DETERMINATION OF THE PLASTIC STRENGTH OF CARBONATES …

European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012)

J. Eberhardsteiner et.al. (eds.) Vienna, Austria, September 10-14, 2012

DETERMINATION OF THE PLASTIC STRENGTH OF CARBONATES FROM MICROTOMOGRAPHY AND THE UPSCALING USING

PERCOLATION THEORY

Jie Liu1, Reem Freij-Ayoub2, and Klaus Regenauer-Lieb1,2

1 School of Earth and Environment, the University of Western Australia 35 Stirling Hwy, Crawley WA 6009, Australia

e-mail: [email protected]; [email protected]

2 Earth Science and Resource Engineering, CSIRO 26 Dick Perry Avenue, Kensington, WA 6151, Australia

e-mail: [email protected]; [email protected]

Keywords: plastic strength, carbonate, microtomography, percolation theory, upscaling

Abstract. In this paper we establish a workflow for upscaling of rock properties from microtomogra-phy using percolation theory and focus on the plastic strength of rocks. The novel aspects of this study are: (1) determining the size of the mechanical representative volume element by using upper/lower bound computations based on thermodynamics; (2) using FEM to simulate the rock yielding at micro-scale, two cases of different pressures of linear Drucker-Prager plasticity of rocks are simulated, then the cohesion and the angle of internal friction of the rock are obtained; (3) detecting critical expo-nents of mechanical parameters from a series of derivative models that created by a shrink-ing/expanding algorithm, and then deriving scaling laws. We use a microtomographic data set of a carbonate sample to test the procedures. The preliminary results are promising.

Jie Liu, Reem Freij-Ayoub, and Klaus Regenauer-Lieb

2

1 INTRODUCTION

Understanding the physics of porous rocks is a challenge for the industry with economical interest in hydrocarbon resources, geothermal energy and mining. Microtomography enables the detection of internal structure of rocks on micro- to nano-scales and opens a new way to quantify the relationship between the microstructure and their mechanical and transport prop-erties. Numerical computations of microtomographic data have shown good agreement with experimental data for fluid flow and elastic properties [1, 2]. However, the extension of the methodology to plastic properties is still a poorly understood area. Owing to the finite size of microtomographic images, the higher the resolution, the smaller is the length scale of the sample that is contained in a certain number of pixels or voxels. This leads to the immediate challenge to detect the internal structure of rocks across micro to nano scales and at the same time connect the images to the macro-scale essential for describing the petroleum, geothermal or mining reservoir.

In this paper we establish a workflow for upscaling of rock properties from microtomogra-phy using percolation theory and apply this to the plastic strength of rocks. A microtomo-graphic dataset of a carbonate sample is used to test the workflow.

1.1 Workflow

Our work starts from the gray scale images of microtomography and involves three main components based on the binary data obtained after image segmentation (see Figure 1). The binary data have only two phases -- pores and solid, being represented by 0 and 1, respective-ly. Three main components in the workflow are:

1) geometrical analyses (left column) – it conducts quantitative analysis including stochas-tic analysis of the geometry of the model. General parameters, such as porosity, connectivity, specific surface area, and the fractal dimension are obtained. Stochastic analysis outputs prob-abilities of porosity, percolation, and anisotropy of different sizes [3]. The size of representa-tive volume element (RVE) can be determined when the probabilities converge with the increase of the size analyzed. Permeability of rock is directly related to the geometrical cha-racteristics, thus the geometrical RVE is suitable for computing permeability.

2) mechanical analyses (middle column) – in this component, we determine the size of me-chanical RVE by detecting the mechanical responses of maximum and minimum (upper and lower bounds) entropy productions of models of different sizes [4, 5]. Then the plastic re-sponse of the microstructural model is analyzed over the mechanical RVE. We will explain this component in detail in the next sub-section.

3) extracting critical exponents of key parameters of rock property – here we use a shrink-ing/expanding algorithm [6] to create a series of derivative models with different porosity. From percolation theory, some parameters (including permeability, elastic modulus, yield stress, and more) change exponentially when the porosity is approaching the percolation thre-shold [7 - 10]. The exponential index describing this tendency is the critical exponent of the parameter.

