Estimation of the apertures of water-saturated fractures by nuclear magnetic resonance well logging

Fracture aperture is an important transport property in subsurface hydrology because it influences well productivity and the volume of the water resource. Nuclear magnetic resonance (NMR) well logging measures the hydrogen‐bearing fluid molecules in porous or fractured strata, and the NMR signal intensity increases with the amount of fluid in the sensed region of the NMR sonde. Fluid confined in a large fracture of >>0.2 mm in aperture has T2 (i.e. spin‐spin relaxation time) values as long as those of the bulk fluid. The bulk‐fluid porosity (i.e. porosity calculated using this long T2 component in a T2 histogram data) increases linearly with aperture. Therefore, NMR logging enables quantitative estimation of fracture apertures of >>0.2 mm using the bulk‐fluid porosity data if the calibration of the NMR sonde is performed adequately. We applied NMR logging to a borehole in a Holocene andesite lava at Sumikawa, Japan, to estimate the aperture of open fractures within the lava. A test well of 100 m depth and 20 cm diameter, filled with bentonite drilling mud, was scanned with an NMR sonde to obtain a profile of the porosity and the T2 histogram of the andesite. The bulk‐fluid porosity was calculated from the T2 histogram data, as the porosity at which the T2 value is larger than or equal to a threshold T2 of bulk bentonite mud. The bulk‐fluid porosity of a specific inclined fracture responsible for the total loss of circulation at 61.2 m depth during drilling was calculated assuming a threshold or T2 cut‐off of 33 ms, and again for a cut‐off of 100 ms. Calibration of the NMR sensor in a laboratory and measurement of the fracture dip angle by electrical microimaging logging enabled us to estimate the fracture aperture as 1.7 cm, assuming a T2 cut‐off of 33 ms, or 1.6 cm for a T2 cut‐off of 100 ms. The method of aperture determination described in this study is independent of fluid species and lithology, and is applicable to various hydrogen‐bearing borehole fluids (clean water, mud and oil) and geological settings.     

Matrix diffusion-derived plume attenuation in fractured bedrock

Matrix diffusion can attenuate the rate of plume migration in fractured bedrock relative to the rate of ground water flow for both conservative and nonconservative solutes of interest. In a system of parallel, equally spaced constant aperture fractures subject to steady-state ground water flow and an infinite source width, the degree of plume attenuation increases with time and travel distance, eventually reaching an asymptotic level. The asymptotic degree of plume attenuation in the absence of degradation can be predicted by a plume attenuation factor, beta, which is readily estimated as R' (phi(m)/phi(f)), where R' is the retardation factor in the matrix, phi(m) is the matrix porosity, and phi(f) is the fracture porosity. This dual-porosity relationship can also be thought of as the ratio of primary to secondary porosity. Beta represents the rate of ground water flow in fractures relative to the rate of plume advance. For the conditions examined in this study, beta increases with greater matrix porosity, greater matrix fraction organic carbon, larger fracture spacing, and smaller fracture aperture. These concepts are illustrated using a case study where dense nonaqueous phase liquid in fractured sandstone produced a dissolved-phase trichloroethylene (TCE) plume approximately 300 m in length. Transport parameters such as matrix porosity, fracture porosity, hydraulic gradient, and the matrix retardation factor were characterized at the site through field investigations. In the fractured sandstone bedrock examined in this study, the asymptotic plume attenuation factors (beta values) for conservative and nonconservative solutes (i.e., chloride and TCE) were predicted to be approximately 800 and 12,210, respectively. Quantitative analyses demonstrate that a porous media (single-porosity) solute transport model is not appropriate for simulating contaminant transport in fractured sandstone where matrix diffusion occurs. Rather, simulations need to be conducted with either a discrete fracture model that explicitly incorporates matrix diffusion, or a dual-continuum model that accounts for mass transfer between mobile and immobile zones. Simulations also demonstrate that back diffusion from the matrix to fractures will likely be the time-limiting factor in reaching ground water cleanup goals in some fractured bedrock environments.     

