International Journal for Numerical and Analytical Methods in Geomechanics
Volume 49, Issue 12 pp. 2820-2831
RESEARCH ARTICLE
Open Access

ANN-Based Green's Function Approach for Reservoir Geomechanics

Matheus L. Peres, 

Corresponding Author

Matheus L. Peres

Civil and Environmental Engineering Department, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

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Elisa D. Sotelino, 

Elisa D. Sotelino

Civil and Environmental Engineering Department, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

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Leonardo C. Mesquita, 

Leonardo C. Mesquita

Civil and Environmental Engineering Department, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

Civil Engineering Department, Federal University of Viçosa, Viçosa, Brazil

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First published: 05 June 2025
https://doi.org/10.1002/nag.4012

Funding: The first and last authors of this work were supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.

ABSTRACT

Green's functions have been a tool to analyze important reservoir geomechanics effects like subsidence, compaction, well closure, and permeability variation. However, the classical Green's function is related to linear elastic, homogenous, and geometrically simplified media. The use of auxiliary functions that consider heterogeneous media has the potential to simplify the numerical analyses of some complex problems. This research work proposes a new approach to Green's function method based on artificial neural networks (ANNs) to consider stratified media with direct application in reservoir geomechanics. These solutions obtained through ANN have as output the normalized displacement at any point of the stratified massif due to the application of a point load inside the massif. The input data of the proposed ANN are the material properties of the media, the point load, and the point of interest. By using the developed ANN-based Green's function together with the classical Green's function approach, it is possible to obtain individually the displacement at any point of the massif due to a pore pressure variation within the reservoir, only discretizing the reservoir boundary. This leads to a more efficient method when compared to traditional methods, such as the finite element method. For the points of interest, seafloor, and top and bottom of the reservoir, the numerical example presented was 25 times faster than the classic Green function approach on a computer with 12 threads of 2.6 GHz and 32 GB RAM. This reduction in processing time is crucial for decision-makers to act in field applications.

1 Introduction

Green's functions are traditionally known as solutions of differential equations due to a point disturbance in a medium in several applications and for the differential equations of static equilibrium, which are of interest in this work. These functions are also the basis for classical numerical methods like the boundary element method and meshless methods.

Usual Green's functions are formulated for homogeneous media due to the complexity of obtaining analytical expressions for more generic media. One way to overcome this barrier is to calculate Green's functions numerically. However, for some applications, such as fluid dynamics [1], geophysics [2], and fracture mechanics [3], the need to numerically solve the Green's function and then apply it to the problem in question becomes computationally expensive. A solution for these situations is the use of Artificial Intelligence (AI) as a proxy of Green's functions for more generic material distribution through the media. One class of AI algorithms that can be used for this purpose consists of artificial neural networks (ANNs). Since this is the algorithm selected for this work, the developed method is referred to as the ANN-based Green's functions.

Although Green's functions and AI are two classic research topics, only recently they have been treated coupled. This coupling has contributed to the study of differential equations in general [4-6] and also in specific problems like for example in the wave equation solution [7], in fusion simulation [8], in radiation dose estimation [9] and in coordinates transformation for nonlinear boundary value problems linearization [10].

Despite this, no research works were found that specifically couple AI and Green's function applied to geomechanics. In fact, AI has been widely applied to mechanical and geomechanical problems, through other approaches. A common application of AI in these areas is the replacement of time-consuming classical calculation by an AI model [11]. This approach can be observed in geomechanical problems with different objectives like process optimization of oil and gas production [12, 13], well design [14], well drilling [15], and carbon dioxide injection [16-18]. The AI surrogate models are also used in multiscale material simulation [19-22], coupled flow-geomechanical simulations [23, 24], geomechanical risk and reliability analysis [25-28], and geomechanical inverse analysis [29-34]. The coupling between AI and green's functions has its main characteristics, in comparison to the surrogate model strategy, the ability to apply it to problems with different geometries and material distribution without the need for new training. For this reason, the present research work proposes the use of ANN as a proxy of generic Green's functions for calculating the displacement field in rock masses due to pore pressure variations in geomechanical reservoirs. The use of a simple Green's function for linear elastic homogeneous half-space is a numerical approach to predict the displacement field resulting from reservoir pore pressure variations [35]. However, there are other numerical and analytical classical methods, with their specificities, for this aim. A classical analytical model for subsidence prediction was developed by Geertsma [36-39], based on the nuclei of strain principle introduced by Mindlin and Cheng [40, 41]. Geertsma's model predicts subsidence due to the depletion of cylindrical reservoirs in a linear elastic semi-infinite massif. Van Opstal [42] extended this model to account for a rigid base, representing crystalline rocks or rigid carbonates, while Tempone [43] further refined it to calculate displacement and strain fields throughout the massif.

Building on Geertsma's work, Segall [44] developed an analytical model to evaluate strains and stresses around axisymmetric reservoirs subjected to pore pressure variations. Soltanzadeh and Hawkes [45], based on Eshelby's inclusion theory [46-48], addressed 2D elliptical and rectangular reservoirs. For further geometric generalization, Muñoz and Roehl [49] proposed a model for 3D reservoirs of arbitrary geometry, eliminating the need for singular function integration. These models assume reservoirs embedded in a homogeneous, linear elastic, and semi-infinite medium.

More complex stratigraphic scenarios were addressed by Du and Olson [50], Mehrabian and Abousleiman [51], and Wangen and Halvorsen [52], who analyzed displacement and strain fields in three-layered media influenced by cylindrical reservoirs. Mehrabian and Abousleiman [51] also demonstrated the scalability of their model to multilayered stratigraphy. Fokker and Orlic [53] advanced this approach with a semi-analytical model that incorporates multiple viscoelastic layers.

In comparison with analytical or semi-analytical approaches, numerical methods provide greater flexibility, enabling simulations of complex geometries, material behaviors, and coupled flow-geomechanics problems. The finite element method (FEM) [54-57] is widely used due to its ability to handle nonlinearities such as plasticity [58, 59], viscosity [60, 61], fracturing [62], and fault reactivation [63] in reservoir geomechanics simulation. However, the numerical solution of this type of problem requires discretizing the geometry over large regions beyond the vicinity of the reservoir, significantly increasing the computational cost. Thus, selecting the appropriate method for such geomechanical analysis involves weighing options: methods that better represent geometries, material distribution, and their properties but demand higher computational resources, versus methods that provide simplified but faster results. In this context, the use of ANN-based Green's function for heterogeneous media emerges as an approach that preserves the generalization capability of classical numerical solutions while reducing computational costs.

