ML13309A840
| ML13309A840 | |
| Person / Time | |
|---|---|
| Site: | San Onofre  |
| Issue date: | 10/31/1979 |
| From: | Iwai K, Jamie Wang, Witherspoon P CALIFORNIA, UNIV. OF, BERKELEY, CA, LAWRENCE LIVERMORE NATIONAL LABORATORY |
| To: | |
| Shared Package | |
| ML13302A500 | List:
|
| References | |
| LBL-9557, SAC-23, UC-70, NUDOCS 8103120709 | |
| Download: ML13309A840 (33) | |
Text
LBL-9557 Submitted for publication to Water Resources Research.
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 1 Now with Nakano Corporation, Niigatashi, Japan.
2 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.
8103120709
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. Degermnan.
(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-0 2)..
- 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).
- 5.
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).
-iv
- 10.
Mechanical and Thermal Design Considerations for Radioactive Waste Repos itories in Hard Rock.
Part 1:
An Appraisal of Hard Rock for PotentiaT Underyround 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-1).
- 11.
Full-Scale and Time-Scale Heating Experiments at Stripa:
Preliminary Results by Neville G. W. Cook and Michael Hood. (LBL-7072, SAC-11).
- 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).
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
-vi LIST OF FIGURES
- 1. Effect of cyclic loading on permeability of tension fracture in granite with straight flow.
10
- 2. Effect of cyclic loading on permeability of tension fracture in granite with radial flow.
11
- 3. Mechanical properties of fractured granite sample used in straight flow model............................... 12
- 4. Mechanical properties of fracture used in determining changes in aperture with stress...
13
- 5. Comparison of experiments for straight flow through tension frac ture in granite with cubic law.
18
- 6. Comparison of experimental results for radial flow through tension fracture in granite with cubic law.
20
- 7. Comparison of experimental results for radial flow through tension fracture in basalt with cubic law..
21
- 8. Comparison of experimental results for radial flow through tension fracture in marble with cubic law..
22
- 9. Idealized fracture showing the mating of asperities as the fracture surfaces are closed under stress..
. 24 LIST OF TABLES
- 1. Results of least squares fit for parameters of n and 2br.
. 15
- 2. Results of least squares fit for parameters of f and 2br..
.. 17
-vii NOMENCLATURE b
aperture half width L
bd apparent apperture half width L
br residual aperture half width L
C proportional constant in cubic law 1/LT D
hydraulic diameter (=4b)
L f
fracture surface characteristic factor y
acceleration of gravity L/T 2 h
hydraulic head L
Kf fracture hydraulic conductivity L/T L
length of fracture L
n exponent in fracture flow law Q
flow rate L /T Re Reynolds number re outer radius L
rw wellbore radius L
V fracture deformation L
Vm maximum fracture deformation L
Vr rock deformation L
Vt total deformation L
v flow velocity L/T W
width of fracture L
E height of asperities L
fluid density M/L' friction factor oe axial stress M/LT 2 P
flow viscosity M/LT weighting factor
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 pm. The law may be given in simplified form by Q/ah = C(2b) 3, where Q is the flow rate, Ah 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 using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced and the labor atory setup used radial as well as straight flow geometries. Apertures ranged from 250 pm down to 4 um, 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/f. The factor f varied from 1.04 to 1.65 in these investigations.
The model of a fracture that is being closed under normal stress is vis ualized as being controlled by the strength of the asperities that are in con tact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture de creases. The controlling factor is the magnitude of the aperture and since flow depends on (2b) 3, a slight change in aperture evidently can easily domi nate 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.
-3
- 1. INTRODUCTION The problem of laminar flow in a viscous incompressible fluid in a frac ture has been studied by many workers starting with Boussinesq (1868).
Lomize (1951), Polubarinova-Kochina.(1962), Snow (1965), Romm (1966), Louis (1969),
and Bear (1972) are only a few of the investigators that have derived the ba sic equations describing flow through a fracture. If the flow is laminar and one adopts the analogy of parallel planar plates to represent the fracture surfaces, these workers have shown that the hydraulic conductivity of a frac ture with an aperture 2b is given by*
K
=
(2b) 2pg (1) f 120 If the flow is steady and isothermal, the flux per unit drop in head can be developed from Darcy's law and may be written in simplified form as
= C(2b) 3 (2) iih where C is a constant, which in the case of radial flow is given by C -
)(2nS)
(3) n(r e /r 12 and in the case of straight flow, by C
=
)
(4)
L 12,/
Equation 2 is the basis for what is often called the "cubic law" for flow in a fracture. One must bear in mind that this equation has been derived for an "open" fracture; i.e., the planar surfaces remain parallel and thus are not in contact at any point.
