scholarly journals A Mathematical Model of Gas and Water Flow in a Swelling Geomaterial—Part 1. Verification with Analytical Solution

Minerals ◽  
2019 ◽  
Vol 10 (1) ◽  
pp. 30 ◽  
Author(s):  
Elias Ernest Dagher ◽  
Julio Ángel Infante Sedano ◽  
Thanh Son Nguyen

Gas generation and migration are important processes that must be considered in a safety case for a deep geological repository (DGR) for the long-term containment of radioactive waste. Expansive soils, such as bentonite-based materials, are widely considered as sealing materials. Understanding their long-term performance as barriers to mitigate gas migration is vital in the design and long-term safety assessment of a DGR. Development and the application of numerical models are key to understanding the processes involved in gas migration. This study builds upon the authors’ previous work for developing a hydro-mechanical mathematical model for migration of gas through a low-permeable geomaterial based on the theoretical framework of poromechanics through the contribution of model verification. The study first derives analytical solutions for a 1D steady-state gas flow and 1D transient gas flow problem. Using the finite element method, the model is used to simulate 1D flow through a confined cylindrical sample of near-saturated low-permeable soil under a constant volume boundary stress condition. Verification of the numerical model is performed by comparing the pore-gas pressure evolution and stress evolution to that of the results of the analytical solution. The results of the numerical model closely matched those of the analytical solutions. Future studies will attempt to improve upon the model complexity and investigate processes and material characteristics that can enhance gas migration in a nearly saturated swelling geomaterial.

1988 ◽  
Vol 1 (21) ◽  
pp. 141
Author(s):  
Todd L. Walton ◽  
Philip L.F. Liu ◽  
Edward B. Hands

This paper examines the effects of random and deterministic cycling of wave direction on the updrift beach planform adjacent to a jetty. Results provided using a simplified numerical model cast in dimensionless form indicate the importance of the time series of wave direction in determining design jetty length for a given net sediment transport. Continuous cycling of • wave direction leads to the expected analytical solution. Simplications in the numerical model used restrict the applications to small wave angles, no diffraction, no reflection of waves off structure, no refraction, and no sand bypassing at jetty. The concept can be extended to more sophisticated numerical models.


Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1174 ◽  
Author(s):  
Yutaka Okabe ◽  
Akira Shudo

This is a tutorial for the mathematical model of the spread of epidemic diseases. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. Subsequently, we present the numerical and exact analytical solutions of the SIR model. The analytical solution is emphasized. Additionally, we treat the generalization of the SIR model including births and natural deaths.


2019 ◽  
Vol 30 (4) ◽  
pp. 1677-1688 ◽  
Author(s):  
Arun Appadurai ◽  
Vasudevan Raghavan

Purpose Dynamic separator is an equipment having a rotor and static vanes and is used to separate solids from gas-solids flow based on size. Particle separation in a dynamic separator happens due to complex interchanges between multiple forces exerted in the separation zone. Currently, there is only limited knowledge concerning the working principles of separation. This paper aims to systematically study a dynamic separator using numerical models to get insights into particle separation. Design/methodology/approach The Lagrangian–Eulerian formulation is used to simulate gas-solid flow. Multiple frames of reference using stage interpolation are used to account for rotation. Periodic symmetry in the equipment is exploited to create a simplified numerical model. The predictions from the numerical model are compared against available experimental data. Findings The numerical results indicate that only when particle collision is included, the separation efficiency trend from the experiment is matched by numerical predictions. Further, it is shown that at the same range of rotor speeds where numerical results predict increased separation efficiency, the solid pressure due to particle collision also reaches its maximum value. The gas flow and particle behavior in the separator are explained in detail. Originality/value The importance of particle collision in separation is interesting because traditionally, particle separation is assumed to be influenced by three forces, namely, centrifugal force, drag force and gravity. The numerical results, however, point to the contribution by particle collision, in addition to the above three forces.


2005 ◽  
Vol 13 (8) ◽  
pp. 741-752 ◽  
Author(s):  
Samit Roy ◽  
Yong Wang

The objective of this work was to find an analytical solution to the stresses in the cohesive damage zone and the damage zone length at the interface between a fibre reinforced polymer (FRP) plate and concrete substrate. Analytical solutions have been derived to predict the stress in the cohesive layer when considering the deformation in the stiff substrate. A two-dimensional cohesive layer constitutive model with a prescribed traction-separation (stress-strain) law was constructed using a modified Williams' approach, and analytical solutions derived for the elastic zone as well as the damage zone. Detailed benchmark comparisons of analytical results with finite element predictions for a double cantilever beam specimen were performed for model verification, and issues related to cohesive layer thickness were investigated. It was observed that the assumption of a rigid substrate in analytical modelling can lead to inaccurate analytical prediction of the cohesive damage zone length.


Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. MR129-MR145 ◽  
Author(s):  
Yury Alkhimenkov ◽  
Eva Caspari ◽  
Boris Gurevich ◽  
Nicolás D. Barbosa ◽  
Stanislav Glubokovskikh ◽  
...  

