6. Time-dependent solid volume evolution mechanisms

In open, nonequilibrium systems with rock-fluid interactions, the solid volume is subject to change. It has been shown in a range of experiments how additional strain is accumulated during compaction at constant stress conditions when reactive brines are injected [4, 23]. The solid volume varies when solid mass (Ms) and mineralogical density (rs) change

$$V\_s = \frac{M\_s}{\rho\_s} \tag{31}$$

r<sup>s</sup> ¼ c1r<sup>1</sup> þ c2r<sup>2</sup> þ … þ cnr<sup>n</sup>

It is not always the case that a detailed kinetic chemical model exist tuned to take into account how different mineral mixtures react with fluids in each different case. If the overall density before (rs,0) and after (rs,f ) the chemical experiment are known (from e.g., pycnometry) a reduced mass parameter (m~ ) ranging from 0 to 1 can be defined from the initial and final mass

> m t <sup>~</sup> ðÞ¼ Mkð Þ� <sup>t</sup> Mk,<sup>0</sup> Mk,f � Mk, <sup>0</sup>

An example of a dynamic porosity development analysis is presented here based on material published in 2015 [3]. The results of an experiment performed over 1090 days where a Liegè (Belgium) chalk sample was exposed to hydrostatic stress (11.1 MPa, approximately 5 MPa

and 130�C). Basic sample measurements were performed of dry/saturated mass, pore volume, solid and bulk volume and hence porosity before and after test, hence the mineral density estimated and confirmed using He-pycnometry (Table 1). The bulk core volume was reduced more than 15% and mineral mass is reduced by more than 18% while the density is increased

0.219 M MgCl2-brine (with equal ion strength as seawater) induced dissolution of the calcium carbonate and precipitation of denser Mg-bearing carbonates (e.g., magnesite and dolomite).

Table 1. Basic measurements of the core before and after the 1090 days long-term test [3]. Figure 1 displays SEM images

Dry mass (on scale) 125.57 g 102.64 g �22.93 g Wet weight (saturated) 158.56 g 126.34 g �32.22 g Pore volume 32.99 cm3 23.71 cm3 �9.28 cm3 Solid volume 46.85 cm3 35.53 cm3 �11.32 cm<sup>3</sup> Bulk volume 79.84 cm3 59.23 cm3 �20.61 cm<sup>3</sup> Mineral density (saturation and pycnometer) 2.68 and 2.70 g/cm<sup>3</sup> 2.89 and 2.90 g/cm3 0.21 and 0.20 g/cm<sup>3</sup> Porosity (saturation and pycnometer) 41.3 and 41.7% 40.0 and 40.1% �1.3 and �1.6%

. The stresses are sufficient to induce pore collapse for these chalks, and the

Before test After test (1090 days) Change

As the concentrations of different minerals vary, the changes to r<sup>s</sup> can be estimated.

7. Predicting dynamic porosity evolution: an illustrative example

using mineral k (Mk,<sup>0</sup> and Mk,f , respectively)

of the core material before and after the flow-through test.

from 2.7 to 2.9 g/cm3

Then, the density at any given time may be estimated using

above yield) and continuous flow of 0.219 M MgCl2 (33 and 99 cm<sup>3</sup>

1 ¼ c<sup>1</sup> þ c<sup>2</sup> þ … þ cn (34)

Porosity Evolution during Chemo-Mechanical Compaction

http://dx.doi.org/10.5772/intechopen.72795

rsðÞ¼ t m t ~ ð Þrs,<sup>f</sup> þ ð Þ 1 � m t ~ ð Þ rs,<sup>0</sup> (36)

(35)

215

/day, pore pressure 0.7 MPa

The change in solid volume may be evaluated by

$$
\Delta V\_s(t) = \frac{M\_s(t)}{\rho\_s(t)} - \frac{M\_{s,0}}{\rho\_{s,0}} \tag{32}
$$

Here, the solid volume change is given by the difference between the ratio of the mass and density at a given time and the values before chemo-mechanical processes initiated. The evolution of the solid mass over time is given by the difference between the chemical mass flux in and out of the system, and density changes as new minerals precipitate.

