Theoretical Analysis and Semi-Analytical Formulation for Efficient Thermal-Hydraulic-Mechanical Reservoir Simulation

The research of multiphysical thermal-hydraulic-mechanical (THM) simulation has achieved significant progress in the past decade. Currently, two general approaches for poromechanical simulation co-exist in the reservoir simulation community, namely the stress approach with stress as the primary variable for the mechanical governing equations and the displacement approach with displacement as the primary variable. In this work, we aim to provide a theoretical foundation and a practical semi-analytical solution for the stress approach based on the Navier-Beltrami-Michell Equations. Moreover, we will clarify the relationship (and equivalence) between the two approaches. We have firstly proven the existence and uniqueness of the stress solution of Navier-Beltrami-Michell equation with given pressure and temperature field. Moreover, we have demonstrated the equivalence of the stress formulation to the displacement formulation. Based on Fourier's expansion, we have developed a general semi-analytical solution for thermal-hydraulic-mechanical process. The semi-analytical solution takes the pressure solution from the hydraulic simulation module (or a commercial reservoir simulator) and directly predicts the stress tensor of the multiphysical system. As such, the solution can be programmed fully coupled with the hydraulic simulation module to predict the stress field with varying pressure and temperature of homogeneous poroelastic rocks under given stress boundary conditions. From the work above, we have laid a theoretical foundation for the stress approach. The derived semi-analytical solution of the stress field shows excellent accuracy. The solution has been used to predict the transient stress field of a dual-porosity system during primary depletion. This paper is arguably the first trial to clarify the relationship between the stress approach and the displacement approach. Moreover, the derived semi-analytical solution provides a convenient yet precise way to obtain the stress field without time-consuming numerical simulation.

A SPH-GFDM Coupled Method for Elasticity Analysis

SPH (smoothed particle hydrodynamics) is one of the oldest meshless methods used to simulate mechanics of continuum media. Despite its great advantage over the traditional grid-based method, implementing boundary conditions in SPH is not easy and the accuracy near the boundary is low. When SPH is applied to problems for elasticity, the displacement or stress boundary conditions should be suitably handled in order to achieve fast convergence and acceptable numerical accuracy. The GFDM (generalized finite difference method) can derive explicit formulae for required partial derivatives of field variables. Hence, a SPH–GFDM coupled method is developed to overcome the disadvantage in SPH. This coupled method is applied to 2-D elastic analysis in both symmetric and asymmetric computational domains. The accuracy of this method is demonstrated by the excellent agreement with the results obtained from FEM (finite element method) regardless of the symmetry of the computational domain. When the computational domain is multiply connected, this method needs to be further improved.

Molecular Dynamics Study of Bulk Properties of Polycrystalline NiTi

Molecular dynamics (MD) simulations were carried out to study the bulk polycrystalline properties of NiTi. Thermally driven phase transitions of NiTi between martensite and austenitewere simulated using single crystalline simulation domains. With external stress boundary conditions, MD simulation successfully predicted experimentally observed phase transformation temperatures of bulk polycrystalline. Elastic characteristics of NiTi martensite were simulated using polycrystalline simulation domains that consist of realistic disorientations and grain boundary structures. The existence of grain disorientation and grain boundary lowered the potential energy of the simulation domain, which led to more realistic elastic modulus prediction. Analysis of simulation domains that predicted realistic bulk polycrystalline properties showed that the major difference between single crystalline and polycrystalline structures is atomic stress distribution.

Extension of Boley’s continuum mechanics-based successive approximation method to two-layer rectangular beams

An extension of Boley’s continuum mechanics-based successive approximation method is presented for rectangular beams composed of two isotropic linear elastic layers. The solution is cast into the form of tables, in complete analogy to the tables originally presented by Boley and Tolins for single-layer strips. The first column in these tables corresponds to the classical Bernoulli–Euler theory of beams. The further columns represent comparatively fast converging correction terms of an increasing refinement. Our two-layer formulation automatically satisfies the stress continuity conditions at the interface of the two layers. Enforcing displacement continuity at the interface between the layers, we derive results that do satisfy the equilibrium field equations, the stress continuity conditions at the interface and the stress boundary conditions at the upper and lower edges. When converged, the field constitutive relations and the displacement continuity at the interface between the two layers are also satisfied. We present a compact formulation, which allows writing down the results for more than the three successive steps considered by Boley and Tolins. The elasticity solutions presented subsequently can be used as novel analytic benchmarks for comparison with refined structural mechanics beam theories. Interior solutions for beams with a finite axial extent can be obtained by assigning approximate boundary conditions at the lateral ends. Comparisons to finite element computations for a clamped–clamped beam give strong evidence for the correctness of our analytic results.

