Numerical investigation of the effective Skempton coefficient in fluid-saturated fractured rock

Characterization of hydro-mechanical processes in reservoir rocks is an essential issue for many geo investigations such as characterization of subsurface fluid flow or geothermal exploitation. For geothermal applications, the role of fractures as storage and transport components of a hydraulic system are highly important. In the present contribution we focus on investigating the effective Skempton coefficient of a damaged porous rock analyzing a modified Cryer problem, which provides a simple model of a porous rock containing a storage and transport pat. The effective Skempton coefficient is defined as the ratio of the increase in mean pore pressure induced by change in confining pressure for undrained boundary conditions. Using approaches from computational homogenization, we evaluate the confining pressure as the negative volume average of the total mean stress. Similarly, we compute the effective fluid pressure in terms of the volume-averaged fluid pressure in the rocks. We compare the numerical results to those from typical experiments and highlight the problems with the latter. The proposed concept for determination of an effective Skempton coefficient based on numerically evaluated volume averages helps to generate a better understanding of the process-inherent constituents. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On seismic imaging of fractured geological media

Doklady Mathematics - Seismic waves propagating in a fractured geological medium are numerically simulated. Their dynamic behavior is described using a linear elastic model with an explicit...     

Numerical modeling of fluid flow in anisotropic fractured porous media

A model of double porosity in the case of an anisotropic fractured porous medium is considered (Dmitriev, Maksimov; 2007). A function of fluid exchange between the fractures and porous blocks depending on flow direction is given. The flow function is based on the difference between the pressure gradients. This feature enables one to take into account anisotropic properties of filtration in a more general form. The results of numerical solving a model two-dimensional problem are presented. The computational algorithm is based on a finite-element space approximation and explicit-implicit time approximations.     

Analysis of wave propagation in the fluid-saturated porous media

In the present work, a so-called hybrid two-phase model composed of a materially incompressible solid and a compressible pore fluid is studied. The mechanical behavior is described by the thermodynamically consistent Theory of Porous Media (TPM). Numerical experiments were performed with the space-time coupled discontinuous galerkin (DGT)-method. The existence of two compressional waves (P-waves) and one shear wave (S-wave) was confirmed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the representations of solutions in the theory of fluid-saturated porous media

In the present paper the linear theory of the liquid-saturatedporous medium consisting of a microscopically incompressiblesolid skeleton containing microscopically incompressible liquidis considered. First, the representation of Galerkin type solutionof equations of motion is obtained. Then, the representationtheorem of Galerkin type of system of the equations of steadyoscillations is presented. Finally, the general solution ofthe system of homogeneous equations of the steady oscillationsin terms of one harmonic function and four metaharmonic functionsis established.     

The wavelet multiscale method for inversion of porosity in the fluid-saturated porous media

In this paper, the wavelet multiscale method is applied to the inversion of porosity in the fluid-saturated porous media. The inverse problem is decomposed to multiple scales with wavelet transform and hence the original inverse problem is re-formulated to be a set of sub-inverse problem corresponding to different scales and is solved successively according to the size of scale from the smallest to the largest. On each scale, regularization Gauss–Newton method is carried out, which is stable and fast, until the optimum solution of original inverse problem is found. The results of numerical simulations demonstrate that the method is a widely convergent optimization method and exhibits the advantages of conventional regularization Gauss–Newton method methods on computational efficiency and precision.     

Mathematical simulation of acoustic wave scattering in fractured media

The simulation of acoustic waves in fractured media is considered. A self-consistent field model is proposed that describes the formation of a scattered field and the attenuation of the incident field. For the total field, a wave equation with a complex velocity is derived and the corresponding dispersion equation is studied. A frequency-dependent field damping law and an energy variation law are established. An initial and a boundary value problem for waves in a fractured medium is addressed. A finite-difference scheme for the initial value problem is constructed, and a condition for its stability is established. Numerical results are presented.     

Fuzzy reasoning in the investigation of seismic behavior

In this paper, we propose a fuzzy logic–based mathematical model of a sequence of earthquakes using fuzzy reasoning tools. We formed a set of fuzzy implications in order to study them, and we computed their deviation, so that we could compare them and conclude about the most accurate one. The compositional rule of inference was considered, which is based on the generalized modus ponens scheme. The new fuzzy methodology was used for each implication. The data required for the implications were obtained from the aftershocks of an earthquake with significant effects in a specific area. The magnitude of the aftershocks and their time difference from the main incident provided the values for the new fuzzy algorithm application. Two samples were selected relative with the seismic activity, which occurred the following days. The one sample consisted of 30 values, and the other sample from all the values found in the data archives of the National Observatory of Athens. Results were shown for both samples. So a mathematical technique, which reproduces the incidents using basic information and based on only two parameters, is developed for the simulation of a seismic sequence, which follows a strong earthquake.     

