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Statistical evaluation of hydraulic fracturing stress measurement parameters
Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 26, No. 6, pp. 447-456, 1989
Printed in Great Britain. All rights reserved
0148-9062/89 $3.00+0.00 Copyright ~ 1989 Pergamon Press plc
Statistical Evaluation of Hydraulic Fracturing Stress Measurement Parameters M. Y. LEE? B. C. HAIMSON? Statistical analysis of hydraulic fracturing fieM data enhances the objectivity of determining shut-in and fracture reopening pressures and fracture orientations. Using nonlinear regression analysis (NLRA), we isolate the closedfracture segment of the pressure-time curve by fitting it to an exponential decay model. The shut-in pressure (P,) is expected to be in the range between ~ - l (the onset of the post fracture-closure segmenO and I~,~-" (the pressure level of the best-fitting exponential curve extrapolated to the time of pump shut-off). We employ bilinear regression analysis to improve the objectivity of selecting P, from the pressure-decay rate vs pressure plot. The same analysis helps to identify P, in the pressure vs flowrate record obtained during steprate pressurization of the hydrofractured interval. We identify unambiguously the fracture reopening pressure (Pr) by superposing the ascending portion of the pressure-time curves in the fracture-inducing cycle and a subsequent cycle. P, is the onset of deviation between the two curves, defined as the point where the difference in pressures is larger than a statistical "reference threshold", provided the difference continues to increase from there on. Circular statistics and a sinusoidal curve-fitting method provide the means of objectively delineating the complete trace of induced hydrofractures from fragmented or splaying traces on the impression packer or on the borehole televiewer photograph. Examples of field hydraulic fracturing data and statistically determined pressures and fracture orientations are presented. A startling discovery was made in the course of determining P, with our new technique: it appears to be very nearly equal to the estimated P,. A review of the literature shows that in a great number of cases P, and P, are indeed close to each other, which raises the question whether we are in fact reading the correct fracture reopening pressure.
INTRODUCTION
shut-in pressure (Ps) is the lowest pressure level, during Hydraulic fracturing (or hydrofracturing) consists of the borehole pressure decay, capable of maintaining t h e injecting water at a constant flowrate into a sealed-off hydrofracture open against the far-field least horizontal segment of a vertical drillhole until a critical (or "break- stress. Hence, Ps serves as a direct estimate of Sh [1,2]: down") pressure (Pc) is reached at which the rock Sh = P,. (l) surrounding the hole fails in tension and a fracture develops. In linear elastic, homogeneous, isotropic, and The conventional interpretation of hydrofracturing reintact rock the hydrofracture is expected to be vertical quires the additional assumption that rock is imperand perpendicular to the minimum horizontal stress meable to the injected fluid. This allows the maximum (Sh). An impression packer or other borehole logging horizontal stress (S.) to be calculated directly from the devices (e.g. the borehole televiewer) can be used to Kirsch solution for stresses around a circular hole, and obtain the orientation of the hydrofracture which yields the Terzaghi effective stress law [1,2]: the direction of Sh. Following breakdown, fluid injection ( S . - P 0 ) = T + 3 (Sh -- P 0 ) - - (Pc - P0), (2) is ceased causing the borehole pressure to drop. The tGeological Engineering Program, University of Wisconsin-Madison, 1509 University Avenue, Madison, Wl 53706, U.S.A. 447
where P0 is the pore pressure, and T is the tensile strength of the rock. If subsequent pressurization is carried out in the hydrofractured segment of the hole,
448
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
equation (2) reduces to [3]: (s.
- C o ) = 3 ( s h - Co) - ( e , - t'0),
(3)
where Pr is the fracture-reopening (or "refrac") pressure, and T is assumed to be zero since the tested interval is already fractured. The assumptions here are that the fracture closes completely between pressurizations, and that P, is the pressure level at which the preexisting hydrofracture just begins to open, i.e. it is equal to the minimum compressive hoop stress at the hole wall induced by far-field stresses. In this paper we are mainly concerned with reducing the subjectivity of graphical methods for the determination of hydrofracturing parameters, such as Ps, P, and fracture orientation. We accomplish this by introducing nonlinear and other statistical techniques to digitallyrecorded test data.
tical analysis procedures applied to digitally-recorded field pressure and flowrate data.
Exponential pressure-decay method Pressure decay following pump shut-off is a result of additional extension of the hydrofracture, as well as permeation of the fracturing fluid from the test interval and the hydrofracture into the surrounding rock. As it decreases, the pressure will reach the shut-in pressure value (P,), equal to the far-field fracture-normal stress; further decay will bring about complete fracture closure. Beyond that, any additional pressure drop is mainly the result of radial fluid flow through the borehole wall into the rock. Muskat's [8] model of exponential pressure decay due to such radial flow has been suggested as a method of identifying P, [9]. We express this model as: P=exp(djt+d_,)+Pa~
SHUT-IN PRESSURE
The determination of P, is straightforward when a sharp break is observed in the pressure-time curve after the initial fast pressure decline following pump shut-off. In some cases, however, pressure decay is gradual and P, is indistinct (Fig. 1). Over the years several methods, mostly graphical, have been suggested and employed for estimating P, from pressure-time curves. Three of the most common have been the tangent divergence [4], the tangent intersection [5] and the logarithmic methods [6, 7]. Although the above methods provide reasonable approximations of P, in many situations, ambiguities remain when pressure decay is gradual with no obvious "breaks" or "knees". We have developed and are describing in this paper means of improving the objective determination of P, by employing well-established statis-
for
t>lh,
where P is the pressure in the test interval, dl ( < 0) and d2 are unknown parameters characterizing the pressure decay rate, P,t is the asymptotic pressure level in the test interval, t is the time after pump shut-off and tt is the time at which pure radial flow initiates (complete fracture closure). We first determine p~r~-~, the interval pressure upon fracture closure, by applying nonlinear regression analysis (NLRA) to the decaying portion of the pressure--time curve. Nonlinear regression analysis is a statistical technique for fitting an arbitrary function [for example equation (4)] to a given set of data points [10]. It determines the pressure decay parameters (d~ and d2) and the asymptote (P,~) by minimizing the sum of the squares of errors (SSE) between the recorded data and the predicted pressures based on the model. The fit is evaluated in terms of the residual mean square (RMS)
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20
30
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Tirol (rain) Fig. 1. Pressure-time record during a field hydrofracturing test in the Waterloo quartzite drillhole, W]. Breakdown pressure Pc, and indistinct fracture reopening (P,) and shut-in (Ps) pressures are shown. Cycle 4 is a steprate pressurization test using constant flowrate and self-stabilized pressure.
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
which represents the average deviation of the curve from the fitted model: RMS =
Ii
i- I
( e , - Pp,):/(n - 3)
1
,
(5)
where P~ is the digital pressure data, Pp~ is the NLRA predicted pressures based on the exponential decay model, and n is the number of data used in the regression analysis. An iteration procedure is invoked aimed at excluding the segment of the decaying pressure-time record before fracture closure. Starting with the pressure at the time of pump shut-off (t = 0), data points are removed sequentially with each iteration. This fitting procedure ends when the decreasing RMS-value stabilizes. At this point it is assumed that all the pressure data belonging to the open-fracture time segment (t < tt) have been removed from the curve-fitting process (Fig. 2). The largest pressure value of the fitted pressure--time curve ( P ~ - ] ) is interpreted as the level at which the induced fracture has completely closed; hence, it is the lower limit of the expected shut-in pressure value. The extrapolated pressure level obtained by the fitted exponential curve at time t = 0 ( P ~ - " ) represents the pressure at which pure radial flow would have commenced were the fracture to close instantaneously upon pump shut-off (Fig. 2); it is, thus, the upper limit of the range of values within which P~ is to be found: p ~ d - ] < p, < p~d-., (6) where
pressure estimates based on other techniques [11-13]. Aamodt and Kuriyagawa [9] also recommended using P ~ - " as the shut-in pressure, although their technique for obtaining P ~ - " was somewhat different and less rigorous. We have selected at random a set of six tests out of some 40 hydrofracturing measurements recently conducted in Precambrian quartzite near Waterloo, WI and analyzed the digitally-recorded data using the exponential pressure-decay method. We were able to obtain objective values of P s~ - t and P ~ - " , unaffected by analogue recording scale or any other bias. Among the different methods selected to estimate P, in Table 1, the average shut-in pressure using P ~ - " generally yields smaller standard deviations than when employing P,~-L It appears that P ~ - ~ yields a more consistent value of P, when compared to other methods of shut-in pressure determination than does p,pd-I (Table 1), although the most correct estimates probably lie somewhere between P ~ - I and P ~ - ~ . This finding is strongly supported by laboratory results detailed in a companion paper [14]. Bilinear pressure-decay-rate method (dP /dt vs P)
The functional form of the pressure decay prior to fracture closure is generally unknown. As an approximation, one could assume that this portion of the pressure decay is also represented by an exponential function (L. Tunbridge, personal communication, 1987). Under this assumption the total pressure decay after pump shut-off is composed of two exponential curves:
p ~ - i = exp(dltt + d2) + P,l and P ~ - " = exp(d2) + Pa, In our experience of the last several years, P ~ - ~ is consistently close (usually within +0.5 MPa) to shut-in
7'5~VV 7.0 6.5
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t < h,
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(7)
where t, is the time at which P, is reached; d3(<0) and d4 are unknown parameters characterizing the pressure decay prior to t,; d5 (<0) and d6 are unknown par-
2 Me4.
V
5.5
P = e x p ( d 3 t + d , ) + P~2 for
M~
v
6.0
449
Fig. Shut-in pressure determination using exponential pressure-decay method. Nonfitting data are shown as blank triangles. The stabilized residual mean square (RMS) value determines pressure level at which pure radial flow initiates ( P ~ - ' ) . The extrapolated shut-in pressure (pq, d-u) to time t ffi 0 is also shown.
