Convergence of iterative coupling for coupled flow and geomechanics

In this paper, we study solving iteratively the coupling of flow and mechanics. We demonstrate the stability and convergence of two widely used schemes: the undrained split method and the fixed stress split method. To our knowledge, this is the first time that such results have been rigorously obtained and published in the scientific literature. In addition, we propose a new stress split method, with faster convergence rate than known schemes. These results are specially important today due to the interest in hydraulic fracturing (Dean and Schmidt SPE J. 14:707–714, 2009; Ji et al. SPE J. 14:423–430, 2009; Samier and De Gennaro 2007; Settari and Maurits SPE J. 3:219–226, 1998), in oil and gas shale reservoirs.     

Numerical convergence study of iterative coupling for coupled flow and geomechanics

In this paper, we consider algorithms for modeling complex processes in porous media that include fluid and structure interactions. Numerous field applications would benefit from a better understanding and integration of porous flow and solid deformation. Important applications in environmental and petroleum engineering include carbon sequestration, surface subsidence, pore collapse, cavity generation, hydraulic fracturing, thermal fracturing, wellbore collapse, sand production, fault activation, and waste disposal, while similar issues arise in biosciences and chemical sciences as well. Here, we consider solving iteratively the coupling of flow and mechanics. We employ mixed finite element method for flow and a continuous Galerkin method for elasticity. For single-phase flow, we demonstrate the convergence and convergence rates for two widely used schemes, the undrained split and the fixed stress split. We discuss the extension of the fixed stress iterative coupling scheme to an equation of state compositional flow model coupled with elasticity and a single-phase poroelasticity model on general hexahedral grids. Computational results are presented.     

Transient response of a circular cavity in a poroelastic medium

This paper considers the transient response of a pressurized long cylindrical cavity in an infinite poroelastic medium. To obtain transient solutions, Biot's equations for poroelastodynamics are specialized for this problem. A set of exact general solutions for radial displacement, stresses, pore pressure and discharge are derived in the Laplace transform space by using analytical techniques. Solutions are presented for three different types of prescribed transient radial pressures acting on the surface of a permeable as well as an impermeable cavity surface. Time domain solutions are obtained by inverting Laplace domain solutions using a reliable numerical scheme. A detailed parametric study is presented to illustrate the influence of poroelastic material parameters and hydraulic boundary conditions on the response of the medium. Comparisons are also presented with the corresponding ideal elastic solutions to portray the poroelastic effects. It is noted that the maximum radial displacement and hoop stress at the cavity surface are substantially higher than the classical static solutions and differ considerably from the transient elastic solutions. Time histories and radial variations of displacement, hoop stress, pore pressure and fluid discharge corresponding to a cavity in two representative poroelastic materials are also presented.     

Vertical vibrations of a rigid disk embedded in a poroelastic medium

This paper considers the steady-state vertical vibrations of a rigid circular disk embedded at a finite depth below the free surface of a poroelastic medium. Biot's elastodynamic theory for porous media is used in the analysis. General solutions for axisymmetric poroelastic fields are obtained by using Hankel integral transforms. Analytical solutions for influence functions corresponding to four types of buried axisymmetric excitations are derived. The embedded disk problem is fomulated in terms of a set of coupled integral equations for unknown traction and pore pressure jumps across the disk. The kernel functions of the integral equations are the influence functions corresponding to buried vertical, radial and pore pressure ring loads. The system of integral equations is solved numerically by discretizing the disk into several concentric annular rings. Selected numerical solutions for displacements, vertical stress and pore pressure due to a buried fully flexible disk (uniform pressure) are also presented. The vertical compliances of a rigid disk are examined for different depths of embedment, poroelastic materials and hydraulic boundary conditions. Solutions for traction and pore pressure jumps are also examined. The present results are useful in the study of dynamic response of embedded foundations and anchors in poroelastic soils. Copyright © 1999 John Wiley & Sons, Ltd.     

An enhanced ensemble Kalman filter scheme incorporating model error in sequential coupling between flow and geomechanics

In this work, we construct a new methodology for enhancing the predictive accuracy of sequential methods for coupling flow and geomechanics while preserving low computational cost. The new computational approach is developed within the framework of the fixed-stress split algorithm procedure in conjunction with data assimilation based on the ensemble Kalman filter (EnKF). In this context, we identify the high-fidelity model with the two-way formulation where additional source term appears in the flow equation containing the time derivative of total mean stress. The iterative scheme is then interlaced with data assimilation steps, which also incorporate the modeling error inherent to the EnKF framework. Such a procedure gives rise to an “enhanced one-way formulation,” exhibiting substantial improvement in accuracy compared with the classical one-way method. The governing equations are discretized by mixed finite elements, and numerical simulation of a 2D slab problem between injection and production wells illustrate the tremendous achievement of the method proposed herein.     

