**Apendix: Analytical solutions for plane strain Kristianovic-Geertsma-de Klerk (KGD) hydraulic fractures**

The solution of a plane strain KGD hydraulic fracture in an infinite elastic body depends on the injection rate *Q*0 and on the three material parameters *E*' , *K*' , and *μ*' , which are defined as [21]

*E*' =*E* / 1 - *ν* 2 , *K*' =(32 / *π*)1/2 *K*I*c*, *μ*' =12*μ*

For the plane strain KGD hydraulic fracture, the crack opening *w*(*x*, *t*), crack length (half length) *l*(*t*), and net fluid pressure *p*(*x*, *t*) can be expressed as [21]

$$\begin{aligned} \varphi\_t(\mathbf{x}, t) &= \epsilon(t) \mathsf{L}(t) \mathsf{L}[\xi, \mathsf{F}\_t, P(t)] = \epsilon(t) \mathsf{L}(t) \gamma[P(t)] \overline{\mathsf{L}}(\xi) \\ p(\mathbf{x}, t) &= \epsilon(t) \mathsf{E}^\dagger \Pi[\xi, P(t)] \mathbf{I} \end{aligned} \tag{34}$$

$$l(t) = \gamma[P(t)] \mathsf{L}(t)$$

where *ξ* = *x* / *l*(*t*) is the scaled coordinate (0≤*ξ* ≤1), ϵ(*t*) is a small dimensionless parameter, *L*(*t*) denotes a length scale of the same order as the fracture length *l*(*t*), *P*(*t*) is the dimensionless evolution parameter, and *γ P*(*t*) is dimensionless fracture length.

The evolution parameter *P*(*t*) can be interpreted as a dimensionless toughness *κ* in the viscosity scaling [21]

$$\kappa = K \left( E^{\top 3} \mu^{\prime} \mathbf{Q}\_{0} \right)^{-1/4} \tag{35}$$

or as a dimensionless viscosity ℳ in the toughness scaling [21]

$$\mathcal{M} = \mu^{\stackrel{\cdot}{\cdot}} \mathrm{E}^{\stackrel{\cdot}{\cdot}} \mathbb{Q}\_0 / \mathrm{K}^{\stackrel{\cdot}{\cdot}} \tag{36}$$

For the toughness scaling, denoted by a subscript *k*, the small parameter ϵ(*t*) and the length scale *L*(*t*) take the explicit forms [21]

$$L\_{\vec{k}}(t) = \left(\mathbf{K}^{\prime 4} / \boldsymbol{E}^{\prime 4} \mathbf{Q}\_{0} t\right)^{1/3}, \ L\_{\vec{k}}(t) = \left(\boldsymbol{E}^{\prime} \mathbf{Q}\_{0} t / \boldsymbol{K}^{\prime}\right)^{2/3} \tag{37}$$

The solution for the zero viscosity case is given by [21]

assigned to the appropriate nodes of the proposed element to describe the fluid flow within the crack and its contribution to the crack deformation. Verification of the user-defined element has been made by comparing the FEM predictions with the analytical solutions available in the literature. The preliminary result presented here is a first attempt to the promising

**Apendix: Analytical solutions for plane strain Kristianovic-Geertsma-de**

The solution of a plane strain KGD hydraulic fracture in an infinite elastic body depends on

For the plane strain KGD hydraulic fracture, the crack opening *w*(*x*, *t*), crack length (half

ϵ

Π *ξ*, *P*(*t*)

ϵ

denotes a length scale of the same order as the fracture length *l*(*t*), *P*(*t*) is the dimensionless

The evolution parameter *P*(*t*) can be interpreted as a dimensionless toughness *κ* in the viscosity

, *K*'

(*t*)*L*(*t*)*γ P*(*t*) Ω¯(*ξ*)

, and *μ*'

(*t*) is a small dimensionless parameter, *L*(*t*)

*Q*0)-1/4 (35)

*Q*0 / *K*' (36)

ϵ

)2/3 (37)

(*t*) and the length

, which are defined as

(34)

application of the XFEM to the hydraulic fracture simulation.

the injection rate *Q*0 and on the three material parameters *E*'

*K*I*c*, *μ*'

ϵ

=12*μ*

(*t*)*L*(*t*)Ω *ξ*, *P*(*t*) =

ϵ(*t*)*E*'

*l*(*t*)=*γ P*(*t*) *L*(*t*)

*p*(*x*, *t*)=

evolution parameter, and *γ P*(*t*) is dimensionless fracture length.

or as a dimensionless viscosity ℳ in the toughness scaling [21]

