Noelle E. Odling

SCALING OF FRACTURE SYSTEMS IN GEOLOGICAL MEDIA

Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

Scaling and connectivity of joint systems in sandstones from western Norway

Abstract The scaling properties of a joint system in Devonian sandstones in western Norway have been investigated using seven maps, covering areas from 18 to 720 m across, which were generated by mapping in the field and from low-level aerial photography taken from different heights. Each map represent a scale ‘window’ on the fracture population, bounded by resolution at small scales and the sample size at large scales. A power-law relationship between fracture trace length and critical observation height (maximum height at which a trace can be identified) is derived and used to create a statistical model for the resolution effects. The model indicates that a continuous smooth curve without a straight-line segment on a log-log cumulative frequency distribution plot does not necessarily rule out a power law as the underlying population distribution. Together, the maps indicate a powerlaw trace-length distribution with an exponent of −2.1. This power law may be valid over four or more orders of magnitude, with natural lower cut-off of around 1 m. The exponent is significantly different from −2.0 (strictly selfsimilar case) and is reflected in a decrease in the abundance of fractures with length comparable to map size, as map scale decreases. The fracture trace-length distribution results in a decrease in apparent connectivity, with decreasing scale. High resolution (large-scale) maps are well connected while the lowest resolution map (smallest scale) is unconnected. Fractures in the smallest scale map are, however, connected by small fractures below the limit of resolution, represented by the largest scale map. The variation in apparent connectivity with scale has implications for fluid flow. When fractures are open to fluid flow the scaling properties of apparent connectivity imply that, beyond a certain scale, the size of fracture controlling flow will be scale-independent. In this fracture system, this appears to occur in sample areas of around 300 m across.

Network properties of a two-dimensional natural fracture pattern

The influence that fractures exert on the permeability of a fractured rock is, to a large extent, controlled by the nature of the network formed by the fracture system. Here, the network properties of a two-dimensional natural pattern, mapped from the surface of a sandstone layer, are investigated and compared to those of realizations of spatially randomly distributed line segments with similar orientation and length distributions and line segment density (line length per unit area) to the natural pattern. These patterns are composed of clusters of varying size and shape, made up of interconnected fracture traces or line segments. Comparing the natural pattern with the realizations, the natural pattern was found to contain roughly half the number of clusters while the mass (total line length) of the largest cluster is approximately double that of the realizations. The size of the largest cluster controls the connectivity of the patterns, as can be seen by comparing the largest cluster of the natural pattern, which connects all four sides of the region, with those of the realizations, which are unconnected or connect only two sides. Cluster scaling characteristics were found to be similar in the natural pattern and the realizations and show a crossover from a dimension of one (their topological dimension) to two (the dimension of the embedding medium) at a point that corresponds to the fracture spacing. An investigation of the self-similarity dimension, using the box-counting method, showed similar characteristics with a broad transition zone between one- and two-dimensional behaviour at smaller box sizes. The patterns are therefore found to be non-fractal. The effect of the spatial distribution shown by the natural pattern is thus to modify the manner in which fractures are distributed among clusters, increasing connectivity (and permeability in the case of open fractures), but does not affect the cluster scaling characteristics or the self-similarity dimension of the fracture patterns.

A “Conductance” Mesh Approach to the Permeability of Natural and Simulated Fracture Patterns

The “conductance” mesh method is used to determine the permeability properties of natural and simulated, two-dimensional, fracture patterns. In the method, the fractures and their matrix are discretized on a grid of 200 × 200 elements. The permeability of the whole pattern in the direction of the pressure gradient is determined from the total flow through the network. By applying the model in varying orientations, a picture of the directional permeability is constructed. Using the model, the effects of varying matrix permeability and fracture aperture on the global permeability of a natural fracture pattern and 10 simulations are investigated. The simulated fracture patterns were found to increasingly underestimate the permeability of the natural pattern as the permeability contrast between rock matrix and fractures increased. The lower permeability of the simulations was found to relate to their poorer connectivity, as reflected in the sizes of their “backbones” (comprising all paths across the region along the fracture network) and in the number and type of fracture intersections. The results show that the effect of the natural pattern spatial distribution is to promote fracture intersection, thereby increasing connectivity and permeability.

