An equivalent medium model for wave simulation in fractured porous rocks

Seismic wave propagation in reservoir rocks is often strongly affected by fractures and micropores. Elastic properties of fractured reservoirs are studied using a fractured porous rock model, in which fractures are considered to be embedded in a homogeneous porous background. The paper presents an equivalent media model for fractured porous rocks. Fractures are described in a stress‐strain relationship in terms of fracture‐induced anisotropy. The equations of poroelasticity are used to describe the background porous matrix and the contents of the fractures are inserted into a matrix. Based on the fractured equivalent‐medium theory and Biot's equations of poroelasticity, two sets of porosity are considered in a constitutive equation. The porous matrix permeability and fracture permeability are analysed by using the continuum media seepage theory in equations of motion. We then design a fractured porous equivalent medium and derive the modified effective constants for low‐frequency elastic constants due to the presence of fractures. The expressions of elastic constants are concise and are directly related to the properties of the main porous matrix, the inserted fractures and the pore fluid. The phase velocity and attenuation of the fractured porous equivalent media are investigated based on this model. Numerical simulations are performed. We show that the fractures and pores strongly influence wave propagation, induce anisotropy and cause poroelastic behaviour in the wavefields. We observe that the presence of fractures gives rise to changes in phase velocity and attenuation, especially for the slow P‐wave in the direction parallel to the fracture plane.     

Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media

We introduce two new channeling indicators D_ic and D_cc based on the Lagrangian distribution of flow rates. On the basis of the participation ratio, these indicators characterize the extremes of both the flow-tube width distribution and the flow rate variation along flow lines. The participation ratio is an indicator biased toward the larger values of a distribution and is equal to the normalized ratio of the square of the first-order moment to the second-order moment. Compared with other existing indicators, they advantageously provide additional information on the flow channel geometry, are consistently applicable to both porous and fractured media, and are generally less variable for media generated using the same parameters than other indicators. Based on their computation for a broad range of porous and fracture permeability fields, we show that they consistently characterize two different geometric properties of channels. D_ic gives a characteristic scale of low-flow zones in porous media and a characteristic distance between effectively flowing structures in fractured cases. D_cc gives a characteristic scale of the extension of high-flow zones in porous media and a characteristic channel length in fractured media. D_ic is mostly determined by channel density and permeability variability. D_cc is, however, more affected by the nature of the correlation structure like the presence of permeability channels or fractures in porous media and the length distribution in fracture networks.     

Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes

— It is well known that there is no “universal” permeability-porosity relationship valid in all porous media. However, the evolution of permeability and porosity in rocks can be constrained provided that the processes changing the pore space are known. In this paper, we review observations of the relationship between permeability and porosity during rock evolution and interpret them in terms of creation/destruction of effectively and non-effectively conducting pore space. We focus on laboratory processes, namely, plastic compaction of aggregates, elastic-brittle deformation of granular rocks, dilatant and thermal microcracking of dense rocks, chemically driven processes, as a way to approach naturally occurring geological processes. In particular, the chemically driven processes and their corresponding evolution permeability-porosity relationships are discussed in relation to sedimentary rocks diagenesis.     

Mixing-induced precipitation and porosity evolution in porous media

The dynamics of porosity evolution are explored during mineral precipitation that is induced by the mixing of two fluids of different compositions. During mineral precipitation in geological formations, the physical parameters that characterize the porous matrix, such as porosity and specific surface area, can change significantly. A series of coupled equations that determine the changes in porosity is outlined and solved for a 2D model domain using a finite element scheme. Using model parameters equivalent to those for calcite precipitation in a saline system, the evolution of porosity is examined for two types of porous media: (1) an initially homogeneous system and (2) a heterogeneous system containing high porosity regions that serve initially as preferential flow paths. In addition, the influence of two different expressions that relate specific surface area to porosity is explored. The simulations in both domains indicated that porosity was reduced primarily in the regions in which significant degrees of mixing occurred. Although an effective barrier was created in these regions, the fluids bypassed the clogged areas allowing precipitation to continue farther “downstream”. Furthermore, mixing-induced precipitation can account for systems in which some high porosity regions are filled while others remain almost unchanged. Thus, mixing-induced precipitation represents a viable mechanism for the infilling of pores in fractured and porous rocks. The simulations also demonstrate that the choice of functional form for specific surface area plays an important role in controlling porosity patterns by influencing both the kinetics of precipitation and the permeability of the porous medium. As specific surface area is currently one of the least constrained parameters in models of porosity evolution, this result highlights the need for future experimental studies in this field of research.     