450
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
Table 1. Waterloo quartzite hole--shut-in pressures determined by different techniques using NLRA in six randomly-selected tests
Depth
p~-;
p~r~-o
p~,-,
(m)
Cycle
(MPa)
(MPa)
(MPa)
50.3
2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4
6.3 5.4
6.7 6.0
6.5 6.0
3.3 3.2
3.8 3.7
3.7 3.6
4.3 4.1
5.9 5.7
5.5 5.3
4.2 3.7
5.7 4.7
5.8 5.4
7.2 8.3
7.4 8.5
7.3 8.3
9.6 9.2
10.7 10.0
10.5 10.0
66.5
81.5 129.2 183.4 206.9
p~
p~8-~
p~-~
(MPa)
(MPa)
(MPa)
5.3
5.8(___0.4) 6.0(4-0.5)
4.4
3.8(4-0.5)
3.9(4-0.3)
6.0
5.2(+0.7)
5.7(+0.2)
6.6
5.4(+1.1)
5.8(+0.6)
7.8
7.8(+0.0)
7.9(+0.1)
10.2
10.0(-I-0.4)
10.3(_-4-0.1)
P ~ - ~ and p , ~ - u are the lower and the upper limits of shut-in pressures estimated from the exponential pressure-decay (epd) method. p~,-z~ = shut-in pressure estimated from the pressure-decay-rate (pdr) method using bilinear (11) best-fit. P~ = shut-in pressure estimated from the pressure-flowrate (pq) method. p~vg-l = average shut-in pressure using P ~ - ' , e ~ r - n and P~. p~,g-u = average shut-in pressure using p ~ - u , p ~ , - , and P~. Numbers in parentheses represent standard deviations of the average shut-in pressure.
ameters representing pressure drop after t,; Pa2 and Pa3 are the unknown asymptotic pressure levels, respectively. By taking time derivatives of equations (7), a bilinear relationship between dP/dt and P is obtained (L. Tunbridge, personal communication, 1987):
dP/dt = d3 exp(d3 t + d4) =d3(P-Pa2)
for
P>P~,
dP/dt = d5 exp(ds/+ d6) =ds(P-Pa3)
for
P
(8)
A typical dP/dt vs P plot applied to a field record shows two linear segments connected by a transition zone, within which the shut-in pressure is found (Fig. 3). In order to single out P~ objectively we define the bestfitting bilinear curve by applying NLRA to the set of dP/dt vs P data. We start by assigning an initial transition point which divides the data into two discernable groups. Within each group a best-fitting straight line is obtained and the sum of squared errors (SSE) between the data points and the fitted straight line is calculated. Using an iterative process the nonlinear
0.20
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i 35
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.
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' 4.5
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i 55
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6 i5
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7.5
Preuure [MPa] Fig. 3. Shut-in pressure determination using the bilinear pressure-d~ay-rate (dP/dt vs P) method. The bilinear curve resulting
from NLRA determines P ~ ' - " .
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
regression analysis effectively determines the best-fitting transition point (P~r-") that gives rise to the smallest of all the calculated SSE for both straight-line approximations (Fig. 3). The calculated p ~ - H is unaffected by any other factors except the digitized test data, and is thus unambiguous and given to confirmation by independent workers. Table 1 indicates that, using the bilinear pressure-decay-rate method, the shut-in pressures (P~'-") in our set of tests are comparable to shut-in pressure values obtained by two alternative methods, and are in very close proximity to P~-U. Laboratory testing has also confirmed the reliability of the bilinear technique [14].
Pressure-flowrate method (P-Q) The pressure-flowrate method requires a special procedure to be followed during the hydrofracturing test. We typically carry this out after several conventional pressurization cycles (see for example Fig. 1). The hydrofractured interval is first pressurized at a very low flowrate (Q) until the pressure (P) stabilizes. The flowrate is then raised to a new constant level and maintained until a corresponding constant borehole pressure is recorded. The process is continued for several steps of flowrate (typically 6-12). A pressure vs flowrate (P-Q) plot is typified by two dominant slopes (Fig. 4). The interpretation of this plot is that at pressures below P~ the hydrofracture remains closed and the P-Q slope is considerably higher than that achieved after the fracture reopens. A transition zone is often observed between the two main slopes. The selection of Ps within the transition zone can be subjective if the change in the P-Q slope is not abrupt. Nonlinear regression analysis helps singleout P~ objectively by best-fitting a bilinear curve which can be formulated as: P =b, + b:Q for Q <<.Qs, P=b3+b4Q
for
Q>Qs,
(9)
where P is the borehole pressure (function of Q--the flowrate), and b~ (i = 1 to 4) and Q, (flowrate when the fracture opens) are unknown parameters. We apply NLRA to equation (9) to obtain the unknown parameters and to determine the shut-in pressure, P~ = b, + b2Q~. A typical bilinear P-Q regression in a digitallyrecorded steprate pressure test is shown in Fig. 4. Table 1 summarizes the results of our selected set of tests in which steprate pressure cycles were carried out. The shut-in pressures derived from this method were generally close to the shut-in pressures obtained from the exponential pressure-decay [within 13(+ 8)% of P ~ - u] and the bilinear pressure-decay-rate methods [within
12(+7)% of P~'-"]. In general, the application of NLRA in P, estimation using the described techniques yields unambiguous values, which are both reasonable and independently verifiable by anyone who has access to the field digital data and are, therefore, amendable to standardization.
FRACTURE REOPENING PRESSURE Objective determination The fracture reopening (refrac) pressure (Pr) in a pressure-time record has been defined as the pressure in a subsequent cycle (commonly cycle 2 or 3) at which the slope of the ascending portion departs from that of the initial fracture-inducing curve, provided the flowrate is kept constant throughout [15, 16]. Conventionally, Pr is determined graphically by superposing the fractureinducing pressure--time curve over that of a subsequent pressurization cycle. This procedure can be subjective, since in most cases, the change in slopes is not sufficiently distinct (Fig. 5). We have introduced a statistical technique of superposing the curves and selecting Pr such that the uncertainties involved in the graphical pro-
8
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ft.
,5PQ
Ps
ft. 3-
2-
0
o
I
2'0
I
451
4'0
I
I
60
FLowrote (cc/sec)
Fig. 4. Shut-in pressure determination using a pressure--flowrate ( P - ~ ) plot obtained from steprate pressurization. The bilinear curve resulting from NLRA determines P,~.
4_._
L E E and H A I M S O N :
HYDRAULIC
FRACTURING
PARAMETERS
15815m,
14,
W1 h o k e , W i s c
/
13. 12, 11 10 &. :E
9 8
I-
o. n
6 5-
4321. 0
I
©
20
4
60
8
100
120
140
0
i
20
i
/ 40
I
i 60
I
i
i
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i
lOO
80
I
I
i
12o
140
Time (see)
Time (see)
Fig. 5. Fracture reopening pressure (P,) using graphical technique of superposition.
cedure are minimized even when the onset of the deviation between the two curves is ill-defined (Fig. 5). The procedure involves the comparison of digitized pressure-time data belonging to a segment of the ascending portion in the fracturing cycle to data points in the repressurization cycle having the same elapsed time. The superposition of the two curves is accomplished by minimizing the sum of squared pressure errors. In order to evaluate the fit of the two curves, the average pressure difference and its standard deviation in a selected segment are calculated. Here, we define the "reference threshold" as the average pressure difference added to the value of two standard deviations. We interpret the
onset of deviation between the two curves as the starting point of continuous pressure difference that is larger than the "reference threshold". The rationale behind this is that any difference greater than the defined "reference threshold" can be regarded as statistically significant with approx. 95% confidence level. If the determined onset of deviation is found inside the initial segment (Fig. 6a), the "reference threshold" value is probably overestimated, since nonmatching data are included in the comparison between the two curves. If so, the segment is shortened at the high end and the iteration procedure is repeated until the onset of deviation is found just outside the segment (Fig. 6b). We define this
15 81.Sm,W1
14-
hoLe,Wisc.
1312-
Cycle 1 : ",,~: .
11-
Cycle 2
CycLe 1
,t
"...j Cycle
.
#
2
• $
10-
9-
s
Seg 1
8-
p2-_. p 3 = p r --r r ~
1
7-
Pr
6-
Seg 2 5-
Seg 3
Y
: '
321-
m
0
= 0
20
40
60
Time (sec) (a)
80
10
t00
120
0
I
!
40
I
i 60
i
i !
80
100
120
Time (sic) (b}
Fig. 6. Fracture reopening pressure determination using "reference threshold" method: (a) initial segment 1 yields unsatisfactory
matching, since P~ is inside the segment; (b) segment 2 determines Pr; reduced segment 3 confirms Pr.
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
point as the fracture reopening pressure (P~). A further reduction in the segment selected for comparison will yield an onset-of-deviation point close to the originally selected P~. We applied our statistical method to a set of field hydrofracturing records obtained in Waterloo, WI and the results are presented in Table 2. The values of the "reference threshold" ranging from 0.1 to 0.6 MPa represent individual uncertainties of the determined refrac pressures. We could obtain Pr unambiguously in all cases for which the same flowrate was used in the compared pressurization cycles. However, we could not obtain P, in the third pressurization cycle of the test conducted at depth 50.3 m. The slope of the ascending portion was higher than that of the fracture cycle since a larger flowrate was used during repressurization.
