*2.6. Expression Analysis of the HhMYB Genes by qRT-PCR*

Total RNA was extracted with a total RNA extraction kit (Shanghai PuDi Biotech Co., Ltd., Shanghai, China), and cDNA was synthesized with HiScript® III 1st Strand cDNA Synthesis kit (Vazyme, Nanjing, China), both procedures following their respective manufacturer recommendations. The Genscript tool (https://www.genscript.com/tools/ pcr-primers-designer, accessed on 20 March 2021) was used for designing primers for qRT-PCR (Table S1). qRT-PCR was performed using a StepOnePlus TM Real-Time PCR system (Applied Biosystems, Foster City, CA, USA). The reaction mixture contained 2 μL diluted cDNA, 10 μL 2 × SYBR Green Master Mix (Bimake, Houston, TX, USA), 0.4 μL ROX (Dye I), 1 μL each of forward and reverse primer, and ddH2O to a total volume of 20 μL. The PCR procedure utilized an initial denaturation step set as 95 ◦C for 10 min, followed by 40 cycles of 95 ◦C for 15 s, 60 ◦C for 30 s, and 72 ◦C for 30 s. Melting curve analyses of the amplified products were conducted at 60–95 ◦C. The *Actin* was selected as a reference gene: forward primer: 5 -GGCACCTCTCAACCCCAAGG-3 , reverse primer: 3 -GAGAGAACGGCCTGGATGGC-5 [5]. Three technical replicates were prepared for each extract, and the quantitative results were analyzed using the 2−ΔΔCT method.


**3. Methodology**

Latest investigations classify hydro cyclones based on the particle size of which 50% reports to the overflow and 50% to the underflow, or the so-called D50c point [9]. All equations used in Sections 3 and 4 have been taken from Richard A. Arterburn et al. [9]. Studies have shown that this classification remains persistent through a range of cyclone diameters and applications. The separation a cyclone can achieve can be approximated using the Eq. (1). The D50c (base) for a given diameter cyclone is multiplied by a series of correction factors designated by C1, C2, and C3 (Eq. (1)).

$$\text{D50}\_c(application) = \text{D50}\_c(base) \times \text{C}\_1 \times \text{C}\_2 \times \text{C}\_3,\tag{1}$$

where D50c (base) is the micron size that a standard cyclone can achieve under baseline conditions and D50c (application) is the filtering potential for a particular application.

$$D\mathbf{50}\_{\epsilon}(base) = 2.84 \,\mathrm{xD}^{0.66},\tag{2}$$

where D is the cyclone diameter in centimeters.

The first correlation factor C1 in Eq. (3) refers to the influence of the concentration of solids contained in the feed. The higher the concentration the coarser the separation. This correlation is a factor of slurry viscosity and particle size and shape. Variables such as liquid viscosity also affect this correlation.

$$C\_1 = \left(\frac{(\text{53-96}\_{\text{solid}})}{\text{53}}\right)^{-1.43},\tag{3}$$

where %solids is the percent solid by volume of cyclone feed.

The second correlation C2 in Eq. (4) is for the influence of pressure drop in the cyclone, measured by taking the difference between feed pressure and overflow pressure. It is recommended that the pressure drop varies between 40 kPa and 70 kPa. This is to limit energy usage as well as equipment wear. As a result, a higher pressure drop would equate in finer separation.

$$\mathbf{C}\_2 = \mathbf{3.27} \times \Delta P^{-0.28},\tag{4}$$

where ΔP is the pressure drop in kPa.

The third correlation C3 in Eq. (5) corrects the influence of specific gravity on the solids and liquid inside the cyclone. Stoke's law has been used to determine particle diameters which would produce the same terminal settling velocity for a particle of known specific gravity in a liquid.

$$\mathbf{C}\_3 = \left(\frac{\mathbf{1.65}}{(\boldsymbol{\gamma}\_s - \boldsymbol{\gamma}\_l)}\right)^{0.5},\tag{5}$$

where *γs* is the specific gravity for solids and *γl* is the specific gravity for liquids. D50c (application) may be formulated as the product of the selected minimum size of separation and the associated multiplier for the percentage of solids passing

through the overflow (Eq. (6)).

$$\text{D50}\_c(approx) = \text{microsize} \text{x} \\ \text{multiplier}, \tag{6}$$

where the multiplier is defined form the **Table 2** below, taken from [9]. The result is that for a identified D50c (application) micron size, all the particles less than that will go into the overflow and all the particles bigger that that size will discharge to the underflow.

