Interpretation of well-cell pressures on stretched hexagonal grids in numerical reservoir simulation

Peaceman’s equivalent well-cell radius for 2D Cartesian grids has been generalized to 2D uniform Voronoi grids consisting of stretched hexagons in an isotropic medium. An analytical expression for the equivalent well-cell radius for infinitely fine grids is derived. The derivation is performed by comparison of analytical and numerical solution for boundary value problems with one or two wells. Since the well-cell radius varies slowly with the grid fineness, the found formula can be considered representative for all grid sizes.     

低渗透油藏水平井注CO2井网优化与影响因素

针对低渗透油藏直井井网CO₂驱油的不足,以大庆某低渗区块为例,建立反七点水平井井网CO₂驱油地质模型。在物性参数相态拟合的基础上,对反七点水平井和五点直井两种不同井网模式的开发效果进行了对比,对反七点水平井井网参数进行了优化,同时对储层非均质(渗透率、储层非均质性、裂缝的发育情况等)进行了影响因素分析。研究表明：利用水平井注CO₂可以适当地增大井距、排距,减少井的数量,降低钻井成本;在生产制度相同的情况下水平井井网的开发效果要比直井井网的开发效果好;在水平井井网中,采出程度随着油井到水平井的垂直距离的比值变化而变化,当比值为1.4时采出程度最大;水平井注CO₂更适合低渗透正韵律储层;在储层裂缝发育的情况下,裂缝条数越多,CO₂在储层中的不均匀推进现象越明显,最终采出程度越低。     

Interface error analysis for numerical wave propagation

The numerical error associated with finite-difference simulation of wave propagation in discontinuous media consists of two components. The first component is a higher-order error that leads to grid dispersion; it can be controlled by higher-order methods. The second component results from misalignment between numerical grids and material interfaces. We provide an explicit estimate of the interface misalignment error for the second order in time and space staggered finite-difference scheme applied to the acoustic wave equation. Our analysis, confirmed by numerical experiments, demonstrates that the interface error results in a first-order time shift proportional to the distance between the interface and computational grids. A 2D experiment shows that the interface error cannot be suppressed by higher-order methods and indicates that our 1D analysis gives a good prediction about the behavior of the numerical solution in higher dimensions.      

颗粒离散单元法动力人工边界设置方法

采用颗粒离散单元法进行动力计算时,人工截断边界上需设置吸收边界条件,以防止波的反射。鉴于颗粒离散单元数值计算模型的人工边界上颗粒单元半径大小不一、边界面凸凹不平,在连续介质的黏性、黏弹性、自由场边界条件方程基础之上,推导出适用于离散介质的等效方程。在离散介质的黏性边界条件等效方程中引入微调系数,提出比值迭代法以快速确定其最优值,以实现对波的最佳吸收。采用二维颗粒离散单元计算软件PFC2D,分别建立黏性、黏弹性、自由场边界条件相关数值分析模型,探讨颗粒分布模式对黏性边界上颗粒单元半径、速度分布及比值迭代过程的影响;采用外源波动算例及经典Lamb问题算例验证黏弹性边界设置方法的正确性;通过隧洞算例检验提出的自由场边界条件设置方法的正确性。     

Open-source MATLAB implementation of consistent discretisations on complex grids

Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose novel mathematical methods but instead presents an open-source Matlab? toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation. The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the toolkit contains add-on modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale methods and adjoint methods for use in optimisation of rates and placement of wells.     

Generalized sequential kriging (GSK) for heterogeneous cell size: property modeling without matrices

Sequential kriging avoids the use of matrices and resolves the issue of unstable solutions. It allows for stepwise ways to get joint estimations and cosimulations that are equivalent to the simultaneous solution. The approach is proposed as the solution for geocellular modeling with variable cell size from heterogeneous structural properties (HSPs) as required for modeling with structural constraints. Rock properties are controlled by structural domains, regions, and structural geology parameters. In some cases, rock properties are cross-correlated to formation thickness, curvature of structures, and other structural attributes. Cell thickness may be proportional to formation thickness and may enter as a conditioning property in the estimation of rock property parameters for simulation. In addition, cell volume controls the upscaling of covariance structures (i.e., regularized variograms). Structural properties are priorly modeled. Perturbation response functions (PRFs) are computed for each cell vs all possible sample point locations to facilitate sequential kriging. Upscaled PRFs are modified following conditional updating after each new data value is included in the estimation of parameters. Generalized sequential kriging is expected to become the main tool for real-time spatial modeling of 3D cellular models with HSP. In addition, some new developments related to the sequential kriging algorithm are included. Sequential kriging can be used for the estimation of parameters for simulation in the so-called unstructured grids.     

