Galina V. Reshetova

Local time-space mesh refinement for simulation of elastic wave propagation in multi-scale media

This paper presents an original approach to local time-space grid refinement for the numerical simulation of wave propagation in models with localized clusters of micro-heterogeneities. The main features of the algorithm are-the application of temporal and spatial refinement on two different surfaces;-the use of the embedded-stencil technique for the refinement of grid step with respect to time;-the use of the Fast Fourier Transform (FFT)-based interpolation to couple variables for spatial mesh refinement. The latter makes it possible to perform filtration of high spatial frequencies, which provides stability in the proposed finite-difference schemes.In the present work, the technique is implemented for the finite-difference simulation of seismic wave propagation and the interaction of such waves with fluid-filled fractures and cavities of carbonate reservoirs. However, this approach is easy to adapt and/or combine with other numerical techniques, such as finite elements, discontinuous Galerkin method, or finite volumes used for approximation of various types of linear and nonlinear hyperbolic equations.

Finite-difference algorithm with local time-space grid refinement for simulation of waves

This paper presents a new approach to a local time-space grid refinement for a staggered-grid finite-difference simulation of waves. The approach is based on approximation of a wave equation at the interface where two grids are coupled. As no interpolation or projection techniques are used, the finite-difference scheme preserves second order of convergence. We have proved that this approach is low-reflecting, the artificial reflections are about 10 − 4 of an incident wave. We have also shown that if a successive refinement is applied, i.e. temporal and spatial steps are refined at different interfaces, this approach is stable.

Simulation of seismic waves propagation in multiscale media: impact of cavernous/fractured reservoirs

In order to simulate the interaction of seismic waves with cavernous/fractured reservoirs, a finite-difference technique based on locally refined time-and-space grids is used. The need to use these grids is due primarily to the differing scale of heterogeneities in the reference medium and the reservoir. Domain Decomposition methods allow for the separation of the target area into subdomains containing the reference medium (coarse grid) and reservoir (fine grid). Computations for each subdomain can be carried out in parallel. The data exchange between each subdomain within a group is done using MPI through nonblocking iSend/iReceive commands. The data exchange between the two groups is done simultaneously by coupling the coarse and fine grids. The results of a numerical simulation of a carbonate reservoir are presented and discussed.

Impact of Cavernous/Fractured Reservoirs to Scattered Seismic Waves In 3D Heterogeneous Media: Accurate Numerical Simulation And Field Study

In order to simulate the interaction of seismic waves with cavernous/fractured reservoirs, a finite-difference technique based on locally refined time-and-space grids is used. The need to use these grids is due primarily to the differing scale of heterogeneities in the reference medium and the reservoir. Domain Decomposition methods is used in order to split the target area into subdomains containing the reference medium (coarse grid) and reservoir (fine grid). Computations for each subdomain is carried out in parallel. The data exchange between each subdomain within a group is done using MPI through nonblocking iSend/iReceive commands. The data exchange between the two groups is done simultaneously by coupling the coarse and fine grids and is implemented via Master Processors assigned to each . The results of a numerical simulation of a carbonate reservoir are presented and discussed.

Numerical simulation of 3D acoustic logging

A Finite-difference method for simulation of sonic waves propagating in a vicinity of a borehole filled with fluid and surrounded by a 3D heterogeneous elastic medium is developed. The method utilizes an explicit second order of approximation FD scheme on staggered grids that approximates the elastodynamic system of equations in cylindrical coordinates. A computational domain is surrounded by a special Perfectly Matched Layer for cylindrical coordinate system designed to attenuate waves reflected from outer boundaries. Parallelization is based on a domain decomposition approach and implemented with the help of MPI library. Results of numerical experiments are presented.

Effect of CT image size and resolution on the accuracy of rock property estimates

In order to study the effect of the micro-CT scan resolution and size on the accuracy of up-scaled digital rock property estimation of core samples Bentheimer sandstone images with the resolution varying from 0.9 μm to 24 μm are used. We statistically show that the correlation length of the pore-to-matrix distribution can be reliably determined for the images with the resolution finer than 9 voxels per correlation length and the representative volume for this property is about 153 correlation length. Similar resolution values for the statistically representative volume are also valid for the estimation of the total porosity, specific surface area, mean curvature and topology of the pore space. Only the total porosity and the number of isolated pores are stably recovered, whereas geometry and the topological measures of the pore space are strongly affected by the resolution change. We also simulate fluid flow in the pore space and estimate permeability and tortuosity of the sample. The results demonstrate that the representative volume for the transport property calculation should be greater than 50 correlation lengths of pore-to-matrix distribution. On the other hand, permeability estimation based on the statistical analysis of equivalent realizations shows some weak influence of the resolution on the transport properties. The reason for this might be that the characteristic scale of the particular physical processes may affect the result stronger than the model (image) scale.