**Table 2.** *% of solids passing through overflow and the correspondent multiplier.* *Performance Analysis and Modeling of Microplastic Separation through Hydro Cyclones DOI: http://dx.doi.org/10.5772/intechopen.99447*

#### **Table 3.**

*Rietema and Bradleys standard relations for hydro cyclone dimensions.*

**Figure 2.** *Hydro cyclone dimension nomenclature and position.*

Other cyclone geometric variables such as Di, Do, L and angle, have been found through the standard cyclone dimension relationships of the Rietema and Bradley hydro cyclones [8]. **Table 3** shows the correspondent relations (**Figure 2**).

## **4. Modeling**

In order to create a model of a hydro cyclone that provides an good representation of its behavior and dimensions is critical. The minimum size of microplastics was assumed to be 5 μm; the density of the plastics going inside the hydro cyclone was assumed to be 1500 kg/m3 (which is the average of densities between the most common plastics and the % that are present in the environment); the % volume of solids (microplastics) in the fluid (water) going inside the hydro cyclone; the pressure drop was considered to be 50 kPa as it is standard for most hydro cyclones; the % of solids passing through the overflow which relates the multiplier; the Rietema standard cyclone dimension relations where chosen because considered a better fit for the application (**Table 4**).

Initial variables were calculated using the subsequent equations:

$$
\rho\_{average} = (\rho\_{oldis} + \rho\_{water})/2 = 1248.5 \,\text{kg}/m^3,\tag{7}
$$

$$\chi\_s = \frac{\rho\_{solid}}{\rho\_{water}} = \mathbf{1.50},\tag{8}$$

$$
\Delta P = P \text{--} P\_{drop} = \text{51}kPa,\tag{9}
$$

$$\text{D50}\_c(\text{application}) = d\_{\text{solid}} \text{x}\\ \text{Multiplier} = \text{13.9 } \mu m,\tag{10}$$

$$\mathbf{C}\_{1} = \left(\frac{(\mathbf{53-9\phi\_{solid}})}{\mathbf{53}}\right)^{-1.43} = \mathbf{1.03},\tag{11}$$

$$\mathbf{C}\_2 = \mathbf{3.27} \times \Delta P^{-0.28} = \mathbf{1.09},\tag{12}$$

$$\mathbf{C}\_{3} = \left(\frac{\mathbf{1.65}}{(\boldsymbol{\chi}\_{\rm s} - \boldsymbol{\chi}\_{\rm l})}\right)^{0.5} = \mathbf{1.81},\tag{13}$$

$$D50\_c(base) = \frac{D50\_c(application)}{\text{C}\_1 \times \text{C}\_2 \times \text{C}\_3} = 26.60,\tag{14}$$

$$D = \left(\frac{\text{D50}\_c(base)}{2.84}\right)^{\left(\frac{1}{0.66}\right)} = 29.65 \text{ cm},\tag{15}$$

Using the Rietema relations the rest of the hydro cyclone dimensions can be calculated.

$$\text{Di} = D \times 0.28 = 8.30 \text{ cm},\tag{16}$$

$$Do = D \times 0.34 = 10.08 \text{ cm},\tag{17}$$

$$L = D \times \dots = 148.28 \text{ cm},\tag{18}$$

$$Da = Do = \mathbf{10.08} \, cm,\tag{19}$$

The dimensions of the apex diameter (Da) are a result of investigations done by [8]. Which result in Da being the equivalent to Do. This is to optimize the flow in the hydro cyclone.

**Table 4.**

*First set of variables for average conditions for microplastic separation.*

**Name Value Equation** C1 1.348 (11) C2 1.087 (12) C3 2.846 (13) D50c (base) 31.91 (14) D 39.07 (15) Di 10.94 (16) Do 13.28 (17) L 195.3 (18) Da 13.28 (19)

*Performance Analysis and Modeling of Microplastic Separation through Hydro Cyclones DOI: http://dx.doi.org/10.5772/intechopen.99447*

#### **Table 5.**

*Results for particle density 1200 kg/m3 .*

If the density of plastics going into the hydro cyclone is assumed to be 1200 kg/m3 (which is the average of densities between the most common plastics between 900 kg/m3 and 1400 kg/m3 ), to achieve the same performance as in the 1500 kg/m3 case, the dimensions of the hydro cyclone would have to increase as it would take longer to separate the particles. The results are shown in **Table 5**. These show an increase in size for the diameter (D) of 50% and of 30% for the length (L). Although this may seem like a large increase the design can still be manufactured, as the dimensions are relatively similar.