- See Nomenclature section for definitions of all terms.
-4 The law for flow in fractures can be generalized in terms of the Reynolds number, Re, and the friction factor T (Lomize, 1951; Romm, 1966; Louis, 1969).
These workers have shown that if one introduces the factor D equal to four times the hydraulic radius, then Re Dvp (5) and
=
D (Vh)
(6)
(v /2g)
The cubic law for laminar flow in an open fracture then reduces to the simple relationship 96 (7)
The first comprehensive work on flow through open fractures was by Lomize (1951).
He used parallel glass plates and demonstrated the validity of the cubic law as long -as the flow was laminar. He also investigated the effects of changing the fracture walls from smooth to rough and finally to models with different fracture shapes. He introduced the concept of defining the rough
- ness, e,
in terms of the absolute height of the asperities and developed the empirical equation W 6= 1 +6.0 1.5(
which is valid for c/2b > 0.065. Lomize (1951) also considered the effect of flow through fractures with planar but nonparallel (converging or diverging) sides.
We shall simplify (8) by rewriting the equation in the form 96f (9)
Re
-5 where f is a factor that accounts for deviations from the ideal conditions that were assumed in deriving (7).
In the case of roughness, f > 1. The cubic law then becomes:
= C (2b)3 (10) ah f
Lomize (1951) developed a flow regime chart with several serniempirical flow laws to take into account the effects of roughness and turbulent flow in open fractures. Louis (1969) has independently performed experiments similar to Lomize's and reached essentially the same conclusions. A number of other workers (Baker, 1955; Huitt, 1956; Parrish, 1963; and Rayneau, 1972) have also investigated the effects of roughness on flow in open fractures.
In his comprehensive treatise on flow in fractured rocks, Romm (1966) presented the results of very careful laboratory studies on flow phenomena in fine (10 to 100 wm) and superfine (0.25 to 4.3 un) fractures. His superfine fractures were made of optically smooth glass and were carefully constructed so as to be open fractures. He demonstrated the validity of the cubic law for laminar flow in both fine and superfine fractures using various fluids and also verified the presence of a boundary layer 0.015 um in thickness. He concluded that laminar flow in a fracture obeys the cubic law at least down to apertures of 0.2 wm. The critical Reynolds number at the transition from laminar to turbulent flow was found to be 2400. This same critical value has also been reported by Lomize (1951) and Louis (1969).
Sharp (1970) performed flow tests in the laboratory with a natural rock fracture in a hard granite porphyry whose matrix permeability was extremely
-6 small.
He started the flow tests from a so-called "closed" condition which was achieved by placing the fracture in a horizontal position so that it would "close" under the weight of the upper half of the rock sample. Water flow through the fracture in this "closed" condition indicated the aperture was about 270 pm, and flow tests were then conducted by jacking the fracture open until a maximum aperture of 1540 pm was reached.
Sharp (1970) and Sharp and Maini (1972) proposed an empirical flow law for natural fractures that was based on the "effective" aperture, which is the difference in the opening from the initial condition, and the net flow rate obtained from a calculation of the measured flow rate minus that observed under the initial "closed" condi tion. They dispute the validity of the cubic law and have suggested that for their particular conditions the exponent in equation 2 should be 2. In review iny these results, Gale (1975) has pointed out that the cubic law is still valid if one correlates their flow rates corresponding to the apertures that were actually present.
All of these investigations have been concerned with open fractures and, of course, one will encounter many situations in the field where the fractures are not open. Usually, fracture surfaces have some degree of contact and the effective aperture will depend upon the normal stresses acting across the dis continuity. Under these conditions, flow is still possible but since part of the flow path is now blocked by asperities, will the cubic law still hold?
This is an important question in many fields where flow through fracture systems must be considered. It is of critical importance in the underground isolation of radioactive waste where one is concerned with the problem of
-7 investigating nearly impermeable rocks. To our knowledge, the validity of the cubic law under these conditions has not been investigated.
We have therefore undertaken a laboratory investigation of this problem using fractures created in three different rock types.