In comparison with the homogeneous Green's function approach, the use of Green's functions for heterogeneous media can reduce the computational cost for reservoir geomechanical problems by avoiding the need to solve a nonlinear system of equations. Furthermore, the proposed ANN-based Green's function approach also reduces the computational effort because it only demands the reservoir discretization, while the traditional linear elastic Green's function approach demands the discretization of the reservoir and the regions around it. Thus, this research work improves the Green's function approach using ANN in order to achieve a Green's function for stratified media. The ANNs used are multilayer Perceptron type (a feedforward neural network), and their training was based on a database composed of the numerical solutions of several problems that used Green's functions applied to heterogeneous stratified media.

2 Green's Function Approach for Reservoir Geomechanics

The Green's function approach for reservoir geomechanics is based on the reciprocity theorem, which relates the solution of two systems, one auxiliary system and one system of interest. The auxiliary system has as solution a Green's function, and the system of interest is the reservoir geomechanics problem to be solved. For the sake of illustration, without loss of generality, consider the Green's function for a force applied in a linear elastic semi-infinite medium (Melan Fundamental Solution), shown in Appendix A, and the problem of interest is a geomechanical reservoir with a pore-pressure variation located in a stratified zone embedded in a semi-infinite medium (Figure 1).

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FIGURE 1
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Combining auxiliary problem with reservoir geomechanics.

Although Melan's fundamental solution considers a semi-infinite homogeneous linear elastic medium, it can be used by the Green's function method to solve heterogeneous problems composed of elastic materials, linear and non-linear, and visco-plastic materials. To do this, it is necessary to understand the concept of complementary tension for situations of heterogeneity and visco-plasticity, obtained from constitutive relationships.

2.1 Linear Poroelasticity Applied to Heterogeneous Problems

The stress field is calculated based on the principle of equivalent stress, for which the expression in Equation (1) applies.
Δ σ i j = C i j k l ε k l − α δ i j Δ P $$\begin{equation}\Delta {{\sigma }_{ij}} = {{\mathbb{C}}_{ijkl}}\ {{\varepsilon }_{kl}} - \alpha {{\delta }_{ij}}\ \Delta {\mathrm{P}}\end{equation}$$ (1)
The tensors Δ σ i j $\Delta {{\sigma }_{ij}}$ and ε k l ${{\varepsilon }_{kl}}$ are, respectively, the stress change and the strain due to the pore pressure variation Δ P $\Delta {\mathrm{P}}$ . C i j k l ${{\mathbb{C}}_{ijkl}}$ is a fourth-order elastic constitutive tensor, α $\alpha $ is the biot coefficient and δ i j ${{\delta }_{ij}}$ is the identity tensor. If the term C i j k l e ε k l $\mathbb{C}_{ijkl}^e{{\varepsilon }_{kl}}$ is added and subtracted to Equation (1), it can be rewritten as in Equation (2):
Δ σ i j = C i j k l e ε k l − A i j · Δ P p − C i j k l e − C i j k l x i ε k l , $$\begin{equation}\Delta {{\sigma }_{ij}} = \mathbb{C}_{ijkl}^e\ {{\varepsilon }_{kl}} - {{A}_{ij}} \cdot \Delta {{P}_p} - \left( {\mathbb{C}_{ijkl}^e - {{\mathbb{C}}_{ijkl}}\left( {{{x}_i}} \right)} \right){{\varepsilon }_{kl}},\end{equation}$$ (2)
in which C i j k l e $\mathbb{C}_{ijkl}^e$ is the fourth-order elastic constitutive tensor of the auxiliary system. The third term of the right-hand side of Equation (2) is defined as the complementary stress field (Equation 3).
Δ σ C i j = C i j k l e − C i j k l x i ε k l $$\begin{equation}\Delta {{\sigma }_C}_{ij} = \left( {\mathbb{C}_{ijkl}^e - {{\mathbb{C}}_{ijkl}}\left( {{{x}_i}} \right)} \right)\ {{\varepsilon }_{kl}}\end{equation}$$ (3)
Considering Equation (3), Equation (2) can be rewritten as Equation (4):
Δ σ i j = C i j k l e ε k l − A i j · Δ P p − Δ σ C i j $$\begin{equation}\Delta {{\sigma }_{ij}} = \mathbb{C}_{ijkl}^e\ {{\varepsilon }_{kl}} - {{A}_{ij}} \cdot \Delta {{P}_p} - \Delta {{\sigma }_C}_{ij}\end{equation}$$ (4)

2.2 Poro-visco-plasticity

Reservoir porepressure variation can cause permanent deformation, so the linear elasticity relation in Equation (1) should be modified to account for this characteristic, which leads to Equation (5).
Δ σ i j = C i j k l ε k l − ε k l p − α δ i j Δ P , $$\begin{equation}\Delta {{\sigma }_{ij}} = {{\mathbb{C}}_{ijkl}}\ \left( {{{\varepsilon }_{kl}} - {{\varepsilon }_{kl}}^p} \right) - \alpha {{\delta }_{ij}}\ \Delta {\mathrm{P}},\end{equation}$$ (5)
in which ε k l p ${{\varepsilon }_{kl}}^p$ is the plastic strain at a time t i ${{t}_i}$ of the production history calculated by Equation (6):
ε k l p = ∫ 0 t i γ . p M i j dt , $$\begin{equation}{{\varepsilon }_{kl}}^p = \mathop \int \limits_0^{{{t}_i}} {{\mathop \gamma \limits^{.} }^p}\ {{{\mathrm{M}}}_{ij}}{\mathrm{dt}},\end{equation}$$ (6)
in which γ . p ${{\mathop \gamma \limits^{.} }^p}$ defines the equivalent plastic strain rate, which is a function of the current equivalent plastic strain and current stress field. M i j ${{{\mathrm{M}}}_{ij}}$ is the flow direction, which is a function of the current stress field.
Equation (5) can be rewritten as Equation (4), but for that, the complementary stress field should be stated for poro-visco-plasticity:
Δ σ C i j = C i j k l ε k l p $$\begin{equation}\Delta {{\sigma }_C}_{ij} = {{\mathbb{C}}_{ijkl}}\ {{\varepsilon }_{kl}}^p\end{equation}$$ (7)