A constitutive model for water flow in unsaturated fractured rocks

A conceptual model for describing effective saturation in fractured hard rock is presented. The fracture network and the rock matrix are considered as an equivalent continuum medium where each fracture is conceptualized as a porous medium of granular structure and the rock matrix is assumed to be impermeable. The proposed model is based on the representation of a rough‐walled fracture by an equivalent porous medium, which is described using classical constitutive models. A simple closed‐form equation for the effective saturation is obtained when the van Genuchten model is used to describe saturation inside fractures and fractal laws are assumed for both aperture and number of fractures. The relative hydraulic conductivity for the fractured rock is predicted from a simple relation derived by Liu and Bodvarsson. The proposed constitutive model contains three independent parameters, which may be obtained by fitting the proposed effective saturation curve to experimental data. Two of the model parameters have physical meaning and can be identified with the reciprocal of the air entry pressure values in the fractures of minimum and maximum apertures. Effective saturation and relative hydraulic conductivity curves match fairly well the simulated constitutive relations obtained by Liu and Bodvarsson. Copyright © 2008 John Wiley & Sons, Ltd.     

The Applicability of Incoherent Array Processing to IMS Seismic Arrays

The seismic arrays of the International Monitoring System (IMS) for the Comprehensive Nuclear-Test-Ban Treaty (CTBT) are highly diverse in size and configuration, with apertures ranging from under 1 km to over 60 km. Large and medium aperture arrays with large inter-site spacings complicate the detection and estimation of high-frequency phases lacking coherence between sensors. Pipeline detection algorithms often miss such phases, since they only consider frequencies low enough to allow coherent array processing, and phases that are detected are often attributed qualitatively incorrect backazimuth and slowness estimates. This can result in missed events, due to either a lack of contributing phases or by corruption of event hypotheses by spurious detections. It has been demonstrated previously that continuous spectral estimation can both detect and estimate phases on the largest aperture arrays, with arrivals identified as local maxima on beams of transformed spectrograms. The estimation procedure in effect measures group velocity rather than phase velocity, as is the case for classical f–k analysis, and the ability to estimate slowness vectors requires sufficiently large inter-sensor distances to resolve time-delays between pulses with a period of the order 4–5 s. Spectrogram beampacking works well on five IMS arrays with apertures over 20 km (NOA, AKASG, YKA, WRA, and KURK) without additional post-processing. Seven arrays with 10–20 km aperture (MJAR, ESDC, ILAR, KSRS, CMAR, ASAR, and EKA) can provide robust parameter estimates subject to a smoothing of the resulting slowness grids, most effectively achieved by convolving the measured slowness grids with the array response function for a 4 or 5 s period signal. Even for medium aperture arrays which can provide high-quality coherent slowness estimates, a complementary spectrogram beampacking procedure could act as a quality control by providing non-aliased estimates when the coherent slowness grids display significant sidelobes. The detection part of the algorithm is applicable to all IMS arrays, with spectrogram-based processing offering a potential reduction in the false alarm rate for high-frequency signals. Significantly, the local maxima of the scalar functions derived from the transformed spectrogram beams are robust estimates of the signal onset time. High-frequency energy is of greater importance for lower event magnitudes and in the cavity decoupling detection evasion scenario. There is a need to characterize both propagation paths with low attenuation of high-frequency energy and situations in which parameter estimation on array stations fails.     

Influence of small-scale heterogeneities on contaminant transport in fractured crystalline rock

We present a sequence of purely advective transport models that demonstrate the influence of small-scale geometric inhomogeneities on contaminant transport in fractured crystalline rock. Special weight is placed on the role of statistically generated variable fracture apertures. The fracture network geometry and the aperture distribution are based on information from an in situ radionuclide retardation experiment performed at Grimsel test site (Swiss Alps). The obtained breakthrough curves are fitted with the advection dispersion equation and continuous-time random walks (CTRW). CTRW is found to provide superior fits to the late-arrival tailing and is also found to show a good correlation with the velocity distributions obtained from the hydraulic models. The impact of small-scale heterogeneities, both in fracture geometry and aperture, on transport is shown to be considerable.     

