2. Constitutive equations for porosity evolution

The pore volume fraction, the pore size distribution, and the mineral surfaces are key parameters to ensure safe disposal of radioactive waste and captured CO2, and to understand how ores' deposit evolves with time. In petroleum sciences, chemo-mechanical processes are important to accurately predict the porosity since it is inside the pores where hydrocarbons are stored, and it is through the pores, the hydrocarbons are being produced by miscible and immiscible fluid migration across reactive mineral surfaces that again are subject to change. Both pore volume and production rate are crucial to determine the recoverable hydrocarbon potential.

tion and flow of reactive 0.219 M MgCl2 brine at 130C (Table 1).

202 Porosity - Process, Technologies and Applications

(constant diameter) such that lengths relate to volumes.

Figure 1. (a) SEM image of an unaltered chalk (Liegè, Belgium [3]). Calcite grains partially organized in coccolith rings and foraminifers. (b) Reworked Liegè chalk from the same core as (a) after 1090 days of continuous mechanical compac-

Reactive pore fluids in nonequilibrium with their host rocks lead to dissolution and precipitation transforming the mineral assembly into another, see for example, [3–5]. Dissolution and

Figure 2. (a) Triaxial cell setup controlling axial and radial stress, the pore pressure, flow rate, and temperature. (b) Additive partitioning of the total bulk strain into a pore and solid volume component. Here, uniaxial strain is assumed The basic equations that are used to quantify the porosity evolution through time are presented. The analysis is based on the work presented in Nermoen, et al. [3]. The overall bulk volume of a bi-phase material equals the sum of the solid volume and pore volume

$$\mathbf{V\_b = V\_s + V\_p} \tag{1}$$

Any changes in solid volume and pore volume lead to changes in the bulk

$$
\Delta V\_b = \Delta V\_s + \Delta V\_p \tag{2}
$$

The pore volume, and hence the porosity, itself is not a conserved quantity. In that case, the bulk volume (size of the object of study) and the solid volume evolution have to be used. Since the volumes are additive by nature, the changes in pore volume can be calculated

$$
\Delta V\_p = \Delta V\_b + \Delta V\_s \tag{3}
$$

At any given time through dynamic porosity evolution, the porosity is given by

$$
\phi = \frac{V\_p}{V\_b} = 1 - \frac{V\_s}{V\_b} \tag{4}
$$

When both the bulk volume and the pore volume change dynamically from known measurements before the experiment starts (Vb,<sup>0</sup> and Vp, <sup>0</sup> are known), then the time-evolution of the porosity is given by

$$\phi(t) = \frac{V\_{p,0} + \Delta V\_p(t)}{V\_{b,0} + \Delta V\_b(t)}\tag{5}$$

mineral framework [6]). In open systems subjected reactive flow, both mass and density of the core material may change because of mineral reactions. To evaluate the relative importance of how evolution mechanisms of the solid volume and pore volume dictate the porosity in real systems, a rigorous definition of stresses and strains are required. The aim is to pave the way for quantitative analyses of how stresses impact strains, and how strains impact the porosity

The stress tensor describes the stresses (force per unit area) in a solid porous body. For

Shear and normal components are abbreviated τij and σij, respectively, with ij ¼ f g z;r; θ denoting the axial (z), radial (r), and tangential (θ) direction. Compressive stresses and inward deformation are defined positive. When there is no net translational or rotational force acting in the solid body (i.e., τzr ¼ τrz, τ<sup>z</sup><sup>θ</sup> ¼ τθz, and τ<sup>r</sup><sup>θ</sup> ¼ τθr), only six independent stress tensor components apply. For a cylindrical core plug stressed in a triaxial cell, the tangential stress equals the radial, and the principal stress directions coincide with the imposed z and r directed stress such that the shear stresses are zero. The stress tensor may, therefore, be expressed through the orthogonal principal

> σz σr

In reservoir systems, however, all stress components may apply, and as such, the off-diagonal elements of the stress tensor are nonzero. However, in these cases, the stress tensor can be rotated such that the principal stress notation can be obtained. It is customary procedure to arrange the first, second, and third principal stress directions as σ<sup>1</sup> > σ<sup>2</sup> > σ3, where σ<sup>1</sup> is typically in the vertical direction (weight dominated), and, consequently, the σ<sup>2</sup> and σ<sup>3</sup> are horizontal (often abbreviated σ<sup>H</sup> as the highest horizontal stress and σ<sup>h</sup> is the least horizontal

In porous rocks, it is the effective stresses introduced by [7] that drive deformation. The external load applied onto a material that consists of solids and voids is balanced by the interparticle contacts in force networks (material framework) and a fraction α of the pore pressure. Drained conditions apply to the cases where fluids are allowed to escape to keep the pore pressure constant (hence constant effective stress), differ from undrained conditions in which the pore pressure increases because of compaction (thereby reducing the effective stress).