With any two critical exponents and/or the fractal dimension, scaling laws are defined. Scaling laws, the percolation threshold, and some other parameters such as crossover length (refer to [6]) are constraints used for upscaling properties from micro-scale to large scale. The workflow is suitable for properties of fluid flow and mechanics. In our previous study [3, 6], the determination of geometrical RVE and the upscaling of permeability are verified using a sandstone sample. In this paper we focus on deriving mechanical material parameters with the workflow.

Jie Liu,

Figure 1: Workflo

1.2 Focus of this paper

The first focus of this paper is the determination of modynamic upper and lower bound principles postulquantify whether our computations of have converged such that they can deliver homogenized valuesconstant displacement boundary condition; lower bound corresponds to boundary condition. Theoretically, upper and lower bounds els of small size; the difference diminishes of mechanical RVE is deterworkflow, we calculate mechanical properties such afor simplicity that the elastic response can be used to derive the size of the mechanical

Figure 2: Thermodynamic homogenization of upper and lower bound

, Reem Freij-Ayoub, and Klaus Regenauer-Lieb

3

flow for upscaling of rock properties from microtomography

The first focus of this paper is the determination of the mechanical RVE. modynamic upper and lower bound principles postulated in Regenauer-Lieb et al. [4, 5

computations of the mechanical response of models with dihave converged such that they can deliver homogenized values. Upper bound corresponds to constant displacement boundary condition; lower bound corresponds to boundary condition. Theoretically, upper and lower bounds give different responses for

; the difference diminishes as the model size increases (see Figure 2).of mechanical RVE is determined from the converging solutions. Toworkflow, we calculate mechanical properties such as elasticity and plasticity and we assume for simplicity that the elastic response can be used to derive the size of the mechanical

Figure 2: Thermodynamic homogenization of upper and lower bounds of microstructures (from [4

of rock properties from microtomography

mechanical RVE. We use the ther-Lieb et al. [4, 5] to

mechanical response of models with different sizes Upper bound corresponds to a

constant displacement boundary condition; lower bound corresponds to a constant force fferent responses for mod-

(see Figure 2). The size To demonstrate the

s elasticity and plasticity and we assume for simplicity that the elastic response can be used to derive the size of the mechanical RVE.

of microstructures (from [4]).


4

The second focus is the computation of plastic strength of microstructures. We use linear Drucker-Prager plasticity and the finite element method to simulate the rock yielding for a mechanical RVE. In order to conduct cohesion and the angle of friction of rock samples, two cases of different pressures are simulated. Using the relationship of

�� = ��tan� + , (1)

where �� and �� are yield stress and normal stress computed from Drucker-Prager plasticity, c and� are cohesion and the angle of internal friction of the rock, with two groups of �� and �� results of the model, c and � are deduced.

The third focus of this paper is to extract critical exponent of yield stress. Shrink-ing/expanding algorithms [6] are used to create a series of derivative models with different volume fractions. The percolation threshold � , which is the lowest volume fraction when there exists an unbroken structure connecting at least two opposite outside boundaries for a finite volume, can be detected from these derivative models. Then the derivative models close to the percolation threshold are used to simulate the deformation and yielding. With a series of results of yield stress of models, it is possible to fit the critical exponent of yield stress �� in the form of

�� = �� − � ��, (2)

when the volume fraction � is approaching the percolation threshold. �� is a scale-independent parameter describing the change of yield stress for all scales.

2 IMPLEMENTATIONS AND RESULTS

We use a publicly available microtomographic dataset of a carbonate sample to test the workflow. The dataset was downloaded from http://www3.imperial.ac.uk/earthscienceanden gineering/research/perm/porescalemodelling. It is a segmented binary RAW format dataset of the size of 400-cube voxels. The resolution of a voxel is 2.9 µm [11]. The porosity is 23.3% and the 3D rendering of the structure can be seen in Figure 3.