2.3 Reciprocity Theorem Extended to Heterogeneous Porous Media

The presented constitutive equations and the complementary stress field can be used to extend the reciprocity theorem to heterogeneous porous media. Defining System number 1 as the auxiliary system and system number 2 as the system of interest, Equation (8) is obtained applying the mechanical equilibrium condition to the auxiliary system with the virtual displacement field corresponding to the exact solution of the system of interest and vice-versa, in which the superscript denotes the system number.
∫ Ω t Δ σ i j 1 ε i j 2 d V = ∫ Ω t f i 1 u i 2 d V , ∫ Ω t Δ σ i j 2 ε i j 1 d V = ∫ Ω t f i 2 u i 1 d V . $$\begin{equation} \def\eqcellsep{&}\begin{array} {l}\displaystyle \mathop \int \limits_{{{{{\Omega}}}_t}} {{\Delta}}\sigma _{ij}^{\left( 1 \right)}\varepsilon _{ij}^{\left( 2 \right)}{\mathrm{\ d}}V = \displaystyle \mathop \int \limits_{{{{{\Omega}}}_t}} f_i^{\left( 1 \right)}\ u_i^{\left( 2 \right)}{\mathrm{\ d}}V,\\[20pt] \displaystyle \mathop \int \limits_{{{{{\Omega}}}_t}} {{\Delta}}\sigma _{ij}^{\left( 2 \right)}\varepsilon _{ij}^{\left( 1 \right)}{\mathrm{\ d}}V = \displaystyle \mathop \int \limits_{{{{{\Omega}}}_t}} f_i^{\left( 2 \right)}\ u_i^{\left( 1 \right)}{\mathrm{\ d}}V. \end{array} \end{equation}$$ (8)
The vectors f i ${{f}_i}$ and u i ${{u}_i}$ represent external force and displacement fields, respectively, the stress change in problem number 1 is Δ σ i j ( 1 ) ${{\Delta}}\sigma _{ij}^{( 1 )}$ and is associated with a point load within the domain or on the boundary. Problem number 2 has the unknown stress field Δ σ i j ( 2 ) ${{\Delta}}\sigma _{ij}^{( 2 )}$ which is related to the strain field ε i j ( 2 ) $\varepsilon _{ij}^{( 2 )}$ and a complementary stress ( Δ σ C i j ( 2 ) ) $( {{{\Delta}}{{\sigma }_C}_{ij}^{( 2 )}} )$ . Considering the constitutive relations previously presented:
∫ Ω t C i j k l e 1 ε k l 1 ε i j 2 d V = ∫ Ω t f i 1 u i 2 d V , ∫ Ω t C i j k l e 2 ε k l 2 ε i j 1 d V = ∫ Ω R α Δ P 2 δ i j ε i j 1 dV + ∫ Ω C Δ σ C i j 2 ε i j 1 d V $$\begin{equation} \def\eqcellsep{&}\begin{array} {l} \displaystyle \mathop \int \limits_{{{{{\Omega}}}_t}} \mathbb{C}{{_{ijkl}^e}^{\left( 1 \right)}}\varepsilon _{kl}^{\left( 1 \right)}\varepsilon _{ij}^{\left( 2 \right)}{\mathrm{\ d}}V = \displaystyle \mathop \int \limits_{{{{{\Omega}}}_t}} f_i^{\left( 1 \right)}\ u_i^{\left( 2 \right)}{\mathrm{\ d}}V,\\[20pt] \displaystyle \mathop \int \limits_{{{{{\Omega}}}_t}} \mathbb{C}{{_{ijkl}^e}^{\left( 2 \right)}}\varepsilon _{kl}^{\left( 2 \right)}\varepsilon _{ij}^{\left( 1 \right)}{\mathrm{\ d}}V = \displaystyle \mathop \int \limits_{{{{{\Omega}}}_R}} \alpha {{\Delta}}{{{\mathrm{P}}}^{\left( 2 \right)}}{{\delta }_{ij}}\varepsilon _{ij}^{\left( 1 \right)}{\mathrm{\ dV}} + \displaystyle \mathop \int \limits_{{{{{\Omega}}}_C}} {{\Delta}}{{\sigma }_C}_{ij}^{\left( 2 \right)}\ \varepsilon _{ij}^{\left( 1 \right)}{\mathrm{d}}V \end{array} \end{equation}$$ (9)
as C i j k l e ( 1 ) $\mathbb{C}{{_{ijkl}^e}^{( 1 )}}$ is symmetrical and, consequently, it makes the left-hand sides of the two equations in the system (9) equal to each other. Thus, equating the right-hand side of the two equations of the system, results in the expression (10)
∫ Ω t f i 1 u i 2 d V = ∫ Ω R α Δ P 2 δ i j ε i j 1 dV + ∫ Ω C Δ σ C i j 2 ε i j 1 d V $$\begin{equation} \displaystyle \mathop \int \limits_{{{{{\Omega}}}_t}} f_i^{\left( 1 \right)}\ u_i^{\left( 2 \right)}{\mathrm{d}}V = \displaystyle \mathop \int \limits_{{{{{\Omega}}}_R}} \alpha \Delta {{{\mathrm{P}}}^{\left( 2 \right)}}{{\delta }_{ij}}\varepsilon _{ij}^{\left( 1 \right)}{\mathrm{\ dV}} + \displaystyle \mathop \int \limits_{{{{{\Omega}}}_C}} \Delta {{\sigma }_C}_{ij}^{\left( 2 \right)}\ \varepsilon _{ij}^{\left( 1 \right)}{\mathrm{d}}V\end{equation}$$ (10)
Knowing that f i ( 1 ) $f_i^{( 1 )}$ is a generalized force, such as a Dirac delta, with direction k $k$ , and eliminating superscripts, without risk of ambiguity, Equation (3) can be rewritten as
u k X i = ∫ Ω R α x i Δ P x i δ m n ε m n k x i , X i d V + ∫ Ω C Δ σ C m n x i ε m n k x i , X i d V , $$\begin{align} {{u}_k}\ \left( {{{X}_i}} \right) &= \displaystyle \mathop \int \limits_{{{{{\Omega}}}_R}} \alpha \left( {{{x}_i}} \right)\Delta {\mathrm{P}}\left( {{{x}_i}} \right){{\delta }_{mn}}{{\varepsilon }_{mnk}}\left( {{{x}_i},{{X}_i}} \right)\ dV \nonumber\\ & \quad + \displaystyle \mathop \int \limits_{{{{{\Omega}}}_C}} \Delta {{\sigma }_C}_{mn}\left( {{{x}_i}} \right)\ {{\varepsilon }_{mnk}}\left( {{{x}_i},{{X}_i}} \right)dV,\end{align}$$ (11)
where u k ( X i ) ${{u}_k}( {{{X}_i}} )$ is the displacement vector at position X i , ${{X}_i},$ α ( x i ) $\alpha ( {{{x}_i}} )$ , Δ P ( x i ) $\Delta P( {{{x}_i}} )$ , Δ σ C m n ( x i ) ${{\Delta}}{{\sigma }_C}_{mn}( {{{x}_i}} )$ and ε j l ( x i ) ${{\varepsilon }_{jl}}( {{{x}_i}} )$ are, respectively, the Biot coefficient, the pore pressure variation, the complementary stress tensor, and the strain variation tensor at position x i ${{x}_i}$ of the real problem. ε m n k ∗ ( x i , X i ) ${{\varepsilon }_{mnk}}^{\mathrm{*}}( {{{x}_i},{{X}_i}} )$ are the strains at the position X i ${{X}_i}$ due to a point load with direction k $k$ applied at x i ${{x}_i}$ . Furthermore, the domain Ω R ${{{{\Omega}}}_R}$ represents the region of the model where there is pore pressure variation, and the domain Ω C ${{{{\Omega}}}_C}$ symbolizes the stratified region where the reservoir is located, referred to in this work as a complementary region.