Simultaneously, seepage forces arising from differences in fluid pressure expose a net force onto the solid framework (see Figure 3). In partially consolidated systems, in which the cross

σzz τzr τ<sup>z</sup><sup>θ</sup> τrz σrr τ<sup>r</sup><sup>θ</sup> τθ<sup>z</sup> τθ<sup>r</sup> σθθ 3 7

<sup>5</sup>: (11)

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205

� � (12)

cylindrical core plug, it is convenient to express the stress tensor σ as

σij ¼

stress). σ<sup>h</sup> and σ<sup>H</sup> depend upon Poisson ratio and tectonic regional stresses.

2 6 4

chemo-mechanical compaction.

3.1. The stress tensor in porous materials

stresses vector with two components

3.2. Effective stress

Using Eq. (3) enables the determination of the porosity from known quantities

$$\phi(t) = \frac{V\_{p,0} + \Delta V\_b(t) - \Delta V\_s(t)}{V\_{b,0} + \Delta V\_b(t)}\tag{6}$$

This equation is useful when determining pore volume evolution when considering mechanical and chemical processes that occur at reactive rock-fluid systems exposed to elevated stresses. To simplify the porosity evolution equation further, the volumetric strain and the initial porosity before chemo-mechanical processes occur are introduced

$$
\varepsilon\_{\rm rel}(t) = -\frac{\Delta V\_b(t)}{V\_{b,0}} \text{ and } \phi\_0 = \frac{V\_{p,0}}{V\_{b,0}} \tag{7}
$$

The minus sign in the volumetric strain here are in line with the definition in geotechnical engineering that inward deformation is positive, often different from other fields of sciences. Dividing by the initial bulk volume and employing the definitions Eq. (6) become

$$\phi(t) = \frac{\phi\_0 - \varepsilon\_{\rm vol} - \Delta V\_s(t) / V\_{b,0}}{1 - \varepsilon\_{\rm vol}} \tag{8}$$

Eq. 8 is used to analyze how the pore volume fraction changes as the overall volume and the solid volume changes through time. Typically, it is easier to quantify the changes in the solid volume and total volume because of conservation of mass, but this does not generally apply. In other cases, when the pore volume and solid volume are known, the porosity can be calculated from

$$\phi(t) = \frac{V\_{p,0} + \Delta V\_p(t)}{V\_{p,0} + V\_{s,0} + \Delta V\_p(t) + \Delta V\_s(t)}.\tag{9}$$

This equation could be used when the volumes of injected and produced fluid volumes are monitored and solid volume change can be back-calculated from ion chromatography (IC) of produced fluids. If, however, the bulk volume (e.g., 4D seismic) and the pore volume were obtained from monitoring the injected and produced fluid volumes, the porosity is as follows:

$$\phi(t) = \frac{V\_{p,0} + \Delta V\_p(t)}{V\_{b,0} + \Delta V\_b(t)}.\tag{10}$$

### 3. Volumetric strain by imposed stress

In compressive hydrostatic systems, the porous rocks deform by reducing its bulk volume. This may affect the porosity through, for example, Eq. (8). In closed systems, in which the mass and density of the minerals are conserved, the bulk volume reduction equals the pore volume reduction. This is facilitated by grains moving relative to each other, and/or by pressure solution (dissolution of stressed grain contacts and precipitation in unstressed parts of the mineral framework [6]). In open systems subjected reactive flow, both mass and density of the core material may change because of mineral reactions. To evaluate the relative importance of how evolution mechanisms of the solid volume and pore volume dictate the porosity in real systems, a rigorous definition of stresses and strains are required. The aim is to pave the way for quantitative analyses of how stresses impact strains, and how strains impact the porosity chemo-mechanical compaction.

#### 3.1. The stress tensor in porous materials

ϕðÞ¼ t

Using Eq. (3) enables the determination of the porosity from known quantities

ϕðÞ¼ t

204 Porosity - Process, Technologies and Applications

initial porosity before chemo-mechanical processes occur are introduced

εvolðÞ¼� t

ϕðÞ¼ t

3. Volumetric strain by imposed stress

Vp, <sup>0</sup> þ ΔVpð Þt

Vp, <sup>0</sup> þ ΔVbðÞ� t ΔVsð Þt

and <sup>ϕ</sup><sup>0</sup> <sup>¼</sup> Vp, <sup>0</sup>

Vb,<sup>0</sup>

Vp, <sup>0</sup> <sup>þ</sup> Vs, <sup>0</sup> <sup>þ</sup> <sup>Δ</sup>Vpð Þþ <sup>t</sup> <sup>Δ</sup>Vsð Þ<sup>t</sup> : (9)

Vb,<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>Vbð Þ<sup>t</sup> : (10)

This equation is useful when determining pore volume evolution when considering mechanical and chemical processes that occur at reactive rock-fluid systems exposed to elevated stresses. To simplify the porosity evolution equation further, the volumetric strain and the

> ΔVbð Þt Vb,<sup>0</sup>

Dividing by the initial bulk volume and employing the definitions Eq. (6) become

The minus sign in the volumetric strain here are in line with the definition in geotechnical engineering that inward deformation is positive, often different from other fields of sciences.