Figure 3: 3D rendering of the carbonate sample of 400-cube voxels

2.1 Mechanical RVE

Sixteen cubic volumes of sizes from 40-cube to 400-cube are analyzed (voxel as unit in the following unless specified). Each volume that is smaller than 400-cube is arbitrarily selected from the 400-cube model but it is ensured the porosity is 23±0.5%.

Two kinds of meshes are used for these volumes for finite element computing. Hexahedral elements are used for small volumes (side-length L < 100), which are easy to create and the computing time is acceptable. Tetrahedral elements are used for large volumes (L > 100). This

Jie Liu,

entails extra procedures and manual time. For the volume of L = 100,While creating tetrahedral element meshsion and the fineness of the mesh and reducing the computing time.

For all these computing models, we (refer to Figure 4a) � = �0, �rection; surfaces � = �� and compressive pressure loads areresponding to upper and lower boundis 0.01% of the L. The pressure value on the boundary is 0.05 GPa. conditions and loads are chosen such that noThe input elastic parameters of solid are: Young’s modulus 0.2. Abaqus® is selected to performcomputing models are shown in Figure 4b and 4c.

Figure 4: (a) A computing volume and its coordinate system; (b) ments; (c) mesh of the 100

Young’s modulus and Poisson’s ratio of the computed model put of Abaqus® by using the equation

where each stress component components ��, however, are average displacements on surfacethe model. It gives us different results of es � = ��, � = �� and � = ��

modulus calculated from the displacement onIn contrast, Poisson’s ratio is more reasonable from the surfaces of use the averaged Poisson’s ratio

Figure 5 shows the tendency of lower bounds of different volume sizes. lower bounds are lower than the upper boundcept 2 values of L for Poisson’s ratioper and lower bound principlesConvergence can be seen for the volume size larger than 320. Thus, the mechanical RVE is determined as 320-cube voxels for GPa and � ≅ 0.26. Thus the elastic modulus of this and the Poisson’s ratio has increased

(a)


5

procedures and manual work to create but can dramatically reduce the computing = 100, we used both to compare the difference caused by mesh.

tetrahedral element meshes, we considered the balance of keeping the precsion and the fineness of the mesh and reducing the computing time.

For all these computing models, we used the same boundary constraints� = �0, and � = �0 are displacement constrained

and � = �� are free. Constant normal convergent pressure loads are applied on the surface � = �� for each computing model, co

responding to upper and lower bounds, respectively. The displacement value on the boundary . The pressure value on the boundary is 0.05 GPa. Displacement

are chosen such that no large deformation or plastic deformation occurThe input elastic parameters of solid are: Young’s modulus � = 50 GPa, Poisson’s ratio

selected to perform all the finite element computations. computing models are shown in Figure 4b and 4c.

and its coordinate system; (b) mesh of the 100-cube volume, hexahedral elments; (c) mesh of the 100-cube volume, tetrahedral elements.

Young’s modulus and Poisson’s ratio of the computed model are calculated from the by using the equation

�� =�

!�1 + �� − ��##$��%,

each stress component �� is the average value of all elements inare average displacements on surfaces divided by the side

It gives us different results of � and � when use displacements on different surfa��. Corresponding to the boundary conditions

calculated from the displacement on the surface of � = �� is reasonable and reliable. In contrast, Poisson’s ratio is more reasonable from the surfaces of � = ��

d Poisson’s ratio of the results from these two surfaces. shows the tendency of the elastic modulus and Poisson’s ratio of

different volume sizes. Although they are not ideal as indicated in are lower than the upper bounds from a certain minimum volume onwards

for Poisson’s ratio in Figure 5. These two exceptions do not refute the uper and lower bound principles according to the physical definition of Poisson’s ratioConvergence can be seen for the volume size larger than 320. Thus, the mechanical RVE is

cube voxels for this carbonate sample. The convergent valueelastic modulus of this porous structure is 64%

oisson’s ratio has increased more than 20%.

(b) (c)

work to create but can dramatically reduce the computing we used both to compare the difference caused by mesh.

