ML13309A839
| ML13309A839 | |
| Person / Time | |
|---|---|
| Site: | San Onofre  |
| Issue date: | 11/30/1970 |
| From: | Wilson C, Witherspoon P CALIFORNIA, UNIV. OF, BERKELEY, CA |
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| Shared Package | |
| ML13302A500 | List:
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| References | |
| NUDOCS 8103120704 | |
| Download: ML13309A839 (18) | |
Text
Reference B-1 Pages 55 through 62, 165 and 168 AN INVESTIGATION OF LA4INAR FLOW IN FRACTURED POROUS ROCKS by Charles R. Wilson and Paul A. Witherspoon Department of Civil Engineering University of California Berkeley November 1970 91 08120704
55
=~
1-- fK b b.
+ K (c b. + bc.) + K cc I
2D xx m xz m
i i
zz mi
+
K b b.
+
K (cb. + b c.)
+
K c c.
xx ma xz mm3 mm3z
+
K bb
+ K (cb + b c ) + Kzz cc j
These solutions are valid only for elements whose nodes both i and j lie on a boundary, and node m does not lie on a boundary. For an element with only one node on the boundary, the integration along the length of the boundary is
- zero, the velocity term drops out, and the standard internal equation results.
For nodes along boundaries of zero flow, V will be zero and the equations reduce to the standard internal equation.
A copy of the finite element computer program produced from these theoretical considera tions is presented in Appendix A.
- 2.
Verification and Use of the Triangular Element Program This program was verified by creating several simple problems which could be checked by hand.
First, flow in a straight parallel-wall fracture was calculated, and the results were in good agreement with hand calcul-tions.
- Next, a system with one fracture sloping upward at +4.50, and a second fracture intersecting near the midpoint and branching down at
-9.0' was checked (Fig. II-10).
50.2
-2 2 5 Qn2b)=
.02 3
i----
out = 50cm i2 =.01
=Q0c.Ha O
t2.
out2 in center of intersection Fig. 11-10:
Plan for second check of computer program.
Elements are numbered 1 to 7. Lengths and heads 6 in~cm units.
57 is less than about 200.
Laninar flow in any conduit is always characterized by a linear relationship between velocity and gradient:
V C
where Z is measured along the path and the proportionality factor c, called the hydraulic conductivity, is a function of conduit geometry and fluid properties. For smooth wall parallel plate.flow the hydraulic con 2
ductivity is equal to b y/l121.
Hence for a flow channel of any shape, if velocities are in the laminar range an aperture can be calculated for a smooth parallel plate conduit which will offer the same resistance to flow as the arbitrarily shaped channel.
Therefore, in the laminar regime, any real fracture with its many contortions and wall asperities will behave in &n overall sense, but not in terms of internal detail, as a parallel plate with a correctly chosen effective aperture beff' A simple model of a fracture with varying aperture is presented in Fig. II-11 where three separate ape:tures are encountered along the length of the fracture.
From continuity considerations similar to thoze used in the calculations accompanying Fig. II-10, the total flow was calculated by hand to be 0.0207 cm 3/sec.
By assigning element apertures equal to the local width of the fracture, and setting the permeability ellipsoid orien tation angle a = 0 in all elements, the finite element result of 0.0201 3
cm /sec compares well with the hand calculated value.
From these same continuity considerations, the effective permeability for a fracture of three different apertures may be expressed as:
7*
58 b3= 0.01 C m 0.025 ryj b0.045 cm =b.
e = 50 cmi Fig. 11-11.
Fracture with irregular aperture.
L Fig. 11-12.
A wedge-shaped fracture.
o NTERSECTI a= 0 az 01
,ctl-I frctues f identical Fig. 11-13.
Idea3.z d flow 1incs in intes o rieactation of on xinc aperture and fegi
- s.
- o. gives ori aor
59 z
+
12
+
1 3(1-8 3
beff 1
12 3
++
3 bl3 b2 b3 whence b eff 0.0161 for the system of
- Fig.
Calculating bff from the computer results yields:
1 3
12VIQT L 0.0159 cm beff yn It is interesting to note that at point A in Fig. 11-11,,the head has dropped from 100 cm to 95.4 cm, and at point B it has further dropped to 94.5, leaving the remaining head of 44.5 cm to be lost in the short length of narrow channel.
In this sYstc-,- 89% of the headluss occurs in olyg1%
of naro lchn
- b.
of the smaller channel and the cubed rela only 21% of the length becau',e of h mle tionship between aperture and ff hon voluote.
it is evidnt from equation 11-28 that for a flow channel of more than three apertures along its length, the effective aperture may be exp as:
n b
3f (11-29) beff i
i=l b 3 Or, if the aperture varies continuously along the length of the
- fracture, the effective aperture may be expressed as:
60 L
If d-9 (11-30) b eff L
d di The aperture b is now expressed as a function of path length k along the fracture, where L is the total path length. This derivation assumes that locally the flow lines will be essentially horizontal and will not hold in those cases where apertures change abruptly along the length of the fracture.