> <sup>ϕ</sup>ðÞ¼ <sup>t</sup> <sup>ϕ</sup><sup>0</sup> � <sup>ε</sup>vol � <sup>Δ</sup>Vsð Þ<sup>t</sup> <sup>=</sup>Vb, <sup>0</sup> 1 � εvol

Eq. 8 is used to analyze how the pore volume fraction changes as the overall volume and the solid volume changes through time. Typically, it is easier to quantify the changes in the solid volume and total volume because of conservation of mass, but this does not generally apply. In other cases, when the pore volume and solid volume are known, the porosity can be calculated from

This equation could be used when the volumes of injected and produced fluid volumes are monitored and solid volume change can be back-calculated from ion chromatography (IC) of produced fluids. If, however, the bulk volume (e.g., 4D seismic) and the pore volume were obtained from monitoring the injected and produced fluid volumes, the porosity is as follows:

In compressive hydrostatic systems, the porous rocks deform by reducing its bulk volume. This may affect the porosity through, for example, Eq. (8). In closed systems, in which the mass and density of the minerals are conserved, the bulk volume reduction equals the pore volume reduction. This is facilitated by grains moving relative to each other, and/or by pressure solution (dissolution of stressed grain contacts and precipitation in unstressed parts of the

ϕðÞ¼ t

Vp,<sup>0</sup> þ ΔVpð Þt

Vp,<sup>0</sup> þ ΔVpð Þt

Vb, <sup>0</sup> <sup>þ</sup> <sup>Δ</sup>Vbð Þ<sup>t</sup> (5)

Vb, <sup>0</sup> <sup>þ</sup> <sup>Δ</sup>Vbð Þ<sup>t</sup> (6)

(7)

(8)

The stress tensor describes the stresses (force per unit area) in a solid porous body. For cylindrical core plug, it is convenient to express the stress tensor σ as

$$
\sigma\_{\vec{\eta}} = \begin{bmatrix}
\sigma\_{zz} & \tau\_{zr} & \tau\_{z\theta} \\
\tau\_{rz} & \sigma\_{\theta\tau} & \tau\_{r\theta} \\
\tau\_{\theta z} & \tau\_{\theta r} & \sigma\_{\theta\theta}
\end{bmatrix}.
\tag{11}
$$

Shear and normal components are abbreviated τij and σij, respectively, with ij ¼ f g z;r; θ denoting the axial (z), radial (r), and tangential (θ) direction. Compressive stresses and inward deformation are defined positive. When there is no net translational or rotational force acting in the solid body (i.e., τzr ¼ τrz, τ<sup>z</sup><sup>θ</sup> ¼ τθz, and τ<sup>r</sup><sup>θ</sup> ¼ τθr), only six independent stress tensor components apply. For a cylindrical core plug stressed in a triaxial cell, the tangential stress equals the radial, and the principal stress directions coincide with the imposed z and r directed stress such that the shear stresses are zero. The stress tensor may, therefore, be expressed through the orthogonal principal stresses vector with two components

$$\begin{bmatrix} \sigma\_z \\ \sigma\_r \end{bmatrix} \tag{12}$$

In reservoir systems, however, all stress components may apply, and as such, the off-diagonal elements of the stress tensor are nonzero. However, in these cases, the stress tensor can be rotated such that the principal stress notation can be obtained. It is customary procedure to arrange the first, second, and third principal stress directions as σ<sup>1</sup> > σ<sup>2</sup> > σ3, where σ<sup>1</sup> is typically in the vertical direction (weight dominated), and, consequently, the σ<sup>2</sup> and σ<sup>3</sup> are horizontal (often abbreviated σ<sup>H</sup> as the highest horizontal stress and σ<sup>h</sup> is the least horizontal stress). σ<sup>h</sup> and σ<sup>H</sup> depend upon Poisson ratio and tectonic regional stresses.

#### 3.2. Effective stress

In porous rocks, it is the effective stresses introduced by [7] that drive deformation. The external load applied onto a material that consists of solids and voids is balanced by the interparticle contacts in force networks (material framework) and a fraction α of the pore pressure. Drained conditions apply to the cases where fluids are allowed to escape to keep the pore pressure constant (hence constant effective stress), differ from undrained conditions in which the pore pressure increases because of compaction (thereby reducing the effective stress).