Compressional- and shear-wave velocities as a function of confining stress in dry sandstones

A laboratory study was carried out to investigate the influence of confining stress on compressional- and shear-wave velocities for a set of rock samples from gas-producing sandstone reservoirs in the Cooper Basin, South Australia. The suite of samples consists of 22 consolidated sublitharenites with helium porosity ranging from 2.6% to 16.6%. We used a pulse-echo technique to measure compressional- and shear-wave velocities on dry samples (cylindrical 4.6 × 2 cm) at room temperature and at elevated confining stress (≤ 60 MPa). Compressional- and shear-wave velocities in samples increase non-linearly with confining stress. A regression equation of the form V = A ? Be^?DP gives a good fit to the measured velocities with improved prediction of velocities at high confining stresses compared with equations suggested by other studies. The predicted microcrack-closure stresses of the samples show values ranging from 70 MPa to 95 MPa and insignificant correlation with porosity, permeability or clay content. There is a positive correlation between change in velocity with core porosity and permeability, but this association is weak and diminishes with increasing confining stress. Experimental results show that pore geometry, grain-contact type, and distribution and location of clay particles may be more significant than total porosity and clay content in describing the stress sensitivity of sandstones at in situ reservoir effective stress. The stress dependence of Cooper Basin sandstones is very large compared with data from other studies. The implication of our study for hydrocarbon exploration is that where the in situ reservoir effective stress is much less than the microcrack-closure stress of the reservoir rocks, the variation of reservoir effective stress could cause significant changes in velocity of the reservoir rocks. The velocity changes induced by effective stress in highly stress-sensitive rocks can be detected at sonic-log and probably surface-seismic frequencies.     

Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry

The semi-empirical Kozeny–Carman (KC) equation is the most famous permeability–porosity relation, which is widely used in the field of flow in porous media and is the starting point for many other permeability models. However, this relation has many limitations from its inception, and the KC constant is an empirical parameter which was proved to be not a constant. In this paper, we briefly reviewed the KC equation, its modifications and various models for the KC constant. We then derived an analytical expression for the permeability in homogeneous porous media based on the fractal characters of porous media and capillary model. The proposed model is expressed as a function of fractal dimensions, porosity and maximum pore size. The analytical KC constant with no empirical constant is obtained from the assumption of square geometrical model. Furthermore, a distinct linear scaling law between the dimensionless permeability and porosity is found. It is also shown that our analytical permeability is more closely related to the microstructures (fractal dimensions, porosity and maximum pore size), compared to those obtained from conventional methods and models.     

基于三相孔隙弹性理论的火山岩气层岩石物理响应特征分析

Unlike previous theories with velocity and/or elastic modulus averaging, we use a three-phase porous rock physics model developed by Santos for analyzing the seismic response of two immiscible fluids in saturated porous media. Considering reservoir reference pressure and coupling drag of two fluids in pores, the effects of frequency, porosity, and gas saturation on the phase velocities of the P-and S-waves are discussed in detail under field conditions. The effects of porosity and gas saturation on Vp/Vs are also provided. The data for our numerical experiments are from a sample of deep volcanic rock from Daqing. The numerical results show that the frequency dispersion effect can be ignored for deep volcanic rocks with low porosity and low permeability. It is concluded that for deep volcanic rocks the effect of gas content in pores on Vp/Vs is negligible but the effect of porosity is significant when there is a certain amount of water contained in the pores. The accurate estimate of lithology and porosity in this case is relatively more important.     