As an example, the results of equation 11-30 will be compared to a coMputerZ solution for the case of a wedge-shaped fracture (Fig. 11-12).
For this fracture. equation 11-30 may be written:
L dZ 3
o beff L
di, (mi + c) 0 where if 26 is the angle of opening, then m 2 tan e, c b,
aperture at smaller end, and mL + c aperure at large~r end.
The solution to this equation is:
2mLb 2b 2 2b1 2b2 (11-31) eff b
- b 2 b1 + b2 2
1 For example, if b1 = 0.205 cm, b2 = 0.415 cm, L = 20 cm, and 0 = 0018' then from equation 11-31, beff 0.286 cm.. The computer, using 40 elements, obtained a value of beff =0.280 cm.
Even greater accuracy can be obtained
61 from the finite element method if more elements are used. Note that while equation 11-31 is independent of L and e, in reality e must be rather small for flow to remain essentially horizontal.
A laboratory study of flow in wedge-shaped parallel plate conduits was performed by G. M. Lomize (33).
In one of Lomize's experiments, utilizing water in a conduit with the same dimensions as in the previous example, a flow rate of approximately 3 cm 3/sec per unit plate width was measured under a gradient of 0.02 cm/cm. From these results an effective aperture of beff = 0.27 cm is calculated.
This conduit has been modeled using 40 triangular elements.
Assigning the same gradient of 0.02 cm/cm, and assuming unit density and a viscosity of 1 centipoise for the water in the test, the flow rate was calculated 3
to be 3.6 cm /sec per unit width. This compares well with the experimental value, the discrepancies being due in a large part to slight roughnesses in the plate walls in Lomize's experiments (they cannot be ideally smooth as is assumed in the mathcmatical model).
Also error occurs because the water temperature-used in the calculations was assumed to be 70*F, which is probably not identical to the experimental conditions.
No information was given by Lomize concerning temperature.
Since errors due to effects of wall roughness and temperature can be accounted for mathematically if these effects are known, in general the results should be as good as or better than those obtained above.
This comparison with Lomize's experimental results indicates that fractures with gradually varying apertures.can be accurately modeled using the two-dimensional finite element program.
Fracture intersections present a difficulty in the application of the triangular element program because hydraulic conductivity is not easily
62 defined in this space. The intersection belongs mutually to two separate fractures which may be of different aperture and hence different permea bilities, and at their juncture neither the magnitude nor the orientation of the permeability ellipsoid is defined.
The magnitude of the permeability which is mathematically identified with each triangular fracture element is related to an aperture assigned to that element. In an intersection or in other areas where the aperture is undefined the permeability is also undefined. Hence for elements in these regions it is necessary to arbitrarily select an "effective aperture" which will give a reasonable permeability value. By varying this effective aperture in intersection elements it is possible to introduce special head2osses into the model to account for interference effects which exist at intersections.
The magnitude of these effects was studied in the laboratory and the results are presented in Chapter III.
They were found to be suffi ciently small at low flow rates to be neglected, and it has been the prac tice in this paper to assign to intersection elements apertures equal to that of the larger fracture. H.owever, the conductivity assigned to each fracture element is arbitrary and if it is desired, this model can be made to account for interference effects by assigning slightly smaller apertures to intersection elements.
The orientation of the permeability ellipsoid in each element is governed by the choice of angle a which determines specifically the orientation of the directional permeabilJ.ty K, within the permeability ellipsoid.
Within a fracture segment it is clear that a should equal the fracture orientation, and that K, should equal b 2y/12p, but within an intersection flow is undergoing abrupt changes in direction and also
165
- 90. Hartsock, J. H. and J. E. Warren, "The effect of horizontal frac turing on the well performance," J. Pet. Tech., p. 1050, 1961.
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- 92. Heitfeld, K. H., "Zur Frage der oberflachennahen Gebirgsauflockerung" (The question of surface loosening of rock masses), Proc. First Cong. of Int. Soc. Rock Mech., Lisbon, v. 1, p. 15, 1966.
- 93. Hobbs, D. W., "The formation of tension joints in sedimentary rocks:
an explanation," Geol. Mag., v. 104, p. 550, 1967.
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- v. 45, p. 1, 1961.
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74, p. 1411, 1963.
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- p.
460, 1951.
- 97.
Hsu, C. C.,
"A simple solution for boundary layer flow of power law fluids past a semi-infinite flat plate," AIChE Jour.,
- v. 15,
- p.
- 367, 1969.
- 98.
Huang W.,
J. M. Robertson and M. B. McPherson, "Some analytical results for plane 900 'bend flow," J. Hydraulics Div,
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- v. 93, no. 6,
- p.
- 169, 1967.
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- v. 210,
- p.
153, 1957.
100.
Huitt, J. L.,
"Fluid flow in simulated fractures," AIChE Jour.,
- v.
2,
- p. 259, 1956.
101.
- Huskey,
- 1.