Simultaneously, seepage forces arising from differences in fluid pressure expose a net force onto the solid framework (see Figure 3). In partially consolidated systems, in which the cross

<sup>Γ</sup> <sup>¼</sup> <sup>1</sup> 2

2 6 4

equal, such that the strain vector for cylindrical cores:

strain can be calculated from the radial and axial strains

terms can be omitted, hence, εvol ≃ε<sup>z</sup> þ 2εr.

nized in the strain tensor

cylinder is given by <sup>V</sup> <sup>¼</sup> <sup>π</sup>D<sup>2</sup>

In three dimensions (cylindrical coordinates), the pairs of shear and normal strains are orga-

εzz Γzr Γ<sup>z</sup><sup>θ</sup> Γrz εrr Γ<sup>r</sup><sup>θ</sup> Γθ<sup>z</sup> Γθ<sup>r</sup> εθθ

Similar to the stress tensor, the shear strains balance each other (Γrz ¼ Γzr, Γθ<sup>z</sup> ¼ Γ<sup>z</sup>θ, Γθ<sup>r</sup> ¼ Γ<sup>r</sup>θ), thereby, reducing the number of parameters to fully describe the deformation of a volume element in 3D from nine to six parameters. In addition, for isotropic materials, the principal strains can also be found by rotating the strain matrix, the same way as the stress matrix, such that the off-diagonal elements vanish (Γij ¼ 0). In addition, the radial and tangential strains are

> εz εr

To estimate the porosity evolution, bulk volumetric strain has to be used. The volumetric strain equals the change in volume divided by the initial volume, which is the first strain invariant, given by <sup>ε</sup>vol ¼ �ΔV=V<sup>0</sup> <sup>¼</sup> Tr <sup>ε</sup>ij � �. The volume strain remains unchanged upon coordinate change (i.e., the volume is the same irrespective of which coordinate system is used). Depending upon the geometry of the setup, the way in which strain measurements and hence the strain tensor components will vary. For cylindrical geometries, in which the volume of a

<sup>ε</sup>vol <sup>¼</sup> <sup>ε</sup><sup>z</sup> <sup>þ</sup> <sup>2</sup>ε<sup>r</sup> <sup>þ</sup> <sup>2</sup>εzε<sup>r</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup>

4. Partitioning time-independent and time-dependent deformation

plasticity and other failure mechanisms before time-dependent models are described.

If the length and diameter of cylindrical cores are being measured continuously, then the volumetric strain can be estimated. Typically, for small strains, the second and third order

The volumetric strain can be split into an immediate strain, occurring when the effective stress is being changed, and time-dependent deformation. The two cases are presented briefly in the following sections, even though this is a large area of research. For the time-independent case, Hooke's law is described before nonlinear models are presented, followed by a short note on

3 7

tanΨ: (15)

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207

<sup>5</sup> (16)

� � (17)

<sup>r</sup> (18)

L=4, where D is the diameter and L is the length, the volumetric

<sup>r</sup> <sup>þ</sup> <sup>ε</sup>zε<sup>2</sup>

Figure 3. Fluid pressure differences (ΔPf) impart forces onto the solid framework through the fluid-to-solid contact areas, which covers only a fraction α of the cross-section (Af!<sup>s</sup> ¼ Atot � Acons).

area is given by the sum of the consolidated area (solid–solid area) and the area of the fluid-tosolid contact area (Atot ¼ Afs þ Ass), the force from the fluid pressure differences (ΔPf ) to the solids is given as Ffs ¼ ΔPf αAtot, where the fraction total area is termed as the Biot coefficient, and can be expressed as α ¼ Afs=Atot. In addition, other definitions of the Biot coefficient may also apply. In weight-dominated reservoir systems of fluid saturated rocks, the solid stress increases with the lithostatic weight. The net effective stress, that is, the stress that drive deformation is given by the differences between lithostatic pressure and the fraction α (the Biot coefficient) of the pore pressure

$$
\sigma\_v' = \sigma\_v - aP\_f \text{ which is greater than } \sigma\_H' = \sigma\_H - aP\_f > \sigma\_h' = \sigma\_h - aP\_f \tag{13}
$$

Here, the largest and smallest horizontal stress is abbreviated with an index H and h, respectively.

In core scale experiments, the directions perpendicular to the z-axis are equal, σ<sup>r</sup> ¼ σθ, implying that full description of the effective stress state of a cylindrical core experiment are given by two effective stresses, σ<sup>0</sup> <sup>r</sup> ¼ σ<sup>r</sup> � αPf and σ<sup>0</sup> <sup>z</sup> ¼ σ<sup>z</sup> � αPf . The stress exerted onto the core in the radial direction is in many (not all) rock mechanical experiments performed by increasing the hydraulic confining pressure of oil surrounding the core encapsulated by a rubber or plastic sleeve while a piston placed on top of the core controls the axial stress.

