5. Time-dependent pore volume reduction processes and compaction

To understand how the mechanical and chemical processes affect the porosity during pore collapse, it is important to take a closer look at how the observed bulk strain can be partitioned. In this section, we consider the simplest possible partition (below) in which the overall strain is partitioned additively into a solid volume and a pore volume component. The relative importance of these mechanisms may be found by the analysis of quantitative measurements of the bulk volume change and the change in the solid volume due to the dissolution/ precipitation as the mineral mass and density change over time, while grains reorganize, crush, and solid contacts evolve. In this case, the observed volumetric change can be partitioned additively via

$$
\varepsilon\_{val} = \varepsilon\_{pure} + \varepsilon\_{solid} \tag{26}
$$

volume changes approximately by 0.01 and 0.005 cm3

εvolðÞ¼ t

the observed strains. Three models that have been used are:

dϕ

simplicity, if we assume n ¼ 1 and ϕ<sup>c</sup> ¼ 0, the volumetric strain is explicitly given as

1 � ϕ<sup>0</sup> � �e<sup>ξ</sup><sup>t</sup> e<sup>ξ</sup><sup>t</sup> � ϕ<sup>0</sup>

dt ¼ �ξ ϕ � <sup>ϕ</sup><sup>c</sup>

where ξ is the proportionality constant, ϕ<sup>c</sup> is a terminal porosity (grain reorganization cannot continue until zero porosity), and the power n is used to model nonlinear behavior. For

This model takes the initial porosity (ϕ0) and bulk volume (V0, <sup>b</sup>), while the porosity rate

The mathematical models aimed to match observed creep data have a long history, and several, more or less physically based models were reported. Generally, these models do not consider the underlying solid and pore volume contribution, but may still satisfactorily match

<sup>ε</sup><sup>P</sup> <sup>¼</sup> AtBe

The model parameters ð Þ <sup>A</sup>; <sup>B</sup> and <sup>t</sup><sup>0</sup> are found when the residual strain RES <sup>¼</sup> <sup>1</sup>

The movement of grains relative to each other at high-mean effective pressures causes pore volumes to collapse. It has been experimentally verified that for chalks, the rate of compaction may sometimes accelerate when the fluid composition change, a process termed water weakening. Water weakening has been used to understand reservoir processes [13–15] and to interpret core experiments as exemplified in Figure 5, where an additional strain of �1% is seen at the first days of seawater (SSW) flow (approximately 2 pore volumes). This process cannot be attributed to chemical reactions leading to solid volume changes, since the ions in the produced effluent water are inadequate to cause any solid volume change from the mass loss and increased density often seen in these cases when chemical reactions occur. As such, the additional bulk strain is caused by pore collapse. This does not exclude how long-term

5.1. Pore collapse and grain reorganization: the constant solid volume case

� <sup>β</sup><sup>t</sup>

tion rate can be proportional to porosity

constant ξ is a free variable.

Power law with cut-off:

De Waal [11]:

Griggs [12]:

εmodel∣ is minimized.

/day, respectively. The porosity reduc-

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211

� �<sup>n</sup> (28)

Porosity Evolution during Chemo-Mechanical Compaction

Vb, <sup>0</sup> <sup>1</sup> � <sup>e</sup>�ξ<sup>t</sup> ð Þ � <sup>1</sup> (29)

�<sup>t</sup> <sup>t</sup> <sup>=</sup> <sup>0</sup> (30a)

N P

<sup>n</sup>∣εexp �

εdW ¼ Alogð Þ Bt þ 1 (30b)

ε<sup>G</sup> ¼ Alogð Þþ t þ 1 Bt (30c)

This does not imply that imply that cross terms do not exist in which: (a) the rate of pore volume reduction is sensitive to the reduction in solid volume and (b) how the solid volume rate may depend on how grains reorganize to change the flow pattern and potentially expose new fresh mineral surfaces to the reactive brine. It is likely to assume that based on the accelerated strain presented in [10] minute changes to the solid volume increase the rate of pore collapse (also seen in [4]).

