and the sample behaves elastically with infinite strength. When C=0, the particles are not consolidated, and the sample fractures with friction as the only remaining force when the normal stress reaches the uniaxial compressive strength. Intermediate values ​​of C result in stress-strain curves that fall between the two end-member scenarios.

[0074] When C is relatively small (e.g., C=0.2), the stress-strain curve approaches the result for C=0 due to the asymptotic nature of equation (8) and shown in Figures 9B and 11B. The simulation results show that the rock fracture behavior is conditioned by the consolidation level, which is parameterized in the cohesive contact model 32.

[0075] Figure 14 shows the normal stress as a function of normal strain for various consolidation levels C. Larger values ​​of C result in greater uniaxial strength and modulus of elasticity.

[0076] Table 6 summarizes the Young's modulus and uniaxial compressive strength based on the results shown in Figure 14. Table 6. Volume and shear modulus of Fontainebleau sandstone [Table 5]

[0077] A higher consolidation level parameter C results in a greater Young's modulus and uniaxial compressive strength. Numerical simulations for this particular Fontainebleau model yield reasonable values ​​for Young's modulus and uniaxial compressive strength compared to referenced experimental data.

[0078] In addition to the level of consolidation, contact friction is another parameter for rock fracture. Applying the so-called "Coulomb friction model," which relates the maximum allowable frictional stress across particle contact to the contact pressure between particles, results in a frictional stress at which contact slippage begins, proportional to the product of the coefficient of friction and the contact pressure. A coefficient of friction of 0.2 is assumed. In addition, non-smooth contact boundaries can also add friction to particle contact.

[0079] Next, referring to Figures 15A–15C, these figures show the measured and simulated bulk moduli for sphere packing (Figure 15A), Grossmont carbonate (Figure 15B), and Fontainebleau sandstone (Figure 15C).

[0080] For sphere packing (Figure 15A), the particle contact model allows for free-particle-particle contact, resulting in bulk moduli that are very close to laboratory data compared to other numerical simulations that only allow fixed-particle-particle contact. Sphere packing is a very good example of free-particle-particle contact behavior. Numerical simulations that are directly based on image voxels and only allow fixed-particle-particle contact tend to overestimate elastic properties.

[0081] The described micromechanical finite element model allows for particle rearrangement, resulting in elastic moduli that are very close to "laboratory" data (solutions from fine-grained dynamic simulations). See Z. Sun, R. Salazar-Tio, L. Duranti, B. Crouse, A. Fager, and G. Balasubramanian, Prediction of Rock Elastic Moduli Based on a Micromechanical Finite Element Model, Comput. Geotech. 135, 104149 (2021).

[0082] For Grossmont carbonate (Figure 15B), all conventional numerical methods and particle contact models assume stationary particle-particle contact and yield similar predictions for the bulk modulus.

[0083] For the Fontainebleau sandstone (Figure 15C), numerical simulations were performed for both free and fixed particle-particle contact, serving as two end-member case scenarios. The actual particle-particle contact behavior of the Fontainebleau sandstone lies between these two end-member scenarios, which is accurately reproduced by our parametric agglomeration contact model using C=0.56. Since sediments are compressed and joined under a variety of geotechnical and geochemical conditions, many sedimentary rock samples can exhibit different levels of consolidation.

[0084] In Figures 15B and 15C, the upper and lower dashed lines serve as a reference for the numerical results. The upper and lower dashed lines correspond to the upper and modified lower Hasin-Shtrikman bounds (soft sand model). The solid lines represent the hard sand model.

[0085] Embodiments of the subject matter and the functional operations described herein may be implemented in digital electronic circuits, tangibly embodied computer software or firmware, computer hardware (including structures disclosed herein and their structural equivalents), or one or more combinations thereof. Embodiments of the subject matter described herein may be implemented as one or more computer programs (i.e., one or more modules of computer program instructions encoded on a tangible, non-temporary program carrier for execution by or control of the operation of a data processing device). The computer storage medium may be a machine-readable storage device, a machine-readable storage board, a random or serial access memory device, or one or more combinations thereof.