Seismic waves may exhibit significant dispersion and attenuation in reservoir rocks due to pore-scale fluid flow. Fluid flow at the microscopic scale is referred to as squirt flow and occurs in very compliant pores, such as grain contacts or microcracks, that are connected to other stiffer pores. We have performed 3D numerical simulations of squirt flow using a finite-element approach. Our 3D numerical models consist of a pore space embedded into a solid grain material. The pore space is represented by a flat cylinder (a compliant crack) whose edge is connected with a torus (a stiff pore). Grains are described as a linear isotropic elastic material, whereas the fluid phase is described by the quasistatic linearized compressible Navier-Stokes momentum equation. We obtain the frequency-dependent effective stiffness of a porous medium and calculate dispersion and attenuation due to fluid flow from a compliant crack to a stiff pore. We compare our numerical results against a published analytical solution for squirt flow and analyze the effects of its assumptions. Previous interpretation of the squirt flow phenomenon based mainly on analytical solutions is verified, and some new physical effects are identified. The numerical and analytical solutions agree only for the simplest model in which the edge of the crack is subjected to zero fluid pressure boundary condition while the stiff pore is absent. For the more realistic model that includes the stiff pore, significant discrepancies are observed. We identify two important aspects that need improvement in the analytical solution: the calculation of the frame stiffness moduli and the frequency dependence of attenuation and dispersion at intermediate frequencies.


Author(s):  
Armando Alexandre ◽  
Yohan Percher ◽  
Thomas Choisnet ◽  
Ricard Buils Urbano ◽  
Robert Harries

Floating wind solutions have developed significantly in the recent years, moving from single demonstrators to having several floating wind pilot wind farms currently under development and even in operation. This is an important step for the industry allowing the market to gain confidence in these solutions for offshore wind. Ideol is a leading floating platform designer and they have been working on a demonstration project for their innovative platform in France. The Floatgen demonstration project consists of a 2MW wind turbine mounted on the Damping Pool platform. During the design phase of the project, the coupled analysis of the full system — turbine, tower, floating platform and moorings needs to be carried out to verify the loading on the turbine and platform, adapt the turbine controller for the floating application and re-design the tower and transition piece. For this project, DNV GL performed the aforementioned analysis in Bladed whilst Ideol performed parallel analysis in OrcaFlex, focusing on the platform and mooring design. It is crucial that both numerical models used in the different software tools and parallel analysis workflows are equivalent and lead to the same overall system behavior. This paper describes the numerical model used for coupled analysis in Bladed and its verification against Ideol’s OrcaFlex model, with emphasis on the aspects related to the platform modelling. For the hydrodynamic loading of the platform, boundary element method was considered together with global and local viscous drag terms. To compare and verify the coupled model results in Bladed to Ideol’s own numerical results, a set of static and dynamic tests were run and the resultant kinematics were compared. Ideol’s model was previously validated against tank test experiments giving confidence in its behavior. The viscous drag coefficients in the Bladed model were adjusted to ensure a good agreement between the kinematics of Ideol’s model of the system and the Bladed model. This paper summarizes the results of this verification exercise, along with some recommendations on areas of further research in the floating wind modelling domain.


2018 ◽  
Vol 482 (1) ◽  
pp. 115-148 ◽  
Author(s):  
E. E. Dagher ◽  
T. S. Nguyen ◽  
J. A. Infante Sedano

AbstractIn a deep geological repository (DGR) for the long-term containment of radioactive waste, gases could be generated through a number of processes. If gas production exceeds the containment capacity of the engineered barriers or host rock, these gases could migrate through these barriers and potentially expose people and the environment to radioactivity. Expansive soils, such as bentonite-based materials, are currently the preferred choice of seal materials. Understanding the long-term performance of these seals as barriers against gas migration is an important component in the design and long-term safety assessment of a DGR. This study proposes a hydro-mechanical linear poro-elastic visco-capillary mathematical model for advective-diffusive controlled two-phase flow through a low-permeability expansive soil. It is based on the theoretical framework of poromechanics, incorporates Darcy's Law for both the porewater and poregas, and a modified Bishop's effective stress principle. Using the finite element method (FEM), the model was used to numerically simulate 1D flow through a low-permeability expansive soil. The results were verified against experimental results found in the current literature. Parametric studies were performed to determine the influence on the flow behaviour. Based on the results, the mathematical model looks promising and will be improved to model flow through preferential pathways.