Numerical investigation of the Liebau phenomenon

The Liebau phenomenon is the occurrence of valveless pumping through the application of a periodic force at a place which lies asymmetric with respect to system configuration. This paper is concerned with two different physical configurations and respective models. Comparison and derivation among the models is discussed. Accurate numerical schemes which solve these models are presented. By means of numerical simulations it is investigated under which conditions valveless pumping takes place.     

Numerical investigation of the Liebau phenomenon

The Liebau phenomenon is the occurrence of valveless pumping through the application of a periodic force at a place which lies asymmetric with respect to system configuration. This paper is concerned with two different physical configurations and respective models. Comparison and derivation among the models is discussed. Accurate numerical schemes which solve these models are presented. By means of numerical simulations it is investigated under which conditions valveless pumping takes place.     

Numerical modeling of the pressure coefficient of the circular cylinder

The article examines the possibilities of numerical solution of chimney load from the effect of wind. The shear-stress transport (SST) k-ω turbulence model in ANSYS Fluent software is used to evaluate the task of the flow around the circumference of the rough cylinder. Calculations are performed on two different meshes that lead to the solution using wall function and near wall modeling. These two solution approaches in terms of defining wall roughness are presented in the paper by evaluating of the time dependence of the mean pressure coefficient distribution at the circumference, drag coefficient, and lift coefficient. The accuracy of the calculations is verified with parameters determined according to valid standards.     

On attenuation in layered biot media and in effective models of them

Effective models of periodic layered porous Biot media possessing viscosity and relaxation are established and investigated. These models correspond to generalized Biot media with equations containing, as a rule, exponential kernels of relaxation and viscosity. Such kernels occur even if relaxation is absent in the initial medium. Inequalities that must be satisfield by the parameters of the kernels are established with the help of energy studies. Special cases in which the effective models possess no viscosity or no relaxation are stated. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 244–262. Translated by L. A. Molotkov     

Numerical identification of the leading coefficient of a parabolic equation

For a multidimensional parabolic equation, we study the problem of finding the leading coefficient, which is assumed to depend only on time, on the basis of additional information about the solution at an interior point of the computational domain. For the approximate solution of the nonlinear inverse problem, we construct linearized approximations in time with the use of ordinary finite-element approximations with respect to space. The numerical algorithm is based on a special decomposition of the approximate solution for which the transition to the next time level is carried out by solving two standard elliptic problems. The capabilities of the suggested numerical algorithm are illustrated by the results of numerical solution of a model inverse two-dimensional problem.     

Numerical simulation of seismic activity by the grid-characteristic method

Seismic activity in homogeneous and layered enclosing rock masses is studied. A numerical mechanical-mathematical model of a hypocenter is proposed that describes the whole range of elastic perturbations propagating from the hypocenter. Synthetic beachball plots computed for various fault plane orientations are compared with the analytical solution in the case of homogeneous rock. A detailed analysis of wave patterns and synthetic seismograms is performed to compare seismic activities in homogeneous and layered enclosing rock masses. The influence exerted by individual components of a seismic perturbation on the stability of quarry walls is analyzed. The grid-characteristic method is used on three-dimensional parallelepipedal and curvilinear structured grids with boundary conditions set on the boundaries of the integration domain and with well-defined contact conditions specified in explicit form.     

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

Seismic data is modeled in the high‐frequency approximation, using the techniques of microlocal analysis. We consider general, anisotropic elastic media. Our methods are designed to allow for the formation of caustics. The data is modeled in two ways. First, we give a microlocal treatment of the Kirchhoff approximation, where the medium is assumed to be piecewise smooth, and reflection and transmission occur at interfaces. Second, we give a refined view on the Born approximation based upon a linearization of the scattering process in the medium parameters around a smooth background medium. The joint formulation of Born and Kirchhoff scattering allows us to take into account general scatterers as well as the nonlinear dependence of reflection coefficients on the medium parameters. The latter allows the treatment of scattering up to grazing angles. The outcome of the analysis is a characterization of the singular part of seismic data. We obtain a set of pseudodifferential operators that annihilate the data. In the process we construct a Fourier integral operator and a reflectivity function such that the data can be represented by this operator acting on the reflectivity function. In our construction this Fourier integral operator becomes invertible. We give the conditions for invertibility for general acquisition geometry. The result is also of interest for inverse scattering in acoustic media. © 2002 John Wiley & Sons, Inc.     

Numerical investigation of the generalized lubrication equation

Explicit, implicit-explicit and Crank-Nicolson implicit-explicit numerical schemes for solving the generalized lubrication equation are derived. We prove that the implicit-explicit and Crank-Nicolson implicit-explicit numerical schemes are unconditionally stable. Numerical solutions obtained from both schemes are compared. Initial curves with both zero and finite contact angles are considered.     

Numerical simulation of the two-phase fluid filtration in heterogeneous media

Under study is numerical solution of the problems of two-phase filtration. The formulation of the problem is given in terms of velocity, pressure and saturation. To approximate the velocity and pressure, the mixed finite elementmethod is used. The flux schemes are applied for discretization of the convection term in the saturation equation. We present the results of numerical solution of a model problem for heterogeneous media.