- 2. LABORATORY PROCEDURES Homogeneous intact samples of basalt, granite, and marble were chosen for investigation. All samples came from quarries in California and have matrix permeabilities that are so low that, for the purposes of this study, they can be neglected. The samples were prepared with radial and straight-flow geome tries for the flowing fluid, which was filtered water at ambient conditions.
Details of the sample preparation and laboratory procedures are given by Iwai (1976) and will only be summarized here.
For the radial flow work, cylinders of intact rock with a diameter of 0.15 m were diamond-cored from each rock sample in lengths ranging from 0.17 to 0.21 m. A horizontal fracture was created in each sample using a modified form of the "Brazilian" loading method. In general, this method produced an excellent fracture surface that was essentially orthogonal to the cylindrical axis. A center hole 0.022 m in diameter provided access for outward radial flow of water.
A Riehle testing machine provided axial loads up to 20 MPa. The loading piston of this testing machine was also attached to the upper half of the radial flow samples so that when the piston was raised sufficiently, an open fracture (no points of contact between the two surfaces) of known aperture
-8 could be obtained. Three LVDT's (linear variable differential transducers),
placed 1200 apart and mounted so as to straddle the fracture, were capable of detecting aperture changes as small as 0.4 um.
For the straight flow work, a horizontal tension fracture was created in a rectangular block cut from the granite sample with width = 0.121 m, length =
0.207 m, and height = 0.155 m. By sealing both sides of the fracture along the 0.207 m length of the block and keeping the ends open, a straight flow field was established. A groove across the middle of the fracture plane provided for even distribution of water across the width of the fracture. A 0.08 m hole drilled into this groove from below provided the inlet connection for water to reach tne fracture. Two LVDT's straddled the fracture on one side of the rec tangular block and one LVDT was similarly mounted on the opposite side.
Before testing, the flow system was carefully checked for leaks with the new tension fracture subjected to maximum load conditions. Then a series of flow measurements were made for Run 1, both during loading to maximum stress and also during unloading. The fracture was next loaded and unloaded to the same maximum stress without making flow measurements. On the fifth loading cycle, Run 2 data were collected. Runs 1 and 2 used a head difference of h =_
20 m of water. After these high pressure experiments, Run 3 was carried out using Lh = 0.5 m of water to reduce any tendency for the development of turbulent conditions.
One of the most critical problems was that of determining the fracture aperture., Fig. 1 shows an example of flow rates as measured with the straight flow granite sample when normal stresses up to 17 MPa were employed
in an attempt to close the fracture. The effect of repeated loading cycles was to reduce the flow rate per unit head difference, Q/Ah, to a minimum of 5.33 x 10-10 m2/sec, but the fracture could not be closed completely. Assum ing that equation 2 is valid for a nearly closed fracture such as this, the residual aperture was computed to be 6.7 Pm under these conditions. Fig. 2 shows similar results for flow rates obtained on a tension fracture in the same granite with radial flow.
When deformations across the fractures were measured as a function of stress, a highly nonlinear behavior was observed. This is shown on Fig. 3 for the first loading cycle with the straight flow granite sample. AVtl is the total deformation as measured on one side, and &Vt,2 and AVt, 3 are the total deformations as measured on the opposite side. Knowing AVr, which was meas ured on an intact sample of granite, the net deformation of the fracture was computed from 1
t.2V
+ AV 3
V AV=
1~ [AV
+ tzt2 At z3] - AV (11) 2I I't,1 2
r The apparent aperture at any stress level, 2bd, could then be determined by reference to the maximum fracture deformation, AVm (see Fig. 4), using 2bd =
V - 1V (12)
Deformations were carefully measured during loading and unloading cycles so that equation 12 could be used regardless of the effects of hysteresis which are clearly evident on Figs. 1 and 2. The same procedure was used to calcu late aV for the radial flow samples except that a simple average of the AVt values was used in this case because the LVDT's were equally spaced around the circumference.
1 -6 Runno.
I 2
Loading Unloading 0
e
-TW E
0-e 2Q o
10 A
10-I I
0 5
10 15 Axial Stress, cre (MPa)
XBL 797-7577A Fig. 1. Effect of cyclic loading on permeability of tension fracture in granite with straight flow (after Iwai, 1976).
10--51 Run no.
I 2
Loading 0--
10 -6 Unloading -- +--
--- A--
Oe io -7 10 "1
4!4 Ci E
10-8 10-10 10
-1 1
1 1_II 0
5 10 15 20 Axial Stress,oa (MPa)
XBL 797-7578 A Fig. 2. Effect of cyclic loading on permeability of tension fracture in granite with radial flow (after Iwai, 1976).