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.     

Two-phase flow properties of a horizontal fracture: The effect of aperture distribution

A systematic numerical method has been presented to investigate the constitutive relationships between two-phase flow properties of horizontal fractures and aperture distributions. Based on fractal geometry, single rough-walled fractures are generated numerically by modified successive random addition (SRA) method and then aperture distributions with truncated Gaussian distribution are formed by shear displacement between lower and upper surfaces. (The truncated Gaussian distribution is used to describe aperture evolution under different normal stresses.) According to the assumption of two-dimensional porous media and local parallel plate model, invasion percolation approach is employed to model the two-phase flow displacement (imbibition) in generated horizontal fractures, in which capillary forces are dominant over viscous and gravity forces. For truncated Gaussian distributions, constitutive relationships from numerical simulation are compared to closed-form relationships and a good agreement is obtained. The simulation results indicate strong phase interference with the sum of two phase relative permeability values being less than one in the intermediate saturations. It is found that fracture properties related to residual saturations depend on spatial correlation of aperture distributions. Based on the simulation results, we proposed an empirical relationship between the fracture residual-saturation-rated parameters and the corresponding aperture distributions.     

Density contrast effects on tracer dispersion in variable aperture fractures

We investigate the development of preferential flow paths and anomalous dispersion resulting from weak density contrasts in the course of tracer experiments in a tortuous natural fracture. The processes are first documented by the non-invasive measurement of the fracture aperture and of the time-resolved distribution of the tracer using Positron Emission Projection Imaging. Then, numerical simulations of the three-dimensional tracer transport in the fracture are performed to explore the parameters that control the development and the persistence of the tracer localization, as a function of the density contrast between the tracer and the resident solution. Results reveal that density contrasts representative of what could be expected in borehole and laboratory tracer tests can induce irreversible localization along preferential channels. As density contrast increases, the correlation between velocity and aperture distributions vanishes whereas (i) velocity field increasingly correlates with the fracture median plan elevation and (ii) the longitudinal dispersion coefficient increases. The anomalous velocity distribution may persist well after the injection stops due to the occurrence of tracer trapped zone.     

The fractal geometry of flow paths in natural fractures in rock and the approach to percolation

The distributions of contact areas in single, natural fractures in quartz monzonite (Stripa granite) are found to have fractal dimensions which decrease fromD=2.00 to values nearD=1.96 as stress normal to the fractures is increased from 3 MPa up to 85 MPa. The effect of stress on fluid flow is studied in the same samples. Fluid transport through a fracture depends on two properties of the fracture void space geometry. the void aperture; and the tortuosity of the flow paths, determined through the distribution of contact area. Each of these quantities change under stress and contribute to changes observed in the flow rate. A general flow law is presented which separates these different effects. The effects of tortuosity on flow are largely governed by the proximity of the flow path distribution to a percolation threshold. A fractal model of correlated continuum percolation is presented which quantitatively reproduces the flow path geometries. The fractal dimension in this model is fit to the measured fractal dimensions of the flow systems to determine how far the flow systems are above the percolation threshold.     

Numerical examination of the factors controlling DNAPL migration through a single fracture

The migration of five dense nonaqueous phase liquids (DNAPLs) through a single fracture in a clay aquitard was numerically simulated with the use of a compositional simulator. The effects of fracture aperture, fracture dip, matrix porosity, and matrix organic carbon content on the migration of chlorobenzene, 1,2-dichloroethylene, trichloroethylene, tetra-chloroethylene, and 1,2-dibromoethane were examined. Boundary conditions were chosen such that DNAPL entry into the system was allowed to vary according to the stresses applied. The aperture is the most important factor of those studied controlling the migration rate of DNAPL through a single fracture embedded in a clay matrix. Loss of mass to the matrix through diffusion does not significantly retard the migration rate of the DNAPL, particularly in larger aperture fractures (e.g., 50 microm). With time, the ratio of diffusive loss to the matrix to DNAPL flux into the fracture approaches an asymptotic value lower than unity. The implication is that matrix diffusion cannot arrest the migration of DNAPL in a single fracture. The complex relationships between density, viscosity, and solubility that, to some extent, govern the migration of DNAPL through these systems prevent accurate predictions without the use of numerical models. The contamination potential of the migrating DNAPL is significantly increased through the transfer of mass to the matrix. The occurrence of opposite concentration gradients within the matrix can cause dissolved phase contamination to exist in the system for more than 1000 years after the DNAPL has been completely removed from the fracture.     