2021 ◽  
Vol 1 ◽  
pp. 165-167
Author(s):  
Tuanny Cajuhi ◽  
Jobst Maßmann ◽  
Gesa Ziefle

Abstract. Salt, crystalline and clay formations are under discussion as potential host rocks for storage of heat-generating radioactive waste. Each of these rocks has a different structure and composition, and consequently a different material behavior. The latter needs to be studied and evaluated with respect to the main aim: to find a place to store the waste in a safe and sustainable manner. Several requirements in the context of the safety of a repository need to be fulfilled, concerning the long-term as well as the operational phase. One key point in this matter is the integrity, which refers to retention of the isolating rock zone's containment capabilities. With the focus on some experimental and numerical investigations on the excavation influenced near-field behavior of Opalinus clay (OPA), this contribution aims to illustrate an example for the role of numerical modeling in safety assessment. Once, e.g. anthropogenic action such as excavation starts, the natural state of equilibrium in the formation is disturbed. Trying to restore it, the rock deforms (convergence) and/or releases energy in other ways such as cracking. This could lead to loss of integrity since crack nucleation and propagation can affect the mechanical stability and create paths to transport contaminants. During operation in the excavated rock, environmental changes, e.g. temperature and humidity, further affect its behavior. The understanding of these dynamic phenomena ideally needs to occur at the in situ scale; however, performing an experiment in the spatial and time scales of interest is not always possible. For this reason, the in situ problem needs to be formulated, abstracted and mathematically modeled. The interpretation of the results must take place with simplifying assumptions and complementary laboratory scale experiments can be used to improve understanding of the system. The real problem is approached stepwise, each step associated to the size of the model and its complexity. The gradually obtained knowledge is necessary to achieve a better understanding of the process and to evaluate the capacities and limitations of the models. This contribution aims at showing the basic practical steps for numerical modeling with particular focus on the preparation and interpretation of the models and results, e.g. model calibration, verification and validation. As an example, the OPA at the Mont Terri site is chosen. The material parameters are obtained either experimentally or from the literature. We choose and perform laboratory scale simulations that are related to nearly the same mechanism as in the in situ scale. To have a first impression on the latter, a simplified, large-scale numerical model is prepared. The mechanism in study is drying and wetting, which is associated with shrinkage and swelling. We analyze the pore pressure and stress development in both scales. Thus, hydraulic mechanically coupled approaches are essential. The concept of effective stress is used, which combines the contributions of the solid and fluid phases (gas and liquid). In the current modeling approach, the gas pressure remains constant (atmospheric pressure) and during drying, the liquid pressure induces capillary pressure development and decrease of saturation. The laboratory scale simulation is important to evaluate the model of choice and to assess potential numerical problems. Furthermore, it can be used to perform a sensitivity study of material and numerical parameters. This step is necessary during the development or extension of numerical models as well as to evaluate their applicability on new research questions. The simplified in situ scale numerical model is then extended. In this phase the numerical model is evaluated once again, especially with respect to its complexity. Furthermore, specific questions related to this scale are posed: overall behavior of the rock, influence of the excavation, seasonal and long-term effects. In this contribution we deal with the long-term cyclic deformation (CD-A) experiment. The CD-A experiment has been taking place in the Mont Terri Rock Laboratory since October 2019. It consists of twin niches, a closed and an open niche, subjected to either high air humidity or seasonal humidity changes leading to saturation/desaturation during summer/winter in the OPA, respectively. Several parameters are periodically or continuously measured, including relative air humidity, convergence and crack development. We attempt to transfer the knowledge and numerical models developed in the small scale to the large scale and to evaluate the possibilities and limitations of the chosen approaches by comparing the numerical and experimental results.


2007 ◽  
Vol 581 ◽  
pp. 79-95 ◽  
Author(s):  
ADRIAN FARCAS ◽  
ANDREW W. WOODS

This paper presents some simple analytical and numerical models which describe the dynamics of gas flowing from a multilayered low-permeability porous rock into a fracture. The models account for the vertical flow between relatively high- and low-permeability layers. The motion of gas in a permeable rock is governed by a nonlinear diffusion equation for the gas pressure. We analyse the gas flow described by this equation in both bounded and unbounded domains. In both cases simple scalings laws are developed to determine the fluxes and the dimensions of the regions within the rock which may be depleted over a given time scale. These are compared with the results of a full numerical model.


2004 ◽  
Vol 41 (5) ◽  
pp. 921-942 ◽  
Author(s):  
Dan Yang ◽  
Ernest Naesgaard ◽  
Peter M Byrne ◽  
Korhan Adalier ◽  
Tarek Abdoun

Dynamic soil–structure interaction analyses were carried out for the seismic retrofit design of the immersed George Massey Tunnel, both to predict and study soil liquefaction and related tunnel movements and to design ground improvement. The proposed ground improvement included ground densification using vibroreplacement stone columns along both sides of the tunnel and seismic gravel drains adjacent to the outer edge of the densified zones. The den sification and drainage were proposed to locally mitigate soil liquefaction and reduce displacements of the tunnel to tolerable levels. Centrifuge model tests with base shaking to simulate earthquake effects were conducted to verify and calibrate the numerical models. This included simulating the effects of ground densification and drainage on reme diating tunnel movements. This paper presents the principal results from the dynamic analyses, the centrifuge model design and testing procedure, the class A predictions of the centrifuge tests, and discussions of the centrifuge test results and numerical model calibrations.Key words: immersed tunnel, seismic retrofit, soil liquefaction, design verification, centrifuge testing, numerical calibration.


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