#### 3.3. Defining strain

The most commonly used definition of strain, applicable to small finite deformations, is presented here. It is acknowledged that other definitions of strain also exist in the scientific literature. The strain at any time is given by the ratio of elongation divided by the initial length

$$\varepsilon(t) = -\frac{L(t) - L\_0}{L\_0} = -\frac{\Delta L}{L\_0} \text{ and } \varepsilon\_{vol} = -\frac{\Delta V}{V\_0} \tag{14}$$

Stresses may deform Earth materials so that two initially orthogonal directions change by an angle Ψ. This change in angle is related to the shear strain Γ as

Porosity Evolution during Chemo-Mechanical Compaction http://dx.doi.org/10.5772/intechopen.72795 207

$$
\Gamma = \frac{1}{2} \tan \Psi. \tag{15}
$$

In three dimensions (cylindrical coordinates), the pairs of shear and normal strains are organized in the strain tensor

$$\begin{bmatrix} \varepsilon\_{zz} & \Gamma\_{zr} & \Gamma\_{z\theta} \\ \Gamma\_{rz} & \varepsilon\_{rr} & \Gamma\_{r\theta} \\ \Gamma\_{\theta z} & \Gamma\_{\theta r} & \varepsilon\_{\theta\theta} \end{bmatrix} \tag{16}$$

Similar to the stress tensor, the shear strains balance each other (Γrz ¼ Γzr, Γθ<sup>z</sup> ¼ Γ<sup>z</sup>θ, Γθ<sup>r</sup> ¼ Γ<sup>r</sup>θ), thereby, reducing the number of parameters to fully describe the deformation of a volume element in 3D from nine to six parameters. In addition, for isotropic materials, the principal strains can also be found by rotating the strain matrix, the same way as the stress matrix, such that the off-diagonal elements vanish (Γij ¼ 0). In addition, the radial and tangential strains are equal, such that the strain vector for cylindrical cores:

area is given by the sum of the consolidated area (solid–solid area) and the area of the fluid-tosolid contact area (Atot ¼ Afs þ Ass), the force from the fluid pressure differences (ΔPf ) to the solids is given as Ffs ¼ ΔPf αAtot, where the fraction total area is termed as the Biot coefficient, and can be expressed as α ¼ Afs=Atot. In addition, other definitions of the Biot coefficient may also apply. In weight-dominated reservoir systems of fluid saturated rocks, the solid stress increases with the lithostatic weight. The net effective stress, that is, the stress that drive deformation is given by the differences between lithostatic pressure and the fraction α (the

Figure 3. Fluid pressure differences (ΔPf) impart forces onto the solid framework through the fluid-to-solid contact areas,

Here, the largest and smallest horizontal stress is abbreviated with an index H and h, respec-

In core scale experiments, the directions perpendicular to the z-axis are equal, σ<sup>r</sup> ¼ σθ, implying that full description of the effective stress state of a cylindrical core experiment are given by

radial direction is in many (not all) rock mechanical experiments performed by increasing the hydraulic confining pressure of oil surrounding the core encapsulated by a rubber or plastic

The most commonly used definition of strain, applicable to small finite deformations, is presented here. It is acknowledged that other definitions of strain also exist in the scientific literature. The strain at any time is given by the ratio of elongation divided by the initial length

> ¼ � <sup>Δ</sup><sup>L</sup> L0

Stresses may deform Earth materials so that two initially orthogonal directions change by an

<sup>H</sup> ¼ σ<sup>H</sup> � αPf > σ<sup>0</sup>

and <sup>ε</sup>vol ¼ � <sup>Δ</sup><sup>V</sup>

V0

<sup>z</sup> ¼ σ<sup>z</sup> � αPf . The stress exerted onto the core in the

<sup>h</sup> ¼ σ<sup>h</sup> � αPf (13)

(14)

Biot coefficient) of the pore pressure

206 Porosity - Process, Technologies and Applications

<sup>v</sup> ¼ σ<sup>v</sup> � αPf which is greater than σ<sup>0</sup>

which covers only a fraction α of the cross-section (Af!<sup>s</sup> ¼ Atot � Acons).

<sup>r</sup> ¼ σ<sup>r</sup> � αPf and σ<sup>0</sup>

εðÞ¼� t

angle Ψ. This change in angle is related to the shear strain Γ as

sleeve while a piston placed on top of the core controls the axial stress.