Given the simple partitioning above, a model can be developed to describe the observed creep curve with a few physical parameters (see Eqs. (20)–(23) in [10]). In this model, overall volumetric strain is additively partitioned into a pore and solid volume component in which the pore volume equals, Vp ¼ ϕVb. Extending the rate of change in bulk volume is by the pore volume change (via using the product rule) and the solid volume change rate

$$\frac{dV\_b}{dt} = \frac{dV\_p}{dt} + \frac{dV\_s}{dt} = \phi \frac{dV\_b}{dt} + V\_b \frac{d\phi}{dt} - \beta \tag{27}$$

The solid volume rate is assumed to be constant (β, in cm3 /day determined from ion chromatography data). For Mons chalk at 130�C and 92�C at 1 PV/day of 0.219 MgCl2 brine, the solid volume changes approximately by 0.01 and 0.005 cm3 /day, respectively. The porosity reduction rate can be proportional to porosity

$$\frac{d\phi}{dt} = -\xi \left(\phi - \phi\_c\right)''\tag{28}$$

where ξ is the proportionality constant, ϕ<sup>c</sup> is a terminal porosity (grain reorganization cannot continue until zero porosity), and the power n is used to model nonlinear behavior. For simplicity, if we assume n ¼ 1 and ϕ<sup>c</sup> ¼ 0, the volumetric strain is explicitly given as

$$\varepsilon\_{\rm vol}(t) = \frac{(1 - \phi\_0)e^{\xi t}}{e^{\xi t} - \phi\_0} - \frac{\beta t}{V\_{b,0}(1 - e^{-\xi t})} - 1 \tag{29}$$

This model takes the initial porosity (ϕ0) and bulk volume (V0, <sup>b</sup>), while the porosity rate constant ξ is a free variable.

The mathematical models aimed to match observed creep data have a long history, and several, more or less physically based models were reported. Generally, these models do not consider the underlying solid and pore volume contribution, but may still satisfactorily match the observed strains. Three models that have been used are:

Power law with cut-off:

$$
\varepsilon\_P = A t^B e^{-t/\sqrt{\varepsilon\_0}} \tag{30a}
$$

De Waal [11]:

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.

additively via

pore collapse (also seen in [4]).

210 Porosity - Process, Technologies and Applications

5. Time-dependent pore volume reduction processes and compaction

To understand how the mechanical and chemical processes affect the porosity during pore collapse, it is important to take a closer look at how the observed bulk strain can be partitioned. In this section, we consider the simplest possible partition (below) in which the overall strain is partitioned additively into a solid volume and a pore volume component. The relative importance of these mechanisms may be found by the analysis of quantitative measurements of the bulk volume change and the change in the solid volume due to the dissolution/ precipitation as the mineral mass and density change over time, while grains reorganize, crush, and solid contacts evolve. In this case, the observed volumetric change can be partitioned

This does not imply that imply that cross terms do not exist in which: (a) the rate of pore volume reduction is sensitive to the reduction in solid volume and (b) how the solid volume rate may depend on how grains reorganize to change the flow pattern and potentially expose new fresh mineral surfaces to the reactive brine. It is likely to assume that based on the accelerated strain presented in [10] minute changes to the solid volume increase the rate of

Given the simple partitioning above, a model can be developed to describe the observed creep curve with a few physical parameters (see Eqs. (20)–(23) in [10]). In this model, overall volumetric strain is additively partitioned into a pore and solid volume component in which the pore volume equals, Vp ¼ ϕVb. Extending the rate of change in bulk volume is by the pore

dt <sup>¼</sup> <sup>ϕ</sup> dVb

tography data). For Mons chalk at 130�C and 92�C at 1 PV/day of 0.219 MgCl2 brine, the solid

dt <sup>þ</sup> Vb

dϕ

dt � <sup>β</sup> (27)

/day determined from ion chroma-

volume change (via using the product rule) and the solid volume change rate

dVs

dVb dt <sup>¼</sup> dVp dt þ

The solid volume rate is assumed to be constant (β, in cm3

εvol ¼ εpore þ εsolid (26)

$$
\varepsilon\_{dW} = A \log(Bt + 1) \tag{30b}
$$

Griggs [12]:

$$
\varepsilon\_G = A \log(t+1) + Bt\tag{30c}
$$

The model parameters ð Þ <sup>A</sup>; <sup>B</sup> and <sup>t</sup><sup>0</sup> are found when the residual strain RES <sup>¼</sup> <sup>1</sup> N P <sup>n</sup>∣εexp � εmodel∣ is minimized.

#### 5.1. Pore collapse and grain reorganization: the constant solid volume case

The movement of grains relative to each other at high-mean effective pressures causes pore volumes to collapse. It has been experimentally verified that for chalks, the rate of compaction may sometimes accelerate when the fluid composition change, a process termed water weakening. Water weakening has been used to understand reservoir processes [13–15] and to interpret core experiments as exemplified in Figure 5, where an additional strain of �1% is seen at the first days of seawater (SSW) flow (approximately 2 pore volumes). This process cannot be attributed to chemical reactions leading to solid volume changes, since the ions in the produced effluent water are inadequate to cause any solid volume change from the mass loss and increased density often seen in these cases when chemical reactions occur. As such, the additional bulk strain is caused by pore collapse. This does not exclude how long-term

Figure 5. Axial creep strain over time at uniaxial strain condition performed on a Kansas chalk sample. The injection of seawater (SSW) leads to accelerated creep. The accelerated creep period is associated with the loss of sulfate ions in the effluent samples. Mixing CO2 into the SSW from 120 days and onwards does not induce additional strain (from [20]).