A fractal model for streaming potential coefficient in porous media

Streaming potential is the result of coupling between a fluid flow and an electric current in porous rocks. The modified Helmholtz–Smoluchowski equation derived for capillary tubes is mostly used to determine the streaming potential coefficient of porous media. However, to the best of our knowledge, the fractal geometry theory is not yet applied to analyse the streaming potential in porous media. In this article, a fractal model for the streaming potential coefficient in porous media is developed based on the fractal theory of porous media and on the streaming potential in a capillary. The proposed model is expressed in terms of the zeta potential at the solid?liquid interface, the minimum and maximum pore/capillary radii, the fractal dimension, and the porosity of porous media. The model is also examined by using another capillary size distribution available in published articles. The results obtained from the model using two different capillary size distributions are in good agreement with each other. The model predictions are then compared with experimental data in the literature and those based on the modified Helmholtz–Smoluchowski equation. It is shown that the predictions from the proposed fractal model are in good agreement with experimental data. In addition, the proposed model is able to reproduce the same result as the Helmholtz–Smoluchowski equation, particularly for high fluid conductivity or large grain diameters. Other factors influencing the streaming potential coefficient in porous media are also analysed.     

THE STUDY OF STREAMING POTENTIALS ON FISSURED MEDIA MODELS*

In massive rocks ground waters mainly flow in fracture zones whose permeability greatly changes depending on their filling. When studying ground-water flow in fissures, the results of observations of the electric fields of filtrational origin—which, in this case, considerably differ from those in porous media—can be used. Therefore the authors have made experiments on fissured media models. The measurements have been made in a special filtration tube with the fissured media simulated by a system of quartz glass plates. The spacings between plates were regarded as fissure widths. The observations have been made in fully open “fissures” and in those partially filled with sand or sandy-clay material. These experiments have resulted in establishing a dependence between the values of streaming potentials and pressure drops. The SP values have particularly been found to decrease with the opening of “fissures”. The most intensive electro-filtrational fields were observed at 40 per cent filling with medium grained sand. Additions of argillaceous material to the sand filler brought about sharp reductions in the intensity of electro-filtrational potentials.     

Image-based relative permeability upscaling from the pore scale

High resolution images acquired from X-ray μ-CT are able to map the internal structure of porous media on which multiphase flow properties can be computed. While the resolution of a few micrometers may be sufficient for capturing the pore space of many sandstones, most carbonates exhibit a large amount of microporosity; pores which are below the image resolution and are not resolved at specific resolution. Neglecting the effect of micropores on fluid flow and transport properties of these rocks can cause erroneous results in particular at partial saturations. Current image-based pore scale models typically only consider macropores for simulating fluid flow. In this paper, we quantify the effect of microporosity on the effective permeability of the wetting phase for heterogeneous model structures with varying amount of micro-to-macro porosity. A multi-scale numerical approach is proposed to couple an average effect of micropores with an explicit representation of macropores. The Brinkman equation is solved using a lattice Boltzmann formulation to facilitate the coupling of Darcy and Stokes equations in micropores and macropores, respectively. The results show good agreement between the fine scale solution and the results of the upscaled models in which microporous regions are homogenised. The paper analyses in particular the choice of the momentum sink parameter at low wetting phase saturations. It is shown that this parameter can be found using either a flux-based calculation of permeability of microporous regions or chosen purely on the basis of the effective permeability of these regions.     

储层岩石流动电位频散特性的数学模拟

利用储层岩石流动电位的频散特性评价复杂储层已经成为勘探地球物理领域关注的热点,但是目前还没有形成基于储层岩石储渗特性及电化学性质的具有普遍指导意义的理论方法和数学模型.本文利用微观毛管理论,通过随时间谐变条件下渗流场和电流场的耦合模型,建立了描述储层岩石流动电位频散特性的数学方法,定量分析了频率域储层岩石动态渗透率、动电耦合系数和流动电位耦合系数随储层岩石孔隙度、溶液浓度和阳离子交换量的变化规律.研究结果表明:储层岩石流动电位频散特性是储层流体惯性力与流体黏滞力相互作用的结果.储层岩石孔隙度越大,储层维持流体原有运动状态的能力越大,临界频率越小;储层岩石的溶液浓度和阳离子交换量对临界频率没有影响.储层岩石的孔隙度越大,流体流动能力越强,流动电位各耦合系数的数值越大;溶液浓度越小或阳离子交换量越大,孔隙固液界面的双电层作用越强,各耦合系数的数值越大.     