The applicability of t', to the estimation of Sn Using the values obtained statistically for P~ we calculated the inferred hydraulic fracturing tensile strength of the rock (T = Pc - P~) based on equations (2) and (3), and found it to be in the range of 4.5-10 MPa (Table 2). This averages 7(+ 2) MPa, which is about one half of the 12MPa determined in laboratory hydrofracturing tests for the same rock [15]. The argument can be made that the difference between these values may be accounted for by a possible size effect, and that the refrac pressures determined from the "reference threshold" method appear reasonable. On the other hand, we note that there is no significant
453
difference between the average P~ and P~ [10(+4)%; see Table 2]. This may indicate that the state of stress is such that the maximum horizontal effective stress is approximately equal to twice the minimum horizontal effective stress: ( S , - Po)/(Sh--Po)=2. However, a substantial number of field hydrofracturing tests record P, values that are suspiciously close to P~. Table 3 lists some 15 hydrofracturing sets of tests selected at random from all over the world, and conducted at depths reaching 1500 m, in rocks as diverse as granite, sandstone, salt and basalt, in which the difference between the two pressures is no larger than 25%. Could it be that the relation between the principal horizontal stress magnitudes is so consistent in different parts of the world, irrespective of regional geology, tectonics and rock type? Or is it perhaps that Pr obtained from field data is not the same as the expected pressure, based on the theoretical relation given in equation (3)? Does the pressure read from test records represent a later event, such as the reopening of the entire fracture, rather than the reopening of just the fracture lip at the drillhole wall? Preliminary laboratory experiments [14] also suggest that the recorded Pr may be more closely associated with P~ than with the minimum tangential stress at the borehole wall as postulated in equation (3). The uncertainty surrounding the use of field-recorded P~ to determine Sa requires additional research to determine how to define the point on the pressure-time cUrve that corresponds to the theoretical interpretation of the refrac pressure. It is recommended that in the interim
Table 2. Waterloo quartzite hole---fracture reopening pressures using the "'reference threshold" method and comparison with Pc and Ps values in six randomly-selected tests Depth (m) 50.3
66.5
81.5
129.2
183.4
206.9
Cycle
Q (cm3/sec)
Pc (MPa)
1 2 3
12 11 27
I0.0
1 2 3
39 36 35
14.3
1
37
13.6
2 3
34 35
1 2 3
33 30 35
14.8
1 2 3
36 35 36
15.0
1 2 3
35 35 36
16.9
P, (MPa)
p,~g (MPa)
p~vs-u (MPa)
Error T (%) (MPa)
5.6
6.0
6.7
4.4
4.3
3.9
10.3
10.0
5.4
5.7
5.3
8.2
6.5
5.8
12.1
8.3
8.4
7.9
6.3
6.6
10.3 16.5 Mean 9.5 SD 3.9
4.9 7.1 2.2
5.6(0.12)
3.7 (0.08) 4.8 (0.10)
5.3 (0.04) 5.5 (0.08) 6.6 (0.30) 6.4 (0.27) 8.3 (0.20) 8.5 (0.17) 12.0 (0.41) 12.0 (0.57) 12.0
Q is the flowrate, P~ is the breakdown pressure, Pr is the fracture reopening pressure. p~vg is the average Pr between cycles 2 and 3. Error = 100 x [p~vs _ p,vs-ul/p~vs-u; T ~ P c - p~,s. Numbers in parentheses represent the "reference threshold" used to define P~; SD is the standard deviation.
454
LEE and HAIMSON: HYDRAULIC FRACTURING PARAMETERS
Table 3. Comparison of PT and P~ values obtained during hydrofracturing in various rock types around the world Depth No. of Site and range tests Comparison rock type (m) (n) (%) Reference a 50-210 6 11 (+5) b 266-849 7 22 ( + 10) [17] c 80-230 14 19 (+ 16) [17] d 97-961 I1 18(+13) [18] e 140-290 5 24 ( _+7) [19] f 954-1477 5 2 ( __+1) [20] g 76-218 14 22 ( +__13) [21] h 1002-1042 6 6 (-+-4) [22] i 108-1284 8 25 (+ 19) [23] j 60-124 16 26 (+ 14) [24] k 24-50 15 3 ( + 3) [25] 1 77--429 16 18 (+ 15) [26] m 80-710 9 14(+11) [12] n 18-85 4 8 ( + 7) [27] o 214-612 11 15(+14) [28] Pr is the fracture reopening pressure, P~ is the shut-in pressure, n is the number of tests conducted. Comparison = (100/n) x ~" (IP', - P'rl/P~). i=1
Sites and rock types: (a) Waterloo, WI, quartzite; (b) XTLR hole, CA, sedimentary rock; (c) Mojave Desert, CA, quartz monzonite; (d) Monticello, SC, gneiss and schist; (e) Jordanelle, UT, andesite; (f) Paradox Basin, UT, rock salt; (g) Rocky Mountain, GA, shale, sandstone and siltstone; (h) Hanford, WA, basalt; (i) Moodus, CT, gneiss; (j) Niagara Falls, Canada, limestone, sandstone and shale; (k) Alsace, France, rock salt; (1) Okabe, Japan, mudstone and sandstone; (m) Jianchuan, China, sandstone; (n) Yixian, China, diorite: and (o) Palekhori. Cyprus, diabase. m o r e emphasis should be placed on the use o f the original equation (2) and, where appropriate, the fracture pressurization m e t h o d [I1, 29] should be applied.
FRACTURE DELINEATION An impression packer, a borehole televiewer, or other logging devices can be used to determine the orientation o f fractures on the borehole wall. This information is
crucial to defining the m situ stress tensor. However, it is sometimes difficult to uniquely identify the fracture plane if its traces on the impression packer or televiewer" p h o t o g r a p h are only partially visible or meandering. We have employed, and are presenting herein, techniques based on sinusoidal curve-fitting and circular statistics to delineate accurate orientations that can be inferred from incomplete traces o f inclined and vertical fractures, respectively.
Inclined fracture The intersection o f a drillhole with an inclined fracture takes the form o f an ellipse, which appears as a complete cycle o f a sinusoidal curve on the u n w r a p p e d image o f an impression packer (or on a borehole televiewer picture [30]). The dip direction o f inclined fractures is determined by the azimuth o f the lower peak o f the sinusoidal curve; the dip angle is calculated from the arctangent o f the sinusoidal curve amplitude divided by the borehole radius. However, it is difficult to delineate an inclined fracture if its traces are only partially visible, and not describing a unique plane. Uncertainties with respect to the peak direction and the height o f the fracture could result in significant errors in fracture configuration. To minimize these difficulties, we utilize a sinusoidal curve-fitting method. A 2-D inclined fracture intersecting a borehole leaves the following sinusoidal signature on the borehole wall: D = e. + e_, sin(E + e3),
where e~, e2 and e3 are u n k n o w n parameters depending on fracture configuration, and D and E are coordinates o f the traces in terms o f depth and azimuthal angle along the drilihole axis, respectively. The traces obtained in field experiments are digitized in terms o f D - E coordinates (Fig. 7a). The set o f data is fitted to equation (10) using N L R A in order to
." ":...
"... ...." Ee.__.__.~_ / (a)
•
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f
\
!
,,.
"
i
(b} i
t
Er
(10)
--"
'
Fig. 7. (a) Sinusoidal regression method applied to an inclined fracture trace; (b) circular statistics applied to a vertical fracture trace.
LEE and HAIMSON: HYDRAULIC FRACTURING PARAMETERS
455
133"
190* 211.5
227.4
l
i
Fitted
~°•l
inclined 212.0
227.6
8
ver t i c o k frocture
D i p dir. 3 4 5 * +- 1.5 ° Stril~e 4 8 0 ~ : 4.,~ ° •
Dip 5 5 ° t 1.5 ° 227.8
I
I
i
t
S
E
N
W
Azimuth (o)
212.5
I
O
E
N
I
I
W
s
Azimuth (b)
Fig. 8. Unwrapped impression packer image showing: (a) digitally-tracedincomplete inclined fracture with the optimum fracture plane obtainedby sinusoidalcurve-fitting;(b) digitally-traceden 6chelonfracturewith the mean directionand standard deviation of the fracture determined by circular statistics.
determine the three unknown parameters. This results in an objective definition of fracture strike and dip. The degree of uncertainty of the fracture orientation depends on the planar features of the fracture and the amount of available traces. The average dip direction and its uncertainty are calculated from the parameter e3 and its standard deviation. The average dip angle and its uncertainty are calculated from parameters el and ez and their standard deviations. An example of fracture delineation based on sinusoidal curve-fitting is presented along with the digitized fracture traces in Fig. 8a. Although the traces are splaying and incomplete, we were able to delineate the mean fracture plane with an uncertainty of only + 1.5 °.
Vertical fracture When the trace of a fracture is vertical (parallel to hole axis) the sinusoidal model [equation (10)] becomes invalid. The image of an induced vertical hydrofracture obtained on an impression packer or a borehole televiewer picture is not always in the form of two straight lines. The traces may be discontinuous, en 6chelon and off-centre. In order to remove subjectivity in the determination of fracture orientation, we have employed circular statistics [31]. Figure 7b shows the major steps used to delineate the vertical hydrofracture. First, fracture traces are digitized in terms of azimuthal angle with respect to a reference direction (e.g. North). Each angle (Ej) corresponding to the digitized trace can be represented by a unit vector on a circle. This results in two groups of vectors representing the two halves of the vertical fracture. Since the strike of a fracture is independent of sense (E; = E~ + 180°) we can reduce the range of possible strikes to 0-180 ° by rotating one of the groups by 180 °. Using vectorial summation the direction and
length of the resultant vector can be calculated. The mean orientation of the fracture (Eo) is defined as the resultant vector direction, and the standard deviation (SD) of the mean direction is obtained from the length of the resultant vector: Eo = cos-' [X/L], X=(/=~ cosEi)/n,
SD = [ - 2 In(L)]-2/2, Y=(i=~ sinEi)/n.