L tð Þ� L<sup>0</sup> L0

σ0

two effective stresses, σ<sup>0</sup>

3.3. Defining strain

tively.

$$
\begin{bmatrix} \mathcal{E}\_{\mathcal{E}} \\ \mathcal{E}\_{\mathcal{E}} \end{bmatrix} \tag{17}
$$

To estimate the porosity evolution, bulk volumetric strain has to be used. The volumetric strain equals the change in volume divided by the initial volume, which is the first strain invariant, given by <sup>ε</sup>vol ¼ �ΔV=V<sup>0</sup> <sup>¼</sup> Tr <sup>ε</sup>ij � �. The volume strain remains unchanged upon coordinate change (i.e., the volume is the same irrespective of which coordinate system is used). Depending upon the geometry of the setup, the way in which strain measurements and hence the strain tensor components will vary. For cylindrical geometries, in which the volume of a cylinder is given by <sup>V</sup> <sup>¼</sup> <sup>π</sup>D<sup>2</sup> L=4, where D is the diameter and L is the length, the volumetric strain can be calculated from the radial and axial strains

$$
\varepsilon\_{\rm vol} = \varepsilon\_z + 2\varepsilon\_r + 2\varepsilon\_z\varepsilon\_r + \varepsilon\_r^2 + \varepsilon\_z\varepsilon\_r^2 \tag{18}
$$

If the length and diameter of cylindrical cores are being measured continuously, then the volumetric strain can be estimated. Typically, for small strains, the second and third order terms can be omitted, hence, εvol ≃ε<sup>z</sup> þ 2εr.

### 4. Partitioning time-independent and time-dependent deformation

The volumetric strain can be split into an immediate strain, occurring when the effective stress is being changed, and time-dependent deformation. The two cases are presented briefly in the following sections, even though this is a large area of research. For the time-independent case, Hooke's law is described before nonlinear models are presented, followed by a short note on plasticity and other failure mechanisms before time-dependent models are described.

#### 4.1. Elastic strain: linear elasticity

Hooke's law is the simplest relation to describe the relation between the stress-strain tensors. It assumes that the deformation is immediate, linear, and reversible. In continuous media, for small stress and strain increments in the linear limit, the εij and σij are described by the compliance (stiffness) fourth order tensor cijkl. In 3D systems, it consists of 81 real numbers, and the tensorial equation attains the compact form σij ¼ cijklεkl, where the indexes i, j, k, l represent the three spatial dimensions ½ � x; y; z in Carthesian co-ordinate systems, and ½ � z;r; θ in cylindrical systems. In the case, when the rotational forces balances, which applies to most continuum mechanical cases, the number of stiffness parameters describing the stress-strain relation reduces to 27. In the case of isotropic materials, the number of elastic parameters that describe the stress-strain relation of a volume element along the principal directions is further reduced to the Young's modulus (E) and Poisson's ratio (ν) via the matrix equation

$$
\begin{bmatrix} \varepsilon\_z \\ \varepsilon\_r \\ \varepsilon\_\theta \end{bmatrix} = \frac{1}{E} \begin{bmatrix} 1 & -\nu & -\nu \\ -\nu & 1 & -\nu \\ -\nu & -\nu & 1 \end{bmatrix} \begin{bmatrix} \sigma'\_z \\ \sigma'\_r \\ \sigma'\_\theta \end{bmatrix} \tag{19}
$$

In Eq. (23), the underlying assumption is that Young's modulus and Poisson ratio remain fixed. Furthermore, when pore pressure is included, the effective stress changes due to both

It is assumed that the Biot stress coefficient remains fixed during loading. Using these definitions into Eq. (32) enables us to fully describe the relation between the stress, pore pressure,

> Eδε<sup>r</sup> ¼ ð Þ 1 � ν δσ<sup>r</sup> � νδσ<sup>z</sup> þ ð Þ 2ν � 1 αδPf Eδε<sup>z</sup> ¼ δσ<sup>z</sup> � 2νδσ<sup>r</sup> þ ð Þ 2ν � 1 αδPf

For a highly porous chalk, a nonzero component of the observed strain is always irreversible when the load is released as exemplified by [8] and [9] where irreversible plasticity is seen also within the 'elastic' phase of the QP-plot (Figure 4). As such, loading and unloading may display history dependence in the elastic parameters. This may be caused by the way in which the porous material is being held together and the relative importance of the different forces responsible for determining the stiffness of the chalk. Now, at increasing stresses beyond 'elasticity', the type of irreversible deformation that develops, depend on the state of stress, as illustrated in Figure 4. Here, the mean effective stress is plotted on the horizontal axis and the deviatoric stress along the vertical axis. Considering cylindrical cases, the deviatoric stress equals Q ¼ σ<sup>z</sup> � σ<sup>r</sup> and the mean effective stress is P ¼ ð Þ σ<sup>z</sup> þ 2σ<sup>r</sup> =3 � αPf: Since both Q and

Figure 4. The failure envelope (solid line) shows that at which stresses plasticity and irreversible deformation occur (numbers are not applicable). Mean effective stress (P) and Q is the deviatoric stress, hence the end cap depends solely

<sup>z</sup> ¼ δσ<sup>z</sup> � αδPf (24)

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209

<sup>r</sup> ¼ δσ<sup>r</sup> � αδPf and δσ<sup>0</sup>

axial and radial stress and pore pressure

4.3. Plasticity and irreversible deformation

and strain in Hooke's law

upon the material and not the geometry.