L. and P.
B. Crawford, "Performance of petroleum reservoirs containing vertical fractures in the matrix," SPE Jour.,
- p.
- 221, 1967.
.102.
- Irmay, S.,
"Flow of liquid through cracked media," Bull. Res.
Council of Isroel.,
- VSA, no.
1,
- p.
84, 1955.
103. Javandel, I.
and P.
A. Witherspoon, "Application of the finite element method to transient flow in porous media," Soc. Pet. Eng.
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168 129.
Lewis, D. C. and R. H. Burgy, "Hydraulic characteristics of frac tured and jointed rock," Groundw-atcr, v. 2, no. 3, p. 4, July 1964.
130.
Lewis, D. C., G. J. Kriz and R. H. Burgy, "Tracer dilution sam pling technique to determine hydraulic conductivity of fractured rock," Water Resources Research, v.
2,
- p.
533, 1966.
131.
Liakopoulos, A. C., "Darcy's coefficient of permeability as sym metric tensor of second rank," Bull. IASH, v. 10, no. 3, p. 41, 1965.
132.
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133.
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134.
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- p. 235, 1969.
135.
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139.
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Appendix C.
Back Up for Figure 4.6 Figure 4.6 represents sinply a cross plotting of normalized flow Q/A H versus axial stress Ge (Figure 1 of Reference) and aperture 2b versus normalized flow graphs (Figure 5 of Reference).
For granite these curves are presented on Figures 1 and 5 in Witherspoon et al., (October 1979) attached as refer ence C-1 and for marble and basalt they are presented on Figures 4.40 and 4.45 attached from Iwai's thesis and Figures 7 and 8 in Witherspoon et al.,
(October 1979).
These figures are labeled C-1 through C-6 and are used to explain the cross plotting conpleted to develop Figure 4.6.
Specifically, for granite a range of axial stress values was obtained from the data on Figure C-1 for a selected value of Q/A h.
The value of aperature correspond ing to the same value of Q/Ah was then obtained fron the 2b vs. Q/Ah graph at the top of Fig. C-2.
The value of 2b (aperture) and range of Ge (axial strss) was then plotted with aperature as the ordinate and conpressive stress (axial stress) as the abscissa. Similar cross plotting was cxrpleted for marble using Figures C-3 and C-4 and for basalt using Figures C-5 and C-6.
The range of values shown on Figure 4-6 approximately envelopes the data resulting from this cross plotting.
10-10 Runno.
I 2
Loading
... A-Unloading p
-A UI 10 -T Y
W-1 20 A
A
-- ~----
E.
10 10I 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).
Figure C-1 Granite
-18 A
Granite 102 (Straight flow) f 1.0
- 10,
E c1J 10 I 10-10 10-9 10-8 IO 10-6 10-5 Q/Ah (m2 /sec)
Source head 20m 0.5m Run no.
I 2
3 10 Loading o
A a
Unloading a
102 10' f 1.21 f =1.0 4
10 jo-1 10-2 10-'
10o 101 102 103 Re XBL 797-7563A Fig. 5. Comparison of experiment for straight flow through tension fracture in granite with cubic law.
Figure C-2 Granite
165.
.C2 Looding Unlooding Ist Run 0
2nd Run A
-3 0c Marble 0
19.0 Cm 104 15 cm
-6 10 0 5
10 15 20 Axial Stress,Oe(MN/m 2 )
Fig. 4.45. Effect of cyclic loading on permeability of tension fracture in marble with radial flow.
Figure C-3 Marble
-22 A
Marble 02 (Radial flow)
-10 E
"-Run 3
Run 2 (1t
- A f=10o 10- 10 10 0
IO 106 0-5 O/Ah (m2/sec)
Source head 20m 0.5m A
Run no.
I 2
3 10 o
Loading o
a 0 A*
Unloading A
a 0
102 101210-00-'
0 Re
's/,f =1. 36 f =1.0 10-1 10 2 10 IO 1 100 10' 102 10, 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.
Figure C-4 Marble
158.
-2 4' Basalt 16.0 cm 10 2
10
-5 Pa th 10 Run Looding Uniooding I st
-a-2 nd
-- A----
0 5
IO 15 20 Axiol Stress, q (MN/rn 2)
Fig. 4.40. E.ffect of cyclic loading on permeability of tension fracture in basalt with r-adial f low.
Figure C-5 Basalt
-21 10 3 1
1 "ti l
l I'
'1 A
Basalt (Radial flow) 102 f =1 0 Run 3:
0, 1
IG 10-10 I-108 I0 106 10 5 10 Q/6h (m 2 /sec)
IO 1
I i ij I
li ii j
I I
I lilii I
I l I llyI Ij I
I IIII.
's Source head o
20m 0.5m o
Run no.
I 2
3 10 Loading o
a Unloading U
102 -
0 e 101 f = 1.65 1001 10-2 101 100 10 102 10 Re XBL 797-7565A 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.
Figure C-6 Basalt
IL L
fi
,I It l
i
':LT
- I zo.
1I'
~
sit