[0086] Computer programs, also called or described as programs, software, software applications, modules, software modules, scripts, or code, can be written in any form of a programming language, including compiled or interpreted languages, or declarative or procedural languages, and can be deployed as standalone programs or in any form including modules, components, subroutines, or other units suitable for use in a computing environment. Computer programs may, but are not required, correspond to files in a file system. Programs may be stored in part of files that hold other programs or data (e.g., in a markup language document, in a single file dedicated to the program, or in one or more scripts stored in a group of coordinated files, e.g., files that store one or more modules, subprograms, or parts of code). Computer programs can be deployed so that the program runs on one computer, or on multiple computers located in one site, or distributed across multiple sites and interconnected by a data communication network.

[0087] A computer suitable for running computer programs can be based on a general-purpose or dedicated microprocessor, or both, or any other type of central processing unit. Generally, the central processing unit will receive instructions and data from read-only memory or random-access memory, or both. Essential elements of a computer are the central processing unit that issues or executes instructions, as well as one or more memory devices for storing instructions and data. Generally, a computer will also include one or more mass storage devices (e.g., magnetic, magneto-optical disks, or optical disks) for storing data, or will be operablely coupled to receive data from or transfer data to them, although it is not essential for a computer to have such devices.

[0088] Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile memory on media and memory devices, including, for example, semiconductor memory devices (e.g., EPROM, EEPROM, and flash memory devices), magnetic disks (e.g., internal hard disks or removable disks), magneto-optical disks, and CD-ROM and DVD-ROM disks. Processors and memory can be complemented by or incorporated into dedicated logic circuits.

Claims

1. A method implemented by a computer for simulating physical properties of a porous medium, comprising: receiving, by the computer, a micro-CT 3D image capturing a representative elementary volume of the porous medium, wherein the porous medium is defined as having mineral species and fluid species by individual particles and particle-to-particle contacts; labeling, by the computer, the micro-CT 3D image as individual voxels according to mineral and fluid species; labeling, by the computer, the mineral species voxels as belonging to separated and fixed individual particles; converting, by the computer, the labeled voxels into an unstructured conformal mesh representation for all particles; applying the unstructured conformal mesh representation to a parametric cohesive contact engine, wherein the parametric cohesive contact engine executes a parametric cohesive contact model having adjustable parameters and a critical separation δ conditioned according to a consolidation level 0 having a length; and applying. including, wherein the critical separation δ0 is a length.

2. The method according to claim 1, wherein labeling the mineral species voxels as belonging to separated and fixed individual particles is performed by a watershed method.

3. The method according to claim 1, wherein the micro-CT 3D image has a resolution sufficient to identify individual particles and combined pore geometries.

4. The method according to claim 1, wherein the particles are free particle contacts where only friction between the particles is modeled, or fixed particle contacts where the particles are completely fused.