*k* (*t*)=(*K*'4 / *E*'4

*κ* =*K*' (*E*'3 *μ*'

ℳ=*μ*' *E*'3

For the toughness scaling, denoted by a subscript *k*, the small parameter

*Q*0*t*)1/3

, *Lk* (*t*)=(*E*'

*Q*0*t* / *K*'

length) *l*(*t*), and net fluid pressure *p*(*x*, *t*) can be expressed as [21]

**Klerk (KGD) hydraulic fractures**

736 Effective and Sustainable Hydraulic Fracturing

=(32 / *π*)1/2

*w*(*x*, *t*)=

where *ξ* = *x* / *l*(*t*) is the scaled coordinate (0≤*ξ* ≤1),

scale *L*(*t*) take the explicit forms [21]

ϵ

[21] *E*'

=*E* / 1 - *ν* 2

scaling [21]

, *K*'

$$
\gamma\_{k0} = 2 \left/ \pi^{2/3} \right. \tag{38}
$$

$$
\overline{\Omega}\_{k0}(\xi) = \sqrt{1 - \xi^2} \Big/ \pi^{1/3} \tag{39}
$$

$$
\Pi\_{k0} = \pi^{1/3} / 8 \tag{40}
$$

For the viscosity scaling, denoted by a subscript *m*, the small parameter ϵ(*t*) and the length scale *L*(*t*) take the explicit forms [21]

$$\mathbf{e}\_m(t) = \left(\mu^\top/\to^\cdot t\right)^{1/3}, \quad \mathbf{L}\_m(t) = \left(\mathbf{E}^\cdot Q\_0^3 t^4/\mu^\prime\right)^{1/6} \tag{41}$$

The first order approximation of the zero toughness solution is [21]

$$
\gamma\_{m0} \cong 0.616\tag{42}
$$

$$\overline{\Omega}\_{m0}(\boldsymbol{\xi}) = A\_0 \{ \mathbf{1} \cdot \boldsymbol{\xi}^{-2} \}^{2/3} + A\_1^{(1)} \{ \mathbf{1} \cdot \boldsymbol{\xi}^{-2} \}^{5/3} + B^{(1)} \left[ \mathbf{4} \sqrt{\mathbf{1} \cdot \boldsymbol{\xi}^2} + 2 \boldsymbol{\xi}^2 \ln \left| \frac{\mathbf{1} \cdot \boldsymbol{\sqrt{1 \cdot \boldsymbol{\xi}^2}}}{\mathbf{1} \cdot \boldsymbol{\sqrt{1 \cdot \boldsymbol{\xi}^2}}} \right| \right] \tag{43}$$

$$\mathbf{T}\_{m0}^{(1)} = \frac{1}{3\pi} B\left(\frac{1}{2}, \,\,\frac{2}{3}\right) \mathbf{I}\_0 F\_1\left(\mathbf{-}\frac{1}{6}, \,\, 1; \frac{1}{2}; \, \xi^2\right) + \frac{10}{7} A\_1^{(1)} F\_1\left(\mathbf{-}\frac{7}{6}, \, 1; \frac{1}{2}; \, \xi^2\right) + B^{(1)}(2 - \pi \, \, \vert \, \xi \, \vert) \tag{44}$$

where *A*0 =31/2 , *A*1 (1) ≅ - 0.156, and *B*(1) ≅0.0663; *B* is Euler beta function, and *F*1 is hypergeo‐ metric function. Thus, Ω¯ *m*0 (0)≅1.84.

#### **Acknowledgements**

The author would like to thank Dr. Rob Jeffrey for the support of this work. Furthermore, the author thanks CSIRO CESRE for support and for granting permission to publish.

#### **Author details**

Zuorong Chen\*

CSIRO Earth Science & Resource Engineering, Melbourne, Australia

#### **References**

[1] Clifton, R. J, & Abou-sayed, A. S. On the computation of the three-dimensional ge‐ ometry of hydraulic fractures. Proceedings of the SPE Symposium on Low Permea‐ bility Gas Reservoir. Denver, Richardson: Society of Petroleum Engineers; (1979). , 307-313.

[14] Carrier, B, & Granet, S. Numerical modeling of hydraulic fracture problem in perme‐ able medium using cohesive zone model. Engineering Fracture Mechanics. (2012). ,

An ABAQUS Implementation of the XFEM for Hydraulic Fracture Problems

http://dx.doi.org/10.5772/56287

739

[15] Chen, Z. R. Finite element modelling of viscosity-dominated hydraulic fractures.