-12
'a t,2 aVt,.3
& Vt,l I
I I
I I
15-I AVrV 10 I
Ca
(
I I;
II 0
I I III II x
II I; 0
20 40 60 80 100 120 140 160 18 Deformation, 6V (alO04 CMn)
XBL 793-9029A Fig. 3. Mechanical properties of fractured granite sample used in straight flow model (after Iwai. 1976).
-13 20' AVm 15
- 0.
b 0
3' I
0
. 2bd 2br 0 0 20 40 60 80 100 120 Fracture Deformation, AV (A m)
XBL 798-11132 Fig. 4. Mechanical properties of fracture used in determining changes in aperture with stress.
-14
- 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 given aperture, Q was observed to be proportional to Ah under the experimen tal conditions (Iwai, 1976), and thus Darcy's law holds. Therefore, we have a simple functional relation between Q/sh and 2bd*
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. As shown on Fig. 4, the true aperture is the sum of the apparent and residual values, i.e.,
2b =
2bd + 2b (13)
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.
To test the validity of the cubic law, we must treat 2br as an unknown parameter common to all data points. As a first approach, we assumed that the factor f could be set to unity, and rearranged equation 10 to
= C(2bd + 2br )n (14)
The unknown parameters n and 2br were then determined by minimizing X2 =Z log
- log C(2bd + 2b) n 2 w (15) over all the data points from one flow run, which meant measurements during
both the loading and unloading cycles (Figs. 1 and 2).
The symbol w in (15) is a weighting factor that took into account the fact that the values of log Q/Ah were not evenly spread.
The results of the least squares fit for the data using (15) are given in Table 1. The table also includes values for the residual apertures that were calculated directly assuming equation 2 is valid. Note that these results are not significantly different from those obtained by the least squares fit.
The reason is quite evident; the factor n 3. The cubic law is valid for these fracture flow conditions.
Table 1. Results of least squares fit for parameters n and 2br.
Residual Aperture Sample Run Fitted Fitted Calculated(1) n Um Um Granite (2) 1 3.04 9.0 7.9 2
3.03 6.7 6.7 3
3.01 11.6 11.4 Granite ()
1 3.07 5.1 4.4 2
3.04 4.0 3.2 3
3.06 13.1 10.9 Basalt (3 1
3.08 10.5 10.0 2
3.10 10.8 10.4 3
3.02 9.1 9.8 Marble 1
3.06 2.5 4.0 2
3.06 2.2 4.0 3
3.01 18.2 18.1 (1) calculated from equation 2 (2) with straight flow
(
with radial flow.
This is a significant result because the data on Figs. 1 and 2 show evi dence of hysteresis and permanent set.
For example, during the first load ing cycle for the radial flow tests in granite (Fig. 2), a maximum displace ment of aVm = 108.2 pm was obtained, and upon unloading, the deformations stopped with V = 41.4 pm. The cubic law seems to hold regardless of the loading.path and no matter how often the loading process is repeated. Per meability is uniquely defined by the fracture aperture, and one could predict changes due to stress as long as there are no effects of shear movement or weathering.
From a theoretical standpoint, the value of n should be 3, and in the results on Table 1, it is apparent that experimental results differ by no more than 3% from the theoretical value. Note also that all values are greater than 3, which suggests that some other systematic factor is contributing to this difference from theory. We therefore concluded that we were justified in adopting n = 3 and should reexamine the experimental data to determine the importance of the factor f.
To do this, we recast (14) in the form
-h =
(2bd + 2b) 3 (16)
The unknown parameters f and 2br were then determined by minimizing 2
X (log
- -log [T (2bd + 2b r) 3 (17) over all data points in the same fashion as previously. The results are given in Table 2.
-17 We see immediately that the least squares results for the residual aperture are somewhat closer to the values that were calculated directly from equation 2 than is the case on Table 1. We also see that reasonable values are obtained for the factor f and that they are all greater than unity as pre dicted by equation 8.
Table 2. Results of least squares fit for parameters of f and 2br*
Residual Aperture Sample Run Fitted Fitted Calculated()
f um Um Granite (2) 1 1.21 8.8 7..9 2
1.15 6.6 6.7 3
1.04 11.6 11.4 Granite (3) 1 1.49 4.8 4.4 2
1.29 3.8 3.2 3
1.32 12.4.
