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1.
In this paper, we consider the upscaling of Hooke's law and its parameters on the fine scale, to a similar law with upscaled parameters on a larger scale. It is assumed that the fine scale material properties of the rock are imperfectly layered. In the governing equations, the deviations from perfect layering introduce a small parameter that can be used in perturbation series expansions for the stress, the strain, and the displacement. In the approximation of order zero the upscaled compliance matrix contains the well-known Backus parameters; this approximation holds exactly for a perfect layering. However, many natural rock types are imperfectly layered and in that case the approximation of order zero may not be sufficiently accurate. Therefore, we consider also the first order corrections. The derivation and results are presented both for the most general case and for the much simpler case in which the fine scale Poisson ratio may be assumed constant. From thermodynamic principles, it follows that the compliance tensor is symmetric on the fine scale. However, it is shown that the argument for symmetry cannot be extended to upscaled rigidities. One of the most important conclusions is that upscaled compliance tensors are nonsymmetric when there are trends in the deviations from perfect layering. 相似文献
2.
In this paper, we consider the upscaling of Hooke's law and its parameters on the fine scale, to a similar law with upscaled parameters on a larger scale. It is assumed that the fine scale material properties of the rock are imperfectly layered. In the governing equations, the deviations from perfect layering introduce a small parameter that can be used in perturbation series expansions for the stress, the strain, and the displacement. In the approximation of order zero the upscaled compliance matrix contains the well-known Backus parameters; this approximation holds exactly for a perfect layering. However, many natural rock types are imperfectly layered and in that case the approximation of order zero may not be sufficiently accurate. Therefore, we consider also the first order corrections. The derivation and results are presented both for the most general case and for the much simpler case in which the fine scale Poisson ratio may be assumed constant. From thermodynamic principles, it follows that the compliance tensor is symmetric on the fine scale. However, it is shown that the argument for symmetry cannot be extended to upscaled rigidities. One of the most important conclusions is that upscaled compliance tensors are nonsymmetric when there are trends in the deviations from perfect layering. 相似文献
3.
The impact of organic matter on the flow capacity of shale oil rocks is presumably significant, and the knowledge about the representative size is fundamental for the upscaling studies. The error of the experimentally determined permeability values is comparable with the contribution of kerogen to shale permeability, instead a 2D numerical model is employed to explore the normalised equivalent permeability and the representative elementary area (REA) of shale oil rocks in detail incorporating the effects of kerogen. The discussions on the normalised equivalent permeability and the REA are based on the statistical average and standard deviation from 1000 different runs, respectively. The inorganic permeability heterogeneity is introduced based on the assumption of a lognormal pore size distribution and the Monte Carlo sampling method. The effects of kerogen geometric characteristics are incorporated by putting forward several representative cases for comparison. The effects of the organic permeability contrast (ratio of permeability to the inorganic permeability with no heterogeneity), total organic carbon (TOC, volume fraction), inorganic permeability heterogeneity and kerogen geometric characteristics on the normalised equivalent permeability (ratio of the intrinsic equivalent permeability to inorganic permeability with no heterogeneity) and the REA are discussed comprehensively. This work can provide a better understanding of shale oil rocks at the micrometer scale. 相似文献
4.
Large scale geomechanical simulations are being increasingly used to model the compaction of stress dependent reservoirs, predict the long term integrity of under‐ground radioactive waste disposals, and analyse the viability of hot‐dry rock geothermal sites. These large scale simulations require the definition of homogenous mechanical properties for each geomechanical cell whereas the rock properties are expected to vary at a smaller scale. Therefore, this paper proposes a new methodology that makes possible to define the equivalent mechanical properties of the geomechanical cells using the fine scale information given in the geological model. This methodology is implemented on a synthetic reservoir case and two upscaling procedures providing the effective elastic properties of the Hooke's law are tested. The first upscaling procedure is an analytical method for perfectly stratified rock mass, whereas the second procedure computes lower and upper bounds of the equivalent properties with no assumption on the small scale heterogeneity distribution. Both procedures are applied to one geomechanical cell extracted from the reservoir structure. The results show that the analytical and numerical upscaling procedures provide accurate estimations of the effective parameters. Furthermore, a large scale simulation using the homogenized properties of each geomechanical cell calculated with the analytical method demonstrates that the overall behaviour of the reservoir structure is well reproduced for two different loading cases. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
5.