2.4 Numerical Scheme

To solve Equation (11) the continuous media can be discretized into n R $nR$ elements in the reservoir region and n C $nC$ elements in the complementary region, resulting in Equation (12).
u k X i = ∑ p = 1 n R ∫ R E p α x i Δ P x i δ m n ε m n k x i , X i d V + ∑ q = 1 n C ∫ C E q C m n j l e − C m n j l x i ε j l x i ε m n k x i , X i d V $$\begin{align}{{u}_k}\ \left( {{{X}_i}} \right) &= \sum_{{\mathrm{p}} = 1}^{nR} \displaystyle \mathop \int \limits_{R{{E}_p}} \alpha \left( {{{x}_i}} \right)\Delta {\mathrm{P}}\left( {{{x}_i}} \right){{\delta }_{mn}}{{\varepsilon }_{mnk}}\left( {{{x}_i},{{X}_i}} \right)dV \nonumber\\ & \quad + \sum_{{\mathrm{q\ }} = 1}^{nC} \displaystyle \mathop \int \limits_{C{{E}_q}} \left( {\mathbb{C}_{mnjl}^e - {{\mathbb{C}}_{mnjl}}\left( {{{x}_i}} \right)} \right){{\varepsilon }_{jl}}\left( {{{x}_i}} \right){{\varepsilon }_{mnk}}\left( {{{x}_i},{{X}_i}} \right)dV\end{align}$$ (12)
Considering that the pore pressure and the Biot coefficient are uniform in each element, Equation (12) reduces to Equation (13).
u k X i = ∑ p = 1 n R α p Δ P p ∫ R E p δ m n ε m n k x i , X i d V + ∑ q = 1 n C ∫ C E q C m n j l e − C m n j l x i ε j l x i ε m n k x i , X i d V $$\begin{align} {{u}_k}\ \left( {{{X}_i}} \right) &= \sum_{p = 1}^{nR} {{\alpha }_p}{{\Delta}}{{{\mathrm{P}}}_p}\displaystyle\mathop \int \limits_{R{{E}_p}} {{\delta }_{mn}}{{\varepsilon }_{mnk}}\left( {{{x}_i},{{X}_i}} \right)dV \nonumber\\ & \quad + \sum_{{\mathrm{q\ }} = 1}^{nC} \displaystyle \mathop \int \limits_{C{{E}_q}} \left( {\mathbb{C}_{mnjl}^e - {{\mathbb{C}}_{mnjl}}\left( {{{x}_i}} \right)} \right){{\varepsilon }_{jl}}\left( {{{x}_i}} \right){{\varepsilon }_{mnk}}\left( {{{x}_i},{{X}_i}} \right)dV\end{align}$$ (13)
This consideration is realistic since the pore pressure variation information is usually provided by discrete models used to define the flow inside the reservoir. In addition, it is also possible to apply Green's theorem in the integral relative to the reservoir. For the two-dimensional case, the expression (14) is obtained.
u k X i = ∑ p = 1 n R α p Δ P p ∫ ∂ R E p u 1 k x i , X i d y − ∫ ∂ R E p u 2 k x i , X i d x + ∑ q = 1 n C ∫ C E q C m n j l e − C m n j l x i ε j l x i ε m n k x i , X i d V $$\begin{align} {{u}_k}\ \left( {{{X}_i}} \right) &= \sum_{p = 1}^{nR} {{\alpha }_p}{{\Delta}}{{{\mathrm{P}}}_p}\displaystyle \mathop \int \limits_{\partial R{{E}_p}} {{u}_{1k}}\left( {{{x}_i},{{X}_i}} \right)dy - \displaystyle \mathop \int \limits_{\partial R{{E}_p}} {{u}_{2k}}\left( {{{x}_i},{{X}_i}} \right)dx \nonumber\\ & \quad + \sum_{q = 1}^{nC} \displaystyle \mathop \int \limits_{C{{E}_q}} \left( {\mathbb{C}_{mnjl}^e - {{\mathbb{C}}_{mnjl}}\left( {{{x}_i}} \right)} \right){{\varepsilon }_{jl}}\left( {{{x}_i}} \right){{\varepsilon }_{mnk}}\left( {{{x}_i},{{X}_i}} \right)dV\end{align}$$ (14)
The application of Green's theorem is not trivial for the case under study because the solution of Melan's problem is singular at the position where the point load is applied. A collocation method can be used to solve Equation (14) at all points of the discrete model [64]. Thus, Equation (14) can be rewritten in its recurrence form as shown in Equation (15),
u k a + 1 X i = ∑ p = 1 n R α p Δ P p ∫ ∂ R E p u 1 k x i , X i d y − ∫ ∂ R E p u 2 k x i , X i d x + ∑ q = 1 n C ∫ C E q C m n j l e − C m n j l x i ε j l a x i ε m n k x i , X i d V $$\begin{align}{{u}_k}_{a + 1}\ \left( {{{X}_i}} \right) &= \sum_{p = 1}^{nR} {{\alpha }_p}\Delta {{{\mathrm{P}}}_p}\displaystyle \mathop \int \limits_{\partial R{{E}_p}} {{u}_{1k}}\left( {{{x}_i},{{X}_i}} \right)dy - \displaystyle \mathop \int \limits_{\partial R{{E}_p}} {{u}_{2k}}\left( {{{x}_i},{{X}_i}} \right)dx \nonumber\\ & \quad + \sum_{q = 1}^{nC} \displaystyle \mathop \int \limits_{C{{E}_q}} \left( {\mathbb{C}_{mnjl}^e - {{\mathbb{C}}_{mnjl}}\left( {{{x}_i}} \right)} \right){{\varepsilon }_{jl}}_a\left( {{{x}_i}} \right){{\varepsilon }_{mnk}}\left( {{{x}_i},{{X}_i}} \right)dV\end{align}$$ (15)
with the initial solution given by
u k 0 X i = ∑ p = 1 n R α p Δ P p ∫ ∂ R E p u 1 k x i , X i d y − ∫ ∂ R E p u 2 k x i , X i d x $$\begin{equation}{{u}_k}_0\ \left( {{{X}_i}} \right) = \sum_{p = 1}^{nR} {{\alpha }_p}\Delta {{{\mathrm{P}}}_p}\displaystyle \mathop \int \limits_{\partial R{{E}_p}} {{u}_{1k}}\left( {{{x}_i},{{X}_i}} \right)dy - \displaystyle \mathop \int \limits_{\partial R{{E}_p}} {{u}_{2k}}\left( {{{x}_i},{{X}_i}} \right)dx\ \end{equation}$$ (16)

The discretization for the first term of expression (16) can be the same as the one used for flow simulation. Furthermore, the integration region of the second term of the equation can be discretized in traditional polynomial elements as was done by Peres et al. [35].