10.9 BasaltP) 1 1.45 9.8 10.0.
2 1.65 9.9 10.4 3
1.10 8.7 9.8 Marble (3 1
1.36 2.2 4.0 2
1.36 1.8 4.0 3
1.05 18.2 18.1 (1) calculated from equation 2 (2) with straight flow with radial flow Having determined values for the residual aperture at maximum normal stress, it was then possible to examine how well the experimental data agree with the cubic law over the range of flow rates employed. Fig. 5 shows results for the granite sample with straight flow (see Fig. 1).
Fig. 5A presents the data in accordance with equation 10 where the solid line is for
I.C ) 3 I
1 1 11 Ij I ill l
I I11 11 I I I
I II I I I I II 1 11i1
-A Granite (Straight flow) f 1.o
-102 E
L 10' 10 0 1 1 1 l a 1 1.
.e 111 1 1u n
1 1 1 1 1
.. nnII I 1 1 1 11 1,, -
1 1 10-10 10-9 10-10-10-6 10-5 Q/Ah (m2/sec)
B Source head 20m 0.5m Run no.
I 2
3 0
Loading o
a 0 Unloading
- a 102 101 fI L21 00.
10-2 101 Io 10' 102 10 Re XBL 797-7563A Fig. 5. Comparison of experiment for straight flow through tension fracture in granite with cubic law.
-19 the special case of f = 1.0.
For comparison, Fig. 5B presents the data in accordance with equation 9, which is the method used by Lomize (1951), Romm (1966), and Louis (1969).
The effect of changing f from 1.0 to 1.21, which is the range of values shown on Table 2, is indicated. Note on Table 2 that the values of f generally decreased in progressing from Run 1 to Run 3. This may be an indication that the fractures became progressively better mated during the cyclic loading.
Fig. 6 shows results for the same granite but with radial flow (see Fig. 2).
Note that the radial flow data deviate somewhat more from the ideal cubic law (f = 1.0) than the straight flow data shown on Fig. 5. This may be a manifestation that the effects of roughness are more complicated in a radial flow field, as is indicated by the fact that the maximum value of f reached 1.49 (see Table 2).
In a radial flow system, the Reynolds number is difficult to define be cause velocities decrease by a factor of 7 from the inner radius to the outer boundary. The values of Re for radial flow samples were calculated using the inner radius, and are therefore maximum values. Note that in Run 3, data were also collected during the unloading cycle after the fracture surfaces were truly open, i.e., the surfaces were no longer in contact. One sees good agreement with the cubic law over a range of apertures from 4 to 250 wn, regardless of whether the fracture was open or closed.
Fig. 7 shows results for the basalt sample with radial flow. Here the maximum deviation from the ideal cubic law (f = 1.0) was obtained and this is indicated by the dashed line for f = 1.65 on Fig. 7B. The basalt fracture
-20 A
Granite (Rodial flow) 1020 E
fZ=1.0
-E o
-Run 3
10J 10-1 10-10-8 107 10-6 10-5 10 Q/Ah (m2/sec)
's, B
Source head 20m 0.5m
'0 Run no.
I 2
3 o
Loading 0
a s
Unloading I
a 102 10 f I 49 f =1.0 I00 0
10 -11 1
li lt 10-2 10-'
IOo 10' 102 103 Re XBL 797-7564 A 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 when aperture exceeded value indicated by arrow.
-21 A
Basalt (Radial flow) 10 E
f =1.0 L
'Run 3
10 10-10 10-9 10-8 I
10-6 IO 10 Q/Ah (m2 /sec)
B Source head 0
NP 20m 0.5m a
'P Run no.
I 2
3 103 Loading o
a Unloading
£ U
10
\\
102 f =1.65 f =1.0 I00
'4 loo 10-1l 10 I0' 100 101 102 10 Re X8L 797-T565A Fig. 7.
Comparison of experimental results for radial flow through tension fracture in basalt with cubic law.
In Run 3, frac ture surfaces were no longer in contact during unloading when aperture exceeded value indicated by arrow.
-22 A
Marble (Radial flow)
-Run 3
Run 2 10 af=
100 1asi IO-1o 9
0-IO 10 10-6 0
10 Q/&h (m 2/sec)
B Source head 20m 0.5m 1
Run no I
2 3
Loading 0
A a
onUnloading 102 100
- 1. 36 f 1.
100 10-11i 10-2 10-'
100 lot 102 Re XBL 797-7566A 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.