A new method for upscaling fine scale permeability fields to general quadrilateral-shaped coarse cells is presented. The procedure, referred to as the conforming scale up method, applies a triangle-based finite element technique, capable of accurately resolving both the coarse cell geometry and the subgrid heterogeneity, to the solution of the local fine scale problem. An appropriate averaging of this solution provides the equivalent permeability tensor for the coarse scale quadrilateral cell. The general level of accuracy of the technique is demonstrated through application to a number of flow problems. The real strength of the conforming scale up method is demonstrated when the method is applied in conjunction with a flow-based gridding technique. In this case, the approach is shown to provide results that are significantly more accurate than those obtained using standard techniques. 相似文献
6.
Estimating the hydraulic properties of fractured aquifers is challenging due to the complexity of structural discontinuities that can generally be measured at a small scale, either in core or in outcrop, but influence groundwater flow over a range of scales. This modeling study uses fracture scanline data obtained from surface bedrock exposures to derive estimates of permeability that can be used to represent the fractured rock matrix within regional scale flow models. The model is developed using PETREL, which traditionally benefits from high resolution data sets obtained during oil and gas exploration, including for example seismic data, and borehole logging data (both lithological and geophysical). The technique consists of interpreting scanline fracture data, and using these data to generate representative Discrete Fracture Network (DFN) models for each field set. The DFN models are then upscaled to provide an effective hydraulic conductivity tensor that represents the fractured rock matrix. For each field site, the upscaled hydraulic conductivities are compared with estimates derived from pumping tests to validate the model. A hydraulic conductivity field is generated for the study region that captures the spatial variability of fracture networks in pseudo-three dimensions from scanline data. Hydraulic conductivities estimated using this approach compare well with those estimated from pumping test data. The study results suggest that such an approach may be feasible for taking small scale fracture data and upscaling these to represent the aquifer matrix hydraulic properties needed for regional groundwater modeling. 相似文献
7.
The use of limited global information in multiscale simulations is needed when there is no scale separation. Previous approaches entail fine-scale simulations in the computation of the global information. The computation of the global information is expensive. In this paper, we propose the use of approximate global information based on partial upscaling. A requirement for partial homogenization is to capture long-range (non-local) effects present in the fine-scale solution, while homogenizing some of the smallest scales. The local information at these smallest scales is captured in the computation of basis functions. Thus, the proposed approach allows us to avoid the computations at the scales that can be homogenized. This results in coarser problems for the computation of global fields. We analyze the convergence of the proposed method. Mathematical formalism is introduced, which allows estimating the errors due to small scales that are homogenized. The proposed method is applied to simulate two-phase flows in heterogeneous porous media. Numerical results are presented for various permeability fields, including those generated using two-point correlation functions and channelized permeability fields from the SPE Comparative Project (Christie and Blunt, SPE Reserv Evalu Eng 4:308–317, 2001). We consider simple cases where one can identify the scales that can be homogenized. For more general cases, we suggest the use of upscaling on the coarse grid with the size smaller than the target coarse grid where multiscale basis functions are constructed. This intermediate coarse grid renders a partially upscaled solution that contains essential non-local information. Numerical examples demonstrate that the use of approximate global information provides better accuracy than purely local multiscale methods. 相似文献
8.
The aim of upscaling is to determine equivalent homogeneous parameters at a coarse-scale from a spatially oscillating fine-scale parameter distribution. To be able to use a limited number of relatively large grid-blocks in numerical oil reservoir simulators or groundwater models, upscaling of the permeability is frequently applied. The spatial fine-scale permeability distribution is generally obtained from geological and geostatistical models. After upscaling, the coarse-scale permeabilities are incorporated in the relatively large grid-blocks of the numerical model. If the porous rock may be approximated as a periodic medium, upscaling can be performed by the method of homogenization. In this paper the homogenization is performed numerically, which gives rise to an approximation error. The complementarity between two different numerical methods – the conformal-nodal finite element method and the mixed-hybrid finite element method – has been used to quantify this error. These two methods yield respectively upper and lower bounds for the eigenvalues of the coarse-scale permeability tensor. Results of 3D numerical experiments are shown, both for the far field and around wells. 相似文献
9.