Analysis of pile groups in a poroelastic medium subjected to horizontal vibration

This work investigates the dynamic response of pile groups embedded in a poroelastic medium subjected to horizontal loading. The dynamic response is analyzed using the Muki and Sternberg Method. The load transfer problem is formulated in terms of a second-kind Fredholm integral. The dynamic impedance of the pile groups is computed using the pile–soil–pile dynamic interaction factors. The shear force, bending moment and pore pressure is obtained using the superposition method. Numerical results indicate that the pile flexibility ratio and the pile distance have considerable influence on the dynamic response of the piles and the poroelastic medium.     

Numerical evaluation of the suitability of the equivalent porous medium model for characterizing the two-dimensional flow field in a fractured geologic medium

The equivalent porous medium (EPM) method is an efficient approximation method for groundwater yield analysis considering the equivalent permeability in a fractured geologic medium (FGM). The EPM method is widely used in many practical hydrogeological problems from local to regional scales. However, when calculating water head and velocity distributions, the suitability of the EPM model remains insufficiently evaluated. The suitability refers to the head error caused by the application of the EPM model. The smaller the error, the better the suitability. In this study, the influence of fracture geometric attributes on the suitability was quantitatively studied in numerical simulation experiments, and the EPM model simulation results were compared to those obtained with the discrete fracture network (DFN) model. The results indicated that the suitability decreased with increasing fracture spacing. When the fracture spacing was smaller than 0.6 m, the influence of an increase in the fracture spacing on the suitability was obvious. For the same fracture spacing, the suitability generally increased with increasing trace length. When the spatial variability of fracture aperture is not considered, the change of the fracture aperture did not impact the suitability. The fracture orientation slightly impacted the suitability, which can be ignored. An evaluation standard based on fracture parameters is proposed to estimate the suitability of the EPM model, which provides a scientific basis to ascertain whether this model can be applied to a given site to solve the head-related hydrogeological issues with FGMs.

     

One‐dimensional analysis of a poroelastic medium during freezing

A solution to the problem of freezing of a poroelastic material is derived and analysed in the case of one‐dimensional deformation. The solution is sought within the framework of thermo‐poroelasticity, with specific account of the behaviour of freezing materials. The governing equations of the problem can be combined into a pair of coupled partial differential equations for the temperature and the fluid pressure, with particular forms in the freezing and the unfrozen regions. In the freezing region, the equations are highly non‐linear, partly due to the dependence of thermal and hydraulic properties on water saturation, which varies with temperature. Consequently, the solution is obtained through numerical methods, with special attention to the propagation of the freezing front boundary. The response to one‐dimensional freezing is illustrated for the case of cement paste. Finally, the influence on the solution of varying selected parameters is analysed, such as the temperature boundary conditions, the parameters characterizing the geometry of the porous system, the ratio of fluid and thermal diffusivities, and the rate of cooling applied at the freezing end. Copyright © 2008 John Wiley & Sons, Ltd.     

Dynamic response of a non-circular lined tunnel with visco-elastic imperfect interface in the saturated poroelastic medium

A semi-analytical method is proposed to investigate the dynamic response of a non-circular tunnel with visco-elastic imperfect interface in poroelastic medium. Biot’s dynamic theory is used to simulate the saturated poroelastic medium, and the governing equations are solved by reducing them into three Helmholtz equations that the potential functions satisfy. The visco-elastic interface model with elastic and viscosity coefficients is adopted to analyze the interface effect around the non-circular lined tunnel. The analytic solutions of displacements and stresses are expanded in terms of wave functions. Some numerical examples are provided to analyze the effect of visco-elastic interface on the dynamic stress around the tunnel.     