Numerical analysis of bacterial transport in saturated porous media

A mathematical model to describe bacterial transport in saturated porous media is presented. Reversible/irreversible attachment and growth/decay terms were incorporated into the transport model. Additionally, the changes of porosity and permeability due to bacterial deposition and/or growth were accounted for in the model. The predictive model was used to fit the column experimental data from the literature, and the fitting result showed a good match with the data. Based on the parameter values determined from the literature experimental data, numerical experiments were performed to examine bacterial sorption and/or growth during bacterial transport through saturated porous media. In addition, sensitivity analysis was performed to investigate the impact of key model parameters for bacterial transport on the permeability and porosity of porous media. The model results show that the permeability and porosity of porous media could be altered due to bacterial deposition and growth on the solid matrix. However, variation of permeability due to bacterial growth was trivial compared with natural permeability variation. Copyright © 2005 John Wiley & Sons, Ltd.     

Discontinuous Galerkin methods for advective transport in single-continuum models of fractured media

Accurate simulation of flow and transport processes in fractured rocks requires that flow in fractures and shear zones to be coupled with flow in the porous rock matrix. To this end, we will herein consider a single-continuum approach in which both fractures and the porous rock are represented as volumetric objects, i.e., as cells in an unstructured triangular grid with a permeability and a porosity value associated with each cell. Hence, from a numerical point of view, there is no distinction between flow in the fractures and the rock matrix. This enables modelling of realistic cases with very complex structures. To compute single-phase advective transport in such a model, we propose to use a family of higher-order discontinuous Galerkin methods. Single-phase transport equations are hyperbolic and have an inherent causality in the sense that information propagates along streamlines. This causality is preserved in our discontinuous Galerkin discretization. We can therefore use a simple topological sort of the graph of discrete fluxes to reorder the degrees-of-freedom such that the discretized linear system gets a lower block-triangular form, from which the solution can be computed very efficiently using a single-pass forward block substitution. The accuracy and utility of the resulting transport solver is illustrated through several numerical experiments.     

Specification of Matrix Cleanup Goals in Fractured Porous Media

Semianalytical transient solutions have been developed to evaluate what level of fractured porous media (e.g., bedrock or clay) matrix cleanup must be achieved in order to achieve compliance of fracture pore water concentrations within a specified time at specified locations of interest. The developed mathematical solutions account for forward and backward diffusion in a fractured porous medium where the initial condition comprises a spatially uniform, nonzero matrix concentration throughout the domain. Illustrative simulations incorporating the properties of mudstone fractured bedrock demonstrate that the time required to reach a desired fracture pore water concentration is a function of the distance between the point of compliance and the upgradient face of the domain where clean groundwater is inflowing. Shorter distances correspond to reduced times required to reach compliance, implying that shorter treatment zones will respond more favorably to remediation than longer treatment zones in which back‐diffusion dominates the fracture pore water response. For a specified matrix cleanup goal, compliance of fracture pore water concentrations will be reached sooner for decreased fracture spacing, increased fracture aperture, higher matrix fraction organic carbon, lower matrix porosity, shorter aqueous phase decay half‐life, and a higher hydraulic gradient. The parameters dominating the response of the system can be measured using standard field and laboratory techniques.     

Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces

Pore-scale forces have a significant effect on the macroscopic behaviour of multiphase flow through porous media. This paper studies the effect of these forces using a new volume-of-fluid based finite volume method developed for simulating two-phase flow directly on micro-CT images of porous media. An analytical analysis of the relationship between the pore-scale forces and the Darcy-scale pressure drops is presented. We use this analysis to propose unambiguous definitions of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each phase, and then use them to obtain the relative permeability curves. We show that this definition is consistent with conventional laboratory/field measurements by comparing our predictions with experimental relative permeability. We present single and two-phase flow simulations for primary oil injection followed by water injection on a sandpack and a Berea sandstone. The two-phase flow simulations are presented at different capillary numbers which cover the transition from capillary fingering at low capillary numbers to a more viscous fingering displacement pattern at higher capillary numbers, and the effect of capillary number on the relative permeability curves is investigated. Overall, this paper presents a new finite volume-based methodology for the detailed analysis of two-phase flow directly on micro-CT images of porous media and upscaling of the results to the Darcy scale.     