(ll)
In equation (11) X and Y are the mean components of the resultant vector along the 0 and 90 ° directions, respectively; n is the number of the digitized data; and L = (X2+ y:)0.5 is the mean length of the resultant vector. We applied circular statistics to a typical en 6chelon hydrofracture trace obtained in the field• Figure 8b shows the unwrapped digitized impression packer image with the optimum vertical fracture orientation. Visual inspection approximates the mean strike direction at 50-55 ° (_+ 15°). The mean fracture strike based on equation (11) is a well-defined 48 ° (+4.5°). As long as fracture traces are complete and reasonably planar, the use of statistics may seem superfluous. In our experience, however, most impression tests or televiewer photographs yield less than ideal fracture traces. For these cases the statistical techniques described remove subjectivity, offer confidence and uniformity in the estimation of fracture orientation, and enable the standardization of field record interpretation. CONCLUSIONS We have applied statistical data analysis to enhance the objectivity of identifying shut-in and fracture reopen-
456
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
ing pressures and fracture orientation. We introduced general nonlinear regression analysis to the exponential pressure-decay model for estimating shut-in pressures. We employed bilinear regression analysis to assess the shut-in pressure as the inflection point of the bilinear curve best-fitting the variation of the pressure decay rate as a function of pressure. A similar bilinear approximation was used to evaluate the shut-in pressure from pressure vs flowrate plots. An example of field data analysis suggests that all of these regression methods are capable of yielding comparable and reliable shut-in pressure values. We have developed a "reference threshold" method which allows us to select the fracture reopening pressure using numerical superposition of the ascending portions of the pressure-time curves in the fracture-inducing cycle and a subsequent cycle. We employed circular statistics and sinusoidal curvefitting to define the orientation of induced hydraulic fracture planes when the traces obtained are incomplete, en +chelon and/or meandering. The above statistical methods could also serve as a first important step towards standardization of hydraulic fracturing data interpretation due to the objectivity of these techniques. An important byproduct of our research has been the finding that the fracture reopening pressure is almost always close to the field value of the shut-in pressure. This brings into question the validity of the most commonly-used relation for evaluating the maximum horizontal in situ stress. Acknowledgements--The work reported was funded by the National Science Foundation Grants EAR-8708164 and EAR-8720990 and the U.S. Geological Survey Grant 14-08-0001-G1517.
REFERENCES
1. Hubbert M. K. and Willis D. G. Mechanics of hydraulic fracturing. Trans. AIME 210, 153-166 (1957). 2. Haimson B. C. and Fairhurst C. Initiation and extension of hydraulic fracture in rocks. Soc. Petrol. Engrs. J. Sept, 310-318 (1967). 3. Bredehoeft J., Wolff R., Keys W. and Shuter E. Hydraulic fracturing to determine the regional in situ stress field, Piceance Basin, Colorado. Geol. Soc. Am. Bull. 87, 250-258 (1976). 4. Gronseth J. M. and Kry P. R. Instantaneous shut-in pressure and its relationship to the minimum in situ stress. Hydraulic Fracturing Stress Measurements, pp. 55-60. National Academy Press, Washington, D.C. (1983). 5. Enever J. R. and Chopra P. N. Experience with hydraulic fracture stress measurements in granite. Proc. Int. Syrup. on Rock Stress and Rock Stress Measurement, pp. 411-420. Centek, Lule~ (1986). 6. Doe T. W., Hustrulid W. A., Leijon B, Ingvald K. and Strindell L. Determination of the state of stress at the Stripa mine, Sweden. Hydraulic Fracturing Stress Measurements, pp. 119-129. National Academy Press, Washington, D.C. (1983). 7. Zoback M. D, and Haimson B. C. Status of hydraulic fracturing method for in situ stress measurements. Issues in Rock Mechanics Proc. 23rd U.S. Syrup. on Rock Mechanics, pp. 143-156. Soc. of Mining Engineers of AIME, New York (1982). 8. Muskat M. Use of data on the build-up of bottom-hole pressures. AIME Petrol. 123, 44-48 (1937). 9. Aamodt R. and Kuriyagawa M. Measurement of instantaneous shut-in pressure in crystalline rock. Hydraulic Fracturing Stress Measurements. pp. 139-142. National Academy Press, Washington, D.C. (1983). 10. Draper N. R. and Smith H. Applied Regression Analysis, 2nd Edn, pp. 458-462. Wiley, New York (1981).
11. Haimson B. C. and Lee M. Y. The state of stress and natural fractures in a jointed precambrian rhyolite in south-central Wisconsin. Proc. 2 8 t h U.S. Symp. on Rock Mechanics, pp, 231-240. Balkema. Rotterdam (1987). 12. Haimson B. C., Springer J. E.. Lee M. Y., Zoback M. D., L1 F., Zhai Q and Liang H. Results of hydraulic fracturing stress measurements at Jianchuan, Western Yunnan Province, China. USGS Open-file Report 87-477 (1987). 13. Haimson B. C., Tunbridge L., Lee M, Y, and Cooling C. Measurement of rock stresses using the hydraulic fracturing method in Cornwall, U.K.--Part It. Data reduction and stress calculation. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 26, 361-372 (1989). 14. Cheung L. S. and Haimson B. C. Laboratory study of hydraulic fracturing pressure data--how valid is their conventional interpretation? Int. J. Rock Mech. Min Sci. d Geomech. Abstr. 26, 595-604 (1989). 15. Haimson B. C. Near surface and deep hydrofracturing stress measurements in the Waterloo quartzite. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 17, 81-88 (1980). 16. Hickman S. H. and Zoback M. D. The interpretation of hydraulic fracturing pressure-time data from in situ stress determination. Itydraulic Fracturing Stress Measurements, pp. 44-54. National Academy Press, Washington, D.C. (1983). 17. Zoback M. D., Tsukahara H. and Hickman S. Stress measurements near the San Andreas Fault. J. Geophys. Res. 85, 6157-6173 (1980). 18. Zoback M. D. and Hickman S. In situ study of the physical mechanics controlling induced seismicity at Monticello Reservoir, South Carolina. J. Geophys. Res. 87, 6959-6974 (1982). 19. Haimson B. C. and Lee M. Y. Stress measurements at the Jordanelle Dam site, Central Utah, using wireline hydrofracturing. Final Report to USGS (1985). 20. Schnapp M., Kocherhans J. G. and Nelson R. A. Stress measurements in salt in a deep borehole, southeastern Utah. Hydraulic Fracturing Stress Measurements, pp. 79-85. National Academy Press, Washington, D.C. (1983). 21. Barton N. Hydraulic fracturing to estimate minimum stress and rock mass stability at a pumped hydro project, ttydraulic Fracturing Stress Measurements, pp. 61 67. National Academy Press, Washington, D.C. (1983). 22. Haimson B. C. Stress measurements at Hanford, Washington for the design of a nuclear waste repository facility. Proc. 6th Congr. of the International Society .for Rock Mechanics, pp. 119-123. Balkema, Rotterdam (1987). 23. Baumgartner J. and Zoback M. D. Interpretation of hydraulic pressure-time records using interactive analysis methods: application to in situ stress measurements at Moodus, Connecticut. Proc. 2nd Int. Workshop on l~vdraulic Fracturing Stress Measurements, pp. 619 645 (1988). 24. Haimson B. C., Lee C. F. and Huang J. H. S. High horizontal stresses at Niagara Falls, their measurement, and the design of a new hydroelectric plant. Proc. Int. Syrup. on Rock Stress and Rock Stress Measurement, pp. 615-624. CENTEK, Lule~ (1986). 25. Cornet F. Two examples of stress measurements by the H.T.P.F. method. Key Questions m Roek Mechanics: Proc. 29th U.S. Syrup. on Rock Mechanics, pp. 615-623. Balkema, Rotterdam (1988). 26. Tsukahara H. Stress measurements utilizing the hydraulic fracturing technique in the Kanto-Tokai Area, Japan. Hydraulic Fracturing Stress Measurements, pp. 18-27. National Academy Press, Washington, D.C. (1983). 27. Li F., Li Y., Wang E., Zhai Q., Bi S., Zhang J., Liu P., Wei Q. and Zhao S. Experiments of in situ stress measurements using stress relief and hydraulic fracturing techniques. Hvdraulic Fracturing Stress Measurements, pp. 130-134. National Academy Press, Washington, D.C. (1983). 28. Baumgartner J., Rummel F., Haimson B. C. and Lee M. Y. In situ stress measurements and natural fracture logging in Drillhole CY-4, the Troodos Ophiolite, Cyprus. Initial Report of the Cyprus Crustal Study Project, Drillhole CY-4, Canadian Geological Survey (1989). 29. Cornet F. H. and Valette B. In situ stress determination from hydraulic injection test data. J. Geophys. Res. 89, 11,527-11,537 (1984). 30. Zemanek J., Caldwell R. L., Glenn E. E. Jr, Holcomb S. V., Norton L. J. and Straus A. J. D. The borehole televiewer--a new logging concept for fracture location and other types of borehole inspection. J. Petrol. TechnoL 21, 762-774 (1969). 31. Mardia L. The Statistics qf Orientation Data. Academic Press, London (1972).