σ0

By adding up the three equations expressed in the matrix form earlier

$$(\varepsilon\_z + \varepsilon\_r + \varepsilon\_\theta)E = (1 - 2\nu)(\sigma\_z' + \sigma\_r' + \sigma\_\theta'),\tag{20}$$

we may use this equation to define the bulk modulus in hydrostatic tests. When omitting higher order terms in the volumetric strain (Eq. (18)), the left hand side of Eq. (20) equals the volumetric strain. For hydrostatic tests, in which the stresses in all spatial directions equal, σ0 <sup>z</sup> ¼ σ<sup>0</sup> <sup>r</sup> ¼ σ<sup>0</sup> <sup>θ</sup> ¼ σ<sup>0</sup> <sup>p</sup>, then Eq. (20) simplifies to

$$\frac{E}{1-2\nu}\varepsilon\_{\text{vol}} = \sigma\_p' \to \text{K}\varepsilon\_{\text{vol}} = \sigma\_p' \tag{21}$$

In Eq. (21), the bulk modulus (K) is defined. σ<sup>0</sup> <sup>p</sup> is frequently used to define the hydrostatic effective stress. For nonhydrostatic triaxial tests, where σ<sup>z</sup> > σr, and σ<sup>0</sup> <sup>r</sup> ¼ σ<sup>0</sup> <sup>θ</sup> and ε<sup>r</sup> ¼ εθ Hooke's law in Eq. (19) simplifies to

$$\begin{aligned} E\varepsilon\_r &= (1 - \nu)\sigma\_r' - \nu \sigma\_z'\\ E\varepsilon\_z &= \sigma\_z' - 2\nu \sigma\_r' \end{aligned} \tag{22}$$

#### 4.2. The effective stress changes that drive deformation

Within the elastic domain, any change in the effective stress drive deformation in the sample, from here on abbreviated with the δ-symbol used to rewrite Hooke's law at quasistatic changes. The δ-symbol is used to identify the variables that are changing during for example, a loading sequence

$$\begin{aligned} E\delta\varepsilon\_r &= (1 - \nu)\delta\sigma\_r' - \nu\delta\sigma\_z'\\ E\delta\varepsilon\_z &= \delta\sigma\_z' - 2\nu\delta\sigma\_r' \end{aligned} \tag{23}$$

In Eq. (23), the underlying assumption is that Young's modulus and Poisson ratio remain fixed. Furthermore, when pore pressure is included, the effective stress changes due to both axial and radial stress and pore pressure

$$
\sigma\_r' = \delta \sigma\_r - a \delta P\_f \text{ and } \delta \sigma\_z' = \delta \sigma\_z - a \delta P\_f \tag{24}
$$

It is assumed that the Biot stress coefficient remains fixed during loading. Using these definitions into Eq. (32) enables us to fully describe the relation between the stress, pore pressure, and strain in Hooke's law

$$\begin{aligned} E\delta\varepsilon\_{r} &= (1 - \nu)\delta\sigma\_{r} - \nu\delta\sigma\_{z} + (2\nu - 1)a\delta P\_{f} \\ E\delta\varepsilon\_{z} &= \delta\sigma\_{z} - 2\nu\delta\sigma\_{r} + (2\nu - 1)a\delta P\_{f} \end{aligned} \tag{25}$$

#### 4.3. Plasticity and irreversible deformation

4.1. Elastic strain: linear elasticity

208 Porosity - Process, Technologies and Applications

σ0 <sup>z</sup> ¼ σ<sup>0</sup>

<sup>r</sup> ¼ σ<sup>0</sup>

a loading sequence

<sup>θ</sup> ¼ σ<sup>0</sup>

Hooke's law is the simplest relation to describe the relation between the stress-strain tensors. It assumes that the deformation is immediate, linear, and reversible. In continuous media, for small stress and strain increments in the linear limit, the εij and σij are described by the compliance (stiffness) fourth order tensor cijkl. In 3D systems, it consists of 81 real numbers, and the tensorial equation attains the compact form σij ¼ cijklεkl, where the indexes i, j, k, l represent the three spatial dimensions ½ � x; y; z in Carthesian co-ordinate systems, and ½ � z;r; θ in cylindrical systems. In the case, when the rotational forces balances, which applies to most continuum mechanical cases, the number of stiffness parameters describing the stress-strain relation reduces to 27. In the case of isotropic materials, the number of elastic parameters that describe the stress-strain relation of a volume element along the principal directions is further

reduced to the Young's modulus (E) and Poisson's ratio (ν) via the matrix equation