Meet the editors

Dr. Stanislaw P. Stawicki is Professor of Surgery and chair of the Department of Research and Innovation, St. Luke's University Health Network, Bethlehem, Pennsylvania. A specialist in general surgery, surgical critical care and neurocritical care, he co-authored more than 650 scholarly works, including more than 20 books. In addition to local, national, and international medical leadership roles, Dr. Stawicki is a member of numerous

editorial boards. His areas of expertise are diverse and include health security, medical information security, blockchain technology, patient safety, academic leadership, mentorship and leadership development, traumatology, surgical critical care, and sonography.

Dr. Thomas J. Papadimos is Professor of Anesthesiology at The Ohio State University Wexner Medical Center (OSUWMC), Columbus, Ohio, USA. At OSUWMC he has served as the Critical Care Fellowship Director, Vice Chair for Academic Affairs, and is currently the Anesthesiology Clinical Research Fellowship Director. He has also been an associate dean at the University of Toledo College of Medicine and Life Sciences, Toledo, Ohio,

USA. Dr. Papadimos was awarded a Fulbright Scholar Award to teach and mentor at the 2nd Department of Anesthesiology in October–December 2021 at the University of Athens, Greece. He is a retired US Navy Medical Officer and has authored or co-authored more than 400 papers, book chapters, books, and abstracts.

Sagar C. Galwankar, MBBS, DNB, MPH, MBA, FAAEM, FAIM, graduated with an MBBS and MD from Pune University, India. He is residency trained and board-certified in internal medicine and emergency medicine in India and the United States, respectively. He holds prestigious fellowships from the Royal College of Physicians, UK, the American College of Academic International Medicine, and the Academic College of Emergency Experts,

India. He additionally obtained an MPH and MBA in the United States. His academic career over the last two decades is studded with remarkable achievements resulting from innovative strategies that led to the creation of organizations, publication of landmark papers, and commendations with prestigious citations and honors for his works that have impacted the academic medicine community globally. He has played a defining role in founding and building internationally recognized, interdisciplinary indexed journals. He championed the establishment and growth of academic emergency medicine in India which is a benchmark in the history of developing emergency medicine in the world's largest and most diverse democracy in the world. He is the CEO of the INDUSEM Health and Medicine Collaborative and heads the World Academic Council of Emergency Medicine (WACEM). Additionally, he serves as the Global Executive Director for the American College of Academic International Medicine (ACAIM). He is Founding Director for Research in Emergency Medicine at Florida State University (FSU) and serves as a core faculty for the FSU Emergency Medicine Residency Program at Sarasota Memorial Hospital, Florida.

Dr. Andrew C. Miller obtained a BS from St. Louis University, Missouri, an MD from Southern Illinois University School of Medicine, and is currently pursuing an MSc at Southeast Missouri State University. He completed residencies in emergency medicine and internal medicine at the State University of New York (SUNY) Downstate Health Sciences University where he also served as chief resident. He then completed fellowships in pul-

monary diseases at the University of Pittsburgh Medical Center, and Critical Care Medicine at the National Institutes of Health Clinical Center. He currently practices academic emergency medicine, has a track record of publishing original research in high-impact journals, and has been the recipient of numerous local, regional, and national awards.

Dr. Michael S. Firstenberg is a thoracic surgeon at the St. Elizabeth Medical Center (Ascension), Appleton, Wisconsin. He attended Case Western Reserve University Medical School, Cleveland, OH, received his general surgery training at University Hospitals in Cleveland, and completed thoracic surgery fellowships at The Ohio State University and The Cleveland Clinic. He is an active member of the Society of Thoracic Surgeons (STS),

American Association of Thoracic Surgeons (AATS), American College of Cardiology (ACC), and American College of Academic International Medicine (ACAIM), for which he served as president in 2021–2022. He has authored more than 250 peer-reviewed manuscripts, abstracts, and book chapters and has edited several textbooks and lectured worldwide on topics ranging from medical leadership, COVID-19, endocarditis, and extra-corporeal membrane oxygenation (ECMO).


