The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering

In this paper, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed. Both macroscopically homogeneous and heterogeneous rocks are considered. The latter are characterized by large-scale spatial variations and correlations in their effective properties and include rocks that may be characterized by several distinct degrees of porosity, a well-known example of which is a fractured rock with two degrees of porosity---those of the pores and of the fractures. First, the diagenetic processes that give rise to the present reservoir rocks are discussed and a few geometrical models of such processes are described. Then, measurement and characterization of important properties, such as pore-size distribution, pore-space topology, and pore surface roughness, and morphological properties of fracture networks are discussed. It is shown that fractal and percolation concepts play important roles in the characterization of rocks, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Next, various structural models of homogeneous and heterogeneous rock are discussed, and theoretical and computer simulation approaches to flow, dispersion, and displacement in such systems are reviewed. Two different modeling approaches to these phenomena are compared. The first approach is based on the classical equations of transport supplemented with constitutive equations describing the transport and other important coefficients and parameters. These are called the continuum models. The second approach is based on network models of pore space and fractured rocks; it models the phenomena at the smallest scale, a pore or fracture, and then employs large-scale simulation and modern concepts of the statistical physics of disordered systems, such as scaling and universality, to obtain the macroscopic properties of the system. The fundamental roles of the interconnectivity of the rock and its wetting properties in dispersion and two-phase flows, and those of microscopic and macroscopic heterogeneities in miscible displacements are emphasized. Two important conceptual advances for modeling fractured rocks and studying flow phenomena in porous media are also discussed. The first, based on cellular automata, can in principle be used for computing macroscopic properties of flow phenomena in any porous medium, regardless of the complexity of its structure. The second, simulated annealing, borrowed from optimization processes and the statistical mechanics of spin glasses, is used for finding the optimum structure of a fractured reservoir that honors a limited amount of experimental data.

Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing

Fluid flow through rock joints is commonly described by the parallel plate model where the volume flow rate varies as the cube of the joint aperture. However, deviations from this model are expected because real joint surfaces are rough and contact each other at discrete points. To examine this problem further, a computer simulation of flow between rough surfaces was done. Realistic rough surfaces were generated numerically using a fractal model of surface topography. Pairs of these surfaces were placed together to form a “joint” with a random aperture distribution. Reynolds equation, which describes laminar flow between slightly nonplanar and nonparallel surfaces, was solved on the two-dimensional aperture mesh by the finite-difference method. The solution is the local volume flow rate through the joint. This solution was used directly in the cubic law to get the so-called “hydraulic aperture.” For various surface roughnesses (fractal dimensions) the hydraulic aperture was compared to the mean separation of the surfaces. At large separations the surface topography has little effect. At small separations the flow is tortuous, tending to be channeled through high-aperture regions. The parameter most affecting fluid flow through rough joints is the ratio of the mean separation between the surfaces to the root-mean-square surface height. This parameter describes the distance the surface asperities protrude into the fluid and accounts for most of the disagreement with the parallel plate model. Variations in the fractal dimension produce only a second-order effect on the fluid flow. For the range of joint closures expected during elastic deformation these results show that the actual flow rate between rough surfaces is about 70–90% of that predicted by the parallel plate model.

Fluid flow through rock joints: The effect of surface roughness

Natural fractures have long been suspected as a factor in production from shale reservoirs because gas and oil production commonly exceeds the rates expected from low-porosity and low-permeability shale host rock. Many shale outcrops, cores, and image logs contain fractures or fracture traces, and microseismic event patterns associated with hydraulic-fracture stimulation have been ascribed to natural fracture reactivation. Here we review previous work, and present new core and outcrop data from 18 shale plays that reveal common types of shale fractures and their mineralization, orientation, and size patterns. A wide range of shales have a common suite of types and configurations of fractures: those at high angle to bedding, faults, bed-parallel fractures, early compacted fractures, and fractures associated with concretions. These fractures differ markedly in their prevalence and arrangement within each shale play, however, constituting different fracture stratigraphies—differences that depend on interface and mechanical properties governed by depositional, diagenetic, and structural setting. Several mechanisms may act independently or in combination to cause fracture growth, including differential compaction, local and regional stress changes associated with tectonic events, strain accommodation around large structures, catagenesis, and uplift. Fracture systems in shales are heterogeneous; they can enhance or detract from producibility, augment or reduce rock strength and the propensity to interact with hydraulic-fracture stimulation. Burial history and fracture diagenesis influence fracture attributes and may provide more information for fracture prediction than is commonly appreciated. The role of microfractures in production from shale is currently poorly understood yet potentially critical; we identify a need for further work in this field and on the role of natural fractures generally.