2 6 4

ð Þ ε<sup>z</sup> þ ε<sup>r</sup> þ εθ E ¼ ð Þ 1 � 2ν σ<sup>0</sup>

εvol ¼ σ<sup>0</sup>

Eε<sup>r</sup> ¼ ð Þ 1 � ν σ<sup>0</sup>

Eε<sup>z</sup> ¼ σ<sup>0</sup>

Eδε<sup>r</sup> ¼ ð Þ 1 � ν δσ<sup>0</sup>

Eδε<sup>z</sup> ¼ δσ<sup>0</sup>

we may use this equation to define the bulk modulus in hydrostatic tests. When omitting higher order terms in the volumetric strain (Eq. (18)), the left hand side of Eq. (20) equals the volumetric strain. For hydrostatic tests, in which the stresses in all spatial directions equal,

1 �ν �ν �ν 1 �ν �ν �ν 1

3 7 5

<sup>z</sup> þ σ<sup>0</sup>

<sup>p</sup> ! Kεvol ¼ σ<sup>0</sup>

<sup>r</sup> � νσ<sup>0</sup> z

<sup>r</sup> � νδσ<sup>0</sup> z

<sup>z</sup> � 2νδσ<sup>0</sup> r

<sup>z</sup> � 2νσ<sup>0</sup> r

Within the elastic domain, any change in the effective stress drive deformation in the sample, from here on abbreviated with the δ-symbol used to rewrite Hooke's law at quasistatic changes. The δ-symbol is used to identify the variables that are changing during for example,

<sup>r</sup> þ σ<sup>0</sup> θ

2 6 4

σ0 z σ0 r σ0 θ 3 7

<sup>5</sup> (19)

<sup>p</sup> (21)

<sup>r</sup> ¼ σ<sup>0</sup>

<sup>θ</sup> and ε<sup>r</sup> ¼ εθ

(22)

(23)

<sup>p</sup> is frequently used to define the hydrostatic

� �, (20)

εz εr εθ

By adding up the three equations expressed in the matrix form earlier

E 1 � 2ν

effective stress. For nonhydrostatic triaxial tests, where σ<sup>z</sup> > σr, and σ<sup>0</sup>

2 6 4

<sup>p</sup>, then Eq. (20) simplifies to

In Eq. (21), the bulk modulus (K) is defined. σ<sup>0</sup>

4.2. The effective stress changes that drive deformation

Hooke's law in Eq. (19) simplifies to

For a highly porous chalk, a nonzero component of the observed strain is always irreversible when the load is released as exemplified by [8] and [9] where irreversible plasticity is seen also within the 'elastic' phase of the QP-plot (Figure 4). As such, loading and unloading may display history dependence in the elastic parameters. This may be caused by the way in which the porous material is being held together and the relative importance of the different forces responsible for determining the stiffness of the chalk. Now, at increasing stresses beyond 'elasticity', the type of irreversible deformation that develops, depend on the state of stress, as illustrated in Figure 4. Here, the mean effective stress is plotted on the horizontal axis and the deviatoric stress along the vertical axis. Considering cylindrical cases, the deviatoric stress equals Q ¼ σ<sup>z</sup> � σ<sup>r</sup> and the mean effective stress is P ¼ ð Þ σ<sup>z</sup> þ 2σ<sup>r</sup> =3 � αPf: Since both Q and

Figure 4. The failure envelope (solid line) shows that at which stresses plasticity and irreversible deformation occur (numbers are not applicable). Mean effective stress (P) and Q is the deviatoric stress, hence the end cap depends solely upon the material and not the geometry.

P are invariant, the results of core data can be used at any case in which the material is the same: (1) for hydrostatic systems, Q ¼ 0, and pore collapse occurs when the mean effective stress exceeds a certain threshold; (2) tensile fractures develop at negative values of P which can be found for high fluid pressures, or in Brazilian tests; and (3) shear failure occurs when the deviatoric stress exceeds a certain value. For frictional materials, it is typical to observe that the deviatoric stress required to induce shear failure is increasing with increasing mean effective stress. For Coulomb materials, this relation is proportional, and the slope is related to the frictional coefficient. Chalks have been found to be satisfactory described with such a frictional coefficient, while clays behave differently. The way in which the irreversible deformation affects the porosity evolution differs from case to case. Within shear zones, the porosity may both increase, because of dilation and de-compaction when tightly packed grains reorganize or reduce because of grain crushing when the imposed forces exceed a certain level.