[16] Belytschko, T, & Black, T. Elastic crack growth in finite elements with minimal re‐ meshing. International Journal for Numerical Methods in Engineering. (1999). , 45(5),

[17] Moes, N, Dolbow, J, & Belytschko, T. A finite element method for crack growth with‐ out remeshing. International Journal for Numerical Methods in Engineering. (1999). ,

[18] Lecampion, B. An extended finite element method for hydraulic fracture problems. Communications in Numerical Methods in Engineering. (2009). , 25(2), 121-133. [19] Dahi-taleghani, A, & Olson, J. E. Numerical modeling of multistranded hydraulic fracture propagation: accounting for the interaction between induced and natural

[21] Detournay, E. Propagation regimes of fluid-driven fractures in impermeable rocks.

Journal of Petroleum Science and Engineering. (2012).

fractures. SPE Journal. (2011). , 16(3), 575-581.

[20] ABAQUSABAQUS documentation version 6.(2011). , 11-1.

International Journal of Geomechanics. (2004). , 4(1), 35-45.

79-312.

601-620.

46(1), 131-150.

**References**

307-313.

738 Effective and Sustainable Hydraulic Fracturing

[1] Clifton, R. J, & Abou-sayed, A. S. On the computation of the three-dimensional ge‐ ometry of hydraulic fractures. Proceedings of the SPE Symposium on Low Permea‐ bility Gas Reservoir. Denver, Richardson: Society of Petroleum Engineers; (1979). ,

[2] Settari, A, & Cleary, M. P. Dimensional simulation of hydraulic fracturing. Journal of

[3] Vandamme, L, & Curran, J. H. A. Dimensional hydraulic fracturing simulator. Inter‐ national Journal for Numerical Methods in Engineering. (1989). , 28(4), 909-927.

[4] Advani, S. H, Lee, T. S, & Lee, J. K. Dimensional modeling of hydraulic fractures in layered media.1. finite-element formulations. Journal of Energy Resources Technolo‐

[5] Valko, P, & Economides, M. J. Propagation of hydraulically induced fractures- a con‐ tinuum damage mechanics approach. International Journal of Rock Mechanics and

[6] Ouyang, S, Carey, G. F, & Yew, C. H. An adaptive finite element scheme for hydraul‐ ic fracturing with proppant transport. International Journal for Numerical Methods

[7] Papanastasiou, P. An efficient algorithm for propagating fluid-driven fractures.

[8] Zhang, X, Detournay, E, & Jeffrey, R. Propagation of a penny-shaped hydraulic frac‐ ture parallel to the free-surface of an elastic half-space. International Journal of Frac‐

[9] Adachi, J, Siebrits, E, Peirce, A, & Desroches, J. Computer simulation of hydraulic fractures. International Journal of Rock Mechanics and Mining Sciences. (2007). ,

[10] Lecamplon, B, & Detournay, E. An implicit algorithm for the propagation of a hy‐ draulic fracture with a fluid lag. Computer Methods in Applied Mechanics and Engi‐

[11] Peirce, A, & Detournay, E. An implicit level set method for modeling hydraulically driven fractures. Computer Methods in Applied Mechanics and Engineering. (2008).

[12] Chen, Z. R, Bunger, A. P, Zhang, X, & Jeffrey, R. G. Cohesive zone finite elementbased modeling of hydraulic fractures. Acta Mechanica Solida Sinica. (2009). , 22(5),

[13] Dean, R. H, & Schmidt, J. H. Hydraulic fracture predictions with a fully coupled geo‐

mechanical reservoir simulator. SPE Journal. (2009). , 14(4), 707-714.

Petroleum Technology. (1984). , 36(8), 1177-1190.

gy-Transactions of the ASME. (1990). , 112(1), 1-9.

Computational Mechanics. (1999). , 24(4), 258-267.

in Fluids. (1997). , 24(7), 645-670.

ture. (2002). , 115(2), 125-158.

neering. (2007). , 196-4863.

44(5), 739-757.

443-452.

Mining Sciences & Geomechanics. (1994). , 31(3), 221-229.

[21] Detournay, E. Propagation regimes of fluid-driven fractures in impermeable rocks. International Journal of Geomechanics. (2004). , 4(1), 35-45.

**Chapter 37**

**Stress Intensity Factor Determination for**

**Discontinuity Method with Applications to**

**Propagation Paths**

http://dx.doi.org/10.5772/56308

**Abstract**

Farrokh Sheibani and Jon Olson

Additional information is available at the end of the chapter

SIF distribution along the upper front of the fracture.