A Scheme for Initial Beam Deployment for the International Monitoring System Arrays

v--vThe International Monitoring System (IMS) includes a diverse set of seismic arrays with different configurations. These configurations have apertures ranging from less than 1 to more than 25 km and minimum interelement spacings varying from 0.1 to 3.6 km. This paper presents a scheme for initial beam deployment for this variety of seismic arrays. Beamforming is equivalent to a spatiotemporal bandpass filter of which passband is defined by the minimum and maximum wavenumbers, which are functions of the geometry configuration of the array. Deployment of steered-beams for signal detection is based on the wavenumber resolution of the array, slowness and frequency distributions of seismic phases, and coherence properties of seismic signals and noises among sensors. Within the wavenumber passband, all possible slowness values are determined by the resolution for each frequency band, and those that are outside the range of seismological interest are excluded. The appropriate azimuthal distribution for each selected slowness is determined from the azimuthal resolution. Using this approach, detection beams for each array are rationally deployed in the slowness-azimuth and frequency domain.     

Horizontal rough fracture type curves for groundwater movement

Abstract

Type curves are derived analytically for radial flow in rough horizontal fractures toward a well. The basic assumptions are that there is no turbulent flow near the borehole and the well storage is ignored. The basis of the methodology is to write explicit expressions for the continuity and cubic law flow equations, which are combined using a Boltzmann transformation leading to a simple ordinary differential equation for groundwater movement. Solutions are presented as a set of type curves for different fracture apertures. It is observed that the solutions provide a method of uniquely identifying fracture hydraulic parameters when the fracture is smooth, but pose ambiguity for rough fracture parameter estimations. However, large time portions of these type curves appear as straight lines on semi-logarithmic paper, which provides a unique way for rough fracture parameter determination. Identification of the fracture parameters, namely, the aperture and relative roughness, is possible in a unique manner with the use of these lines and the dimensionless time drawdown concept. The cubic law is the asymptotic behaviour, either for large times or large fracture apertures. Prior to this asymptotic part, there is a non-cubic portion which gives rise to systematic deviations from the cubic law. The technique presented is useful, especially for evaluating pumping tests from a single major fracture isolated by packers.     

Variable-density groundwater flow and solute transport in irregular 2D fracture networks

Numerical simulations of variable-density flow and solute transport have been conducted to investigate dense plume migration for various configurations of 2D fracture networks. For orthogonal fractures, simulations demonstrate that dispersive mixing in fractures with small aperture does not stabilize vertical plume migration in fractures with large aperture. Simulations in non-orthogonal 2D fracture networks indicate that convection cells form and that they overlap both the porous matrix and fractures. Thus, transport rates in convection cells depend on matrix and fracture flow properties. A series of simulations in statistically equivalent networks of fractures with irregular orientation show that the migration of a dense plume is highly sensitive to the geometry of the network. If fractures in a random network are connected equidistantly to the solute source, few equidistantly distributed fractures favor density-driven transport. On the other hand, numerous fractures have a stabilizing effect, especially if diffusive transport rates are high. A sensitivity analysis for a network with few equidistantly distributed fractures shows that low fracture aperture, low matrix permeability and high matrix porosity impede density-driven transport because these parameters reduce groundwater flow velocities in both the matrix and the fractures. Enhanced molecular diffusion slows down density-driven transport because it favors solute diffusion from the fractures into the low-permeability porous matrix where groundwater velocities are smaller. For the configurations tested, variable-density flow and solute transport are most sensitive to the permeability and porosity of the matrix, which are properties that can be determined more accurately than the geometry and hydraulic properties of the fracture network, which have a smaller impact on density-driven transport.     