Printed in Great Britain. All rights reserved
0148-9062/89 $3.00+0.00 Copyright ~ 1989 Pergamon Press plc
Statistical Evaluation of Hydraulic Fracturing Stress Measurement Parameters M. Y. LEE? B. C. HAIMSON? Statistical analysis of hydraulic fracturing fieM data enhances the objectivity of determining shut-in and fracture reopening pressures and fracture orientations. Using nonlinear regression analysis (NLRA), we isolate the closedfracture segment of the pressure-time curve by fitting it to an exponential decay model. The shut-in pressure (P,) is expected to be in the range between ~ - l (the onset of the post fracture-closure segmenO and I~,~-" (the pressure level of the best-fitting exponential curve extrapolated to the time of pump shut-off). We employ bilinear regression analysis to improve the objectivity of selecting P, from the pressure-decay rate vs pressure plot. The same analysis helps to identify P, in the pressure vs flowrate record obtained during steprate pressurization of the hydrofractured interval. We identify unambiguously the fracture reopening pressure (Pr) by superposing the ascending portion of the pressure-time curves in the fracture-inducing cycle and a subsequent cycle. P, is the onset of deviation between the two curves, defined as the point where the difference in pressures is larger than a statistical "reference threshold", provided the difference continues to increase from there on. Circular statistics and a sinusoidal curve-fitting method provide the means of objectively delineating the complete trace of induced hydrofractures from fragmented or splaying traces on the impression packer or on the borehole televiewer photograph. Examples of field hydraulic fracturing data and statistically determined pressures and fracture orientations are presented. A startling discovery was made in the course of determining P, with our new technique: it appears to be very nearly equal to the estimated P,. A review of the literature shows that in a great number of cases P, and P, are indeed close to each other, which raises the question whether we are in fact reading the correct fracture reopening pressure.
INTRODUCTION
shut-in pressure (Ps) is the lowest pressure level, during Hydraulic fracturing (or hydrofracturing) consists of the borehole pressure decay, capable of maintaining t h e injecting water at a constant flowrate into a sealed-off hydrofracture open against the far-field least horizontal segment of a vertical drillhole until a critical (or "break- stress. Hence, Ps serves as a direct estimate of Sh [1,2]: down") pressure (Pc) is reached at which the rock Sh = P,. (l) surrounding the hole fails in tension and a fracture develops. In linear elastic, homogeneous, isotropic, and The conventional interpretation of hydrofracturing reintact rock the hydrofracture is expected to be vertical quires the additional assumption that rock is imperand perpendicular to the minimum horizontal stress meable to the injected fluid. This allows the maximum (Sh). An impression packer or other borehole logging horizontal stress (S.) to be calculated directly from the devices (e.g. the borehole televiewer) can be used to Kirsch solution for stresses around a circular hole, and obtain the orientation of the hydrofracture which yields the Terzaghi effective stress law [1,2]: the direction of Sh. Following breakdown, fluid injection ( S . - P 0 ) = T + 3 (Sh -- P 0 ) - - (Pc - P0), (2) is ceased causing the borehole pressure to drop. The tGeological Engineering Program, University of Wisconsin-Madison, 1509 University Avenue, Madison, Wl 53706, U.S.A. 447
where P0 is the pore pressure, and T is the tensile strength of the rock. If subsequent pressurization is carried out in the hydrofractured segment of the hole,
448
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
equation (2) reduces to [3]: (s.
- C o ) = 3 ( s h - Co) - ( e , - t'0),
(3)
where Pr is the fracture-reopening (or "refrac") pressure, and T is assumed to be zero since the tested interval is already fractured. The assumptions here are that the fracture closes completely between pressurizations, and that P, is the pressure level at which the preexisting hydrofracture just begins to open, i.e. it is equal to the minimum compressive hoop stress at the hole wall induced by far-field stresses. In this paper we are mainly concerned with reducing the subjectivity of graphical methods for the determination of hydrofracturing parameters, such as Ps, P, and fracture orientation. We accomplish this by introducing nonlinear and other statistical techniques to digitallyrecorded test data.
tical analysis procedures applied to digitally-recorded field pressure and flowrate data.
Exponential pressure-decay method Pressure decay following pump shut-off is a result of additional extension of the hydrofracture, as well as permeation of the fracturing fluid from the test interval and the hydrofracture into the surrounding rock. As it decreases, the pressure will reach the shut-in pressure value (P,), equal to the far-field fracture-normal stress; further decay will bring about complete fracture closure. Beyond that, any additional pressure drop is mainly the result of radial fluid flow through the borehole wall into the rock. Muskat's [8] model of exponential pressure decay due to such radial flow has been suggested as a method of identifying P, [9]. We express this model as: P=exp(djt+d_,)+Pa~
SHUT-IN PRESSURE
The determination of P, is straightforward when a sharp break is observed in the pressure-time curve after the initial fast pressure decline following pump shut-off. In some cases, however, pressure decay is gradual and P, is indistinct (Fig. 1). Over the years several methods, mostly graphical, have been suggested and employed for estimating P, from pressure-time curves. Three of the most common have been the tangent divergence [4], the tangent intersection [5] and the logarithmic methods [6, 7]. Although the above methods provide reasonable approximations of P, in many situations, ambiguities remain when pressure decay is gradual with no obvious "breaks" or "knees". We have developed and are describing in this paper means of improving the objective determination of P, by employing well-established statis-
for
t>lh,
where P is the pressure in the test interval, dl ( < 0) and d2 are unknown parameters characterizing the pressure decay rate, P,t is the asymptotic pressure level in the test interval, t is the time after pump shut-off and tt is the time at which pure radial flow initiates (complete fracture closure). We first determine p~r~-~, the interval pressure upon fracture closure, by applying nonlinear regression analysis (NLRA) to the decaying portion of the pressure--time curve. Nonlinear regression analysis is a statistical technique for fitting an arbitrary function [for example equation (4)] to a given set of data points [10]. It determines the pressure decay parameters (d~ and d2) and the asymptote (P,~) by minimizing the sum of the squares of errors (SSE) between the recorded data and the predicted pressures based on the model. The fit is evaluated in terms of the residual mean square (RMS)
1614-
"l o 0.
{L
el.Sin,w1 hole ,Wisc.
1210 8 6 42J
O2o
~
6o
~
80 0
10
(4)
20
30
40
Tirol (rain) Fig. 1. Pressure-time record during a field hydrofracturing test in the Waterloo quartzite drillhole, W]. Breakdown pressure Pc, and indistinct fracture reopening (P,) and shut-in (Ps) pressures are shown. Cycle 4 is a steprate pressurization test using constant flowrate and self-stabilized pressure.
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
which represents the average deviation of the curve from the fitted model: RMS =
Ii
i- I
( e , - Pp,):/(n - 3)
1
,
(5)
where P~ is the digital pressure data, Pp~ is the NLRA predicted pressures based on the exponential decay model, and n is the number of data used in the regression analysis. An iteration procedure is invoked aimed at excluding the segment of the decaying pressure-time record before fracture closure. Starting with the pressure at the time of pump shut-off (t = 0), data points are removed sequentially with each iteration. This fitting procedure ends when the decreasing RMS-value stabilizes. At this point it is assumed that all the pressure data belonging to the open-fracture time segment (t < tt) have been removed from the curve-fitting process (Fig. 2). The largest pressure value of the fitted pressure--time curve ( P ~ - ] ) is interpreted as the level at which the induced fracture has completely closed; hence, it is the lower limit of the expected shut-in pressure value. The extrapolated pressure level obtained by the fitted exponential curve at time t = 0 ( P ~ - " ) represents the pressure at which pure radial flow would have commenced were the fracture to close instantaneously upon pump shut-off (Fig. 2); it is, thus, the upper limit of the range of values within which P~ is to be found: p ~ d - ] < p, < p~d-., (6) where
pressure estimates based on other techniques [11-13]. Aamodt and Kuriyagawa [9] also recommended using P ~ - " as the shut-in pressure, although their technique for obtaining P ~ - " was somewhat different and less rigorous. We have selected at random a set of six tests out of some 40 hydrofracturing measurements recently conducted in Precambrian quartzite near Waterloo, WI and analyzed the digitally-recorded data using the exponential pressure-decay method. We were able to obtain objective values of P s~ - t and P ~ - " , unaffected by analogue recording scale or any other bias. Among the different methods selected to estimate P, in Table 1, the average shut-in pressure using P ~ - " generally yields smaller standard deviations than when employing P,~-L It appears that P ~ - ~ yields a more consistent value of P, when compared to other methods of shut-in pressure determination than does p,pd-I (Table 1), although the most correct estimates probably lie somewhere between P ~ - I and P ~ - ~ . This finding is strongly supported by laboratory results detailed in a companion paper [14]. Bilinear pressure-decay-rate method (dP /dt vs P)
The functional form of the pressure decay prior to fracture closure is generally unknown. As an approximation, one could assume that this portion of the pressure decay is also represented by an exponential function (L. Tunbridge, personal communication, 1987). Under this assumption the total pressure decay after pump shut-off is composed of two exponential curves:
p ~ - i = exp(dltt + d2) + P,l and P ~ - " = exp(d2) + Pa, In our experience of the last several years, P ~ - ~ is consistently close (usually within +0.5 MPa) to shut-in
7'5~VV 7.0 6.5
~
t < h,
P = exp(dst + dr) + Pa3 for
t > t,,
V
"
Mm
Me-
......,.
.
~'.0
/
~
81 lSm, Wl ~ol.e,Wise.
(tt,p~l.t)
3.5
.