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,0 and Vp, 0 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, τzθ ¼ τθz, and τrθ 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 σ1 > σ2 > σ3, where σ1 is typically in the vertical direction (weight dominated), and, consequently, the σ2 and σ3 are horizontal (often abbreviated σH as the highest horizontal stress and σh 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 τzθ τrz σrr τrθ τθz τθr σθθ 3 7

5: (11)

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205

� � (12)

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

σij ¼

stress). σh and σH 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, 0 þ ΔVpð Þt

Vp, 0 þ ΔVbðÞ� t ΔVsð Þt

and ϕ0 ¼ Vp, 0

Vb,0

Vp, 0 þ Vs, 0 þ ΔVpð Þþ t ΔVsð Þt : (9)

Vb,0 þ ΔVbð Þt : (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,0

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.

> ϕðÞ¼ t ϕ0 � εvol � ΔVsð Þt =Vb, 0 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,0 þ ΔVpð Þt

Vp,0 þ ΔVpð Þt

Vb, 0 þ ΔVbð Þt (5)

Vb, 0 þ ΔVbð Þt (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, τzθ ¼ τθz, and τrθ 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 σ1 > σ2 > σ3, where σ1 is typically in the vertical direction (weight dominated), and, consequently, the σ2 and σ3 are horizontal (often abbreviated σH as the highest horizontal stress and σh is the least horizontal stress). σh and σH 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

Γ ¼ 1 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 ≃εz þ 2εr.

nized in the strain tensor

cylinder is given by V ¼ πD2

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

εzz Γzr Γzθ Γrz εrr Γrθ Γθz Γθr εθθ

Similar to the stress tensor, the shear strains balance each other (Γrz ¼ Γzr, Γθz ¼ Γzθ, Γθr ¼ Γrθ), 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 εvol ¼ �ΔV=V0 ¼ Tr εij � �. 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

εvol ¼ εz þ 2εr þ 2εzεr þ ε2

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

5 (16)

� � (17)

r (18)

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

r þ εzε2

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!s 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, σr ¼ σθ, implying that full description of the effective stress state of a cylindrical core experiment are given by two effective stresses, σ0 r ¼ σr � αPf and σ0 z ¼ σz � 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

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

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, Γθz ¼ Γzθ, Γθr ¼ Γrθ), 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, σr 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 � ΔL L0

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

H ¼ σH � αPf > σ0

and εvol ¼ � ΔV

V0

z ¼ σz � αPf . The stress exerted onto the core in the

h ¼ σh � αPf (13)

(14)

Biot coefficient) of the pore pressure

206 Porosity - Process, Technologies and Applications

v ¼ σv � αPf which is greater than σ0

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

r ¼ σr � αPf and σ0

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� L0 L0

σ0

two effective stresses, σ0

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 εvol ¼ �ΔV=V0 ¼ Tr εij � �. 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 V ¼ πD2 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 ≃εz 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,

> Eδεr ¼ ð Þ 1 � ν δσr � νδσz þ ð Þ 2ν � 1 αδPf Eδεz ¼ δσz � 2νδσr 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 σz � σr and the mean effective stress is P ¼ ð Þ σz þ 2σr =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

z ¼ δσz � αδPf (24)

Porosity Evolution during Chemo-Mechanical Compaction

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

(25)

209

r ¼ δσr � αδPf and δσ0

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 z ¼ σ0 r ¼ σ0 θ ¼ σ0 p, 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. σ0 p is frequently used to define the hydrostatic effective stress. For nonhydrostatic triaxial tests, where σz > σr, and σ0 r ¼ σ0 θ and εr 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 z ¼ σ0

r ¼ σ0

a loading sequence

θ ¼ σ0

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 εz þ εr þ εθ E ¼ ð Þ 1 � 2ν σ0

εvol ¼ σ0

Eεr ¼ ð Þ 1 � ν σ0

Eεz ¼ σ0

Eδεr ¼ ð Þ 1 � ν δσ0

Eδεz ¼ δσ0

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

z þ σ0

p ! Kεvol ¼ σ0

r � νσ0 z

r � νδσ0 z

z � 2νδσ0 r

z � 2νσ0 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,

r þ σ0 θ

2 6 4

σ0 z σ0 r σ0 θ 3 7

5 (19)

p (21)

r ¼ σ0

θ and εr ¼ εθ

(22)

(23)

p 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 σz > σr, and σ0

2 6 4

p, then Eq. (20) simplifies to

In Eq. (21), the bulk modulus (K) is defined. σ0

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 σz � σr and the mean effective stress is P ¼ ð Þ σz þ 2σr =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.