Demonstration and application of NMM‐based fractured porous medium flow model

In natural rock masses, the shapes of three‐dimensional (3‐D) blocks cut by arbitrary fracture networks may be very complex. Owing to the geometric complexity and difficulty of mesh discretization of 3‐D blocks and fracture facets, explicit consideration of fracture networks in flow analysis of fractured porous medium (FPM) is very challenging. Using the numerical manifold method based on independent covers (NMMIC), an FPM flow model was proposed that can deal with very complex 3‐D fracture networks. In this paper, the convergence of NMMIC was first demonstrated. The theoretical basis of the arbitrary refinement of computational meshes was proven. Moreover, three peculiarities of NMMIC meshes, that is, arbitrary shape, arbitrary connection, and arbitrary refinement of independent covers, were concluded. Finally, some two‐dimensional (2‐D) tunnel flow examples were analyzed and the numerical results were compared with the analytical results. 3‐D examples with complex fracture distributions were also analyzed. In addition, the computational scale of the developed program was tested by increasing the number of computational elements. The results show that our model can accurately analyze the groundwater flow of rocks surrounding tunnels with complex fracture distributions.     

Application of the third medium method for frictionless contact problems in geomechanics

Traditional approaches in contact mechanics demand complicated search algorithms at the interface between the contacting bodies. Recently, a new contact method based on the concept of a third medium has been developed, which overcomes the drawbacks of conventional contact mechanics techniques. This new scheme is based on a space filling mesh, in which the contacting bodies can move and interact. The ability and accuracy of this method in predicting displacements, as well as the contact forces, is validated by solving selected numerical examples. The potential merits of this method for analysing geotechnical problems by the finite element method are addressed.     

Finite element analysis for a finite conductivity fracture in an infinite poroelastic medium

In the technology of oil recovery, oil production rate can be increased by generation of a vertical conductive fracture adjacent to the well-bore. In this paper the seepage flow and isothermal deformation in both the oil formation and the fracture are studied by modelling the formation as a two-dimensional infinite poroelastic medium and the conductive fracture as a one-dimensional poroelastic material, saturated by a one-phase compressible fluid. The plane strain condition is employed. Solutions for a growing conductive fracture and a stationary conductive fracture in the infinite medium are obtained by means of the finite element method based on a variational principle for the formation which can impose the governing equations of the fracture. Infinite elements are used outside the finite element domain. Numerical results indicate that the injection rate, the applied pressure and the crack mouth opening displacement at the well-bore oscillate during the propagation of the conductive fracture. The production rate of a well with the conductive fracture is compared with that of a well without the conductive fracture. Finally, a new definition of the conductivity coefficient for the conductive fracture is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Scattering of elastic wave by a partially sealed cylindrical shell embedded in poroelastic medium

Scattering of an elastic wave by a cylindrical shell embedded in poroelastic medium is investigated theoretically with the assumption that the shell material is also a porous elastic medium. The porous medium is modellized via Biot's theory and the scattering by cylindrical shell is expressed by the definition of scattering matrix. The normal mode expansion technique is employed for analysing the scattering field, and the asymptotic solutions of displacements, stresses and pore pressure are derived. Two limiting cases‐scattering by a poroelastic cylinder in Biot medium and a elastic cylindrical shell in elastic medium are obtained from the general solutions. The dispersion curves of displacement amplitude at the interface of shell and medium is compared with the case of elastic shell. The scattering amplitude associated with the fast, slow and transverse waves are identified by numerical simulation. Furthermore, the influence of the poroelastic property of shell material on scattering amplitude is analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

Dimensional reduction of a fractured medium for a polymer EOR model

Computational Geosciences - Dimensional reduction strategy is an effective approach to derive reliable conceptual models to describe flow in fractured porous media. The fracture aperture is several...     

Elastic waves along a cylindrical borehole in a poroelastic medium saturated by two immiscible fluids

The propagation of elastic waves along a cylindrical borehole filled with/without liquid and embedded in an infinite porous medium saturated by two immiscible fluids has been studied. The theory of porous media saturated by two immiscible fluids developed by Tuncay and Corapcioglu (1997) is employed. Frequency equations determining the phase velocity of axial symmetric waves are obtained. It is found that the surface waves along cylindrical borehole are dispersive. The dispersion equation of Rayleigh-type surface waves along the boundary of a poroelastic solid half-space saturated by two immiscible fluids is also obtained. Some special cases have been deduced and the dispersion curves are obtained numerically for a peculiar model. It is found that the density of fluids affects the Rayleigh mode.     