Contaminant transport in fractured rocks with significant matrix permeability, using natural fracture geometries

Abstract Some results from numerical models of flow and contaminant transport in fractured permeable rocks, where fractures are more conductive than rock matrix, are described. The 2D flow field in the fractured and permeable rock matrix is calculated using a finite difference, ‘conductance mesh’ method, and the contaminant transport is simulated by particle tracking methods using an advection-biased, random walk technique. The model is applied to simulated and naturally occurring fracture patterns. The simulated pattern is an en echelon array of unconnected fractures, as an example of a common, naturally occurring fracture geometry. Two natural fracture patterns are used: one of unconnected, sub-parallel fractures and one with oblique fracture sets which is well connected. Commonly occurring matrix permeability and fracture aperture values are chosen. The simulations show that the presence of fractures creates complex and heterogeneous flow fields and contaminant distribution in the permeable rock matrix. The modelling results have shown that some effects are non-intuitive and therefore difficult to foresee without the help of a model. With respect to contaminant transport rates and plume heterogeneity, it was found that fracture connectivity (crucial when the matrix is impermeable) can play a secondary role to fracture orientation and density. Connected fracture systems can produce smooth break-through curves of contaminants summed over, for example, a bore-hole length, whereas in detail the contaminant plume is spatially highly heterogeneous. Close to a constant-pressure boundary (e.g. an extraction bore-hole), flow and contaminants can be channelled by fractures. Thus observations at a bore-hole may suggest that contaminants are largely confined to the fracture system, when, in fact, significant contamination resides in the matrix.

Predicting the three-dimensional population characteristics of fault zones: a study using stochastic models

Major fault zones are surrounded by damage zones composed of minor faults that, in siliclastic rocks, often form significant barriers to fluid flow. Information on fault damage zone architecture is usually available only as 2D maps, or as 1D line samples or well logs. However, the accurate determination of the 3D fault population characteristics is crucial for flow prediction. In this paper, stochastic models of fault damage zones are generated by incorporating the statistical properties of fault populations (power law length and throw distributions, orientation distribution) and different spatial distributions, including randomly located, simple and hierarchical clustering of faults. These damage zone models are used to investigate the characteristics of 2D and 1D samples, which were found to depend on the 3D power law length exponent, D3, and the spatial distribution of the parent 3D population. Observed 1D samples may fail to show power law characteristics and, therefore, a lack of power law behaviour need not imply a non-power law parent population. The simple rules in which observed 2D and 1D samples follow power laws with exponents D2=D3−1 and D1=D3−2, respectively, are not always obeyed. Clustering tends to reduce the difference between these exponents to less than their simple integer values, most markedly for the simple clustering model. The hierarchical clustering model, in which small faults are clustered around larger faults throughout the fault damage zone and which most closely resembles nature, suggests that the simple rule D2=D3−1 is obeyed with only small deviations but that 1D samples may depart from the simple rule, D1=D3−2, by as much as 0.25.

A Forward Modeling Approach for Interpreting Impeller Flow Logs

A rigorous and practical approach for interpretation of impeller flow log data to determine vertical variations in hydraulic conductivity is presented and applied to two well logs from a Chalk aquifer in England. Impeller flow logging involves measuring vertical flow speed in a pumped well and using changes in flow with depth to infer the locations and magnitudes of inflows into the well. However, the measured flow logs are typically noisy, which leads to spurious hydraulic conductivity values where simplistic interpretation approaches are applied. In this study, a new method for interpretation is presented, which first defines a series of physical models for hydraulic conductivity variation with depth and then fits the models to the data, using a regression technique. Some of the models will be rejected as they are physically unrealistic. The best model is then selected from the remaining models using a maximum likelihood approach. This balances model complexity against fit, for example, using Akaikes Information Criterion.