**1. Introduction**

Due to the rapid development of Internet technology, network terminals have been widely spread. However, traditional network architectures have failed to adapt to future advances in communication and Internet technologies, resulting in heterogeneous networks. As a result, the existing network infrastructure was unable to keep up with the rapid changes of the Internet. A key feature of traditional network architectures is that the data and control planes are tightly coupled, which has some limitations. For example, if you want to change the network configuration, you need to configure each device independently across the entire network which is a daunting task.

Similarly, vendors are reluctant to provide the internal details of the device to developers and users, as changes in the configuration of existing networking devices can lead to malfunctions in the network. In addition, the protocol is strongly built into the firmware of network devices. These limitations hinder network innovation due to proprietary hardware and lack of testing for innovative networking solutions due to their distributed nature. It also increases the management workload and the overall cost of network management.

On the other hand, Software Defined Networking (SDN) [1–5] has revolutionized network management by separating data and control planes. The data plane is composed of forwarding devices, for example routers, switches, etc. Its main functions are forwarding the packets according to the policies of the controller. If the destination of the arrived packets is not found in the forwarding devices, then those packets are sent to the controller by the data plane. The control plane, however, is implemented through intelligent SDN controllers such as OpenDaylight (ODL), Open Networking Operating System (ONOS), POX and RYU [6]. Control plane obtains the status of the underlying network and defines the policies for the packets arriving on the forwarding devices. It then pushes the updated rules to the data plane. The separation of data and control planes has shifted network complexity from networking devices to smart SDN controllers. Thus, the network can be programmed through the application running on the controller and the underlying network is abstracted from the applications [7]. The innovative concept presented by SDN has the great advantage of flexible and efficient network configuration, network management and operation. Therefore, SDN is expected to be an excellent choice for the next generation of telecommunication networks and Internet technologies. Because of these benefits, large information technology organizations such as Facebook, Amazon, and Google have implemented SDN to connect remote data centers [8, 9].

The internet has grown in recent years. As a result, there is a huge increase in the amount of network traffic. Because the accuracy of machine learning algorithms depends mainly on the availability of historical data. There is therefore an increasing tendency towards the use of machine learning techniques. Because the accuracy of machine learning algorithms increases with sufficient data. For this reason, researchers now prefer to apply machine learning solutions because, once trained on the available data, the trained model generates accurate results on the new data through learning experience. The introduction of 5G heterogeneous networks and the rapid ubiquitous use and growth of Internet data processing requirements are rapidly increasing as a result of a dramatic increase in the number of connected devices. For example, the heterogeneous IoT devices in 5G runs different protocols and various technologies results in increasing the traffic load [10]. In addition, there is a need for self-organization and demand-based networks to deal with huge amounts of data. SDN was therefore at the heart of the growing needs of such applications due to their programming, orchestration, and automation characteristics [11].

The SDN has been successfully deployed in data centers and enterprise traffic engineering networks across remote data centers. However, the adoption of SDN in the modern and global Internet still presents a number of challenges that need further investigation. As the internet is scaling and the traffic on the underlying network is dynamically changing. The application of an optimum policy for the underlying network should therefore be adapted in line with the radical changes in the internet. One of the problems in SDN is the configuration of the control plane, because the manual configuration is a costly task, because the traditional SDN approach [12–15] is not optimal in selecting the optimum policy for the underlying network. In addition, repeatedly reconfiguring the policy according to changes in the network will require the control plane to be reconfigured. One of the main issues, therefore, is the automatic orchestration of the control plane [16]. Because the rigid configuration of the control plane will have problems in the optimal configuration of the policy.

