Validity of Cubic Law for fluid flow in a deformable rock fracture

TL;DR: 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.

Abstract: 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.

Summary

3. DISCUSSION OF RESULTS

• At any given stress, the experimentally determined quantities were the apparent aperture, the flow rate, and the difference in hydraulic head.
• For a iven aperture, Q was observed to be proportional to Ah under the experimen tal conditions (Iwai, 1976) , and thus Darcy's law holds.
• The apparent aperture could not be used directly in equations 2 or 10 to check the validity of the cubic law because the measured flow rates depend on the true aperture, 2b, which could not be measured directly.
• One might wish to estimate the residual value using equation 2 but this would introduce a bias toward the assumed cubic law, especially for the small apertures at large stresses.

Figures

Fig. 6. Comparison of experimental results for radial flow through tension fracture in granite with cubic law. In Run 3, frac ture surfaces were no longer in contact during unloading

Table 1. Results of least squares fit for parameters n and 2b r.

Fig. 9. Idealized fracture showing the mating of asperities as the fracture surfaces are closed under stress.

Fig. 3. Mechanical properties of fractured granite sample used in straight flow model (after Iwai, 1976).

Table 2. Results of least squares fit for parameters of f and 2b r.

Fig. 8. Comparison of experimental results for radial flow through tension fracture in marble with cubic law. In Runs 2 and 3, fracture surfaces were no longer in contact during unloading when aperture exceeded value indicated by arrow.

Fig. 4. Mechanical properties of fracture used in determining changes in aperture with stress.

Fig. 2. Effect of cyclic loading on permeability of tension fracture in granite with radial flow (after Iwai, 1976).

Fig. 5. Comparison of experiment for straight flow through tension fracture in granite with cubic law.

Fig. 4. Mechanical properties of fracture used in determining changes in aperture with stress.

Full text

Lawrence Berkeley National Laboratory
Lawrence Berkeley National Laboratory
Title
VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK
FRACTURE
Permalink
https://escholarship.org/uc/item/2276m4b5
Author
Witherspoon, P.A.
Publication Date
1979-10-01
Peer reviewed
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University of California

Submitted for publication to Water Resources Research.
LBL-9557
SAC-23
UC-70
VALIDITY OF CUBIC LAW FOR FLUID FLOW
IN A DEFORMABLE ROCK FRACTURE
P. A. Witherspoon, J. S. Y. Wang, K. Iwail, and J. E. Gale
2
Department of Materials Science and Mineral Engineering
University of California, Berkeley
and
Lawrence Berkeley Laboratory
University of California
Berkeley, California
October 1979
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1
Now with Nakano Corporation, Niigatashi, Japan.
Now at the Department of Earth Sciences, University of Waterloo, Waterloo,
Ontario, Canada.
This report was prepared by the Lawrence Berkeley Laboratory under the
University of California contract W-7405-ENG-48 with the Department of
Energy. Funding for this project is administered by the Office of Nuclear
Waste Isolation at Battelle Memorial Institute.
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PREFACE
This report is one of a series documenting the results of the Swedish-
American cooperative research program in which the cooperating scientists
explore the geological, geophysical, hydrological, geochemical, and structural
effects anticipated from the use of a large crystalline rock mass as a geo-
logic repository for nuclear waste. This program has been sponsored by the
Swedish Nuclear Power Utilities through the Swedish Nuclear Fuel Supply Com-
pany (SKBF), and the U. S. Department of Energy (DOE) through the Lawrence
Berkeley Laboratory (LBL).
The principal investigators are L. B. Nilsson and 0. Degerman for SKBF,
and N. G. W. Cook, P. A. Witherspoon, and J. E. Gale for LBL. Other partici-
pants will appear as authors of the individual reports.
Previous technical reports in this series are listed below.
1.
Swedish-American Cooperative Program on Radioactive Waste Storage in
Mined Caverns by P. A. Witherspoon and 0. Degerman. (LBL-7049, SAC-01).
2.
Large Scale Permeability Test of the Granite in the Stripa Mine and Ther-
mal Conductivity Test by Lars Lundstrom and Haken Stille. (LBL-7052,
SAC-02).
3. The Mechanical Properties of the Stripa Granite by Graham Swan.
(LBL-7074, SAC-03).
4.
Stress Measurements in the Stripa Granite by H. Carlsson. (LBL-7078,
SAC-04).
b. Borehole Drilling and Related Activities at the Stripa Mine by Pavel J.
Kurfurst, T. Hugo-Persson, and G. Rudolph. (LBL-7080, SAC-05).
6. A Pilot Heater Test in the Stripa Granite by Hans Carlsson. (LBL-7086,
SAC-06).
7. An Analysis of Measured Values for the State of Stress in the Earth's
Crust by Dennis B. Jamison and Neville G. W. Cook.
(LBL-7071,
SAC-07).
8. Mining Methods Used in the Underground Tunnels and Test Rooms at Stripa
by B. Andersson and P. A. Halen.
(LBL-7081,
SAC-08).
9. Theoretical Temperature Fields for the Stripa Heater Project by Tin Chan,
Neville G. W. Cook, and Chin Fu Tsang. (LBL-7082, SAC-09).