**Three-Dimensional Crack Using the Displacement**

**Hydraulic Fracture Height Growth and Non-Planar**

Stress intensity factor determination plays a central role in linearly elastic fracture mechanics (LEFM) problems. Fracture propagation is controlled by the stress field near the crack tip. Because this stress field is asymptotic dominant or singular, it is characterized by the stress intensity factor (SIF). Since many rock types show brittle elastic behaviour under hydrocarbon reservoir conditions, LEFM can be satisfactorily used for studying hydraulic fracture devel‐ opment. The purpose of this paper is to describe a numerical method to evaluate the stress intensity factor in Mode I, II and III at the tip of an arbitrarily-shaped, embedded cracks. The stress intensity factor is evaluated directly based on displacement discontinuities (DD) using a three-dimensional displacement discontinuity, boundary element method based on the equations of proposed in [1]. The boundary element formulation incorporates the fundamental closed-form analytical solution to a rectangular discontinuity in a homogenous, isotropic and linearly elastic half space. The accuracy of the stress intensity factor calculation is satisfactorily examined for rectangular, penny-shaped and elliptical planar cracks. Accurate and fast evaluation of the stress intensity factor for planar cracks shows the proposed procedure is robust for SIF calculation and crack propagation purposes. The empirical constant proposed by [2] relating crack tip element displacement discontinuity and SIF values provides surpris‐ ingly accurate results for planar cracks with limited numbers of constant DD elements. Using the described numerical model, we study how fracturing from misaligned horizontal well‐ bores might results in non-uniform height growth of the hydraulic fracture by evaluating of

> © 2013 Sheibani and Olson; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

> © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

**Stress Intensity Factor Determination for Three-Dimensional Crack Using the Displacement Discontinuity Method with Applications to Hydraulic Fracture Height Growth and Non-Planar Propagation Paths**

Farrokh Sheibani and Jon Olson

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56308


**3. Flaws in use of discounted cash flow models for evaluation of engineering projects**

The net present value method principally does not contain any methodological assumptions or deficiencies that could impact the result of evaluation. However, the results are harder to understand in comparison to the percentage from the internal rate of return method. It would not be very meaningful if the investor were to determine that the net present value of a project or an investment were, for example, 25,000 EUR. It certainly may be difficult to compare this result with alternative investments, like real estate investment, bonds or cash deposits. It would be certainly more understandable if the result were expressed as a percentage of internal rate of return. This is probably the main reason that the internal rate of return method is often more popular.

In contrast to the net present value method, the theoretical findings identify numerous limitations connected with the internal rate of return method. The most significant are the following:


**3. Rarita–Schwinger Lagrangian in 3D**

Commonly, *supergravity* [15, 19] with one supersymmetry generator, i.e. N =1, represents the simplest consistent coupling of a Rarita–Schwinger (RS) spin– 3 2 field [35] to gravity.

[23] reads

The corresponding *manifestly Hermitian* RS type Lagrangian three–form of Howe and Tucker

S-Duality in Topological Supergravity 41

2

which is nothing more than the gauge covariant derivative of a spinor-valued one-form Ψ.

As in the case of the Rarita-Schwinger Lagrangian *L*RS, it is manifestly Hermitian when the additional quadratic derivative terms carry *s*1 and *s*2 as dimensionless coupling constants. In order to supersymmetrize this action, it will be coupled to topological gravity later on.

 + *i*

*L*Ψ = *L*RS + *s*1 *D*Ψ ∧ ∗(*D*Ψ) + *s*2 *D*Ψ ∧ *γ* ∧ ∗(*γ* ∧ *D*Ψ). (22)

+ *D ∂L*Ψ

*∂*Ψ − (*eα*�*D*Ψ) ∧

*D*Ψ ∧ *eα*�∗(*D*Ψ) − (*eα*�*D*Ψ) ∧ ∗

*∂L*Ψ

*D*Ψ ∧ *γ* ∧ *γ*

4 *m*Ψ ∧ *γ* ∧ Ψ, (20)

*∂Tα* , (23)

*∂L*Ψ *∂D*Ψ, (24)

. (25)

*∂D*Ψ − (*eα*�*D*Ψ) ∧

*D*Ψ

. (26)

*γα*Γ*α* ∧ Ψ, (21)

Ψ ∧ *D*Ψ − Ψ ∧ *D*Ψ

including, however, a mass term. Here minimal coupling to gravity is achieved via

*D*Ψ = *d*Ψ − 1

Only in 3D, however, there exists a generalization given by the following expression

By definition, the energy-momentum current two-form Σ*α* of matter is given by

*δϑα* = *∂L*Ψ *∂ϑα*

where the second term accounts for the possibility of a non-minimal coupling to torsion via Pauli type terms, cf. Eq. (5.1.8) of Ref. [22]. According to the Noether theorem, the energy-momentum current two-form of matter Σ*α* without Pauli terms can be rewritten as