深埋软弱岩层钻孔围岩应变软化弹塑性分析

为解决在软弱岩层中钻孔易塌孔、成孔难的工程问题,认为岩体黏聚力沿塑性区呈线性弱化,得出应变软化条件下钻孔围岩二次应力、位移和塑性区分布表达式。利用FLAC3D内嵌FISH语言,自定义岩体黏聚力,对钻孔围岩弹塑性分布进行数值模拟分析。研究表明,理想弹塑性模型下的塑性区半径公式为文中解黏聚力co=cs时的特解;考虑黏聚力弱化特性后,相同地应力条件下塑性区半径显著扩大,为修正Fenner解的1.4倍,钻孔围岩最大切向应力位于距钻孔中心约2.9倍钻孔半径处,应力集中系数小于2。成果应用于四川某矿井钻孔工程,表明当加固半径约为3倍钻孔半径时,成孔率在90%以上,取得较好技术经济效果。     

基于地质剖面构建三维地质模型的方法研究

地质剖面是三维地质建模的重要数据源,运用地质剖面构建三维地质模型的方法应用较为广泛。在模型构建之前,统一确定模数据的坐标系和比例足,建立原始资料数据库。构建三维地质模型的关键是不同地质界面。本文详细介绍了模型边界面、断层面、地层界面、岩体界面等4种主要地质界面的构建流程与方法,尤其对褶皱构造、地层界面的断层效应、复杂岩体界面等的构建进行了重点阐述按模型边界面(模型的底界面和四周边界面)、DEM面、断层面、其他地质界面的顺序依次构建地质界面,构建断层面和其他地质界面时严格按先新后老的顺序。运用已构建好的地质界面按先新后老的顺序逐个、依次建立单个地质体,再将所有地质体的面模型组合成整个模型的面模型。通过对面模型进行网格(实体)填充和对网格赋予相应的属性值,最终构建三维地质模型。     

Finite element simulations of logging-while-drilling and extra-deep azimuthal resistivity measurements using non-fitting grids

We propose a discretization technique using non-fitting grids to simulate magnetic field-based resistivity logging measurements. Non-fitting grids are convenient because they are simpler to generate and handle than fitting grids when the geometry is complex. On the other side, fitting grids have been historically preferred because they offer additional accuracy for a fixed problem size in the general case. In this work, we analyse the use of non-fitting grids to simulate the response of logging instruments that are based on magnetic field resistivity measurements using 2.5D Maxwell’s equations. We provide various examples demonstrating that, for these applications, if the finite element matrix coefficients are properly integrated, the accuracy loss due to the use of non-fitting grids is negligible compared to the case where fitting grids are employed.     

考虑井阻及涂抹作用的非饱和土竖井地基固结解析解

以往的非饱和土竖井地基研究中未同时考虑竖井的井阻和涂抹作用,大部分按理想竖井进行研究,然而井阻和涂抹作用是影响非饱和土竖井地基固结的重要因素.针对这种情况,本文基于Fredlund非饱和土一维固结理论及等应变假设,引入变量将超孔隙压力耦合控制方程组转化为等价的线性偏微分方程组,考虑涂抹和井阻条件,并采用分离变量法和待定...     

Pathline tracing on fully unstructured control-volume grids

The trend toward unstructured grids in subsurface flow modeling has prompted interest in the issue of streamline or pathline tracing on unstructured grids. Streamline tracing on unstructured grids is problematic because a continuous velocity field is required for the calculation, while numerical solutions to the groundwater flow equations provide velocity in discretized form only. A method for calculating flow streamlines or pathlines from a finite-volume flow solution is presented. The method uses an unconstrained least squares method on interior cells and a constrained least squares method on boundary cells to approximate cell-centered velocities, which can then be continuously interpolated to any point in the domain of interest. Two-dimensional tests demonstrate that the method correctly reproduces uniform and corner-to-corner flow on fully unstructured grids. In three dimensions using regular hexahedral grids, the method agrees well with established semianalytical methods. Tests also demonstrate that the method produces physically realistic results on fully unstructured three-dimensional grids.     

Simulations of 3D DC borehole resistivity measurements with a goal-oriented <Emphasis Type="Italic">hp</Emphasis> finite-element method. Part II: through-casing resistivity instruments

We simulate direct current (DC) borehole resistivity measurements acquired in steel-cased deviated wells for the assessment of rock formation properties. The assumed data acquisition configuration considers one current (emitter) and three voltage (collector) electrodes that are utilized to measure the second difference of the electric potential along the well trajectory. We assume a homogeneous, 1.27-cm-thick steel casing with resistivity equal to 10^− 5 Ω· m. Simulations are performed with two different numerical methodologies. The first one is based on transferring two-dimensional (2D) axisymmetric optimal grids to a three-dimensional (3D) simulation software. The second one automatically produces optimal 3D grids yielded by a 3D self-adaptive goal-oriented algorithm. Both methodologies utilize high-order finite elements (FE) that are specially well-suited for problems with high-contrast coefficients and rapid spatial variations of the electric field, as it occurs in simulations that involve steel-cased wells. The method based on transferring 2D-optimal grids is efficient in terms of CPU time (few seconds per logging position). Unfortunately, it may produce inaccurate 3D simulations in deviated wells, even though the error remains below 1% for the axisymmetric (vertical) well. The method based on optimal 3D grids, although less efficient in terms of CPU time (few hours per logging position), produces more accurate results that are validated by a built-in a posteriori error estimator. This paper provides the first existing simulations of through-casing resistivity measurements in deviated wells. Simulated resistivity measurements indicate that, for a 30° deviated well, measurements in conductive layers 0.01 Ω· m) are similar to those obtained in vertical wells. However, in resistive layers (10,000 Ω· m), we observe 100% larger readings in the 30° deviated well. This difference becomes 3,000% for the case of a 60° deviated well. For this highly-deviated well, readings corresponding to the conductive formation layer are about 30% smaller in magnitude than those in a vertical well. Shoulder effects significantly vary in deviated wells.     