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10.
Mechanical and Thermal Design Considerations for Radioactive Waste Repos-
itories in Hard Rock. Part I: An Appraisal of Hard Rock for Potential
Underground Repositories of Radioactive Wastes by Neville G. W. Cook;
Part II; In Situ Heating Experiments in Hard Rock: Their Objectives and
Design by Neville G. W. Cook and Paul A. Witherspoon. (LBL-7073,
SAC-10).
11.
Full-Scale and Time-Scale Heating Experiments at Stripa: Preliminary
Results by Neville G. W. Cook and Michael Hood. (LBL-7072,
SAC-ll).
12.
Geochemistry and Isotope Hydrology of Groundwaters in the Stripa Granite:
Results and Preliminary Interpretation by P. Fritz, J. F.Barker, and
J.'E. Gale. (LBL 8285,
SAC-12).
13.
Electrical Heaters for Thermomechanical Tests at the Stripa Mine by R. H.
Burleigh, E. P. Binnall, A. 0. DuBois, D. U. Norgren, and A. R.Ortiz.
(LBL-7063,
SAC-13).
14.
Data Acquisition, Handling, and Display for the Heater Experiments at
Stripa by Maurice B. McEvoy. (LBL-7062,
SAC-14).
15.
An Approach to the Fracture Hydrology at Stripa: Preliminary Results by
J. E. Gale and P. A. Witherspoon. (LBL-7079,
SAC-15).
16.
Preliminary Report on Geophysical and Mechanical Borehole Measurements at
Stripa by P. Nelson, B. Paulsson, R. Rachiele, L. Andersson, T. Schrauf,
W. Hustrulid, 0. Duran, and K. A. Magnusson. (LBL-8280,
SAC-16).
17.
Observations of a Potential Size-Effect in Experimental Determination of
The Hydraulic Properties of Fractures by P. A. Witherspoon, C. H. Amick,
J. E. Gale, and K. Iwai. (LBL-8571,
SAC-17).
18.
Rock Mass Characterization for Storage of Nuclear Waste in Granite by
P. A. Witherspoon, P. Nelson, T. Doe, R. Thorpe, B. Paulsson, J. Gale,
and C. Forster. (LBL-8570,
SAC-18).
19.
Fracture Detection in Crystalline Rock Using Shear Waves by K. H. Waters,
S. P. Palmer, and W. E. Farrell. (LBL-7051,
SAC-19).
20.
Characterization of Discontinuities in the Stripa Granite - Time-Scale
Heater Experiment by Richard Thorpe. (LBL-7083,
SAC-20).
21.
Geology and Fracture System at Stripa by A. Olkiewicz, J. E. Gale,
R. Thorpe, and B. Paulsson. (LBL-8907,
SAC-21).
22.
Calculated Thermally Induced Displacements and Stresses for Heater Experi-
ments at Stripa by T. Chan and N. G. W. Cook. (LBL-7061,
SAC-22).

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TABLE OF CONTENTS
LIST OF FIGURES vi
LIST OF TABLES vi
NOMENCLATURE vii
ABSTRACT 1
1. INTRODUCTION 3
2.
LABORATORY PROCEDURES 7
3. DISCUSSION OF RESULTS 14
4.
CONCLUSIONS 26
5. ACKNOWLEDGMENTS , 27
6. REFERENCES 27
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