*∂L*Ψ

see Eq. (5.4.11) of Ref. [22] for details. This equivalent equation often is more convenient, since it involves only partial derivatives of the matter fields and avoids the intricate treatment of a possible dependence of the matter Lagrangian on the Hodge dual. Taking into account

Σ*α* := *δL*Ψ

*∂*Ψ − (*eα*�Ψ) ∧

*D*Ψ ∧ *γα* ∧ ∗(*γ* ∧ *D*Ψ) − (*eα*�*D*Ψ) ∧ ∗

on the coframe *ϑα*, they provides no contribution to the energy-momentum current.

Since the kinetic terms in the Rarita-Schwinger type Lagrangian *L*RS do not depend explicitly

*ηαβγτβγ* = (−1)*s*

2

*δL*Ψ *δ*Γ *α*

*∂L*Ψ

*m*Ψ ∧ *γα*Ψ + *s*1

*τ α* := 1 2

*L*RS = *i* 4

**3.1 Energy-momentum and spin currents**

Σ*α* := *eα*�*L*Ψ − (*eα*�Ψ) ∧

the identities of Appendix B, we find

Σ*α* = − *i* 4

> +*s*2

The 3-dual of the spin current is defined by

The Rarita-Schwinger [35] type spinor-valued one-form1

$$\Psi = \Psi\_i d\mathbf{x}^i = \Psi\_a \theta^a \tag{13}$$

can be written holononically and anholononically. However, it does not depend on the coframe, inasmuch Ψ*α* := *eα*�Ψ involves the inverse tetrad. In 3D, we adhere to the conventions that the holonomic indices run from *i*, *j*, *k*,... = 0, 1, 2, whereas *α*, *β*,... = 0,ˆ 1, ˆ 2ˆ for the anholonomic indices.

We are going to provide a brief summary of the spinors that will be used in three dimensions: As well known, the covering group of the rotation group *SO*(3) is isomorphic to the unitary group *SU*(2). Since an element of *SU*(2) can be parameterized by three numbers, the most convenient basis of the Lie algebra are the familiar Pauli spin matrices:

$$
\sigma^1 = \begin{pmatrix} 0 \ 1 \\ 1 \ 0 \end{pmatrix}, \qquad \sigma^2 = \begin{pmatrix} 0 \ -i \\ i \ 0 \end{pmatrix}, \qquad \sigma^3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \tag{14}
$$

These matrices satisfy the following Lie algebra:

$$\mathbb{E}\left[\sigma^{\alpha}, \sigma^{\beta}\right] = 2i\eta^{a\beta\gamma}\sigma\_{\gamma}.\tag{15}$$

However, for Lorentzian signature *s* = 1, the covering group of *SO*(1, 2) is isomorphic to the real group *SL*(2, IR). Then the generators of *SL*(2, IR) may be realized by the matrices

$$
\gamma\_0 = i\sigma^2, \quad \gamma\_1 = \sigma^1, \quad \gamma\_2 = \sigma^3. \tag{16}
$$

These real matrices [27] satisfying

$$
\gamma\_{\mathfrak{a}} \gamma\_{\mathfrak{F}} = \mathfrak{g}\_{a\mathfrak{F}} + \eta\_{a\mathfrak{F}\mathfrak{v}} \gamma^{\nu} \tag{17}
$$

also provide a realization of the Clifford algebra

$$
\gamma\_{\mathfrak{a}}\gamma\_{\mathfrak{F}} + \gamma\_{\mathfrak{F}}\gamma\_{\mathfrak{a}} = \mathfrak{Z}g\_{\mathfrak{a}\mathfrak{F}}\tag{18}
$$

in 3D. In addition, the coframe basis *ϑα* converts into one Clifford algebra value one-form

$$
\gamma = \gamma\_\mathfrak{a} \mathfrak{e}^\mathfrak{a} \tag{19}
$$

Then Ψ will become real two-component spinors, with the Dirac adjoint defined by Ψ := Ψ†*γ*0.

$$
\overline{\Psi} \wedge \Psi = 0, \quad \overline{\Psi} \wedge \gamma\_5 \gamma^a \Psi = 0, \quad \overline{\Psi} \wedge \gamma\_5 \Psi = 0
$$

1 In four dimensions (4D), the Rarita–Schwinger field Ψ := Ψ*αϑα* entering Eq. (13) is a *Majorana spinor* valued one-form. As it is well known [34], it satisfies the Majorana condition, i.e. Ψ = *C*Ψ*t* , where *C* is the charge conjugation matrix given by *C* = −*iγ*0 satisfying *C*† = *C*−1 , *Ct* = −*C* and *C*−1*γαC* = − (*γα*) *t* . Consequently,

For the real Majorana representation all *γα* are purely imaginary and the components of the gravitino vector–spinor consequently are all real [30].