### *Management of Software-Defined Networking Powered by Artificial Intelligence DOI: http://dx.doi.org/10.5772/intechopen.97197*

Another issue is the end-to-end (E2E) quality-of-service (QoS) performance of heterogeneous network providers. If the same provider manages SDN controllers, user applications and forwarding devices on the enterprise network, then, the network status of the underlying devices is readily available for upper-layer applications. However, the Internet consists of different providers where end-users, applications and service providers are often heterogeneous. As a result, the status of the network is not directly available for applications running on the upper layers.

Several solutions have been proposed to address the issue of the allocation of E2E resources [17–23]. However, they depend on the traditional and manual configuration of the control plane. i.e., once a policy has been defined for the underlying network. The behavior of the network is then controlled accordingly, regardless of the scale of the network or the dynamic changes. The policy of controlling the network is therefore not always optimal. Moreover, these solutions do not provide effective management of the SDN due to scaling up, increasing network complexity and dynamic changes. There is therefore a need to find a global optimal solution with an excellent value for the objective functions. We therefore propose a software-defined networking management system powered by AI (SDNMS-PAI) architecture to auto-configure policy management and E2E resource allocation.

The advantage of AI based architecture is that the AI agent will interact with the underlying network through the SDN controller for pushing the global optimal policy flow rules in the forwarding devices. The controller will share the network status information with the AI agent and based on real time status of the network the AI agent will find the most appropriate actions to be taken. The actions will be pushed as the flow rules in the forwarding devices. AI can be used to bring a closedloop control of the SDN. The closed-loop control incorporates collection of data, analytics, and subsequent actions that are all based on the results of the analytics [24]. All components of the closed loop can be improved and enhanced by means of AI to improve the speed, accuracy and, ultimately, the effectiveness of the closed loop control.

The main contributions of this chapter are summarized as follows:


*<IP_ADDRESS> Vagal (parasympathetic) innervation*

The vagal nerve transfers information between the internal organs and the brainstem. It contains both afferent and efferent nerve fibers and innervates the entire

*Influence of Gut Microbiota on Behavior and Its Disturbances DOI: http://dx.doi.org/10.5772/intechopen.85317*

gut with exception of the distal third of the colon. Vagal afferents terminate in the nucleus of the solitary tract (NTS) whose impulses go up through the parabrachial nucleus (PBN) to the thalamus, limbic system and insula [33, 34]. Spinal fibers pass up via the spinothalamic tract and the dorsal spine columns. Respectively, the spinothalamic pathway goes ascendingly to the thalamus, and the dorsal columns give projections to the gracile nucleus and cuneate nucleus in the upper medulla. Efferent impulses from the last rostral medullar structures reach the thalamus through the medial lemniscus. In turn, thalamic projections ascend to the primary somatosensory cortex and insula [30]. Vagus motor nuclei are represented by nucleus ambiguus (NA) and dorsal motor nucleus (DMN). The DMN is a source of efferents to the smooth muscles of the gut that form synapses with the neurons of the MP [7, 33].

## **3.2 Gut microbiota influences and human brain function**


**2. Basic theory and properties of derivative spectrophotometry**

Derivative spectrophotometry is a technique which is based on derivative spectra of a basic, zero-order spectrum. The results of derivatisation of function described a run of absorbance curve is called the derivative spectrum and can be expressed as:

$$\mathbf{^nD\_{x,\lambda} = d^n A / d\lambda^n = f(\lambda) \quad \text{or} \quad \mathbf{^nD\_{x,v} = d^n A / d\mathbf{v^n} = f(v)}$$

where: n – derivative order, nDx,λ or nDx,ν represents value of n-order derivative of an analyte (x) at analytical wavelength (λ ) or at wavelength number (ν), A- absorbance.

Derivative spectrophotometry keeps all features of classical spectrophotometry: Lambert-Beer law and law of additivity.

Lambert-Beer low in its differential form is expressed as:

$$\prescript{}{}{}{}{D} = \frac{d\prescript{}{}{}{}{A}}{}{} = \frac{d\prescript{}{}{}{\mathcal{E}}}{}{}{}{\mathcal{E}} \prescript{}{}{\cdot} \cdot \prescript{}{\cdot} \cdot \prescript{}{\cdot}$$

Where ε-molar absorption coefficient (cm-1mol-1l) , c – concentration of analyte (mol l-1), lthickness of solution layer (cm).