10.
Sabine Attinger 《Computational Geosciences》2003,7(4):253-273
This paper focuses on heterogeneous soil conductivities and on the impact their resolution has on a solution of the piezometric
head equation: owing to spatial variations of the conductivity, the flow properties at larger scales differ from those found
for experiments performed at smaller scales. The method of coarse graining is proposed in order to upscale the piezometric
head equation on arbitrary intermediate scales. At intermediate scales large scale fluctuations of the conductivities are
resolved, whereas small scale fluctuations are smoothed by a partialy spatial filtering procedure. The filtering procedure
is performed in Fourier space with the aid of a low-frequency cut-off function. We derive the partially upscaled head equations.
In these equations, the impact of the small scale variability is modeled by scale dependent effective conductivities which
are determined by additional differential equations. Explicit results for the scale dependent conductivity values are presented
in lowest order perturbation theory. The perturbation theory contributions are summed up with using a renormalisation group
analysis yielding explicit results for the effective conductivity in isotropic media. Therefore, the results are also valid
for highly heterogeneous media. The results are compared with numerical simulations performed by Dykaar and Kitanidis (1992).
The method of coarse graining combined by a renormalisation group analysis offers a tool to derive exact and explicit expressions
for resolution dependent conductivity values. It is, e.g., relevant for the interpretation of measurement data on different
scales and for reduction of grid-block resolution in numerical modeling.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
11.
Identifying the Representative Elementary Volume for Permeability in Heterolithic Deposits Using Numerical Rock Models 总被引：1，自引：0，他引：1
Using a range of realistic 3D numerical lithofacies (dm-scale) models of ripple laminated sandstone intercalated with mudstone
we evaluate how single-phase permeability varies as a function of sample support. The models represent a range of mudstone
content which is typical for tidal deposits. Furthermore, the spatial distribution of flow barriers (i.e. mudstone) is not
random, but governed by sedimentological rules giving a variable anisotropy ratio as a function of mudstone content. Both
vertical and horizontal permeability are found to vary at small sample volumes, but these fluctuations reduce as the sample
volume increases. The vertical permeability increases while the horizontal permeability is nearly constant as a function of
sample support for small mudstone contents. For higher mudstone content, the horizontal permeability decreases while the vertical
permeability is nearly constant as a function of sample support. We propose a criterion, based on a normalised standard deviation,
to determine the Representative Elementary Volume (REV). The size of the REV is dependent on both the property measured (vertical
and horizontal permeability) and the correlation lengths of the lithological elements (i.e. lithofacies). Based on this we
identify three flow upscaling regimes that each require a different method for upscaling: (1) layered systems where the arithmetic
and harmonic averages are appropriate, (2) systems close to the percolation threshold where a percolation model should be
used, and (3) discontinuous systems where an effective medium method provides the best estimate of permeability. The work
gives, by using numerical experiments on a range of heterogeneous systems, a new insight in determination of the REV for permeability
at the lithofacies scale and its relation to sedimentological parameters. 相似文献
12.
The compaction of highly heterogeneous poroelastic reservoirs with the geology characterized by long‐range correlations displaying fractal character is investigated within the framework of the stochastic computational modelling. The influence of reservoir heterogeneity upon the magnitude of the stresses induced in the porous matrix during fluid withdrawal and rock consolidation is analysed by performing ensemble averages over realizations of a log‐normally distributed stationary random hydraulic conductivity field. Considering the statistical distribution of this parameter characterized by a coefficient of variation governing the magnitude of heterogeneity and a correlation function which decays with a power‐law scaling behaviour we show that the combination of these two effects result in an increase in the magnitude of effective stresses of the rock during reservoir depletion. Further, within the framework of a perturbation analysis we show that the randomness in the hydraulic conductivity gives rise to non‐linear corrections in the upscaled poroelastic equations. These corrections are illustrated by a self‐consistent recursive hierarchy of solutions of the stochastic poroelastic equations parametrized by a scale parameter representing the fluctuating log‐conductivity standard deviation. A classical example of land subsidence caused by fluid extraction of a weak reservoir is numerically simulated by performing Monte Carlo simulations in conjunction with finite elements discretizations of the poroelastic equations associated with an ensemble of geologies. Numerical results illustrate the effects of the spatial variability and fractal character of the permeability distribution upon the evolution of the Mohr–Coulomb function of the rock. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
13.