4 Will-be-set-by-IN-TECH

can be written holononically and anholononically. However, it does not depend on the coframe, inasmuch Ψ*α* := *eα*�Ψ involves the inverse tetrad. In 3D, we adhere to the conventions that the holonomic indices run from *i*, *j*, *k*,... = 0, 1, 2, whereas *α*, *β*,... = 0,

We are going to provide a brief summary of the spinors that will be used in three dimensions: As well known, the covering group of the rotation group *SO*(3) is isomorphic to the unitary group *SU*(2). Since an element of *SU*(2) can be parameterized by three numbers, the most

> 0 −*i i* 0

However, for Lorentzian signature *s* = 1, the covering group of *SO*(1, 2) is isomorphic to the

real group *SL*(2, IR). Then the generators of *SL*(2, IR) may be realized by the matrices

in 3D. In addition, the coframe basis *ϑα* converts into one Clifford algebra value one-form

Then Ψ will become real two-component spinors, with the Dirac adjoint defined by Ψ :=

1 In four dimensions (4D), the Rarita–Schwinger field Ψ := Ψ*αϑα* entering Eq. (13) is a *Majorana spinor* valued one-form. As it is well known [34], it satisfies the Majorana condition, i.e. Ψ = *C*Ψ*t*

is the charge conjugation matrix given by *C* = −*iγ*0 satisfying *C*† = *C*−1 , *Ct* = −*C* and *C*−1*γαC* =

Ψ ∧ Ψ = 0 , Ψ ∧ *γ*5*γα*Ψ = 0 , Ψ ∧ *γ*5Ψ = 0 For the real Majorana representation all *γα* are purely imaginary and the components of the gravitino

, *σ*3 =

*γ*0 = *iσ*2, *γ*1 = *σ*1, *γ*2 = *σ*3. (16)

*γαγβ* = *gαβ* + *ηαβνγν* (17)

*γαγβ* + *γβγα* = 2*gαβ* (18)

*γ* = *γαϑα* (19)

convenient basis of the Lie algebra are the familiar Pauli spin matrices:

, *σ*2 =

 *σα*, *σβ*

Ψ = Ψ*idxi* = Ψ*αϑα* (13)

 1 0 0 −1

= 2*iηαβγσγ*. (15)

ˆ 1, ˆ 2ˆ

. (14)

, where *C*

The Rarita-Schwinger [35] type spinor-valued one-form1

for the anholonomic indices.

*σ*1 =

These real matrices [27] satisfying

Ψ†*γ*0.

− (*γα*) *t*

. Consequently,

vector–spinor consequently are all real [30].

These matrices satisfy the following Lie algebra:

also provide a realization of the Clifford algebra

 0 1 1 0  The corresponding *manifestly Hermitian* RS type Lagrangian three–form of Howe and Tucker [23] reads

$$L\_{\rm RS} = \frac{i}{4} \left( \overline{\mathbf{Y}} \wedge D\mathbf{Y} - \mathbf{Y} \wedge \overline{D\mathbf{Y}} \right) + \frac{i}{4} m \overline{\mathbf{Y}} \wedge \gamma \wedge \mathbf{Y},\tag{20}$$

including, however, a mass term. Here minimal coupling to gravity is achieved via

$$d\Psi = d\Psi - \frac{1}{2}\gamma\_a \Gamma^{\star \alpha} \wedge \Psi,\tag{21}$$

which is nothing more than the gauge covariant derivative of a spinor-valued one-form Ψ.

Only in 3D, however, there exists a generalization given by the following expression

$$L\_{\overline{\mathbf{Y}}} = L\_{\overline{\mathbf{RS}}} + s\_1 \overline{D\mathbf{Y}} \wedge \, ^\*(D\mathbf{Y}) + s\_2 \, \overline{D\mathbf{Y}} \wedge \gamma \wedge \, ^\*(\gamma \wedge D\mathbf{Y}).\tag{22}$$

As in the case of the Rarita-Schwinger Lagrangian *L*RS, it is manifestly Hermitian when the additional quadratic derivative terms carry *s*1 and *s*2 as dimensionless coupling constants.

In order to supersymmetrize this action, it will be coupled to topological gravity later on.