Finite difference simulation of elastic wave propagation through 3D heterogeneous multiscale media based on locally refined grids

To simulate the interaction of seismic waves with microheterogeneities (like cavernous/fractured reservoirs), a finite difference technique based on grids locally refined in time and space is used. These grids are used because the scales of heterogeneities in the reference medium and in the reservoir are different. Parallel computations based on domain decomposition of the target area into elementary subdomains in both the reference medium (a coarse grid) and the reservoir (a fine grid) are performed. Each subdomain is assigned to a specific processor unit, which forms two groups: one for the reference medium, and the other for the reservoir. The data exchange between the groups within a processor unit is performed by non-blocking iSend/iReceive MPI commands. The data exchange between the two groups is performed simultaneously with coupling the coarse and a fine grids, and is controlled by a specially chosen processor unit. The results of a numerical simulation for a realistic model of fracture corridors are presented and discussed.

3D finite-difference synthetic acoustic log in cylindrical coordinates

A finite-difference method of numerical simulation of sonic logging has been developed and implemented. The very general statement is considered: the surrounding medium is allowed to be 3D heterogeneous and a source can be located at any point inside or outside the well. To provide a maximally precise description of the sharpest interface of the problem, the interface between the well and surrounding formations, we use cylindrical coordinates with the axis directed along the well. In order to avoid an excessive azimuth inflation of grid cells with an increase of radius we perform periodical refinement of the grid step in the azimuth direction. In order to truncate the area of computations, Perfectly Matched Layer (PML) for cylindrical coordinates is developed and implemented. Its main advantage in comparison with other approaches is an extremely low level of artificial reflections and the absence of necessity to perform splitting of variables in the azimuth direction. Based on this numerical method, a software oriented for the use of parallel computations is developed and implemented under Message Passing Interface Library. Results of numerical experiments for a well with completion embedded within layered elastic background with a vertical planar crack are presented and discussed.

Seismic imaging and statistical analysis of fault facies models

AbstractInterpretation of seismic responses from subsurface fault zones is hampered by the fact that the geologic structure and property distributions of fault zones can generally not be directly observed. This shortcoming curtails the use of seismic data for characterizing internal structure and properties of fault zones, and it has instead promoted the use of interpretation techniques that tend to simplify actual structural complexity by rendering faults as lines and planes rather than volumes of deformed rock. Facilitating the correlation of rock properties and seismic images of fault zones would enable active use of these images for interpreting fault zones, which in turn would improve our ability to assess the impact of fault zones on subsurface fluid flow. We use a combination of 3D fault zone models, based on empirical data and 2D forward seismic modeling to investigate the link between fault zone properties and seismic response. A comparison of spatial statistics from the geologic models and the s...

Parallel numerical simulation of seismic waves propagation with intel math kernel library

This paper describes the implementation of parallel computing to model seismic waves in heterogeneous media based on Laguerre transform with respect to time. The main advantages of the transform are a definite sign of the spatial part of the operator and its independence of the parameter of separation. This property allows one to efficiently organize parallel computations by means of decomposition of the computational domain with successive application of the additive Schwarz method. At each step of the Schwarz alternations, a system of linear algebraic equations in each subdomain is resolved independently of all the others. A proper choice of Domain Decomposition reduces the size of matrices and ensures the use of direct solvers, in particular, the ones based on LU decomposition. Thanks to the independence of the matrix of the parameter of Laguerre transform with respect to time, LU decomposition for each subdomain is done only once, saved in the memory and used afterwards for different right-hand sides. A software is being developed for a cluster using hybrid OpenMP and MPI parallelization. At each cluster node, a system of linear algebraic equations with different right-hand sides is solved by the direct sparse solver PARDISO from Intel Math Kernel Library (Intel MKL). The solver is extensively parallelized and optimized for the high performance on many core systems with shared memory. A high performance parallel algorithm to solve the problem has been developed. The algorithm scalability and efficiency is investigated. For a two-dimensional heterogeneous medium, describing a realistic geological structure, which is typical of the North Sea, the results of numerical modeling are presented.