Thermomechanical response of a poroelastic half-space soil medium subjected to time harmonic loads

The thermomechanical responses of a porous elastic medium subjected to time harmonic loads (normal force and thermal source) are investigated analytically in the context of generalised thermoelastic theory with one relaxation time. The material of the foundation, obeying Biot’s dynamic poroelastic theory, is idealised as a uniform, fully saturated poroelastic half-space stratum. The coupled governing equations are established based on Biot’s dynamic poroelastic theory and on generalised thermoelastic theory. Assuming the disturbances to be harmonically time dependent, the general solutions of stress, displacement, temperature distribution and excess pore water pressure are deduced using the Fourier transform, and the transformed solutions are numerically inverted. The differences among the coupled thermo-hydro-mechanical dynamic model (THMD), the hydro-mechanical dynamic model (HMD) and the thermo-elastic dynamic model (TMD) are discussed. In addition, the effects of the thermal loading frequency on the displacement, stress, temperature distribution and excess pore water pressure components are analysed in the numerical results.     

衬砌隧道中移动轴向激励作用下两相多孔介质动力响应

为了研究衬砌隧道中地铁振动在饱和地层中的传播情况,采用移动轴向激励模拟技术,建立了隧道-衬砌-两相介质的动力分析模型;利用波函数展开法、傅里叶变换法等,推导了频域内衬砌隧道移动轴向激励作用下两相多孔介质动力响应的解析解,并给出了两相多孔介质临界速度的经验公式;通过离散快速傅里叶逆变换得到了时-空域内两相介质的动力响应。结果表明：无衬砌隧道中移动的轴向常激励作用下,两相介质临界速度只与介质的剪切模量和密度有关,且数值接近其剪切波速的1.1倍;对于衬砌隧道,介质的临界速度随着衬砌剪切模量的增大而增大,随着衬砌密度的增大而减小,衬砌对振动向介质中的传播有一定的削弱作用,衬砌剪切模量与介质剪切模量相差越大,削弱越明显;动力响应频率越接近激振频率,其幅值越大,所对应的临界速度越小。     

Gas sorption and transport in coals: A poroelastic medium approach

In this paper, single-component gas sorption and transient diffusion processes are described within coal matrix exhibiting bimodal pore structure. The coal matrix is treated as a poroelastic medium manifesting swelling and shrinkage effects due to the sorption of gas under effective overburden stress. Gas transport is considered Fickian with molecular (bulk) and surface diffusion processes simultaneously taking place in the macro- and micropores of coal, respectively. The numerical formulation is intended to be explicit in nature to investigate the influences of sorption phenomena on the macropore volumes and on the overall gas transport for the cases of gas uptake by and release from coal.Results of the study show the presence of hysteresis during a sorption–desorption cycle of the gas. It is also found that the overall gas transport takes place at a rate significantly less than that in the macropores only. Thus the existence of a retardation effect in the overall gas transport is concluded. This retardation effect is primarily due to the micropore resistances, in particular gas adsorption, and is independent of the changes in the macropore volumes. It is shown that macroporosity of the coal matrix may change during gas transport due to combined effects of pressure and sorption-induced swelling or shrinkage of the coal. It is estimated that the macroporosity variation is non-uniform in space and time, as it is expected in reality, and typically taking values less than ± 10 percent of the initial porosity.     

A solution around a backfilled cavity in a low‐permeability poroelastic medium with application in in situ heating tests

Thermo‐hydro‐mechanical responses around a cylindrical cavity drilled or excavated in a low‐permeability formation are studied when the cavity is subjected to a time‐dependent thermal loading. The cavity is considered backfilled after it is supported by casing or lining. Solutions of temperature, pore water pressure, stress, and displacement responses are analytically formulated based on Biot's consolidation theory with the assumption that the backfilling material, supporting material, and surrounding low‐permeability formation are poroelastic media. The solution is expressed in Laplace space, and numerical inversion techniques are used to find field variables in the real‐time domain. After the solution is verified with the numerical results, it is applied in a large‐scale in situ heating test – PRACLAY heating test – for a predictive reference calculation and an extensive parametric study. Another medium‐scale in situ heating test – ATLAS III heating test – is also analyzed using the solution, which provides reasonable agreement with measurements. The new analytical solution proves to be a convenient tool for a good understanding of the resulting coupled thermo‐hydro‐mechanical behavior and is therefore valuable for the interpretation of measured data in engineering practices and for a rational design of potential radioactive waste repositories. Copyright © 2016 John Wiley & Sons, Ltd.