~
3.0
2.5
I 0
iO
I
I eO
1
I 120
I
I 160
I
I 200
TIn~l ( s i c )
2.
the
(7)
where t, is the time at which P, is reached; d3(<0) and d4 are unknown parameters characterizing the pressure decay prior to t,; d5 (<0) and d6 are unknown par-
2 Me4.
V
5.5
P = e x p ( d 3 t + d , ) + P~2 for
M~
v
6.0
449
Fig. Shut-in pressure determination using exponential pressure-decay method. Nonfitting data are shown as blank triangles. The stabilized residual mean square (RMS) value determines pressure level at which pure radial flow initiates ( P ~ - ' ) . The extrapolated shut-in pressure (pq, d-u) to time t ffi 0 is also shown.
450
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
Table 1. Waterloo quartzite hole--shut-in pressures determined by different techniques using NLRA in six randomly-selected tests
Depth
p~-;
p~r~-o
p~,-,
(m)
Cycle
(MPa)
(MPa)
(MPa)
50.3
2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4
6.3 5.4
6.7 6.0
6.5 6.0
3.3 3.2
3.8 3.7
3.7 3.6
4.3 4.1
5.9 5.7
5.5 5.3
4.2 3.7
5.7 4.7
5.8 5.4
7.2 8.3
7.4 8.5
7.3 8.3
9.6 9.2
10.7 10.0
10.5 10.0
66.5
81.5 129.2 183.4 206.9
p~
p~8-~
p~-~
(MPa)
(MPa)
(MPa)
5.3
5.8(___0.4) 6.0(4-0.5)
4.4
3.8(4-0.5)
3.9(4-0.3)
6.0
5.2(+0.7)
5.7(+0.2)
6.6
5.4(+1.1)
5.8(+0.6)
7.8
7.8(+0.0)
7.9(+0.1)
10.2
10.0(-I-0.4)
10.3(_-4-0.1)
P ~ - ~ and p , ~ - u are the lower and the upper limits of shut-in pressures estimated from the exponential pressure-decay (epd) method. p~,-z~ = shut-in pressure estimated from the pressure-decay-rate (pdr) method using bilinear (11) best-fit. P~ = shut-in pressure estimated from the pressure-flowrate (pq) method. p~vg-l = average shut-in pressure using P ~ - ' , e ~ r - n and P~. p~,g-u = average shut-in pressure using p ~ - u , p ~ , - , and P~. Numbers in parentheses represent standard deviations of the average shut-in pressure.
ameters representing pressure drop after t,; Pa2 and Pa3 are the unknown asymptotic pressure levels, respectively. By taking time derivatives of equations (7), a bilinear relationship between dP/dt and P is obtained (L. Tunbridge, personal communication, 1987):
dP/dt = d3 exp(d3 t + d4) =d3(P-Pa2)
for
P>P~,
dP/dt = d5 exp(ds/+ d6) =ds(P-Pa3)
for
P
(8)
A typical dP/dt vs P plot applied to a field record shows two linear segments connected by a transition zone, within which the shut-in pressure is found (Fig. 3). In order to single out P~ objectively we define the bestfitting bilinear curve by applying NLRA to the set of dP/dt vs P data. We start by assigning an initial transition point which divides the data into two discernable groups. Within each group a best-fitting straight line is obtained and the sum of squared errors (SSE) between the data points and the fitted straight line is calculated. Using an iterative process the nonlinear
0.20
81.Sm,W1 hole , Wl$c.
0.15
¢,
/
--
0.10 O
p Ix:lr- tL
J.O
0.05
.
25
.........
i 35
.
.
.
'
.
' 4.5
'
i 55
'
6 i5
'
7.5
Preuure [MPa] Fig. 3. Shut-in pressure determination using the bilinear pressure-d~ay-rate (dP/dt vs P) method. The bilinear curve resulting
from NLRA determines P ~ ' - " .
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
regression analysis effectively determines the best-fitting transition point (P~r-") that gives rise to the smallest of all the calculated SSE for both straight-line approximations (Fig. 3). The calculated p ~ - H is unaffected by any other factors except the digitized test data, and is thus unambiguous and given to confirmation by independent workers. Table 1 indicates that, using the bilinear pressure-decay-rate method, the shut-in pressures (P~'-") in our set of tests are comparable to shut-in pressure values obtained by two alternative methods, and are in very close proximity to P~-U. Laboratory testing has also confirmed the reliability of the bilinear technique [14].
Pressure-flowrate method (P-Q) The pressure-flowrate method requires a special procedure to be followed during the hydrofracturing test. We typically carry this out after several conventional pressurization cycles (see for example Fig. 1). The hydrofractured interval is first pressurized at a very low flowrate (Q) until the pressure (P) stabilizes. The flowrate is then raised to a new constant level and maintained until a corresponding constant borehole pressure is recorded. The process is continued for several steps of flowrate (typically 6-12). A pressure vs flowrate (P-Q) plot is typified by two dominant slopes (Fig. 4). The interpretation of this plot is that at pressures below P~ the hydrofracture remains closed and the P-Q slope is considerably higher than that achieved after the fracture reopens. A transition zone is often observed between the two main slopes. The selection of Ps within the transition zone can be subjective if the change in the P-Q slope is not abrupt. Nonlinear regression analysis helps singleout P~ objectively by best-fitting a bilinear curve which can be formulated as: P =b, + b:Q for Q <<.Qs, P=b3+b4Q
for
Q>Qs,
(9)
where P is the borehole pressure (function of Q--the flowrate), and b~ (i = 1 to 4) and Q, (flowrate when the fracture opens) are unknown parameters. We apply NLRA to equation (9) to obtain the unknown parameters and to determine the shut-in pressure, P~ = b, + b2Q~. A typical bilinear P-Q regression in a digitallyrecorded steprate pressure test is shown in Fig. 4. Table 1 summarizes the results of our selected set of tests in which steprate pressure cycles were carried out. The shut-in pressures derived from this method were generally close to the shut-in pressures obtained from the exponential pressure-decay [within 13(+ 8)% of P ~ - u] and the bilinear pressure-decay-rate methods [within
12(+7)% of P~'-"]. In general, the application of NLRA in P, estimation using the described techniques yields unambiguous values, which are both reasonable and independently verifiable by anyone who has access to the field digital data and are, therefore, amendable to standardization.
FRACTURE REOPENING PRESSURE Objective determination The fracture reopening (refrac) pressure (Pr) in a pressure-time record has been defined as the pressure in a subsequent cycle (commonly cycle 2 or 3) at which the slope of the ascending portion departs from that of the initial fracture-inducing curve, provided the flowrate is kept constant throughout [15, 16]. Conventionally, Pr is determined graphically by superposing the fractureinducing pressure--time curve over that of a subsequent pressurization cycle. This procedure can be subjective, since in most cases, the change in slopes is not sufficiently distinct (Fig. 5). We have introduced a statistical technique of superposing the curves and selecting Pr such that the uncertainties involved in the graphical pro-
8
781.Sin,W1 hote,Wisc. 6-
ft.
,5PQ
Ps
ft. 3-
2-
0
o
I
2'0
I
451
4'0
I
I
60
FLowrote (cc/sec)
Fig. 4. Shut-in pressure determination using a pressure--flowrate ( P - ~ ) plot obtained from steprate pressurization. The bilinear curve resulting from NLRA determines P,~.
4_._
L E E and H A I M S O N :
HYDRAULIC
FRACTURING
PARAMETERS
15815m,
14,
W1 h o k e , W i s c
/
13. 12, 11 10 &. :E
9 8
I-
o. n
6 5-
4321. 0
I
©
20
4
60
8
100
120
140
0
i
20
i
/ 40
I
i 60
I
i
i
I
i
lOO
80
I
I
i
12o
140
Time (see)
Time (see)
Fig. 5. Fracture reopening pressure (P,) using graphical technique of superposition.
cedure are minimized even when the onset of the deviation between the two curves is ill-defined (Fig. 5). The procedure involves the comparison of digitized pressure-time data belonging to a segment of the ascending portion in the fracturing cycle to data points in the repressurization cycle having the same elapsed time. The superposition of the two curves is accomplished by minimizing the sum of squared pressure errors. In order to evaluate the fit of the two curves, the average pressure difference and its standard deviation in a selected segment are calculated. Here, we define the "reference threshold" as the average pressure difference added to the value of two standard deviations. We interpret the
onset of deviation between the two curves as the starting point of continuous pressure difference that is larger than the "reference threshold". The rationale behind this is that any difference greater than the defined "reference threshold" can be regarded as statistically significant with approx. 95% confidence level. If the determined onset of deviation is found inside the initial segment (Fig. 6a), the "reference threshold" value is probably overestimated, since nonmatching data are included in the comparison between the two curves. If so, the segment is shortened at the high end and the iteration procedure is repeated until the onset of deviation is found just outside the segment (Fig. 6b). We define this
15 81.Sm,W1
14-
hoLe,Wisc.
1312-
Cycle 1 : ",,~: .
11-
Cycle 2
CycLe 1
,t
"...j Cycle
.
#
2
• $
10-
9-
s
Seg 1
8-
p2-_. p 3 = p r --r r ~
1
7-
Pr
6-
Seg 2 5-
Seg 3
Y
: '
321-
m
0
= 0
20
40
60
Time (sec) (a)
80
10
t00
120
0
I
!
40
I
i 60
i
i !
80
100
120
Time (sic) (b}
Fig. 6. Fracture reopening pressure determination using "reference threshold" method: (a) initial segment 1 yields unsatisfactory
matching, since P~ is inside the segment; (b) segment 2 determines Pr; reduced segment 3 confirms Pr.