Several candidates coexist. The stress tensor in reservoir systems (and core scale experiments) depends on the weight of the overburden (lithostatic weight), side stress (tectonic forces), pore pressure, and the Biot coefficient. The simplest of determining the thermodynamic pressure is using the pore pressure. This way of thinking may seem reasonable at first glance since it is at the interface between the solid and the fluid where the chemical reactions occur. Simultaneously, at the rock-fluid interface, the stresses through the solid framework could also play a role in determining chemical solubility. In that case, the continuum mechanics provide a range of choices for calculating the thermodynamic pressure: (1) the average compressive stress (i.e., the first invariant of the solid framework stress tensor), (2) the principal stresses, thereby leading to different solubility in the different spatial directions. In sedimentary systems, this would often lead to enhanced solubility in the vertical direction as the first principal stress direction is vertical. This may explain the formation of the horizontal stylolites that are sometimes found in calcitic, carbonate, and limestone rocks [6]. (3) The relevant thermodynamic pressure could be related to the stress gradients that have been observed throughout porous materials, termed force chains. At grain-grain contacts, through which the externally imposed loads are being carried, the stresses can be significantly higher than the average. In these

Figure 6. (a) Outlet cation concentration of 0.219 M MgCl2 (dashed line, same ion strength as seawater) flooded through chalk from the Obourg saint vast formation (Mons, Belgium) at 130�C for 0.5, 3.5, and 12.3 MPa effective stress. (b) Calcium production at varying stresses at 130�C and 92�C. The amount of Ca ions in the produced effluent depends on

ss ¼ σ<sup>0</sup>

is the Biot stress coefficient. As such, for unconsolidated sands and calcitic mudstone, in which α > 0:9, this fraction is significant and may be responsible for additional calcite dissolution [10]. As has been shown previously (in e.g., [18, 22] and also before that), the contact area ratio

Even though pressure solution is a process of chemical nature, it does not necessarily change the solid volume since the mass can be conserved (closed system, i.e., no larger scale mass flow) and the same mineral phase is precipitated as the one dissolved (i.e., same density). In that sense, pressure solution contributes to pore volume reduction rather than the solid volume in the strain partitioning presented here. Hence, pressure solution may fall under mechanical compaction even though the underlying mechanisms of pressure solution are

=ð Þ 1 � α , where σ<sup>0</sup> is the effective stress and α

Porosity Evolution during Chemo-Mechanical Compaction

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213

regions, the solid-solid stress is given by σ<sup>0</sup>

time, temperature, and stress (acquired from Figure 6 in [10]).

is linked to the Biot stress coefficient (α).

chemically driven.

chemical reactions can weaken rocks over longer time periods when porous chalk cores are continuously flooded.

To understand the immediate additional deformation (i.e., 1–2 days corresponding to a flow of 1-2 pore volumes as seen here), a grain-level approach is required. The grain-grain friction controls and cement bonds binding neighboring grains together control the relative movement of grains. Friction between grains is given by the frictional coefficient times the normal force, Ffric ¼ μFN. The normal force arises from the externally imposed load and the attractive Van der Waal forces that induce the cohesive forces grains. This has been shown to be reduced by negative disjoining pressures in the overlapping double layer between adjacent mineral grains when surface-active divalent ions adsorb onto the charged chalk surfaces [16–19].

#### 5.2. Pressure solution and other grain-reorganization mechanisms

Pressure and temperature are the state variables that control the Gibbs chemical potential [21]. During diagenesis and burial, the chemical stability of mineral phases is altered as the temperature, hydrostatic and lithostatic pressure increases. Pressure solution of stressed grain contacts, and precipitation in unstressed parts of the rock framework, have been used as one of the primary rock-forming mechanisms during diagenesis. Pressure solution can occur in closed systems, in which the overall mass and density remain fixed. For high Biot coefficients, the local stress at particle contacts may become significant (see Figure 7 in [10]), and thus, a stressdependent production of Ca-ions is observed where more Ca-production for high stress than low stress (see Figure 6 acquired from [10]).