, we considered the balance of keeping the preci-

same boundary constraints. That is, surfaces displacement constrained in normal di-

convergent displacement and computing model, cor-

The displacement value on the boundary Displacement boundary

large deformation or plastic deformation occurs. GPa, Poisson’s ratio � =

Two meshes of the

cube volume, hexahedral ele-

calculated from the out-

(3)

is the average value of all elements in the model, strain divided by the side-length of

nts on different surfac-boundary conditions we used, elastic

is reasonable and reliable. �� and � = ��. We

on’s ratio of the upper and indicated in Figure 2, volume onwards, ex-

These two exceptions do not refute the up-according to the physical definition of Poisson’s ratio.

Convergence can be seen for the volume size larger than 320. Thus, the mechanical RVE is The convergent values are � ≅ 32

porous structure is 64% of the pure solid;

Jie Liu,

Figure 5: Elastic modulus and Poisson’s ratio of two dots or triangles close to each other for L = 100. They are the results from hexahedral element

2.2 Plastic strength

Simulations of plastic response are based on the mechanical RVE analysis – i.e. 320-cube voxel volPrager plasticity as input: 1) the fied as 40° in this study; 2) cohesion, which istic perfectly plastic behaviour

A displacement boundary conditiondisplacement is fixed to 1.6 units tioned in Section 2, two cases of differenangle of internal friction of the microstructure. Case 1 �0, �0 and �0; Case 2 uses a uniaxial strain condition surface are normally constrained.

Finite element computations deliverments under the applied boundary conditions, increment by increment. For each increment of output, we calculate the average values over all elements stress, total strain, and plastic strain. represented by the minimum principal strain and Case 1 demonstrates a typical elastic perfectMPa. Case 2 illustrates a plastic hardening feature and no obvious yield pfied. In this case, the concept of offset yield point can be used, which is or 0.2% of the strain, arbitrarilyconsidered as yield and the correspondin

Figure 6: Averaged minimum principal strain and stress

0

10

20

30

40

50

0 40 80 120 160 200 240

Ela

sti

c m

od

ulu

s (

GP

a)

Model size L (voxel)

Upper bound

Lower bound

0

20

40

60

80

0

Mag

nit

ud

e o

f av

erag

e σ

_min


6

Elastic modulus and Poisson’s ratio of upper and lower bounds of different volume sizes

two dots or triangles close to each other for L = 100. They are the results from hexahedral elementdral elements, respectively.

Simulations of plastic response are based on the mechanical RVE derived from elastic cube voxel volume. Two more parameters are necessary for Drucker

the angle of internal friction of the solid matrixin this study; 2) cohesion, which is specified as 30 MPa. We only considered e

in our computations. boundary condition is used on the surface at � = ��. The magnitude of

units and it allows 0.5% total strain in the z cases of different pressure are analyzed to detect cohesion and

angle of internal friction of the microstructure. Case 1 uses normal constraint on uniaxial strain condition in which all surfaces except the loading

normally constrained. computations deliver the deformation, strain and stress of nodes and/or el

boundary conditions, increment by increment. For each increment of rage values over all elements of the model

stress, total strain, and plastic strain. Figure 6 shows the averaged stress-presented by the minimum principal strain and the minimum stress for C

Case 1 demonstrates a typical elastic perfectly plastic behaviour with theCase 2 illustrates a plastic hardening feature and no obvious yield p

In this case, the concept of offset yield point can be used, which is generally , arbitrarily. This means that if the strain is larger than 0.1%, the model is

considered as yield and the corresponding stress can be defined as yield stress.

Figure 6: Averaged minimum principal strain and stress of Drucker-Prager plasticity

240 280 320 360 400


0

0.1

0.2

0.3

0.4

0 40 80 120 160 200

Po

iss

on

's r

ati

o


0.001 0.002 0.003 0.004 0.005 0.006 0.007

Magnitude of average ε_min

Case 1

Case 2

upper and lower bounds of different volume sizes. There are

two dots or triangles close to each other for L = 100. They are the results from hexahedral elements and tetrahe-

derived from elastic are necessary for Drucker-

of the solid matrix, which is speci-30 MPa. We only considered elas-

. The magnitude of direction. As men-

are analyzed to detect cohesion and the uses normal constraint on surfaces

all surfaces except the loading

of nodes and/or ele-boundary conditions, increment by increment. For each increment of

of the model for components of -strain relationships Case 1 and Case 2.

the yield stress of 23 Case 2 illustrates a plastic hardening feature and no obvious yield point can be identi-

generally set at 0.1% strain is larger than 0.1%, the model is

g stress can be defined as yield stress.