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
point as the fracture reopening pressure (P~). A further reduction in the segment selected for comparison will yield an onset-of-deviation point close to the originally selected P~. We applied our statistical method to a set of field hydrofracturing records obtained in Waterloo, WI and the results are presented in Table 2. The values of the "reference threshold" ranging from 0.1 to 0.6 MPa represent individual uncertainties of the determined refrac pressures. We could obtain Pr unambiguously in all cases for which the same flowrate was used in the compared pressurization cycles. However, we could not obtain P, in the third pressurization cycle of the test conducted at depth 50.3 m. The slope of the ascending portion was higher than that of the fracture cycle since a larger flowrate was used during repressurization.
The applicability of t', to the estimation of Sn Using the values obtained statistically for P~ we calculated the inferred hydraulic fracturing tensile strength of the rock (T = Pc - P~) based on equations (2) and (3), and found it to be in the range of 4.5-10 MPa (Table 2). This averages 7(+ 2) MPa, which is about one half of the 12MPa determined in laboratory hydrofracturing tests for the same rock [15]. The argument can be made that the difference between these values may be accounted for by a possible size effect, and that the refrac pressures determined from the "reference threshold" method appear reasonable. On the other hand, we note that there is no significant
453
difference between the average P~ and P~ [10(+4)%; see Table 2]. This may indicate that the state of stress is such that the maximum horizontal effective stress is approximately equal to twice the minimum horizontal effective stress: ( S , - Po)/(Sh--Po)=2. However, a substantial number of field hydrofracturing tests record P, values that are suspiciously close to P~. Table 3 lists some 15 hydrofracturing sets of tests selected at random from all over the world, and conducted at depths reaching 1500 m, in rocks as diverse as granite, sandstone, salt and basalt, in which the difference between the two pressures is no larger than 25%. Could it be that the relation between the principal horizontal stress magnitudes is so consistent in different parts of the world, irrespective of regional geology, tectonics and rock type? Or is it perhaps that Pr obtained from field data is not the same as the expected pressure, based on the theoretical relation given in equation (3)? Does the pressure read from test records represent a later event, such as the reopening of the entire fracture, rather than the reopening of just the fracture lip at the drillhole wall? Preliminary laboratory experiments [14] also suggest that the recorded Pr may be more closely associated with P~ than with the minimum tangential stress at the borehole wall as postulated in equation (3). The uncertainty surrounding the use of field-recorded P~ to determine Sa requires additional research to determine how to define the point on the pressure-time cUrve that corresponds to the theoretical interpretation of the refrac pressure. It is recommended that in the interim
Table 2. Waterloo quartzite hole---fracture reopening pressures using the "'reference threshold" method and comparison with Pc and Ps values in six randomly-selected tests Depth (m) 50.3
66.5
81.5
129.2
183.4
206.9
Cycle
Q (cm3/sec)
Pc (MPa)
1 2 3
12 11 27
I0.0
1 2 3
39 36 35
14.3
1
37
13.6
2 3
34 35
1 2 3
33 30 35
14.8
1 2 3
36 35 36
15.0
1 2 3
35 35 36
16.9
P, (MPa)
p,~g (MPa)
p~vs-u (MPa)
Error T (%) (MPa)
5.6
6.0
6.7
4.4
4.3
3.9
10.3
10.0
5.4
5.7
5.3
8.2
6.5
5.8
12.1
8.3
8.4
7.9
6.3
6.6
10.3 16.5 Mean 9.5 SD 3.9
4.9 7.1 2.2
5.6(0.12)
3.7 (0.08) 4.8 (0.10)
5.3 (0.04) 5.5 (0.08) 6.6 (0.30) 6.4 (0.27) 8.3 (0.20) 8.5 (0.17) 12.0 (0.41) 12.0 (0.57) 12.0
Q is the flowrate, P~ is the breakdown pressure, Pr is the fracture reopening pressure. p~vg is the average Pr between cycles 2 and 3. Error = 100 x [p~vs _ p,vs-ul/p~vs-u; T ~ P c - p~,s. Numbers in parentheses represent the "reference threshold" used to define P~; SD is the standard deviation.
454
LEE and HAIMSON: HYDRAULIC FRACTURING PARAMETERS
Table 3. Comparison of PT and P~ values obtained during hydrofracturing in various rock types around the world Depth No. of Site and range tests Comparison rock type (m) (n) (%) Reference a 50-210 6 11 (+5) b 266-849 7 22 ( + 10) [17] c 80-230 14 19 (+ 16) [17] d 97-961 I1 18(+13) [18] e 140-290 5 24 ( _+7) [19] f 954-1477 5 2 ( __+1) [20] g 76-218 14 22 ( +__13) [21] h 1002-1042 6 6 (-+-4) [22] i 108-1284 8 25 (+ 19) [23] j 60-124 16 26 (+ 14) [24] k 24-50 15 3 ( + 3) [25] 1 77--429 16 18 (+ 15) [26] m 80-710 9 14(+11) [12] n 18-85 4 8 ( + 7) [27] o 214-612 11 15(+14) [28] Pr is the fracture reopening pressure, P~ is the shut-in pressure, n is the number of tests conducted. Comparison = (100/n) x ~" (IP', - P'rl/P~). i=1
Sites and rock types: (a) Waterloo, WI, quartzite; (b) XTLR hole, CA, sedimentary rock; (c) Mojave Desert, CA, quartz monzonite; (d) Monticello, SC, gneiss and schist; (e) Jordanelle, UT, andesite; (f) Paradox Basin, UT, rock salt; (g) Rocky Mountain, GA, shale, sandstone and siltstone; (h) Hanford, WA, basalt; (i) Moodus, CT, gneiss; (j) Niagara Falls, Canada, limestone, sandstone and shale; (k) Alsace, France, rock salt; (1) Okabe, Japan, mudstone and sandstone; (m) Jianchuan, China, sandstone; (n) Yixian, China, diorite: and (o) Palekhori. Cyprus, diabase. m o r e emphasis should be placed on the use o f the original equation (2) and, where appropriate, the fracture pressurization m e t h o d [I1, 29] should be applied.
FRACTURE DELINEATION An impression packer, a borehole televiewer, or other logging devices can be used to determine the orientation o f fractures on the borehole wall. This information is
crucial to defining the m situ stress tensor. However, it is sometimes difficult to uniquely identify the fracture plane if its traces on the impression packer or televiewer" p h o t o g r a p h are only partially visible or meandering. We have employed, and are presenting herein, techniques based on sinusoidal curve-fitting and circular statistics to delineate accurate orientations that can be inferred from incomplete traces o f inclined and vertical fractures, respectively.
Inclined fracture The intersection o f a drillhole with an inclined fracture takes the form o f an ellipse, which appears as a complete cycle o f a sinusoidal curve on the u n w r a p p e d image o f an impression packer (or on a borehole televiewer picture [30]). The dip direction o f inclined fractures is determined by the azimuth o f the lower peak o f the sinusoidal curve; the dip angle is calculated from the arctangent o f the sinusoidal curve amplitude divided by the borehole radius. However, it is difficult to delineate an inclined fracture if its traces are only partially visible, and not describing a unique plane. Uncertainties with respect to the peak direction and the height o f the fracture could result in significant errors in fracture configuration. To minimize these difficulties, we utilize a sinusoidal curve-fitting method. A 2-D inclined fracture intersecting a borehole leaves the following sinusoidal signature on the borehole wall: D = e. + e_, sin(E + e3),
where e~, e2 and e3 are u n k n o w n parameters depending on fracture configuration, and D and E are coordinates o f the traces in terms o f depth and azimuthal angle along the drilihole axis, respectively. The traces obtained in field experiments are digitized in terms o f D - E coordinates (Fig. 7a). The set o f data is fitted to equation (10) using N L R A in order to
." ":...
"... ...." Ee.__.__.~_ / (a)
•
.i/
f
\
!
,,.
"
i
(b} i
t
Er
(10)
--"
'
Fig. 7. (a) Sinusoidal regression method applied to an inclined fracture trace; (b) circular statistics applied to a vertical fracture trace.
LEE and HAIMSON: HYDRAULIC FRACTURING PARAMETERS
455
133"
190* 211.5
227.4
l
i
Fitted
~°•l
inclined 212.0
227.6
8
ver t i c o k frocture
D i p dir. 3 4 5 * +- 1.5 ° Stril~e 4 8 0 ~ : 4.,~ ° •
Dip 5 5 ° t 1.5 ° 227.8
I
I
i
t
S
E
N
W
Azimuth (o)
212.5
I
O
E
N
I
I
W
s
Azimuth (b)
Fig. 8. Unwrapped impression packer image showing: (a) digitally-tracedincomplete inclined fracture with the optimum fracture plane obtainedby sinusoidalcurve-fitting;(b) digitally-traceden 6chelonfracturewith the mean directionand standard deviation of the fracture determined by circular statistics.
determine the three unknown parameters. This results in an objective definition of fracture strike and dip. The degree of uncertainty of the fracture orientation depends on the planar features of the fracture and the amount of available traces. The average dip direction and its uncertainty are calculated from the parameter e3 and its standard deviation. The average dip angle and its uncertainty are calculated from parameters el and ez and their standard deviations. An example of fracture delineation based on sinusoidal curve-fitting is presented along with the digitized fracture traces in Fig. 8a. Although the traces are splaying and incomplete, we were able to delineate the mean fracture plane with an uncertainty of only + 1.5 °.