### **3.1 Energy-momentum and spin currents**

By definition, the energy-momentum current two-form Σ*α* of matter is given by

$$\Sigma\_{\mathfrak{n}} := \frac{\delta L\_{\Psi}}{\delta \theta^{\mathfrak{a}}} = \frac{\partial L\_{\Psi}}{\partial \theta\_{\mathfrak{a}}} + D \frac{\partial L\_{\Psi}}{\partial T^{\mathfrak{a}}},\tag{23}$$

where the second term accounts for the possibility of a non-minimal coupling to torsion via Pauli type terms, cf. Eq. (5.1.8) of Ref. [22]. According to the Noether theorem, the energy-momentum current two-form of matter Σ*α* without Pauli terms can be rewritten as

$$\Sigma\_{\mathfrak{a}} := e\_{\mathfrak{a}} \rfloor L\mathfrak{y} - (e\_{\mathfrak{a}} \rfloor \overline{\mathfrak{Y}}) \wedge \frac{\partial L\overline{\mathfrak{Y}}}{\partial \overline{\mathfrak{Y}}} - (e\_{\mathfrak{a}} \rfloor \overline{\overline{\mathfrak{Y}}}) \wedge \frac{\partial L\overline{\mathfrak{Y}}}{\partial \overline{\overline{\mathfrak{Y}}}} - (e\_{\mathfrak{a}} \rfloor D\overline{\mathfrak{Y}}) \wedge \frac{\partial L\overline{\mathfrak{Y}}}{\partial D\overline{\mathfrak{Y}}} - (e\_{\mathfrak{a}} \rfloor D\overline{\overline{\mathfrak{Y}}}) \wedge \frac{\partial L\overline{\mathfrak{Y}}}{\partial D\overline{\overline{\mathfrak{Y}}}} \tag{24}$$

see Eq. (5.4.11) of Ref. [22] for details. This equivalent equation often is more convenient, since it involves only partial derivatives of the matter fields and avoids the intricate treatment of a possible dependence of the matter Lagrangian on the Hodge dual. Taking into account the identities of Appendix B, we find

$$\Sigma\_{\mathfrak{a}} = -\frac{i}{4}m\overline{\Psi} \wedge \gamma\_{\mathfrak{a}}\Psi + \mathrm{s}\_{1}\left\{ \overline{D\Psi} \wedge \mathrm{e}\_{\mathfrak{a}} \right\}^{\*} (D\Psi) - (\mathrm{e}\_{\mathfrak{a}} \rfloor D\Psi) \wedge \,^{\*}\left(\overline{D\Psi}\right) \rangle$$

$$+ \mathrm{s}\_{2}\left[ \overline{D\Psi} \wedge \gamma\_{\mathfrak{a}} \wedge \,^{\*}(\gamma \wedge D\Psi) - (\mathrm{e}\_{\mathfrak{a}} \rfloor D\Psi) \wedge \,^{\*}\left(\overline{D\Psi} \wedge \gamma\right) \wedge \gamma\right].\tag{25}$$

Since the kinetic terms in the Rarita-Schwinger type Lagrangian *L*RS do not depend explicitly on the coframe *ϑα*, they provides no contribution to the energy-momentum current.

The 3-dual of the spin current is defined by

$$
\tau\_a^\star := \frac{1}{2} \eta\_{a\beta\gamma} \tau^{\beta\gamma} = \frac{(-1)^s}{2} \frac{\delta L\_{\overline{\Gamma}}}{\delta \Gamma\_a^\star} \,. \tag{26}
$$

Let us restrict for the moment to the usual Rarita-Schwinger Lagrangian *L*RS, or equivalently

S-Duality in Topological Supergravity 43

1 2

becomes massive. Moreover, in Eq. (34) the term in brackets following form the

+ *γα*Ψ

 − *iR*

+ *cγ* ∧

1 4  *i* 2 *D*Ψ + *i* 4

*m*Ψ*γα*Ψ

*αγα*Ψ − *i*

2

*γα*Ψ ∧ Ψ*γα*Ψ = 0, (37)

*c* = −*m*, (38)

*ηα* + *τ α*

*γα*Ψ = 0, (41)

*γ* ∧ *γ* = −2*γαηα*, (39)

*m*2 2 *ηα*  *mTαγα*Ψ +

*mγ* ∧ Ψ ∼= 0 (35)

*mγ* ∧ Ψ

*i* 2

*γα*Ψ = 0, (40)

*mγ* ∧ *D*Ψ

(36)

*α* , are

to *L*Ψ with *s*1 = *s*2 = 0. Then the Rarita-Schwinger equation

*δL*

 *θ*TL *R α* + *θ*T 2 *Tα* + Σ*α*

 *θ*TL *R α* + *θ*T 2 *Tα*

 *i* 2 *D*Ψ + *i* 4

Moreover, in our restricted model with *s*1 = *s*2 = 0 we have to put

of Howe and Tucker [23], we find from Eq. (36) the requirement

 *R α* + *θ*T 2 − *m* 2 *Tα* +

− 1

in order that our Lagrangian becomes supersymmetric.