Prager plasticity for cases 1 & 2

240 280 320 360 400


Upper bound

Lower bound

0.007


7

In the post-processing we determine the angle of friction and cohesion using equation (1). We use von Mises stress and pressure of the direct output of Abaqus® at the yield point as �� and ��. The yield point of Case 1 is identified as shown in Figure 6 the plateau; the yield point of Case 2 can be arbitrarily selected at the strain of 0.1% or 0.2%. We tried both ap-proaches and found that they lead to similar cohesion and friction angle. In fact, even if the von Mises stress and pressure at the strain of 0.5% are used for Case 2, the results are almost the same. More information can be seen in Figure 7. For Case 1, the relationship between von Mises stress and pressure is linear before yielding occurs; both von Mises stress and pressure are constant after yielding. For Case 2, von Mises stress and pressure show a nonlinear rela-tionship at the beginning of the deformation, and then show a linear feature. More importantly, the linear trend is connected to the point after yielding of Case 1. The linear trend implies that any two points after yielding of Case 1 and Case 2 will give us the same result of cohesion and the angle of friction. A fitted line is shown in Figure 7. The slope and the intercept are 0.2565 and 20.2, respectively. Thus the cohesion is 20.2 MPa and the angle of friction is 14.4°. Compared with the input solid parameters, the porous structure causes the reduction of the cohesion to two-thirds, and significantly decreases the angle of friction to one-third.

Figure 7: Relationship of Mises stress and pressure, and the fitting of cohesion and friction angle from points

after yielding of two cases.

2.3 Critical exponent of yield strength

In the study of elastic and plastic percolation, the connectivity of the solid phase is consi-dered. Thus, a series of derivative models are created by shrinking the solid structure. As the volume fraction of solid � is decreasing while shrinking, at some stage the connection is bro-ken and the percolation threshold can be determined [6]. Figure 8 shows the change of the vo-lume fraction of solid with the shrinking steps, and some images of the structure after shrinking are shown. The original model has 76.7% solid volume fraction and obviously is percolating in all three directions. After 20 shrinking steps, the structure with volume fraction less than 7% is not percolating in three directions anymore; after 24 shrinking steps, the struc-ture is not percolating in any direction. Thus, the model of 23 shrinking steps being percolat-ing only in one direction is defined as critical model; its volume fraction of 4.47% is recognized as the percolation threshold of the solid of this sample.

As we can see from Figure 8, there are floating blocks in the models. These floating blocks must be removed, as they do not contribute to the strength of the model but cause numerical singular problems or rigid movement in computations. After removing the floating blocks, the volume fraction is 3.3% for the critical model. Thus the percolation threshold in fact is 3.3 %.

Jie Liu,

Derivative models that are above and close to the percolation threshold (No. 23) will be used to simulate the plastic response for 20 to 22 are not percolating in all 3 directions. In this situation, the boundary loads may be hard to define because there is no solid phase on one or more outside surfaces of the 320-cube volume. No. 18 models are too weak. The meshing procedure breaks tstepped surfaces are replaced by smoothed coarse triangular surfaces. Thus we analyzed drivative models of Nos. 13 to 1shold |� − � | (all after removing f

Figure 8: Volume fraction decreasing while shrinking and some images of the derivative models

We used the boundary conditions of normal displacement being zero on surfaces � = �0 , � = �0 , and normal displacement being 1.6� = �m. The displacement loading allows 0.5% strain in each direction. Stressunder these boundary conditions are inserted. We see that as the distance from thethe yield point increases. Samples with higher solids volume fraction also exhibit a distinct failure point followed by weakening behavior whereas the samples close to tain deformation at almost constant stress after the yield point.