Natural fractures in shale: A review and new observations

The problem of determining the permeability of a rock mass containing a system of finite fractures is highly dependent on both the degree of interconnection between fractures and the heterogeneity of individual fracture characteristics. This paper examines how degree of interconnection affects both magnitude and nature of the fracture permeability. The interconnection between given fracture sets is a complex function of (1) fracture density, i.e., the number of fractures per unit volume, and (2) the fracture extent or size. Unfortunately, neither the density nor the extent of fractures is easily determined from borehole data. However, fracture frequency can be directly measured in a borehole because it is simply the number of fractures intersected per unit length of the borehole. The frequency is a measure of the product of fracture density and size because the probability of a fracture intersecting a borehole is proportional to this product. The effect of the degree of interconnection was investigated by numerically simulating flow in fracture networks where fracture size and density varied inversely, while the product of these two parameters was held fixed. Directional permeabilities of a number of such networks were determined, and the hydraulic behavior of each fracture system was compared to that of an ideal porous medium. The permeability of the rock matrix between the fractures was assumed to be low enough to be negligible. The results show that as fracture length increases, the degree of interconnection increases. Thus, for a given fracture frequency as measured in a borehole, the permeability of the system increases as fracture length is increased, and density is proportionally decreased. Also, fracture systems with shorter, but more dense fractures, behave less like porous media than do systems with longer but less dense fractures. Knowledge of fracture frequency and orientation alone is inadequate when determining permeability or deciding the important question of whether a given system behaves like a porous medium. For many cases where these parameters can be measured in the borehole it is critical to know fracture length in order to predict the degree of fracture network interconnection. However, for certain cases where fractures are larger than some critical size, it may not be necessary to know fracture lengths exactly.

The relationship of the degree of interconnection to permeability in fracture networks

A model for steady fluid flow in three-dimensional, random networks of fractures has been developed. In this model the fractures are disc shaped discontinuities in an impermeable matrix. The fracture discs can be arbitrarily located within the rock volume and can have any desired distribution of aperture, radius orientation, and density. Thus where the disc model is appropriate it is possible to calculate flow through fracture networks which are statistically similar to those that occur in nature. After the boundary conditions and the desired fractures are specified, the intersections (nodes) between these discs (elements) are identified. Then steady flow through the network is calculated using a mixed analytical-numerical technique. In each fracture, analytic equations for flow into or out of each node as a function of the average head at each node are developed. The equations are based on image theory and the assumption that each node is a source (or sink) of uniform strength. A set of mass balance equations is constructed which equate flow into a node from one of its associated fractures to flow out of the node into the other associated fracture. These equations are solved for the average head at each node, and flux between fractures can then be calculated by substituting the average head values back into the analytical equations. The model has been successfully checked against analytical results for several cases of two and three intersecting fractures. We plan to use these techniques to measure the permeability of fracture networks.

A Model for Steady Fluid Flow in Random Three‐Dimensional Networks of Disc‐Shaped Fractures

In the present paper, attempts are made to develop solutions to various models of slug tests that may be applicable in analyzing the results of such tests where existing solutions are inadequate. Various geometries that may be encountered in heterogeneous systems such as fractured rocks are considered. Solutions are presented for linear flow, radial flow with boundaries, two layer, and concentric composite models with different flow geometries between the inner and outer region. Solutions are obtained in Laplace space and inverted back to real space numerically. Type curves are presented for each solution. Analyses of the type curves and derivative response curves reveal that many curves have unique shapes only for certain combination of the flow parameters and the distance. Other sets of type curves are similar in shape, although log-log plots and derivative plots may emphasize some features that may not be apparent in semilog plots. These results show that slug tests suffer problems of nonuniqueness to a greater extent than other well tests.

Analytical models of slug tests

One of the characteristics of flow and transport in fractured rock is that the flow may be largely confined to a poorly connected network of fractures. In order to represent this condition, Lawrence Berkeley Laboratory has been developing a new type of fracture hydrology model called an “equivalent discontinuum” model. In this model we represent the discontinuous nature of the problem through flow on a partially filled lattice. This is done through a statistical inverse technique called “simulated annealing.” The fracture network model is “annealed” by continually modifying a base model, or “template,” so that with each modification, the model behaves more and more like the observed system. This template is constructed using geological and geophysical data to identify the regions that possibly conduct fluid and the probable orientations of channels that conduct fluid. In order to see how the simulated annealing algorithm works, we have developed a synthetic case. In this case, the geometry of the fracture network is completely known, so that the results of annealing to steady state data can be evaluated absolutely. We also analyze field data from the Migration Experiment at the Grimsel Rock Laboratory in Switzerland.

An inverse technique for developing models for fluid flow in fracture systems using simulated annealing

The authors discuss two inverse approaches to construction of fracture-flow models and their application in characterizing a fractured limestone formation. The first approach creates ``equivalent discontinuum`` models that conceptualize the fracture system as a partially filled lattice of conductors that are locally connected or disconnected to reproduce the observed hydrologic behavior. An alternative approach -- i.e., ``variable aperture lattice`` models -- represent the fracture system as a fully filled network composed of conductors of varying apertures or hydraulic conductivities. The fracture apertures are sampled from a specified distribution, usual log-normal, which is consistent with field data. The spatial arrangement of apertures is altered through inverse modeling to fit the available hydrologic data, such as transient pressure and/or tracer data. Unlike traditional discrete fracture-network approaches that rely on fracture geometry to reproduce flow and transport behavior, the inverse methods directly incorporate hydrologic data in deriving the fracture networks and thus naturally emphasize the underlying features that impact fluid flow and transport. However, hydrologic models derived by inversion are nonunique in general. The authors have addressed such nonuniqueness by examining an ensemble of models that satisfy the observation data within acceptable limits. They then determine properties that are shared by the ensemblemore » of models and their associated uncertainties to create a conceptual model of the fracture system. They show the fracture-flow model to be consistent with geophysical imaging.« less

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J.C.S. Long

Papers

The relationship of the degree of interconnection to permeability in fracture networks

A Model for Steady Fluid Flow in Random Three‐Dimensional Networks of Disc‐Shaped Fractures

Analytical models of slug tests

An inverse technique for developing models for fluid flow in fracture systems using simulated annealing

Detailed Characterization of a Fractured Limestone Formation by Use of Stochastic Inverse Approaches