3 ANN-Based Green's Function Approach

The use of a Green's function that considers the media according to the geomechanical problem makes possible the elimination of the second term on the right side of Equation (15), which leads to Equation (17).
u k X i = ∑ p = 1 n R α p Δ P p ∫ ∂ R E p u 1 k x i , X i d y − ∫ ∂ R E p u 2 k x i , X i d x $$\begin{equation}{{u}_k}\ \left( {{{X}_i}} \right) = \sum_{p = 1}^{nR} {{\alpha }_p}\Delta {{{\mathrm{P}}}_p}\displaystyle \mathop \int \limits_{\partial R{{E}_p}} {{u}_{1k}}\left( {{{x}_i},{{X}_i}} \right)dy - \displaystyle \mathop \int \limits_{\partial R{{E}_p}} {{u}_{2k}}\left( {{{x}_i},{{X}_i}} \right)dx\ \end{equation}$$ (17)

Thus, an iterative solution would not be necessary since the displacements for each point would be obtained from the integration of the Green's function over the contour of the elements within the reservoir. In this work, ANNs of the feedforward multilayer perceptron type were trained to obtain Green's functions that consider a stratified media.

This type of ANN has three types of layers: the input, the output, and the hidden layers, as shown in Figure 2. Furthermore, it is based on the mathematical expression given by Equation (18).
a n + 1 = f n + 1 W n + 1 a n + b n + 1 $$\begin{equation}{{a}_{n + 1}} = {{f}_{n + 1}}\ \left( {{{W}_{n + 1}}{{a}_n} + {{b}_{n + 1}}} \right)\end{equation}$$ (18)
in which a n + 1 ${{a}_{n + 1}}$ and a n ${{a}_n}$ are the current and the previous layer output, respectively, f n + 1 ${{f}_{n + 1}}$ and b n + 1 ${{b}_{n + 1}}$ are the activation function and the bias of the current layer and W n + 1 ${{W}_{n + 1}}$ is the weight that connects the prior layer to the current. Consequently, a 0 ${{a}_0}$ indicates the input data.
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FIGURE 2
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Schematic representation of a Multilayer perceptron.

However, it is important to highlight the neural network Green's function, as traditional Green's function presents singularities at the location where the point load is applied.

The concept presented here can be applied to any type of heterogeneous space, simply by defining a way to parameterize the elastic properties and geometry of the layers of the rock mass. As a way of exemplifying the proposed methodology, the stratigraphy was considered in which four layers are arranged horizontally, and is embedded in a semi-infinite medium, resulting in a distribution of up to five different materials, as seen in Figure 3. The number of layers was determined to be possible to differentiate the material from the substrate, the reservoir region, the cap rock, the overburden, and still have a last layer for detailing any of the other regions, if necessary. Thicknesses ( d 1 , d 2 , d 3 , d 4 ) $( {{{d}_1},\ {{d}_2},\ {{d}_3},\ {{d}_4}} )$ , presented in Figure 3, were determined with values whose sum varies between 3 and 8 km in order to be representative of pre-salt stratigraphy. The length L is chosen as 41 km so that its magnitude is much greater than the width of a reservoir. In this way, the trained neural networks are capable of predicting the Green's function for stratigraphies that have these characteristics, which will make possible solving reservoir geomechanics problems whose geometry fits these characteristics, independently the shape and position of the reservoir.

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FIGURE 3
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General heterogeneity distribution for ANN training.

3.1 Input and Output Data

The neural networks used as a proxy of the Green's function for stratified media have as input the following parameters:
  • five Poisson's ratios, one for each layer;
  • five elastic modulus normalized by the elastic modulus of the last layer;
  • four layers thickness, since the last one always has infinite thickness;
  • the depth where the point load is applied; and
  • the relative position between the evaluation point and the point load.
Four ANNs were trained, one for each of the four displacements considered (vertical or horizontal displacement due to vertical or horizontal unit force). In addition, the output data was normalized, following Equation (19), to keep its magnitude close to the unit, thus improving the efficiency of the training process.
u i k ¯ = u i k u k k ∞ $$\begin{equation}\overline {{{u}_{ik}}} = \frac{{{{u}_{ik}}}}{{{{u}_{kk}}^\infty }}\ \end{equation}$$ (19)
where u i k ${{u}_{ik}}$ and u i k ¯ $\overline {{{u}_{ik}}} $ are the displacement and the normalized displacement in i $i$ direction due to a unitary point load in k direction, respectively, and u k k ∞ ${{u}_{kk}}^\infty $ is the displacement in i $i$ direction due to a unitary point load in k direction, considering a homogeneous semi-infinite media.

3.2 Data Set Generation

To train the neural network, it is necessary to have a robust set of input data and their respective output data, which properly characterizes the phenomenon for which the network is going to be used. For the training of the networks used in this research, it was necessary to build a database that had the vertical and horizontal displacements caused by vertical and horizontal unit forces, applied to stratified environments like the one shown in Figure 3. The displacements were numerically calculated for 45 different stratified media, in order to make the ANN capable of generalizing to different stratigraphies.

The displacements used for training the ANN were obtained numerically from Equation (20), which can be obtained based on the reciprocity theorem analogous to Equation (14), but for this case, the system of interest is the stratified model subjected to a point load. For each medium considered, the point load was positioned at 33 different depths and, for each position, the displacements at 500 points in the domain were evaluated, totaling 16,500 sets of data for each of the 45 stratigraphies.
u k 2 X i = u k 1 X i + ∑ q = 1 n C ∫ C E q C m n j l e − C m n j l x i ε j l x i ε m n k 1 x i , X i d V $$\begin{align}&{{u}_k}^{\left( 2 \right)}\ \left( {{{X}_i}} \right) = {{u}_k}^{\left( 1 \right)}\ \left( {{{X}_i}} \right) \nonumber\\ &+ \sum_{q = 1}^{nC} \displaystyle \mathop \int \limits_{C{{E}_q}} \left( {\mathbb{C}_{mnjl}^e - {{\mathbb{C}}_{mnjl}}\left( {{{x}_i}} \right)} \right){{\varepsilon }_{jl}}\left( {{{x}_i}} \right){{\varepsilon }_{mnk}}^{\left( 1 \right)}\left( {{{x}_i},{{X}_i}} \right)dV\end{align}$$ (20)

3.3 Neural Network Training

For the training of the neural networks, data was generated by 45 different models, which were subdivided into 3 groups, that is, a training group, a validation group, and a test group, corresponding to 70%, 15%, and 15% of the total amount of data, respectively. The elastic properties considered vary from 0,1 to 0,4 for Poisson ratio, and from 5 to 60 GPa for the elastic modulus. These ranges were chosen because they encompass the characteristic subsurface elastic material properties.

