致密油气介质中波的控制方程

致密油气介质是一种特殊的多孔介质,在油气勘探开发中占有重要地位.文章建立致密油气介质中波的控制方程,该方程较一般多孔介质波方程,形式大为简化,可用于由地震数据进行物性参数反演.文章首先简单介绍从孔隙尺度上流体的运动方程和固体骨架颗粒运动方程出发,利用体平均定理推导出完备的Biot方程的思路与结果,厘清了其中使用的假设条件.其次以岩石物理测试结果为基础,详细分析了致密油气介质中渗透率的时间变化率的性质,进而利用Kozeny-Carman方程,研究了孔隙度的时间变化率的性质,提出了致密油气介质中孔隙度作为状态变量的一个合理假设.在此基础上,从完备的Biot方程出发,推导出了致密油气介质中波的控制方程.该方程与经典的"弥散黏滞方程"在形式上一致,通过对比,得到了弥散黏滞方程中的系数和有明确物理意义的介质物性参数间的解析关系式.文中通过数值模拟验证了所建立的方程的正确性.基于所建立的方程,研究了单一致密夹层的地震波反射和透射特性.数值模拟结果表明夹层的厚度和衰减特性对于地震波的反射和透射有显著影响,这一认识对油气探测有重要意义.

Seismic wave equations in tight oil/gas sandstone media

Tight oil/gas medium is a special porous medium,which plays a significant role in oil and gas exploration.This paper is devoted to the derivation of wave equations in such a media,which take a much simpler form compared to the general equations in the poroelasticity theory and can be employed for parameter inversion from seismic data.We start with the fluid and solid motion equations at a pore scale,and deduce the complete Biot’s equations by applying the volume averaging technique.The underlying assumptions are carefully clarified.Moreover,time dependence of the permeability in tight oil/gas media is discussed based on available results from rock physical experiments.Leveraging the Kozeny-Carman equation,time dependence of the porosity is theoretically investigated.We derive the wave equations in tight oil/gas media based on the complete Biot’s equations under some reasonable assumptions on the media.The derived wave equations have the similar form as the diffusiveviscous wave equations.A comparison of the two sets of wave equations reveals explicit relations between the coefficients in diffusive-viscous wave equations and the measurable parameters for the tight oil/gas media.The derived equations are validated by numerical results.Based on the derived equations,reflection and transmission properties for a single tight interlayer are investigated.The numerical results demonstrate that the reflection and transmission of the seismic waves are affected by the thickness and attenuation of the interlayer,which is of great significance for the exploration of oil and gas.

BIOLUMINESCENCE TOMOGRAPHY:BIOMEDICAL BACKGROUND,MATHEMATICAL THEORY,AND NUMERICAL APPROXIMATION

Over the last couple of years molecular imaging has been rapidly developed to study physiological and pathological processes in vivo at the cellular and molecular levels. Among molecular imaging modalities, optical imaging stands out for its unique advantages, especially performance and cost-effectiveness. Bioluminescence tomography （BLT） is an emerging optical imaging mode with promising biomedical advantages. In this survey paper, we explain the biomedical significance of BLT, summarize theoretical results on the analysis and numerical solution of a diffusion based BLT model, and comment on a few extensions for the study of BLT.

C^0DISCONTINUOUS GALERKIN METHODS FOR A PLATE FRICTIONAL CONTACT PROBLEM

Numerous C^0 discontinuous Galerkin (DG) schemes for the Kirchhoff plate bending problem are extended to solve a plate frictional contact problem, which is a fourth-order elliptic variational inequality of the second kind. This variational inequality contains a nondifferentiable term due to the frictional contact. We prove that these C^0 DG methods are consis tent and st able, and derive optimal order error estima tes for the quadratic element. A numerical example is presented to show the performance of the C^0 DG methods;and the numerical convergence orders confirm the theoretical prediction.

NUMERICAL ANALYSIS OF ELLIPTIC HEMIVARIATIONAL INEQUALITIES FOR SEMIPERMEABLE MEDIA

In this paper, we consider elliptic hemivariational inequalities arising in applications in semipermeable media. In its general form, the model includes both interior and boundary semipermeability terms. Detailed study is given on the hemivariational inequality in the case of isotropic and homogeneous semipermeable media. Solution existence and uniqueness of the problem are explored. Convergence of the Galerkin method is shown under the basic solution regularity available from the existence result. An optimal order error estimate is derived for the linear finite element solution under suitable solution regularity assumptions. The results can be readily extended to the study of more general hemivariational inequalities for non-isotropic and heterogeneous semipermeable media with interior semipermeability and/or boundary semiperrneability. Numerical examples are presented to show the performance of the finite element approximations;in particular, the theoretically predicted optimal first order convergence in H' norm of the linear element solutions is clearly observed.

Numerical analysis of history-dependent variational-hemivariational inequalities

In this paper,numerical analysis is carried out for a class of history-dependent variationalhemivariational inequalities by arising in contact problems.Three different numerical treatments for temporal discretization are proposed to approximate the continuous model.Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size.A special temporal discretization is introduced for the history-dependent operator,leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size.As for spatial approximation,the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions.Numerical examples are presented to illustrate the theoretical results.

ANALYSIS OF A NUMERICAL METHOD FOR RADIATIVE TRANSFER EQUATION BASED BIOLUMINESCENCE TOMOGRAPHY

In the bioluminescence tomography （BLT） problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal＇s body surface. The BLT problem is ill-posed and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter to balance the accuracy and stability of approximate solutions. In this paper, a parameter-dependent coupled complex boundary method （CCBM） based Tikhonov regularization is applied to the BLT problem governed by the radiative transfer equation （RTE）. By properly adjusting the parameter in the Robin boundary condition, we achieve one important property： the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy. The discrete-ordinate finite-element method is used to compute numerical solutions. Numerical results are provided to illustrate the performance of the proposed method.

Preface--Special issue on“Modern Optimization and Applications 2018”

This special issue contains eight selected papers from the International Workshop on Modern Optimization and Applications,which was held over the three days,16-18 June 2018 at Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing.This conference brought together leading scientists,researchers,and practitioners from the world to exchange and shared ideas and approaches in using modern optimization techniques to model and solve real-world application problems from engineering,industry,and management.A prominent feature of this conference is the mixture of optimization theory,optimization methods,and practice of mathematical optimization.This conference provided a forum for researchers from academia to present their latest theoretical results and for practitioners from industry to describe their real-world applications,and discuss with participants the best way to construct suitable optimization models and how to find algorithms capable of solving these models.

A New Coupled Complex BoundaryMethod for Bioluminescence Tomography

In this paper,we introduce and study a new method for solving inverse source problems,through aworkingmodel that arises in bioluminescence tomography(BLT).In the BLT problem,one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal’s body surface.The BLT problem possesses strong ill-posedness and often the Tikhonov regularization is used to obtain stable approximate solutions.In conventional Tikhonov regularization,it is crucial to choose a proper regularization parameter for trade off between the accuracy and stability of approximate solutions.The new method is based on a combination of the boundary condition and the boundary measurement in a parameter-dependent single complex Robin boundary condition,followed by the Tikhonov regularization.By properly adjusting the parameter in the Robin boundary condition,we achieve two important properties for our new method:first,the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy;second,the convergence order of the regularized solutions reaches one with respect to the noise level.Then,the finite element method is used to compute numerical solutions and a newfinite element error estimate is derived for discrete solutions.These results improve related results found in the existing literature.Several numerical examples are provided to illustrate the theoretical results.