Figure 9: Stress-strain relationships of five derivative models close to the percolation threshold

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2E

Mag

nit

ud

e o

f av

erag

e σσ σσ_

min

P - Pc = 0.1519

P - Pc = 0.1278

P - Pc = 0.0937

P - Pc = 0.0775

P - Pc = 0.0632


8

models that are above and close to the percolation threshold (No. 23) will be used to simulate the plastic response for computing the yield stress. Among these models, No20 to 22 are not percolating in all 3 directions. In this situation, the boundary

define because there is no solid phase on one or more outside surfaces of and No. 19 are percolating in 3 directions but the links in these

models are too weak. The meshing procedure breaks the very weak links while accurate fine stepped surfaces are replaced by smoothed coarse triangular surfaces. Thus we analyzed d

to 17. The differences of volume fractions to the percolation thr(all after removing floating blocks) are in the range of 6.32% to 15.19%.


We used the boundary conditions of normal displacement being zero on surfaces normal displacement being 1.6 on surfaces � =

The displacement loading allows 0.5% strain in each direction. Stressunder these boundary conditions are shown in Figure 9 and three analyzed derivative models

the distance from the percolation threshold increasesSamples with higher solids volume fraction also exhibit a distinct

failure point followed by weakening behavior whereas the samples close to tain deformation at almost constant stress after the yield point.

strain relationships of five derivative models close to the percolation threshold

2E-05 4E-05 6E-05 8E-05

Magnitude of average εεεε_min

Pc = 0.1519

Pc = 0.1278

Pc = 0.0937

Pc = 0.0775

Pc = 0.0632

models that are above and close to the percolation threshold (No. 23) will be the yield stress. Among these models, Nos.

20 to 22 are not percolating in all 3 directions. In this situation, the boundary constraints and define because there is no solid phase on one or more outside surfaces of

but the links in these he very weak links while accurate fine

stepped surfaces are replaced by smoothed coarse triangular surfaces. Thus we analyzed de-. The differences of volume fractions to the percolation thre-

loating blocks) are in the range of 6.32% to 15.19%.


We used the boundary conditions of normal displacement being zero on surfaces � = �0, �� , � = �� , and

The displacement loading allows 0.5% strain in each direction. Stress-strain curves and three analyzed derivative models

increases, the value of Samples with higher solids volume fraction also exhibit a distinct

failure point followed by weakening behavior whereas the samples close to percolation main-

strain relationships of five derivative models close to the percolation threshold

0.0001

Jie Liu,

Figure 10 shows the yield stress versus the difference of volume fractiontion threshold |� − � | in log-cal exponent of yield stress �

sample. The value is much less than 2.5 obtained 1.7 obtained by Sieradzki and Li [8]. Sahimi [10] thought that Sieradzki and Li’s low value is presumably due to the fact that in their experiments samples with not in the critical region close to the range of 0.06 to 0.15, is similar to that of Benguigui et al.

Figure 10: Yield stress

A distinguish feature of our pared with those of previous laboratory models. Results of that significant stress concentrations and irregularly distributed, especially for the models very close to stress concentration determine the eventual yield of the volume. We speculate this one of the main reasons of the low critical exponent of yield stress. This speculatbe investigated further by comparing with models of more regular structures, such as homgeneously packed sand grains

3 CONCLUSIONS

• In this study we have established a workflow of upscaling microtomography. A carbonate sample irogeneous and performs ascal exponent of yield are three

• Mechanical RVE can be determined from lower bounds for a series of models of different sizesboundary condition) causes constant displacement boundary condition). lower bounds give convergent results. The size of sample is 320-cube voxelscal RVE is close to 1 mm

• Plastic strength has beenDrucker-Prager plasticity. By analyzing two responses of Druckerviour under different pressures, cohesion and the angle of friction bonate sample with 23.3% porosity has twoand its angle of friction has reduced to around one

0.2

2Y

ield

str

es

s


9

Figure 10 shows the yield stress versus the difference of volume fraction-log plot. The fitting line gives a power of 1.33. This is the crit�� based on the analyses of derivative models of the carbonate

sample. The value is much less than 2.5 obtained by Benguigui et al. [9] and even less than 1.7 obtained by Sieradzki and Li [8]. Sahimi [10] thought that Sieradzki and Li’s low value is presumably due to the fact that in their experiments samples with |� − �

se to � . In our numerical experiments, |� − �the range of 0.06 to 0.15, is similar to that of Benguigui et al.