The training process traditionally consists of defining weights and biases that produces results that fit the training data. Thus, this process can be understood as an optimization in which the objective function is a measure of error between the output obtained by the training group and the output predicted by the ANN. However, to guarantee extrapolation capacity for other data than the ones from the training group, the early stopping method [65] as a regularization technique was used. This method uses the validation group in each training iteration to verify if overfitting is occurring, that is, it prevents the ANN model from adapting to the training data without adapting to the validation group data. In this article, the error measurement considered is the mean squared error. Thus, the loss function is shown in Equation (21).
L F = 1 n ∑ i = 1 n Y − Y ̂ 2 , $$\begin{equation}LF = \frac{1}{n}\ \sum_{i = 1}^n {{\left( {Y - {{\hat{\rm Y}}}} \right)}^2},\end{equation}$$ (21)

In which Y $Y$ is one of the four displacements, u i k ¯ $\overline {{{u}_{ik}}} $ (depending on which of the four ANNs is being trained), and Y ̂ ${{\hat{\rm Y}}}$ the ANN output. As training consists of reducing the loss function, at the end, Y ̂ ${{\hat{\rm Y}}}$ tend to be quite similar to Y $Y$ . The training follows the ADAM (Adaptive Moment Estimation) algorithm [66], which is a gradient-based algorithm. It was chosen because it has better efficiency when compared to heuristic-based algorithms. It also demands less memory and is invariant to the diagonal rescaling of gradients, being, therefore, suitable for problems that have a large amount of data. To avoid stopping at local minima, a learning rate of 0.001 was used. The optimization process was initialized with weight and bias defined by the algorithm proposed by Glorot and Bengio [67]. This algorithm initializes the weight and bias following a Gaussian distribution with mean value equal to zero and the variance equal to 2 / ( f i n + f o u t ) 2 $\sqrt[2]{{2/( {{{f}_{in}} + {{f}_{out}}} )}}$ , in which f i n ${{f}_{in}}$ is the number of input connections with the first hidden layer and f o u t ${{f}_{out}}$ is the number of output connections with the last hidden layer.

In addition to the training process involving the calibration of weights and biases, it was also necessary to define the number of hidden layers, the number of neurons per hidden layer, and the activation function to be used in the problem. This was achieved using an iterative training process based on Bayesian search [68]. This adopted procedure is basically a Bayesian Optimization in which the objective function is the error metric calculated for the validation data set, and the variables are the activation function, the number of hidden layers, and the number of neurons per hidden layer. The methodology consisted of training different networks, in which, for each trained network, the topology of the next one was defined in such a way that the average absolute error calculated based on a validation database had a greater probability of being smaller than the trained networks until that moment. During this process, it was considered that the neural network would have between three and eight hidden layers and between 1 and 300 neurons per hidden layer. This consideration is closely related to the amount of data for training, and consequently, is related to the amount of stratigraphies generated for training. By considering 45 different stratigraphies and evaluating the displacements at 500 points due to the load applied at 33 different depths, the database set has 742,500 data sets, of which 519,750 were used for training. This amount of data was considered because it is of the same order of magnitude as the number of parameters (weights and bias) that would be necessary to calibrate a neural network that had three hidden layers composed of 200 neurons (567,200).

Finally, the networks that presented the best performance were selected. These neural networks use the Relu function as activation function for the input and for the hidden layers, and a linear function as activation function for the output layer. The number of neurons in each layer and the total number of weights and biases are presented in Table 1. The neural network was trained to be a proxy of u 21 ¯ $\overline {{{u}_{21}}} $ presented the largest number of weights and biases (118,328), which is approximately five times smaller than the amount of data in the data set used for training (519,750). The generalization capacity of the neural network is verified through a numerical example whose parameters are different from those used for training. The training process for each neural network took approximately 24 h on a computer with 12 threads of 2.6 GHz and 32 GB RAM.

TABLE 1. Number of neurons per layer of each trained ANN.
Output Neurons per layer Total n° of weights and bias
u 11 ¯ $\overline {{{u}_{11}}} $ 17–242–104–116–92–1 52,110
u 21 ¯ $\overline {{{u}_{21}}} $ 17–221–94–262–263–1 118,328
u 12 ¯ $\overline {{{u}_{12}}} $ 17–217–198–68–83–72–5–264–1 73,683
u 22 ¯ $\overline {{{u}_{22}}} $ 17–97–244–76–179–1 57,644

4 Numerical Example

In this section, an application of the neural network Green's function is presented. This example consists of a depleting rectangular-shaped reservoir into a four-layered stratigraphy. The upper layer is named overburden, the second layer is the cap rock, the third layer is where the reservoir is embedded, and the fourth layer is the substratum, as shown in Figure 4. The dashed lines in Figure 4 represent the delimitation of the reservoir while the letters “L”, “R”, “T”, and “B” represent the left side, the right side, the top, and the bottom of the reservoir. The entire model is embedded in a semi-infinite media stiffer than the other layers, and all the materials are linear elastic with properties shown in Table 2. The initial pressure decays inside the reservoir, but it remains constant outside, which is a usual behavior of reservoir during oil and gas extraction.

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FIGURE 4
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Stratified media of the numerical application.
TABLE 2. Geomechanics properties for numerical application.
Notation Definition Value Unit
Layer 1
E 1 ${{E}_1}$ Elasticity modulus 34 GPa
ν 1 ${{\nu }_1}$ Poisson's ratio 0.25 —
Layer 2
E 2 ${{E}_2}$ Elasticity modulus 46 GPa
ν 2 ${{\nu }_2}$ Poisson's ratio 0.23 —
Layer 3
E 3 ${{E}_3}$ Elasticity modulus 7 GPa
ν 3 ${{\nu }_3}$ Poisson's ratio 0.33 —
Reservoir
E R ${{E}_R}$ Elasticity modulus 7 GPa
ν R ${{\nu }_R}$ Poisson's ratio 0.33 —
α ${{\alpha}}$ Biot coefficient 0.8 —
Δ P p ${{\Delta}}{{P}_p}$ Pore pressure change −10 MPa
Layer 4
E 4 ${{E}_4}$ Elasticity modulus 38 GPa
ν 4 ${{\nu }_4}$ Poisson's ratio 0.34 —
Semi-infinite media
E ∞ ${{E}_\infty }$ Elasticity modulus 46 GPa
ν ∞ ${{\nu }_\infty }$ Poisson's ratio 0.20 —

This example was solved by the classical Green's function approach for heterogeneous media [35], which applies a homogeneous Green's function, and by the ANN-based Green's function approach, which applies a ANN as proxy of a heterogeneous Green's function. Both results were discussed and compared in relation accuracy and processing time. The discretization considered for the classical approach had 2880 elements and 3066 degrees of freedom, and the numerical integration adopted was a level 3 Bartholomew quadrature [69]. For the ANN-based approach, the model was discretized in 44 linear elements around the reservoir, with a three-point Gauss quadrature rule for numerical integration.