#### 6.1. Mass transfer in open systems

When fluids continuously flow and react with the rock, the mass (and hence the solid volume) changes. The chemical flux can be monitored by evaluating the effluent concentration through difference between the ion concentrations in and out of the volume element (Figure 6a, b). This volume element may, in some cases, be between an injector and a producer in an oil field, or a core scale experiment in the laboratory [3]. The concentration of ions can be measured using ion chromatography, and over a time interval δt the difference in mass is given by

$$\frac{\delta M\_s}{\delta t} = \sum\_j (\mathbf{c}\_{in,j} - \mathbf{c}\_{in,j}) qm\_j \tag{33}$$

In Eq. (33), the factor cin,j � cin,j � � is the difference in the ion concentration of chemical species j (mole/L), q is the flow rate (L/day), and mj is the molar mass of species j (g/mole). Hence, the term δMs=δt is given in g/day. The overall mass is estimated by summing over all measured ions in the chemical interaction, giving a unit (g/day), which can be used further. The total mass evolution of each species is determined by integration. Assessing rock-fluid interactions to real cases, for example during, seawater flooding of the Ekofisk field (North Sea, Norway), chemical reactions have been observed. Here, dissolution of 1–2 wt. % is anticipated from the analysis of the produced water [24–26].

#### 6.2. Method to quantify the solid volume evolution

In Eq. (32), the change in solid volume depends on the change in both mass and in density as the minerals dissolve and precipitate. The overall mineral density, as n minerals dissolve/ precipitate in time is given by

$$
\rho\_s = c\_1 \rho\_1 + c\_2 \rho\_2 + \dots + c\_n \rho\_n
$$

$$
1 = c\_1 + c\_2 + \dots + c\_n \tag{34}
$$

As the concentrations of different minerals vary, the changes to r<sup>s</sup> can be estimated.

It is not always the case that a detailed kinetic chemical model exist tuned to take into account how different mineral mixtures react with fluids in each different case. If the overall density before (rs,0) and after (rs,f ) the chemical experiment are known (from e.g., pycnometry) a reduced mass parameter (m~ ) ranging from 0 to 1 can be defined from the initial and final mass using mineral k (Mk,<sup>0</sup> and Mk,f , respectively)

$$
\tilde{m}\,(t) = \frac{M\_k(t) - M\_{k,0}}{M\_{k,f} - M\_{k,0}} \tag{35}
$$

Then, the density at any given time may be estimated using

6. Time-dependent solid volume evolution mechanisms

The change in solid volume may be evaluated by

6.1. Mass transfer in open systems

214 Porosity - Process, Technologies and Applications

In Eq. (33), the factor cin,j � cin,j

analysis of the produced water [24–26].

precipitate in time is given by

6.2. Method to quantify the solid volume evolution

solid volume varies when solid mass (Ms) and mineralogical density (rs) change

ΔVsðÞ¼ t

flux in and out of the system, and density changes as new minerals precipitate.

ion chromatography, and over a time interval δt the difference in mass is given by

δMs <sup>δ</sup><sup>t</sup> <sup>¼</sup> <sup>X</sup>

In open, nonequilibrium systems with rock-fluid interactions, the solid volume is subject to change. It has been shown in a range of experiments how additional strain is accumulated during compaction at constant stress conditions when reactive brines are injected [4, 23]. The

> Vs <sup>¼</sup> Ms rs

> > Msð Þt <sup>r</sup>sð Þ<sup>t</sup> � Ms, <sup>0</sup> rs, <sup>0</sup>

Here, the solid volume change is given by the difference between the ratio of the mass and density at a given time and the values before chemo-mechanical processes initiated. The evolution of the solid mass over time is given by the difference between the chemical mass

When fluids continuously flow and react with the rock, the mass (and hence the solid volume) changes. The chemical flux can be monitored by evaluating the effluent concentration through difference between the ion concentrations in and out of the volume element (Figure 6a, b). This volume element may, in some cases, be between an injector and a producer in an oil field, or a core scale experiment in the laboratory [3]. The concentration of ions can be measured using

<sup>j</sup> cin,j � cin,j

j (mole/L), q is the flow rate (L/day), and mj is the molar mass of species j (g/mole). Hence, the term δMs=δt is given in g/day. The overall mass is estimated by summing over all measured ions in the chemical interaction, giving a unit (g/day), which can be used further. The total mass evolution of each species is determined by integration. Assessing rock-fluid interactions to real cases, for example during, seawater flooding of the Ekofisk field (North Sea, Norway), chemical reactions have been observed. Here, dissolution of 1–2 wt. % is anticipated from the

In Eq. (32), the change in solid volume depends on the change in both mass and in density as the minerals dissolve and precipitate. The overall mineral density, as n minerals dissolve/

� � is the difference in the ion concentration of chemical species

� �qmj (33)

(31)

(32)

$$
\rho\_s(t) = \check{m}(t)\rho\_{s,\mathbf{f}} + (1 - \check{m}(t))\rho\_{s,0} \tag{36}
$$