It has been a long-standing discussion how to mathematically describe the relevant thermodynamic pressure for accurate determination of the chemical potential from the stress tensor.

Figure 6. (a) Outlet cation concentration of 0.219 M MgCl2 (dashed line, same ion strength as seawater) flooded through chalk from the Obourg saint vast formation (Mons, Belgium) at 130�C for 0.5, 3.5, and 12.3 MPa effective stress. (b) Calcium production at varying stresses at 130�C and 92�C. The amount of Ca ions in the produced effluent depends on time, temperature, and stress (acquired from Figure 6 in [10]).

Several candidates coexist. The stress tensor in reservoir systems (and core scale experiments) depends on the weight of the overburden (lithostatic weight), side stress (tectonic forces), pore pressure, and the Biot coefficient. The simplest of determining the thermodynamic pressure is using the pore pressure. This way of thinking may seem reasonable at first glance since it is at the interface between the solid and the fluid where the chemical reactions occur. Simultaneously, at the rock-fluid interface, the stresses through the solid framework could also play a role in determining chemical solubility. In that case, the continuum mechanics provide a range of choices for calculating the thermodynamic pressure: (1) the average compressive stress (i.e., the first invariant of the solid framework stress tensor), (2) the principal stresses, thereby leading to different solubility in the different spatial directions. In sedimentary systems, this would often lead to enhanced solubility in the vertical direction as the first principal stress direction is vertical. This may explain the formation of the horizontal stylolites that are sometimes found in calcitic, carbonate, and limestone rocks [6]. (3) The relevant thermodynamic pressure could be related to the stress gradients that have been observed throughout porous materials, termed force chains. At grain-grain contacts, through which the externally imposed loads are being carried, the stresses can be significantly higher than the average. In these regions, the solid-solid stress is given by σ<sup>0</sup> ss ¼ σ<sup>0</sup> =ð Þ 1 � α , where σ<sup>0</sup> is the effective stress and α is the Biot stress coefficient. As such, for unconsolidated sands and calcitic mudstone, in which α > 0:9, this fraction is significant and may be responsible for additional calcite dissolution [10]. As has been shown previously (in e.g., [18, 22] and also before that), the contact area ratio is linked to the Biot stress coefficient (α).

chemical reactions can weaken rocks over longer time periods when porous chalk cores are

Figure 5. Axial creep strain over time at uniaxial strain condition performed on a Kansas chalk sample. The injection of seawater (SSW) leads to accelerated creep. The accelerated creep period is associated with the loss of sulfate ions in the effluent samples. Mixing CO2 into the SSW from 120 days and onwards does not induce additional strain (from [20]).

To understand the immediate additional deformation (i.e., 1–2 days corresponding to a flow of 1-2 pore volumes as seen here), a grain-level approach is required. The grain-grain friction controls and cement bonds binding neighboring grains together control the relative movement of grains. Friction between grains is given by the frictional coefficient times the normal force, Ffric ¼ μFN. The normal force arises from the externally imposed load and the attractive Van der Waal forces that induce the cohesive forces grains. This has been shown to be reduced by negative disjoining pressures in the overlapping double layer between adjacent mineral grains

Pressure and temperature are the state variables that control the Gibbs chemical potential [21]. During diagenesis and burial, the chemical stability of mineral phases is altered as the temperature, hydrostatic and lithostatic pressure increases. Pressure solution of stressed grain contacts, and precipitation in unstressed parts of the rock framework, have been used as one of the primary rock-forming mechanisms during diagenesis. Pressure solution can occur in closed systems, in which the overall mass and density remain fixed. For high Biot coefficients, the local stress at particle contacts may become significant (see Figure 7 in [10]), and thus, a stressdependent production of Ca-ions is observed where more Ca-production for high stress than

It has been a long-standing discussion how to mathematically describe the relevant thermodynamic pressure for accurate determination of the chemical potential from the stress tensor.

when surface-active divalent ions adsorb onto the charged chalk surfaces [16–19].

5.2. Pressure solution and other grain-reorganization mechanisms

low stress (see Figure 6 acquired from [10]).

continuously flooded.

212 Porosity - Process, Technologies and Applications

Even though pressure solution is a process of chemical nature, it does not necessarily change the solid volume since the mass can be conserved (closed system, i.e., no larger scale mass flow) and the same mineral phase is precipitated as the one dissolved (i.e., same density). In that sense, pressure solution contributes to pore volume reduction rather than the solid volume in the strain partitioning presented here. Hence, pressure solution may fall under mechanical compaction even though the underlying mechanisms of pressure solution are chemically driven.