-23 surfaces were more undulating than in the case of the granite and marble samples, and one experimental difficulty arose when a small chip of rock broke off around the center borehole. We assumed that rw should be increased from 0.011 m to an estimated value of 0.02 m, but this may not have been the only effect on the flow field.
Fig. 8 shows results for the marble sample with radial flow. Experi mental difficulties at low flow rates during Runs 1 and 2 caused the marked deviations from the cubic law as apertures decreased below 10 in. Some subtle changes in mating of the fracture surfaces also occurred between Runs 2 and 3.
In Run 2, the initial aperture was 25.5 in, but during the unloading cycle, the aperture was opened. Upon reseating in preparation for Run 3, the initial aperture was found to be 123.0 in. Note on Table 2 that f was found to be 1.36 in Run 2 and 1.05 in Run 3. Despite these difficulties with both the basalt and marble samples, it is apparent on Figs. 7 and 8 that good agree ment with the cubic law has been obtained.
The manner in which a fracture closes as the surfaces are forced together under stress raises some important issues. We visualize the process as being controlled by the strength of asperities that can withstand significant stres ses while maintaining space for fluids to continue to flow as the fracture aperture decreases. An idealization of a fracture in the open and "closed" positions is illustrated in Figure 9. Asperities of height E are visualized as coming together in a perfect union that provides fixed areas of contact.
These areas of contact then deform elastically such that the contact area is
-24 OPEN FRACTURE 2b "CLOSED" FRACTURE XSL 797-7589 Fig. 9.
Idealized fracture showing the mating of asperities as the fracture surfaces are closed under stress.
-25 not changed significantly, at least not under the relatively low stress levels employed in this investigation. The only significant change that takes place is that of the closing of the aperture.
As we have seen above, there is a very strong dependence between flow and size of the aperture. As a fracture closes, however, the flow paths are certainly far more complex than the idealization portrayed in Fig. 9. The flow may actually occur through an intricate assembly of interconnected tubes of varying cross-section and separated by the local asperities. In any event, the overall geometrical effect can still be represented by a measure ment of 2b.
Since flow per unit head 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 correlations of the experimental results in passing from an open to a so-called "closed" fracture.
Much more work is needed in understanding the mechanical and hydraulic behavior of a fracture that is deforming under stress. This study has been concerned with fractures under normal stress, whereas the general situation in the field will involve both normal and shear deformations. The actual phy sical process that takes place during the deformation of fracture surfaces and the effect this behavior has on the fluid flow process must be understood.
There is also a problem of a potential size effect that has been observed in determining the hydraulic properties of fractures in rock specimens of differ ent dimensions (Witherspoon et al., 1979).
An understanding of contact area
-26 behavior in fractures will be important in analyzing this problem. The present work was carried out with ambient temperatures, and the flow sensitivity of fractures to very slight changes in aperture indicates that thermal expansion is yet another factor that needs investigation in order to understand fracture behavior at elevated temperatures.
- 4.
CONCLUSIONS The results of this laboratory investigation on tension fractures that were artificially induced in homogeneous samples of granite, basalt, and marble have clearly shown that the cubic law for fluid flow in a fracture, which is given by Q/Lh = (C/f)(2b)3, is valid.
The investigations included radial and straight flow geometries and covered apertures ranging from 250 Wm down to 4 wm and normal stresses up to 20 MPa. The cubic law was found to hold whether the fractures were open or closed, and the results are not depen dent on rock type. The cubic law seems to hold regardless of the loading path and no matter how often the loading process is repeated. Permeability is uniquely defined by the fracture aperture, and one could predict changes due to stress as long as the fracture surfaces are not affected by shear movements or weathering. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and are taken care of by the factor f.
In this investigation, f varied from 1.04 to 1.65.
We visualize the process of a fracture that is closing under normal stress as being controlled by the strength of the asperities that are in con tact. These contact areas are able to withstand significant stresses while
-27 maintaining space for fluids to continue to flow as the fracture aperture de creases. The controlliny factor is the magnitude of the aperture and since flow per unit head 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 correlations of the experimental results in passiny from an open fracture to one where the surfaces are being closed under stress. Much more work is needed in understanding the mechanical and hydraulic behavior of a deformable rock fracture.
- 5.
ACKNOWLEDGMENTS This research was supported by the U. S. Geological Survey under Contract Number 14-08-0001-14583, the National Sciences Foundation under Grant Number GK-42776, and the U. S. Department of Energy under Contract W-7405-ENG-48.
- 6.
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