Figure 10: Yield stress �� versus |� − � | in a log-log plot and the fitted relationship.

A distinguish feature of our models is that our models are strongly heterogeneous, copared with those of previous laboratory models. Results of finite element computing show us

significant stress concentrations are around some weak links. These weak links are rare y distributed, especially for the models very close to � . These few points of

stress concentration determine the eventual yield of the volume. We speculate this of the low critical exponent of yield stress. This speculat

be investigated further by comparing with models of more regular structures, such as hom or mathematically created digital rocks.

established a workflow of upscaling of physical prA carbonate sample is used to test the workflow. The sample is het

as a case study. Mechanical RVE, plastic strength and the critcal exponent of yield are three elements proving the applicability of the work

can be determined from the thermo-dynamical principle of for a series of models of different sizes. Lower bound (or constant force

boundary condition) causes softer response but stronger oscillation than upper bound (or constant displacement boundary condition). As the volume size increas

give convergent results. The size of the mechanical RVE of this carbonate cube voxels. As the resolution is 2.9 µm, the physical size of the mechan

cal RVE is close to 1 mm3. has been analyzed based on microstructures from microtomography for

Prager plasticity. By analyzing two responses of Drucker-Prager plasticnder different pressures, cohesion and the angle of friction are

bonate sample with 23.3% porosity has two-thirds of cohesion of the solid parameter, and its angle of friction has reduced to around one-third of the solid parameter.

Sy = 17.376(P-Pc)1.3343

0.2

2

0.03 0.3

|P - Pc|

Figure 10 shows the yield stress versus the difference of volume fraction and the percola-log plot. The fitting line gives a power of 1.33. This is the criti-

based on the analyses of derivative models of the carbonate by Benguigui et al. [9] and even less than

1.7 obtained by Sieradzki and Li [8]. Sahimi [10] thought that Sieradzki and Li’s low value is | = 0.1~0.23 were � | of models are in

log plot and the fitted relationship.

models is that our models are strongly heterogeneous, com-computing show us

around some weak links. These weak links are rare . These few points of

stress concentration determine the eventual yield of the volume. We speculate this might be of the low critical exponent of yield stress. This speculation needs to

be investigated further by comparing with models of more regular structures, such as homo-

physical properties from . The sample is hete-

Mechanical RVE, plastic strength and the criti-proving the applicability of the work flow.

dynamical principle of upper and . Lower bound (or constant force

than upper bound (or the volume size increases the upper and

mechanical RVE of this carbonate , the physical size of the mechani-

analyzed based on microstructures from microtomography for Prager plastic beha-

are obtained. The car-thirds of cohesion of the solid parameter,

third of the solid parameter. This re-


10

sult is the first step to establish a relationship of porosity and plastic strength. More nu-merical simulations and laboratory experiments are necessary to specify this relationship.

• The critical exponent of the yield strength has been extracted from the derivative models with different volume fractions close to the percolation threshold. This is the first test of rock samples using numerical simulations. The value we have got does not match the re-sult of theoretical deduction and laboratory experiments very well. Whether it is caused by the strong heterogeneity of the structure needs more investigations. This is a valuable pilot study and it paves the road of research of upscaling of physical properties from mi-crotomography of different materials.

ACKNOWLEDGEMENT

We are grateful to Petrobras’ support of this research. We thank Ali Karrech, Thomas Pou-let, and Valeriya Shulakova in CESRE for their help on finite element computing and pre-/post- processing. We also thank Ben Clennell and Thomas Poulet for their comments and suggestions to improve the manuscript.

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