5. Converting the labeled voxelized 3D image is Optimizing elements of the same particles for finite element simulation, wherein the elements on each side of the contact between the particles fit together without voids or overlaps at the contact boundary The method according to claim 1, further comprising **Claim 6** The parametric agglomeration contact engine relates the critical separation δ 0 to the consolidation level defined as C by the following equation **Equation 1** where Δx is a representative length having the same unit as δ 0 The method according to claim 1 **Claim 7** Δx is estimated from the particle size distribution or the particle-particle contact area distribution as extracted from the micro-CT 3D image, the method according to claim 6 **Claim 8** The consolidation level is a dimensionless value and varies from 0 to 1 depending on the increasing level of consolidation, the method according to claim 1 **Claim 9** The consolidation level is defined as C, and for δ 0 = ∞, C = 1, for δ 0 = 0, C = 0, and C represents both poles of fixed and free particle-particle contact, the method according to claim 1 **Claim 10** The porous medium is a porous rock sample, and the method comprises applying the parametric agglomeration contact engine to a finite element solver using specified strain / stress boundary conditions and particle-particle contacts according to the parametric agglomeration contact model determining the contact behavior at different levels of particle solidification to convert the parametric agglomeration contact engine into a net confinement pressure model performing a flow simulation on the net confinement pressure model The method according to claim 1, further comprising **Claim 11** A computer system comprising: one or more processor devices; memory coupled to the one or more processor devices; and storage storing executable computer instructions for performing a fluid simulation of a porous medium, the instructions causing the one or more processors to: receive a micro-CT 3D image capturing a representative elementary volume of the porous medium, the porous medium being defined as having mineral and fluid species by individual particles and particle-to-particle contacts; label the micro-CT 3D image as individual voxels according to mineral and fluid species; label the mineral species voxels as belonging to separate and fixed individual particles; convert the labeled voxels to an unstructured conforming mesh representation for all particles; apply the unstructured conforming mesh representation to a parametric agglomeration contact engine that executes a parametric agglomeration contact model having adjustable parameters and a critical separation δ conditioned according to a consolidation level 0 having a length δ0; and perform such that the critical separation δ0 is a length, computer system. **Claim 12** The computer system of claim 11, wherein the consolidation level is a dimensionless value that varies from 0 to 1 with increasing levels of consolidation. **Claim 13** Converting the labeled voxelized 3D image comprises optimizing the elements of the same particles for finite element simulation, wherein the elements on each side of the contact between the particles conform to each other without voids or overlaps at the contact boundary. The computer system according to claim 11, further comprising instructions for performing **Claim 14** The parametric agglomeration contact engine relates the critical separation δ 0 to the consolidation level C defined as follows by the following equation, **Equation 2** where Δx is a representative length having the same unit as δ 0 The computer system according to claim 11. **Claim 15** The porous medium is a porous rock sample, and the instructions are applying the parametric agglomeration contact engine to a finite element solver using specified strain / stress boundary conditions and particle-particle contacts according to the parametric agglomeration contact model; determining contact behavior at different levels of particle solidification to convert the parametric agglomeration contact engine into a net confinement pressure model; performing a flow simulation on the net confinement pressure model The computer system according to claim 11, further comprising instructions for performing **Claim 16** A computer program product tangibly stored on a computer-readable non-transitory storage device storing executable computer instructions for performing fluid simulations of a porous medium, the instructions causing a computing system to receive a micro-CT 3D image capturing a representative elementary volume of the porous medium, the porous medium being defined as having mineral and fluid species by individual particles and particle-to-particle contacts; label the micro-CT 3D image as individual voxels according to mineral and fluid species; label the mineral species voxels as belonging to separate and fixed individual particles; Converting all of the labeled voxels into an unstructured conforming mesh display for all particles, Applying the unstructured conforming mesh display to a parametric agglomeration contact engine, the parametric agglomeration contact engine having a parametric agglomeration contact model with adjustable parameters and a critical separation δ conditioned according to a consolidation level 0 Executing and applying a parametric agglomeration contact model having a critical separation δ Causing to be performed A computer program product, wherein the critical separation δ0 is a length. **Claim 17** The parametric agglomeration contact engine relates the critical separation δ 0 to the consolidation level defined as C by the following equation **Equation 3** where Δx is a representative length having the same unit as δ 0 The computer program product according to claim 16. **Claim 18** The computer program product according to claim 17, wherein Δx is estimated from the particle size distribution or the particle-particle contact area distribution as extracted from the micro-CT 3D image. **Claim 19** The computer program product according to claim 16, wherein the consolidation level is a dimensionless value and varies from 0 to 1 according to an increasing level of consolidation. **Claim 20** The consolidation level is defined as C, and C = 1 for δ 0 = ∞, C = 0 for δ 0 = 0, and C represents both poles of fixed and free particle-particle contact. The computer program product according to claim 16.

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