Volume changes during fracture and frictional sliding: A review

Summary Volume changes in geologic materials have been measured with strain gauges, cantilever displacement gauges, or through observation of either pore or total volume. When porosity is less than 0.05, compaction is small or absent; apart from elastic strains in the minerals, dilatancy predominates, beginning at 50 to 75 percent of the fracture stress difference. When initial porosity exceeds about 0.05, compaction and dilatancy may overlap. The onset of dilatancy has not been identified, but most of the dilatancy occurs within about 10 percent of the fracture stress difference. In low porosity rocks, dilatancy increases initial porosity by a factor of 2 or more; in porous rocks or granular aggregates the increase is only 20 to 50 percent. However, the actual pore volume increase is larger in rocks of high initial porosity. Hence, earthquake precursors which depend on the magnitude of dilatancy should be more pronounced in porous rocks or in fault gouge. In contrast, precursors which are based on fractional changes in some porosity-related property may be more pronounced in rocks of low initial porosity. Future work is particularly needed on constitutive relations suitable for major classes of rocks, on the effects of stress cycling in porous rocks, on the effects of high temperature and pore fluids on dilatancy and compaction, and on the degree of localization of strain prior to fracture.     

三维层状孔隙介质中弹性波的一种积分表达式I：理论

孔隙介质弹性波传播理论在地球物理勘探、地震工程和岩土动力学等领域有着广泛的应用.而孔隙介质中的弹性波受孔隙度、渗透率、流体黏滞系数等参数的影响,因此研究波场的传播特征将有助于分析和提取这些信息.本文在Biot理论的基础上,针对三维层状孔隙介质模型,利用在合成理论地震图的研究中已经被证实具有稳定、高效且适用范围较广的Luco-Apsel-Chen（LAC）广义反透射方法,给出了弹性波场的一种积分形式的半解析解,可通过数值方法高效、准确地计算层状孔隙介质中的理论波场,所以该积分形式的半解析解可为三维层状孔隙介质波场传播特征的理论数值模拟研究提供一种新的途径和手段.     

Hydrogeology of the Brunswick (Passaic) Formation and Implications for Ground Water Monitoring Practice

Fractured shales of the Brunswick Formation provide a major aquifer in the most industrialized region of New Jersey. Numerous cases of ground water contamination have been documented in this formation. However, effectiveness of monitoring and remediation efforts is often hampered by the use of inappropriate concepts regarding ground water flow controls in this complex aquifer system. One such concept presumes that near-vertical fractures parallel to the strike of beds provide principal passages for the flow and produce an anisotropic response to pumping stress. Field evidence presented in this paper confirms that the Brunswick Formation hosts a gently dipping, multiunit, leaky aquifer system that consists of thin water-bearing units and thick intervening aquitards. The water-bearing units are associated with major bedding partings and/or intensely fractured seams. Layered heterogeneity of such a dipping multiunit aquifer system produces an anisotropic flow pattern with preferential flow along the strike of beds. Within the weathered zone, the permeability of the water-bearing units can be greatly reduced. The commonly used hydrogeologic model of the Brunswick as a one-aquifer system, sometimes with vaguely defined "shallow" and "deep" zones, often leads to the development of inadvertent cross-flows within monitoring wells. If undetected, cross-flows may promote contaminant spread into deeper units and impair the quality of hydrogeologic data. Hydrogeologic characterization of the Brunswick shales at any given site should be aimed primarily at identification of the major water-bearing and aquitard units. Recommended techniques for this characterization include fluid logging and other in-well tests.     

Coupled Models of Brittle-plastic Deformation and Fluid Flow: Approaches,Methods, and Application to Mesoproterozoic Mineralisation at Mount Isa,Australia

Rock deformation has an important effect on the spatial distribution and temporal evolution of permeability in the Earth’s crust. Hydromechanical coupling is of fundamental significance to natural fluid–rock interaction in porous and fractured hydrothermal systems, and in the assessment and production of hydrocarbon resources and geothermal energy. Shearing and fracturing of rocks can lead to the creation or destruction of permeability when fractures or faults form, or when existing structures are reactivated. Changes in stress orientation or fluid pressure can drive rock failure and create dilating fault zones that have the potential to focus fluid flow, or to breach seals above overpressured fluid compartments. Here, numerical models of deformation and fluid flow related to Mesoproterozoic copper mineralisation at Mount Isa, Australia, are presented that show how changes in deformation geometry in multiply deformed geological architectures relate to changes in dilation patterns, fluid pathways and flow geometry. Coupled numerical simulations of deformation and fluid flow can be useful tools to better understand structural control on fluid flow in hydrothermal mineral systems.