裂缝和孔洞型储层孔隙模型的理论进展

对有洞的和裂缝型储层的分析已经成为一个热点,因而孔隙模型的研究近年得到了很好的发展.目前已经用双孔隙和三孔隙模型研究这类储层的特性并寻找估计孔隙指数的方法,以便计算含水饱和度.用串联或并联电阻网络模拟储层表明：双孔隙模型适用于基质与非连通孔洞储层以及裂缝和（或）连通孔洞的储层.三孔隙模型更适用于由基质、裂缝和不连通孔洞组成储层的岩石物理评价,对于当前碳酸盐岩和火成岩以及变质岩储层评价具有明显的指导意义.     

Eddy correlations for water flow in a single fracture with abruptly changing aperture

Fluid flow in single fractures with non‐uniform apertures is an important research subject in many disciplines. The abruptly changing aperture is a special case of such non‐uniformity. This paper simulates water flow in a single fracture with abruptly changing aperture (SF‐ACA) using the Lattice Boltzmann Method (LBM) and the Finite Volume Method (FVM). The flow occurs with the Reynolds number (Re) ranging from 5 to 900 and a ratio of aperture change (E) of 3 (E = D/d, where D and d are the larger and smaller apertures, respectively). For Re values between 5 and 100, both LBM and FVM can successfully simulate the eddy development in the expansion regime of an SF‐ACA. Flow with high Re values (up to 900) is simulated by FVM, which appears to be numerically more stable than LBM for high‐Re flow problems studied here. The flow symmetry in the expansion regime breaks at the Re value between 400 and 500. Our simulation result shows a linear relationship between l₁/d and Re at low Re (5–100) or higher Re (110–900) values, where defined as the length from the location of abrupt expansion to the right edge of the first eddy along the flow direction. If considering the simulation results for the entire simulated range of Re (5–900), the l₁/d–Re relationship is better described by a non‐linear logarithmical function. The l₁/d approaches an asymptotic constant at large Re. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical Study on Behavior of Groundwater Flow in Single Rough Fractures Under Mechanical Effect

This paper numerically investigates the characteristics of groundwater flow in spatially correlated variable aperture fractures under the mechanical effect. Spatially correlated aperture distributions are generated using the geostatistical method (i.e., turning bands algorithm in this study). To represent a nonlinear relationship between the effective normal stress and the fracture aperture, a simple mechanical formula is combined with a local flow model. Numerical results indicate that the groundwater flow is significantly affected by the geometry of aperture distribution, varying with the applied effective normal stress as well as the spatial correlation length of aperture distribution. Moreover, using the flow results simulated in this study, two empirical formulae are proposed: (1) the first one (modified Louis formula) is to represent the relationship between the effective normal stress and the effective permeability of fracture and (2) the second one is to represent the relationship between relative roughness and effective permeability.     

Numerical investigation of alternative fracture stiffness measures and their respective scaling behaviours

We study the mechanical deformation of fractures under normal stress, via tangent and specific fracture stiffnesses, for different length scales using numerical simulations and analytical insights. First, we revisit an equivalent elastic layer model that leads to two expressions: the tangent stiffness is the sum of an “intrinsic” stiffness and the normal stress, and the specific stiffness is the tangent stiffness divided by the fracture aperture at current stress. Second, we simulate the deformation of rough fractures using a boundary element method where fracture surfaces represented by elastic asperities on an elastic half‐space follow a self‐affine distribution. A large number of statistically identical “parent” fractures are generated, from which sub‐fractures of smaller dimensions are extracted. The self‐affine distribution implies that the stress‐free fracture aperture increases with fracture length with a power law in agreement with the chosen Hurst exponent. All simulated fractures exhibit an increase in the specific stiffness with stress and an average decrease with increase in length consistent with field observations. The simulated specific and tangent stiffnesses are well described by the equivalent layer model provided the “intrinsic” stiffness slightly decreases with fracture length following a power law. By combining numerical simulations and the analytical model, the effect of scale and stress on fracture stiffness measures can be easily separated using the concept of “intrinsic” stiffness. We learn that the primary reason for the variability in specific stiffness with length comes from the fact that the typical aperture of the self‐affine fractures itself scales with the length of the fractures.     