Chern-Simons term proportional to *θ*T. In this insertion

(−1)*s θ*TL

 + *θ*T 2 − *m* 2 *Tα* + 1 2 *θ*T 2 + *m*2

*i θ*TL

 *R α* +

in order to eliminate the kinetic *γ* ∧ *D*Ψ terms. Then, using the formula

supersymmetric transformations reads

∼= *iγα*Ψ

−*D iD*Ψ +

= *iγα*Ψ

+*cγ* ∧

*δL δϑα* + *cγ* ∧

By a Fierz rearrangement, i.e.,

vanishing.

*i θ*L +

*θ*TL − 1

*iγα*Ψ ∧

2 *i δL*

*δ*Ψ − 2*D δL*

*δ*Ψ

*mγ* ∧ Ψ

*mγ* ∧ Ψ

terms arising from the energy-momentum current Σ*α*, or likewise from the dual spin *τ*

At first sight, it appears that there is no cosmological constant in order to compensate a similar one arising from the RS mass. However, one should compare the bracket with the second field equation (9) *inserted*, which indeed involves a cosmological term induced by the translational

*i* 2

*δ*Ψ = *D*Ψ +

In view of the definition (21) of the covariant derivative, we find

$$\tau\_a^\* = \frac{(-1)^s}{2} \left\{ \frac{i}{4} \overline{\Psi} \gamma\_a \Psi + \frac{s\_1}{2} \left[ \overline{\Psi} \gamma\_a \wedge \,^\*(D\Psi) + \gamma\_a \Psi \wedge \,^\*\left(\overline{D\Psi}\right) \right] \right. $$

$$+ \frac{s\_2}{2} \left[ \overline{\Psi} \gamma\_a \wedge \,\gamma \wedge \,^\*(\gamma \wedge D\Psi) + \gamma\_a \Psi \wedge \gamma \wedge \,^\*(\gamma \wedge D\Psi) \right] \}. \tag{27}$$

Using the Hermetian properties of the spinor-valued *p*–forms, we finally obtain

$$\tau\_a^\star = \frac{(-1)^s}{2} \left[ \frac{i}{4} \overline{\mathbf{y}} \wedge \gamma\_a \mathbf{y} + s\_1 \overline{\mathbf{y}} \gamma\_a \wedge \,^\*(D\mathbf{y}) + s\_2 \overline{\mathbf{y}} \gamma\_a \wedge \gamma \wedge \,^\*(\gamma \wedge D\mathbf{y}) \right],\tag{28}$$

cf. the identities of Appendix C.

It should be noted that for the pure Rarita-Schwinger Lagrangian with *s*1 = *s*2 = 0, the energy-momentum current is proportional to its dual spin, i.e.

$$
\Sigma\_{\mathfrak{a}} = -(-1)^{s} 2m \tau\_{\mathfrak{a}}^{\star}.\tag{29}
$$


*3.1. Collection of Plants VOCs*

The volatiles mostly are organic compounds with a molecular weight between 100 and 200, such as hydrocarbons, alcohols, ketones, organic acids, nitrogen compounds, and organic sulfur [138]. Most of them have high chemical activity. Different collection methods may directly affect the type and proportion of VOCs, so it is particularly important to choose the appropriate method.

Traditional distillation collection techniques include steam distillation (SD), simultaneous distillation and solvent extraction (SDE), microwave-assisted hydrodistillation extraction (MWHD), ultrasound-assisted extraction (USE), and solid-phase trapping solvent extraction (SPTE). They have certain disadvantages in the isolation and purification of chemical constituents from plants tissues, such as long extraction time, high volumes solvent, and low efficiency [139]. In addition, many natural products are thermally unstable and may degrade during thermal extraction or distillation.

The most mainstream approach is headspace solid-phase microextraction (HS-SPME). It has some advantages over SD, SDE, and SPTE, such as rapid solvent-free extraction, no apparent thermal degradation, less laborious manipulation and sample requirement, and so on [140]. Moreover, due to the relatively low temperature and short headspace solid-phase extraction time, the risk of thermal artifacts is extremely low compared to other techniques [141]. Additionally, it is easy to standardize and fully integrate into the analysis system [142]. Thus, HS-SPME is an ideal technology of plants' VOC collection.