Fractional flow in fractured chalk; a flow and tracer test revisited.

A multi-borehole pumping and tracer test in fractured chalk is revisited and reinterpreted in the light of fractional flow. Pumping test data analyzed using a fractional flow model gives sub-spherical flow dimensions of 2.2-2.4 which are interpreted as due to the partially penetrating nature of the pumped borehole. The fractional flow model offers greater versatility than classical methods for interpreting pumping tests in fractured aquifers but its use has been hampered because the hydraulic parameters derived are hard to interpret. A method is developed to convert apparent transmissivity and storativity (L(4-n)/T and S(2-n)) to conventional transmissivity and storativity (L2/T and dimensionless) for the case where flow dimension, 2<n<3. These parameters may then be used in further applications, facilitating application of the fractional flow model. In the case illustrated, improved fits to drawdown data are obtained and the resultant transmissivities and storativities are found to be lower by 30% and an order of magnitude respectively, than estimates from classical methods. The revised hydraulic parameters are used in a reinterpretation of a tracer test using an analytical dual porosity model of solute transport incorporating matrix diffusion and modified for fractional flow. Model results show smaller fracture apertures, spacings and dispersivities than those when 2D flow is assumed. The pumping and tracer test results and modeling presented illustrate the importance of recognizing the potential fractional nature of flow generated by partially penetrating boreholes in fractured aquifers in estimating aquifer properties and interpreting tracer breakthrough curves.

Borehole water level response to barometric pressure as an indicator of aquifer vulnerability

The response of borehole water levels to barometric pressure is a function of the confining layer and aquifer properties. This study aims to use this response as an aid towards quantitative assessment of groundwater vulnerability, applying the techniques to the confined/semi-confined part of the Chalk Aquifer in East Yorkshire, UK. Time series analysis techniques are applied to data collected from twelve monitoring boreholes to characterize and remove components contributing to the borehole water level signal other than barometric pressure, such as recharge and Earth tides. Barometric response functions are estimated using the cross-spectral deconvolutionaveraging technique performed with up to five overlapping frequency bands. A theoretical model was then fitted to the observed barometric response functions in order to obtain estimates of aquifer and confining layer properties. Derived ranges for pneumatic and hydraulic diffusivities of the confining layer vary over four orders of magnitudes (0.9 to 128.0 m2/day and 10.0 to 5.0×104 m2/day respectively) indicating that the aquifer is nowhere purely confined. Discrepancies between estimates of aquifer transmissivity derived from the barometric response function and pumping tests have been explored using slug tests and results suggest that aquifer model transmissivity are highly sensitive to borehole construction. A simple flow model, constructed to test the potential impact of confining layer heterogeneity on the barometric response function, shows that while high frequencies reflect the immediate vicinity of the borehole, low frequencies detect confining layer properties up to some 500 meters distant from the borehole. A ‘characteristic time scale’ is introduced as a function of derived properties of the confining layer and is used as a quantitative measure of the degree of aquifer confinement. It is concluded that barometric response functions are sensitive to confining layer properties and thus can provide a useful tool for the assessment of aquifer vulnerability.

The scaling of hydraulic conductivity in rock fracture zones

Beyond the scale of a few meters, fracture traces as viewed on aerial photographs represent fracture zones. Thus for bulk fractured rock permeability, the hydraulic properties of fracture zones are important. Here, observations on joint zones in sandstones from western Norway are used to build a quantitative model relating joint zone length and effective hydraulic conductivity. Using simple rules for hydraulic conductors in series and parallel applied to joint zone geometry, the ‘effective’ hydraulic aperture (the aperture of a single parallel-walled channel that conducts the same flow) is estimated for 72 joint zones of known length. A power law relation between trace length, l, and effective hydraulic aperture, be, with an effective exponent of around 0.27 is found which may extend to km scales. This exponent contrasts with values of 0.5 to 2.0 reported for veins.