In structural geology, viscous creep is generally recognized as the major deformation mechanism in the folding of rock layers through geological time scales of hundreds of thousands of years. Moreover, since deformation of rock salt by creep takes already place on relatively small time scales—weeks to months, say—creep is a relevant phenomenon when studying salt mining, notably the convergence of mine cavities and the land subsidence caused by it. While creep is the dominant process on relatively long time scales, elasticity plays a dominant role in processes that take place on relatively short time scales. The elastic response to a stress is a displacement; the shape of the rock is deformed instantaneously with respect to its initial shape. However, the viscous response of a rock to a stress is a relatively low velocity in the order of millimeters per months or years, say. In this paper we consider the two deformation phenomena creep and elasticity. In general, elasticity is a compressible phenomenon, while creep is incompressible. Here we approximate creep by the introduction of a negligibly small amount of compressibility, which makes creep velocity calculations similar to conventional elastic displacement calculations. Using this procedure, a standard finite element package for elasticity can be applied to viscous problems, also in combination with elasticity. The method has been demonstrated to upscaling of creep viscosities. 相似文献
14.
采用Monte Carlo模拟手段,提出描述场地土层特性变异性对传递函数变异性影响的分析方法。选取日本Kik-Net强震数据库中软（FKSH14）、硬（FKSH12）两类场地,建立场地概率模型。应用Monte Carlo技术随机生成50组场地剖面,分别计算场地的传递函数STF及STF标准差,讨论场地土层厚度、剪切波速,以及二者组合情况下场地传递函数的标准差及场地特征频率的变化。结果显示：对于硬土场地而言,场地特征频率标准差相对于软土场地较大,且剪切波速变异性影响比土层厚度变异性的影响略大,而二者组合工况下最大;而软土场地,土层厚度、剪切波速变化工况下场地特征频率的标准差相当,比二者组合工况下略低;软、硬两类场地,土层厚度与剪切波速二者组合工况下STF的标准差比单一量变化情况下略大,但3种工况下场地STF标准差相差不明显;场地STF的标准差在场地自振频率附近的频率段取值较大,极值点与场地STF的极值点相对应。 相似文献
15.
This paper presents a consistent Bayesian solution for data integration and history matching for oil reservoirs while accounting for both model and parameter uncertainties. The developed method uses Gaussian Process Regression to build a permeability map conforming to collected data at well bores. Following that, an augmented Markov Chain Monte Carlo sampler is used to condition the permeability map to dynamic production data. The selected proposal distribution for the Markov Chain Monte Carlo conforms to the Gaussian process regression output. The augmented Markov Chain Monte Carlo sampler allows transition steps between different models of the covariance function, and hence both the parameter and model space are effectively explored. In contrast to single model Markov Chain Monte Carlo samplers, the proposed augmented Markov Chain Monte Carlo sampler eliminates the selection bias of certain covariance structures of the inferred permeability field. The proposed algorithm can be used to account for general model and parameter uncertainties. 相似文献
16.
We derive a macroscopic model for single-phase, incompressible, viscous fluid flow in a porous medium with small cavities called vugs. We model the vuggy medium on the microscopic scale using Stokes equations within the vugular inclusions, Darcy's law within the porous rock, and a Beavers–Joseph–Saffman boundary condition on the interface between the two regions. We assume periodicity of the medium and obtain uniform energy estimates independent of the period. Through a two-scale homogenization limit as the period tends to zero, we obtain a macroscopic Darcy's law governing the medium on larger scales. We also develop some needed generalizations of the two-scale convergence theory needed for our bimodal medium, including a two-scale convergence result on the Darcy–Stokes interface. The macroscopic Darcy permeability is computable from the solution of a cell problem. An analytic solution to this problem in a simple geometry suggests that: (1) flow along vug channels is primarily Poiseuille with a small perturbation related to the Beavers–Joseph slip, and (2) flow that alternates from vug to matrix behaves as if the vugs have infinite permeability. 相似文献
17.