Figure 5 shows the vertical displacement in the model's middle along the depth of the model, with the starting point being at the surface and the end point at a depth −5000 m, which represents the contact between the last layer and the semi-infinite media. The proposed ANN-based Green's function approach presented similar results as those obtained using the classical approach. It was observed two points where the neural network Green's function had reduced precision, at the top (y = −3500 m) and at the bottom of the reservoir (y = −4000 m).

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FIGURE 5
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Vertical displacement through depth over x = 0 (inside reservoir).

Figure 6 shows the vertical displacement along the width of the model at the surface (a), at the top reservoir (b) (y = −3500 m), and at the bottom reservoir (c) (y = −4000 m). The subsidence results show good agreement between the classical and the neural network approaches. For top and bottom reservoir, it was possible to mimic the shape of the function, but it is observed a gap between classical and neural network curves within the reservoir width.

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FIGURE 6
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Vertical displacement through width (a) on the surface, (b) above the reservoir, and (c) below the reservoir.

Figure 7 shows the horizontal displacement along the width of the model at the surface (a), at the top reservoir (b) (y = −3500 m), and at the bottom reservoir (c) (y = −4000 m). It can be observed that there is good agreement between the two approaches at the surface. At the top reservoir, both curves are slightly different, but they present similarities such as a peak at the top corners (with different magnitudes) and a null horizontal displacement at the middle. Observing the horizontal displacement at the bottom of the reservoir, the maximum displacements for this depth are in agreement in the two approaches, and the shape of the curves is similar in both cases.

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FIGURE 7
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Horizontal displacement through model width (a) on the surface, (b) above the reservoir, and (c) below the reservoir.

In summary, the ANN-based Green's function approach can provide displacements consistent with the displacements obtained using the classical approach. The main difference between the two approaches is the CPU time required, which is reduced by 25% in 12 threads of 2.6 GHz hardware for the neural network approach when compared with the classical one. It is important to highlight that to obtain the solution using ANN-based Green's functions, there was also a computational effort to build the database and train each of the four neural networks. Therefore, if it were necessary for each new problem to generate new data sets and carry out new training, the proposed method would be much more expensive. However, the trained neural networks can be used for other problems that have the same characteristics as the dataset for which it was trained. In this way, effective time savings will occur for stratigraphy problems that have horizontal layers and up to five different materials, independently of the position and shape of the reservoir.

5 Conclusion

The use of neural networks as a proxy for Green's function makes it possible to obtain generalized fundamental solution more efficiently when compared to the classical method. Using these neural Green's functions, it is possible to calculate the displacement field due to a reservoir pore pressure variation, discretizing only the boundary reservoir, while the classical approach demands the discretization of the whole massive near the reservoir. Because of this, it is possible to calculate the displacement in each point of the massif without needing to calculate the displacement in other points of the model. Also, in the proposed approach, the iterative process used in the classical approach is not necessary.

The numerical application consisted of the uniform depleting reservoir embedded in a four-layered massive, which is a typical pre-salt field stratigraphy configuration (seafloor, albian, salt, and underburden). The Green's function approach demanded the reservoir boundary discretization in 44 linear unidimensional elements, while the traditional approach demanded a mesh with 2880 elements and 3066 degrees of freedom. The displacements obtained by the two methods were similar. This validates the reliability of the proposed neural network Green's function, since the data used in the application is different from the data used for the training of the neural network. So, it is expected that for other reservoir geomechanics examples, this approach will also produce reliable results.

The proposed neural Green's function approach can be expanded to non-elastic, three-dimensional problems, and to non-horizontal layers. However, to make it feasible, it is necessary to increase the complexity of the ANNs training and develop other parametrization strategies. For instance, the application of ANN-based Green's functions approach for three-dimensional problems with generic material distribution requires the use of convolution neural networks, which makes it possible to apply generic parametrization of spatial properties distribution. For problems that involve non-elastic materials, the training dataset must be composed of numerical solutions of nonlinear media, which can be time-consuming and demand the use of recurrent neural networks to capture trajectory dependence of these materials. However, once the network is trained, the use of the ANN-based Green's function will be identical to what was presented in the example provided, as long as its application has characteristics that follow the same criteria as the training data set.

Finally, it is observed that the proposed method offers a significant computational performance gain, which is highly valuable for obtaining results that support time-sensitive decision-making during reservoir development. Moreover, this computational efficiency becomes a major advantage when performing stochastic or optimization analyses, as such analyses typically require thousands of numerical simulations. While the speedup is a clear benefit of the ANN-based Green's function approach, the most important aspect is understanding the source of this advantage. The performance gain is achieved because the method allows the computation of any component of the displacement vector at any point in the model without the need to calculate the entire displacement field, as is required in the finite element method. Therefore, depending on the number of points of interest to be evaluated, the computational gain can be substantially greater than the gain observed in the numerical example. This characteristic is enabled by the ability to obtain a generic fundamental solution using ANNs, which allows the displacement equation to be formulated solely in terms of input parameters such as the Biot coefficient, pore pressure variation, and the fundamental solutions represented by the neural networks.

Acknowledgments

The authors would like to thank Professor Dr. Yves M. Leroy, from Imperial College London, for his enormous contributions to the Green function approach. The first and last authors of this work were supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.

The Article Processing Charge for the publication of this research was funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) (ROR identifier: 00x0ma614).

    Conflicts of Interest

    The authors declare no conflicts of interest.

    Appendix A: Melan fundamental solution

    The Green function for elastic semi-infinite media ( u n p ( x i , X i ) ${{u}_{np}}( {{{x}_i},{{X}_i}} )$ ) can be decomposed into the sum of two parts, the first referring to the 2D Green Function for elastic infinite media, represented by u n p ( x i , X i ) K ${{u}_{np}}{{( {{{x}_i},{{X}_i}} )}^K}$ , and the second formed by complementary terms, represented by u n p ( x i , X i ) C ${{u}_{np}}{{( {{{x}_i},{{X}_i}} )}^C}$ , X i ${{X}_i}$ denoting the position of the point load and x i ${{x}_i}$ denoting the position where the displacements are evaluated, defined in Figure A.1.
    u n p x i , X i = u n p x i , X i K + u n p x i , X i C , $$\begin{equation}{{u}_{np}}\ \left( {{{x}_i},{{X}_i}} \right) = {{u}_{np}}{{\left( {{{x}_i},{{X}_i}} \right)}^K} + {{u}_{np}}{{\left( {{{x}_i},{{X}_i}} \right)}^C},\end{equation}$$ (A.1)