In Situ Stress Inversion and Distribution Characteristics of Tunnel Based on Numerical Simulation and Neural Network Technology

According to the geological conditions of the study area, the measured data of in situ stress was analyzed and the influence degree of buried depth was obtained. A numerical simulation research model with full consideration of fault structure and surface characteristics is established, and boundary condition functions with variables are used. The neural network optimized by genetic algorithm is used to establish the nonlinear relationship between the measured value and the simulated value of the variable boundary condition, and the optimal boundary condition function is obtained. Finally, the in situ stress in the study area was predicted. Through the analysis of the in situ stress field in the research target area, the stress boundary conditions are provided for the follow-up study, and the practical basis for the division of the dangerous area of the surrounding rock of the deep and long tunnel is provided.

The numerical calculation of statically admissible slip-line field for plane strain compression of a three-layer symmetric strip between rigid plates

This paper present a method to build up statically admissible slip-line field (the field of characteristics) and, as a result, the field of statically admissible stresses of the compression of a three-layer symmetric strip consisting of two different rigid perfectly plastic materials between rough, parallel, rigid plates (for the case: the shear yield stress of the inner layer is greater than that of the outer layer). Under the conditions of sticking regime at bi-material interfaces and sliding occurs at rigid surfaces with maximum friction, the appropriate singularities on the boundary between the two materials have been assumed, then a standard numerical slip-line technique is supplemented with iterative procedure to calculate characteristic and stress fields that satisfy simultaneously the stress boundary conditions as well as the regime of sticking on the bi-material interfaces

An analytical method for determining the non-enclosed elastoplastic interface of a circular hole

Based on the Mohr–Coulomb yield criterion, an analytical method is presented to determine the plastic zone in an infinite plate weakened by a circular hole and subjected to non-hydrostatic stresses at infinity. It is worth noting that this paper considers the more complicated case that the plastic zone cannot completely surround the hole, namely the elastoplastic interface is non-enclosed. Initially, the non-circular elastic zone in the physical plane is mapped onto the outer region of a unit circle in the image plane by the conformal transformation in the complex variable method. Thereby, determining the elastoplastic interface is equivalent to solving the mapping function coefficients. The nonlinear equations for solving the coefficients are established by considering both the stress continuity conditions along the elastoplastic interface and the stress boundary conditions along the elastic part of the hole. Naturally, the problem can be further transformed into an optimization problem, which is ultimately achieved by the differential-evolution algorithm; what is more, an analytical solution with high accuracy is obtained. Based on the programmed computing, the influences of various parameters on the shape and size of the plastic zone are given.

A Mixed Stress/Displacement Approach Model of Homogeneous Shells for Elastodynamic Problems

This paper presents the development of a model of homogeneous, moderately thick shells for elastodynamic problems. The model is obtained by adapting and modifying SAM-H model (stress approach model of homogeneous shells) developed by Domínguez Alvarado and Díaz in (2018) for static problems. In the dynamic version of SAM-H presented herein, displacements and stresses are approximated by polynomials of the out-of-plane coordinate. The stress approximation coincides with the static version of SAM-H when dynamic effects are neglected. The generalized forces and displacements appearing in the approximations are the same as those involved in a classical, moderately thick shell model (CS model) but the stress approximation adopted herein is more complex: the 3D motion equations and the stress boundary conditions at the faces of the shell are verified. The generalized motion and constitutive equations of dynamic SAM-H model are obtained by applying a variant of Euler–Lagrange equation which includes pertinently Hellinger–Reissner functional. In the constitutive equations, Poisson’s effect of out-of-plane normal stresses on in-plane strains is not ignored; this is one important feature of SAM-H. To test the accuracy of dynamic SAM-H model, the following structures were considered: a hollow sphere and a catenoid. In each case, eigenfrequencies are first calculated and then a frequency analysis is performed applying a harmonic load. The results are compared to those of a CS model, MITC6 (mixed interpolation of tensorial components with 6 nodes per element) shell element calculations, and solid finite element computations. In the two problems, CS, MITC6, and dynamic SAM-H models yield accurate eigenfrequencies and eigenmodes. Nevertheless, the frequency analysis performed in each case showed that dynamic SAM-H provides much more accurate amplitudes of stresses and displacements than the CS model and the MITC6 shell finite element technique.