This paper is aimed towards investigating the filtration law of an incompressible viscous Newtonian fluid through a rigid non-inertial porous medium (e.g. a porous medium placed in a centrifuge basket). The filtration law is obtained by upscaling the flow equations at the pore scale. The upscaling technique is the homogenization method of multiple scale expansions which rigorously gives the macroscopic behaviour and the effective properties without any prerequisite on the form of the macroscopic equations. The derived filtration law is similar to Darcy's law, but the tensor of permeability presents the following remarkable properties: it depends upon the angular velocity of the porous matrix, it verifies Hall–Onsager's relationship and it is a non-symmetric tensor. We thus deduce that, under rotation, an isotropic porous medium leads to a non-isotropic effective permeability. In this paper, we present the results of numerical simulations of the flow through rotating porous media. This allows us to highlight the deviations of the flow due to Coriolis effects at both the microscopic scale (i.e. the pore scale), and the macroscopic scale (i.e. the sample scale). The above results confirm that for an isotropic medium, phenomenological laws already proposed in the literature fails at reproducing three-dimensional Coriolis effects in all types of pores geometry. We show that Coriolis effects may lead to significant variations of the permeability measured during centrifuge tests when the inverse Ekman number Ek−1 is 𝒪(1). These variations are estimated to be less than 5% if Ek−1<0.2, which is the case of classical geotechnical centrifuge tests. We finally conclude by showing that available experimental data from tests carried out in centrifuges are not sufficient to determining the effective tensor of permeability of rotating porous media. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
18.
空间尺度转换是近年来区域生态水文研究领域的一个基本研究问题。其需要主要是源于模型的输入数据与所能提供的数据空间尺度不一致以及模型所代表的地表过程空间尺度与所观测的地表过程空间尺度不吻合。综述了目前区域生态水文模拟研究中常用的空间尺度转换研究方法,包括向上尺度转换和向下尺度转换。详细论述了2种向下尺度转换方法: 统计学经验模型和动态模型。前者是通过将GCM大尺度数据与长期的历史观测数据比较从而建立统计学相关模型, 然后利用这个统计学经验模型进行向下的空间尺度转换. 然而动态模型并不直接对GCM数据进行向下尺度的转换,而是对与GCM进行动态耦合的区域气候模型(RCM) 的输出数据进行空间尺度转换. 通常后者所获得的数据精度要比前者高,但是一个主要缺点就是并不是全球所有的研究区域都有对应的RCM。还详细论述了2种向上尺度转换方法: 统计学经验模型和斑块模型。前者是建立一个能代表小尺度信息在大尺度上分布的密度分布概率函数, 然后利用这个函数在所需的大尺度上进行积分而求得大尺度所需的信息。而后者是根据相似性最大化原则将大尺度划分为若干个可操作的小尺度斑块,然后将计算的每个小尺度斑块的信息平均化得到大尺度所需的信息。通常在计算这种斑块化的小尺度信息的时候,对每个小尺度也会采用统计学经验模型来计算代表整个斑块小尺度的信息。建议用斑块模型与统计学经验模型相集合的方法来实现向上的空间尺度转换 相似文献
19.
This paper treats the upscaling of the absolute permeability in a heterogeneous reservoir. By replacing the fine scale permeability tensor with an upscaled, or effective permeability tensor, a modelling error is introduced. An a posteriori error estimate on this modelling error is formulated and tested. An implementation of the theory, based on domain decomposition coupled with a hierarchical representation of the absolute permeability field, is given. As hierarchical basis functions we have chosen the Haar system, which leads to a wavelet representation of the permeability. The wavelet representation offers a natural upscaling technique which resembles the highcut filters commonly used in signal analysis. This procedure represents an adaptive upscaling method. The numerical results show that this method conserves both the dissipation and the mean velocity in the problem fairly well. The a posteriori error estimate on the modelling error coupled with domain decomposition methods constitutes a powerful modelling tool. 相似文献
20.