    1 2D Kelvin's fundamental solution

    Using the Cartesian coordinate system shown in Figure A.1, the displacements at position x i ${{x}_i}$ generated by a horizontal unit point load applied at point X i ${{X}_i}$ are given by:

    Details are in the caption following the image
    FIGURE A.1
    Open in figure viewerPowerPoint
    The 2D coordinate system for point load in semi-infinite media.
    u 11 K = A r x 2 r 2 − 3 − 4 ν · ln r , $$\begin{equation}u_{11}^K = A\left[ {\frac{{{{r}_x}^2}}{{{{r}^2}}} - \left( {3 - 4\nu } \right) \cdot \ln r} \right],\end{equation}$$ (A.2)
    u 21 K = A r x · r y r 2 . $$\begin{equation}u_{21}^K = A\frac{{{{r}_x} \cdot {{r}_y}}}{{{{r}^2}}}\ .\end{equation}$$ (A.3)
    with:
    A = 1 + ν 4 π E 1 − ν . $$\begin{equation}A = \frac{{1 + \nu }}{{4\pi E\left( {1 - \nu } \right)}}\ \ .\end{equation}$$ (A.4)
    Thus, the first part of the Green functions for horizontal direction is defined by:
    u n 1 x i , X i K = u 11 K u 21 K , $$\begin{equation}{{u}_{n1}}{{\left( {{{x}_i},{{X}_i}} \right)}^K} = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {u_{11}^K}\\ {u_{21}^K} \end{array} } \right]\ ,\end{equation}$$ (A.5)
    The displacements at position x i ${{x}_i}$ generated by a vertical unit point load applied at point X i ${{X}_i}$ are given by:
    u 12 K = A r x · r y r 2 , $$\begin{equation}u_{12}^K = A\frac{{{{r}_x} \cdot {{r}_y}}}{{{{r}^2}}},\end{equation}$$ (A.6)
    u 22 K = A r x 2 r 2 − 3 − 4 ν · ln r . $$\begin{equation}u_{22}^K = A\left[ {\frac{{{{r}_x}^2}}{{{{r}^2}}} - \left( {3 - 4\nu } \right) \cdot \ln r} \right].\end{equation}$$ (A.7)
    Thus, the first part of the Green function for vertical direction is defined by:
    u n 2 x i , X i K = u 12 K u 22 K . $$\begin{equation}{{u}_{n2}}{{\left( {{{x}_i},{{X}_i}} \right)}^K} = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {u_{12}^K}\\ {u_{22}^K} \end{array} } \right].\end{equation}$$ (A.8)

    2 Complementary part

    The complementary displacements at position x i ${{x}_i}$ generated by a horizontal unit point load applied at point X i ${{X}_i}$ are given by:
    u 11 C = 3 − 4 ν · ln r − 8 · 1 − ν 2 · ln r + 3 − 4 ν R 2 · r x 2 + 2 · c · h R 2 − 4 · c · x · r x 2 R 4 , $$\begin{eqnarray} {u}_{11}^{C} &=& \left[\left(3-4\nu \right)\cdot \ln r-8\cdot {\left(1-\nu \right)}^{2}\cdot \ln r+\frac{\left(3-4\nu \right)}{{R}^{2}}\right.\nonumber\\ &&\left.\cdot\; {{r}_{x}}^{2}+\frac{2\cdot c\cdot h}{{R}^{2}}-\frac{4\cdot c\cdot x\cdot {{r}_{x}}^{2}}{{R}^{4}}\right], \end{eqnarray}$$ (A.9)
    u 21 C = − A 3 − 4 ν · r x · r y R − 4 · c · x · R y · r x R 4 + 4 · 1 − ν · 1 − 2 ν · θ . $$\begin{eqnarray} {u}_{21}^{C}=-A\left[\left(3-4\nu \right)\cdot \frac{{r}_{x}\cdot {r}_{y}}{R}-\frac{4\cdot c\cdot x\cdot {R}_{y}\cdot {r}_{x}}{{R}^{4}}+4\cdot \left(1-\nu \right)\cdot \left(1-2\nu \right)\cdot \theta \right].\hskip-12pt\nonumber\\ \end{eqnarray}$$ (A.10)
    Thus, the second part of the Green function for horizontal direction is defined by:
    u n 1 x i , X i C = u 11 C u 21 C , $$\begin{equation}{{{\bm{u}}}_{{\bm{n}}1}}{{\left( {{{{\bm{x}}}_{\bm{i}}},{{{\bm{X}}}_{\bm{i}}}} \right)}^{\bm{C}}} = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{\bm{u}}_{11}^{\bm{C}}}\\[6pt] {{\bm{u}}_{21}^{\bm{C}}} \end{array} } \right]\ ,\end{equation}$$ (A.11)
    The displacements at position x i ${{x}_i}$ generated by a vertical unit point load applied at point X i ${{X}_i}$ are given by:
    u 12 C = − A 3 − 4 ν · r x · r y R + 4 · c · x · R y · r x R 4 − 4 · 1 − ν · 1 − 2 ν · θ , $$\begin{eqnarray} {u}_{12}^{C} = -A\left[\left(3-4\nu \right)\cdot \frac{{r}_{x}\cdot {r}_{y}}{R}+\frac{4\cdot c\cdot x\cdot {R}_{y}\cdot {r}_{x}}{{R}^{4}}-4\cdot \left(1-\nu \right)\cdot \left(1-2\nu \right)\cdot \theta \right],\hskip-12pt\nonumber\\ \end{eqnarray}$$ (A.12)
    u 21 C = A − 8 · 1 − ν 2 · ln r + 3 − 4 ν · ln r + 3 − 4 ν R 2 · R y 2 − 2 · c · h R 2 + 4 · c · x · R y 2 R 4 . $$\begin{eqnarray} {u}_{21}^{C} &=& A\left[-8\cdot {\left(1-\nu \right)}^{2}\cdot \ln r+\left(3-4\nu \right)\cdot \ln r+\frac{\left(3-4\nu \right)}{{R}^{2}}\right.\nonumber\\ &&\left.\cdot\; {{R}_{y}}^{2}-\frac{2\cdot c\cdot h}{{R}^{2}}+\frac{4\cdot c\cdot x\cdot {{R}_{y}}^{2}}{{R}^{4}}\right]. \end{eqnarray}$$ (A.13)
    Thus, the second part of the Green function for vertical direction is defined by:
    u n 2 x i , X i C = u 12 C u 22 C . $$\begin{equation}{{u}_{n2}}{{\left( {{{x}_i},{{X}_i}} \right)}^C} = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {u_{12}^C}\\[6pt] {u_{22}^C} \end{array} } \right]{\mathrm{\ \ }}.\end{equation}$$ (A.14)

    Data Availability Statement

    Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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