Derivative spectrum of **n**-component mixture is a sum of derivative spectra of individual components:

$$\mathbf{^nD\_{mi\dot{\alpha}}=nD\_1+nD\_2+...+nD\_n}$$

A new feature of derivative spectrophotometry is a dependence of derivatisation results on geometrical characteristic of starting, zero-order spectrum. A shape and an intensity of the resulted derivative spectrum depend on half- heights width of peak in basic spectrum:

$$\mathbf{^nD} \mathbf{=} \mathbf{P}^n \mathbf{A}\_{\text{max}} \mathbf{L} \mathbf{^1}$$

where Pn- polynomial described run of n-derivative curve, n- derivative order, L- width of half- heights of peak of zero-order spectrum.

Due to this property broad zero-order spectra are quenched with generation of higher orders of derivatives while narrow undergo amplification. If the zero-order spectrum possess two bands A and B which differ from their half- heights width (LB>LA), after a generation of n-order derivative a ratio of derivatives intensity can be expressed as:

$$\mathrm{"{n}D\_{A}/{n}D\_{B} = (L\_{B}/L\_{A})^{n}}$$

This dependence leads to increase in selectivity and/or sensitivity of assay. It allows to use for analytical properties a narrow band, overlapped or completely hooded by a broad ones.

The shape of derivative spectrum is more complicated than its parent one (Fig. 1). New maxima and minima appeared as results of derivatisation. The generation of **n-th** order derivative spectrum produces (n+1) new signals: an intense main signal and weaker bands, so called satellite or wings signals. Position of maxima or minima depend on order of derivative. The main extreme of derivative spectra of even order is situated at the same wavelength as maximum in zero-order spectrum, but for 2, 6 and 10-th order it becomes minimum in

Derivative spectrophotometry is a technique which is based on derivative spectra of a basic, zero-order spectrum. The results of derivatisation of function described a run of absorbance

nDx,λ=dnA/dλn=f(λ) or nDx,ν=dnA/dνn=f(ν) where: n – derivative order, nDx,λ or nDx,ν represents value of n-order derivative of an

Derivative spectrophotometry keeps all features of classical spectrophotometry: Lambert-

*n n dA d D c l d d*

Where ε-molar absorption coefficient (cm-1mol-1l) , c – concentration of analyte (mol l-1), l-

Derivative spectrum of **n**-component mixture is a sum of derivative spectra of individual

nDmix=nD1+nD2+...+nDn A new feature of derivative spectrophotometry is a dependence of derivatisation results on geometrical characteristic of starting, zero-order spectrum. A shape and an intensity of the resulted derivative spectrum depend on half- heights width of peak in basic spectrum:

nD=PnAmaxL-1 where Pn- polynomial described run of n-derivative curve, n- derivative order, L- width of

Due to this property broad zero-order spectra are quenched with generation of higher orders of derivatives while narrow undergo amplification. If the zero-order spectrum possess two bands A and B which differ from their half- heights width (LB>LA), after a

nDA/nDB=(LB/LA)n This dependence leads to increase in selectivity and/or sensitivity of assay. It allows to use for analytical properties a narrow band, overlapped or completely hooded by a broad ones. The shape of derivative spectrum is more complicated than its parent one (Fig. 1). New maxima and minima appeared as results of derivatisation. The generation of **n-th** order derivative spectrum produces (n+1) new signals: an intense main signal and weaker bands, so called satellite or wings signals. Position of maxima or minima depend on order of derivative. The main extreme of derivative spectra of even order is situated at the same wavelength as maximum in zero-order spectrum, but for 2, 6 and 10-th order it becomes minimum in

generation of n-order derivative a ratio of derivatives intensity can be expressed as:

 λ= = ⋅⋅

ε

analyte (x) at analytical wavelength (λ ) or at wavelength number (ν), A- absorbance.

*n n n*

λ

**2. Basic theory and properties of derivative spectrophotometry**

curve is called the derivative spectrum and can be expressed as:

Lambert-Beer low in its differential form is expressed as:

Beer law and law of additivity.

thickness of solution layer (cm).

half- heights of peak of zero-order spectrum.

components:

derivative spectrum and for 4, 8 and 12-th order it remains as a maximum (Fig. 1). The point of initial maximum converts into the point of inflection in derivative spectra of odd order. A narrowing of new signals is observed during generation of consecutive derivative spectra. This feature leads to narrowing bands and as a consequence to separation of overlapped peaks.