基于非结构混合网格求解二维浅水方程的一种数值方法

复杂边界下的流场数值模拟常基于非结构化网格进行求解,建立一种在非结构混合网格上求解水深平均的二维浅水方程的模型,以便精确模拟复杂边界、提高计算效率。该模型时间项离散采用隐格式使得模型具有较好的稳定性,对流项和扩散项分别采用总变差减小(Total Variation Diminishing,TVD)格式和构造辅助点的方法来离散,同时采用水深平均的标准k-ε模型来封闭湍流模型。选用两个经典验证算例检验模型,计算结果表明,基于非结构混合网格开发的模型具有较高的精度,且收敛性能较好。     

冰盖下矩形河道的综合糙率

综合糙率是采用曼宁公式确定河道水位和流量关系的关键参数。在河道冰封期,冰盖的出现增加了流动的阻力,明流条件下确定的综合糙率不再适用,需要重新估算。基于Einstein阻力划分过流断面的原理,冰盖下矩形河道的过水断面可划分为冰盖区、河床区和边壁区。根据总流的连续性方程,在确定各分区糙率系数、水力半径和断面面积的基础上,提出了冰盖下矩形河道综合糙率的计算公式。采用已有的试验水槽测量数据和天然河道实测资料对公式进行了验证,结果表明:公式计算的综合糙率与实测值吻合较好,与Einstein公式和Sabaneev公式相比,计算精度更高;对于冰封水流,宽浅河道采用分区水深代替水力半径进行简化计算的条件有别于明渠水流,在宽深比大于20时,计算结果才满足精度要求。     

Numerical analysis of rafts on visco-elastic media using eigenvector expansions

Problems of a raft on a visco-elastic continuum may be converted to equivalent elastic problems by means of a Laplace transformation. In this paper a numerical solution of the equivalent problem is obtained by finite element or other techniques. This solution is converted into the form of an eigenvector expansion and then transformed into a numerical solution to the original problem by inverting the Laplace transform. This form of solution has the advantage of applying to any visco-elastic model, and of requiring little additional computation as a result of changing the visco-elastic model, the relative raft-soil stiffness or the load pattern. The application of the method is illustrated by results for circular rafts, strip footings of finite length and rectangular rafts, and particular attention is paid to a realistic soil creep function which is asymptotic to a linear function of log time.     

锚杆支护圆形隧洞的等效强度参数及可靠性分析

通过采用均匀化方法,研究了圆形隧洞的锚杆支护特性,将高密度支护模式下的岩石和锚杆复合体考虑成均匀、连续、强度参数增强的等效材料,简化了岩石和锚杆间复杂的力学耦合问题。通过定义锚杆密度参数来反映不同支护模式的特性,建立锚杆密度参数与Mohr-Coulomb屈服准则中主要参数之间的关系,推导出等效弹性模量、等效黏聚力和等效内摩擦角的表达式,并分析比较了隧洞在支护前后的位移情况。结合可靠性理论,采用容许极限位移量作为失稳判据,分析了隧洞在支护前后的可靠性指标与破坏概率,结果表明,文中提出的方法简单可靠,锚杆支护对隧洞的位移限制效果明显,可显著提高隧洞的可靠性。     

基坑出水量计算方法探讨——“面井法”

基于平原区密集分布的农田供水井开采量评价方法——“面井法”,推导出了稳定流状态下,基坑出水量计算的一种新方法——“面井法”,并与“大井法”进行比较.“面井法”计算的基坑水位降深总大于“大井法”计算的基坑水位降深,而出水量则小于“大井法”计算的基坑出水量.“面井法”也定义了基坑等效半径,但形式唯一,简单易记.对于矩形(包括条形、方形)基坑,等效半径计算值总大于“大井法”计算值,而圆形基坑二者相等.由于“面井法”的推导是在整个基坑面积上积分的,其计算结果更能代表整个基坑的渗流特征.     

AUSM格式在二维浅水方程求解中的应用

引入空气动力学中发展起来的AUSM格式,在三角形网格上采用有限体积法,对二维浅水控制方程进行数值求解;引入过程中,对浅水问题中的源项及露滩问题等采用了相应的特殊处理。通过对二维计算常用的溃坝算例及扎龙湿地实际算例的计算结果分析可知:二维浅水问题中引入该计算格式后,计算结果较好且保持了原格式的优良特性,同时也验证了该格式在二维浅水方程中应用的可行性及可靠性。