APPLICATION OF BOREHOLE GEOPHYSICS AT AN EXPERIMENTAL WASTE STORAGE SITE*

A suite of electrical, radiation, and mechanical borehole probes were run in a 76-mm-diameter borehole drilled to a slant depth of 380 m in leptite and granite. The hole is located in Precambrian bedrock in central Sweden where a site is dedicated to in-situ experiments pertaining to the disposal of radioactive wastes. The challenge to borehole logging methods for such site investigations is to resolve geological features and fluid flow parameters in geological sites which are initially chosen for their homogeneity, low porosity, and minimal fracturing. The Stripa borehole is characterized by high electrical resistivity values in the 20–100 kΩm range, by acoustic velocities around 5800 m s^-1 (which is close to laboratory values on intact specimens), and by total porosity of around one volume percent. In this context, probe resolution was adequate to produce interpretable information on almost all of the logs. Two principal rock types were encountered in the hole: granite, of quartz monzonitic composition, and leptite. The granite and leptite intercepts are subdivided into units characterized by mafic mineral content, sulfide mineral content, and electrical and radiation properties. Iron-rich zones in the leptite are highly anomalous on the gamma-gamma and neutron logs; thin mafic zones in the granite can also be distinguished. Occurrences of a few percent pyrite are detected by the electrical, gamma-gamma, and neutron logs. Although overall porosity is quite low throughout the hole, analysis of the resistivity and neutron logs indicates the porosity increases by a few volume percent at fracture zones. The differential resistance and caliper probes detect borehole diameter roughness of less than 1 mm, helping to confirm acoustic waveform anomalies which are indicative of fracture zones. Compres-sional wave transit time and shear-wave interference patterns usually occur coincident with open fractures observed in core, the correlation being especially good at major fracture zones.     

用正演数值计算方法开展双侧向测井对裂缝的响应研究

采用有限元分析软件ANSYS对裂缝的双侧向测井视电阻率与裂缝孔隙度、泥浆电阻率、裂缝倾角和基岩电阻率的关系进行了计算. 在大量正演数据的基础上得出双侧向测井响应反演公式和裂缝孔隙度计算公式,提出更为精细的裂缝倾角的弹性划分模型,用双侧向视电阻率值近似估算裂缝倾角的方法,提高了利用双侧向测井求裂缝产状与裂缝孔隙度（裂缝宽度）的精度.     

Compaction of a Rock Fracture Moderated by Competing Roles of Stress Corrosion and Pressure Solution

Unusually rapid closure of stressed fractures, observed in the initial stages of loading and at low temperatures, is examined using models for subcritical crack growth and pressure solution. The model for stress corrosion examines tensile stress concentrations induced at the Hertzian contact of propping fracture asperities, and mediates fracture growth according to a kinetic rate law. Conversely, pressure solution is described by the rate-limiting process of dissolution, resulting from the elevated stresses realized at the propping asperity contact. Both models are capable of following the observed compaction of fractures in novaculite. However, closure rates predicted for stress corrosion cracking are orders of magnitudes faster than those predicted for pressure dissolution. For consistent kinetic parameters, predictions from stress corrosion better replicate experimental observations, especially in the short-term and at low temperature when mechanical effects are anticipated to dominate. Rates and magnitudes of both stress corrosion and pressure solution are dependent on stresses exerted over propping asperities. Rates of closure due to stress corrosion cracking are shown to be always higher than for pressure solution, except where stress corrosion ceases as contact areas grow, and local stresses drop below an activation threshold. A simple rate law is apparent for the progress of fracture closure, defined in terms of a constant and an exponent applied to the test duration. For current experimental observations, this rate law is shown to replicate early progress data, and shows promise to define the evolution of transport properties of fractures over extended durations.