Vertical fracture When the trace of a fracture is vertical (parallel to hole axis) the sinusoidal model [equation (10)] becomes invalid. The image of an induced vertical hydrofracture obtained on an impression packer or a borehole televiewer picture is not always in the form of two straight lines. The traces may be discontinuous, en 6chelon and off-centre. In order to remove subjectivity in the determination of fracture orientation, we have employed circular statistics [31]. Figure 7b shows the major steps used to delineate the vertical hydrofracture. First, fracture traces are digitized in terms of azimuthal angle with respect to a reference direction (e.g. North). Each angle (Ej) corresponding to the digitized trace can be represented by a unit vector on a circle. This results in two groups of vectors representing the two halves of the vertical fracture. Since the strike of a fracture is independent of sense (E; = E~ + 180°) we can reduce the range of possible strikes to 0-180 ° by rotating one of the groups by 180 °. Using vectorial summation the direction and
length of the resultant vector can be calculated. The mean orientation of the fracture (Eo) is defined as the resultant vector direction, and the standard deviation (SD) of the mean direction is obtained from the length of the resultant vector: Eo = cos-' [X/L], X=(/=~ cosEi)/n,
SD = [ - 2 In(L)]-2/2, Y=(i=~ sinEi)/n.
(ll)
In equation (11) X and Y are the mean components of the resultant vector along the 0 and 90 ° directions, respectively; n is the number of the digitized data; and L = (X2+ y:)0.5 is the mean length of the resultant vector. We applied circular statistics to a typical en 6chelon hydrofracture trace obtained in the field• Figure 8b shows the unwrapped digitized impression packer image with the optimum vertical fracture orientation. Visual inspection approximates the mean strike direction at 50-55 ° (_+ 15°). The mean fracture strike based on equation (11) is a well-defined 48 ° (+4.5°). As long as fracture traces are complete and reasonably planar, the use of statistics may seem superfluous. In our experience, however, most impression tests or televiewer photographs yield less than ideal fracture traces. For these cases the statistical techniques described remove subjectivity, offer confidence and uniformity in the estimation of fracture orientation, and enable the standardization of field record interpretation. CONCLUSIONS We have applied statistical data analysis to enhance the objectivity of identifying shut-in and fracture reopen-
456
LEE and HAIMSON:
HYDRAULIC FRACTURING PARAMETERS
ing pressures and fracture orientation. We introduced general nonlinear regression analysis to the exponential pressure-decay model for estimating shut-in pressures. We employed bilinear regression analysis to assess the shut-in pressure as the inflection point of the bilinear curve best-fitting the variation of the pressure decay rate as a function of pressure. A similar bilinear approximation was used to evaluate the shut-in pressure from pressure vs flowrate plots. An example of field data analysis suggests that all of these regression methods are capable of yielding comparable and reliable shut-in pressure values. We have developed a "reference threshold" method which allows us to select the fracture reopening pressure using numerical superposition of the ascending portions of the pressure-time curves in the fracture-inducing cycle and a subsequent cycle. We employed circular statistics and sinusoidal curvefitting to define the orientation of induced hydraulic fracture planes when the traces obtained are incomplete, en +chelon and/or meandering. The above statistical methods could also serve as a first important step towards standardization of hydraulic fracturing data interpretation due to the objectivity of these techniques. An important byproduct of our research has been the finding that the fracture reopening pressure is almost always close to the field value of the shut-in pressure. This brings into question the validity of the most commonly-used relation for evaluating the maximum horizontal in situ stress. Acknowledgements--The work reported was funded by the National Science Foundation Grants EAR-8708164 and EAR-8720990 and the U.S. Geological Survey Grant 14-08-0001-G1517.
REFERENCES
1. Hubbert M. K. and Willis D. G. Mechanics of hydraulic fracturing. Trans. AIME 210, 153-166 (1957). 2. Haimson B. C. and Fairhurst C. Initiation and extension of hydraulic fracture in rocks. Soc. Petrol. Engrs. J. Sept, 310-318 (1967). 3. Bredehoeft J., Wolff R., Keys W. and Shuter E. Hydraulic fracturing to determine the regional in situ stress field, Piceance Basin, Colorado. Geol. Soc. Am. Bull. 87, 250-258 (1976). 4. Gronseth J. M. and Kry P. R. Instantaneous shut-in pressure and its relationship to the minimum in situ stress. Hydraulic Fracturing Stress Measurements, pp. 55-60. National Academy Press, Washington, D.C. (1983). 5. Enever J. R. and Chopra P. N. Experience with hydraulic fracture stress measurements in granite. Proc. Int. Syrup. on Rock Stress and Rock Stress Measurement, pp. 411-420. Centek, Lule~ (1986). 6. Doe T. W., Hustrulid W. A., Leijon B, Ingvald K. and Strindell L. Determination of the state of stress at the Stripa mine, Sweden. Hydraulic Fracturing Stress Measurements, pp. 119-129. National Academy Press, Washington, D.C. (1983). 7. Zoback M. D, and Haimson B. C. Status of hydraulic fracturing method for in situ stress measurements. Issues in Rock Mechanics Proc. 23rd U.S. Syrup. on Rock Mechanics, pp. 143-156. Soc. of Mining Engineers of AIME, New York (1982). 8. Muskat M. Use of data on the build-up of bottom-hole pressures. AIME Petrol. 123, 44-48 (1937). 9. Aamodt R. and Kuriyagawa M. Measurement of instantaneous shut-in pressure in crystalline rock. Hydraulic Fracturing Stress Measurements. pp. 139-142. National Academy Press, Washington, D.C. (1983). 10. Draper N. R. and Smith H. Applied Regression Analysis, 2nd Edn, pp. 458-462. Wiley, New York (1981).
11. Haimson B. C. and Lee M. Y. The state of stress and natural fractures in a jointed precambrian rhyolite in south-central Wisconsin. Proc. 2 8 t h U.S. Symp. on Rock Mechanics, pp, 231-240. Balkema. Rotterdam (1987). 12. Haimson B. C., Springer J. E.. Lee M. Y., Zoback M. D., L1 F., Zhai Q and Liang H. Results of hydraulic fracturing stress measurements at Jianchuan, Western Yunnan Province, China. USGS Open-file Report 87-477 (1987). 13. Haimson B. C., Tunbridge L., Lee M, Y, and Cooling C. Measurement of rock stresses using the hydraulic fracturing method in Cornwall, U.K.--Part It. Data reduction and stress calculation. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 26, 361-372 (1989). 14. Cheung L. S. and Haimson B. C. Laboratory study of hydraulic fracturing pressure data--how valid is their conventional interpretation? Int. J. Rock Mech. Min Sci. d Geomech. Abstr. 26, 595-604 (1989). 15. Haimson B. C. Near surface and deep hydrofracturing stress measurements in the Waterloo quartzite. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 17, 81-88 (1980). 16. Hickman S. H. and Zoback M. D. The interpretation of hydraulic fracturing pressure-time data from in situ stress determination. Itydraulic Fracturing Stress Measurements, pp. 44-54. National Academy Press, Washington, D.C. (1983). 17. Zoback M. D., Tsukahara H. and Hickman S. Stress measurements near the San Andreas Fault. J. Geophys. Res. 85, 6157-6173 (1980). 18. Zoback M. D. and Hickman S. In situ study of the physical mechanics controlling induced seismicity at Monticello Reservoir, South Carolina. J. Geophys. Res. 87, 6959-6974 (1982). 19. Haimson B. C. and Lee M. Y. Stress measurements at the Jordanelle Dam site, Central Utah, using wireline hydrofracturing. Final Report to USGS (1985). 20. Schnapp M., Kocherhans J. G. and Nelson R. A. Stress measurements in salt in a deep borehole, southeastern Utah. Hydraulic Fracturing Stress Measurements, pp. 79-85. National Academy Press, Washington, D.C. (1983). 21. Barton N. Hydraulic fracturing to estimate minimum stress and rock mass stability at a pumped hydro project, ttydraulic Fracturing Stress Measurements, pp. 61 67. National Academy Press, Washington, D.C. (1983). 22. Haimson B. C. Stress measurements at Hanford, Washington for the design of a nuclear waste repository facility. Proc. 6th Congr. of the International Society .for Rock Mechanics, pp. 119-123. Balkema, Rotterdam (1987). 23. Baumgartner J. and Zoback M. D. Interpretation of hydraulic pressure-time records using interactive analysis methods: application to in situ stress measurements at Moodus, Connecticut. Proc. 2nd Int. Workshop on l~vdraulic Fracturing Stress Measurements, pp. 619 645 (1988). 24. Haimson B. C., Lee C. F. and Huang J. H. S. High horizontal stresses at Niagara Falls, their measurement, and the design of a new hydroelectric plant. Proc. Int. Syrup. on Rock Stress and Rock Stress Measurement, pp. 615-624. CENTEK, Lule~ (1986). 25. Cornet F. Two examples of stress measurements by the H.T.P.F. method. Key Questions m Roek Mechanics: Proc. 29th U.S. Syrup. on Rock Mechanics, pp. 615-623. Balkema, Rotterdam (1988). 26. Tsukahara H. Stress measurements utilizing the hydraulic fracturing technique in the Kanto-Tokai Area, Japan. Hydraulic Fracturing Stress Measurements, pp. 18-27. National Academy Press, Washington, D.C. (1983). 27. Li F., Li Y., Wang E., Zhai Q., Bi S., Zhang J., Liu P., Wei Q. and Zhao S. Experiments of in situ stress measurements using stress relief and hydraulic fracturing techniques. Hvdraulic Fracturing Stress Measurements, pp. 130-134. National Academy Press, Washington, D.C. (1983). 28. Baumgartner J., Rummel F., Haimson B. C. and Lee M. Y. In situ stress measurements and natural fracture logging in Drillhole CY-4, the Troodos Ophiolite, Cyprus. Initial Report of the Cyprus Crustal Study Project, Drillhole CY-4, Canadian Geological Survey (1989). 29. Cornet F. H. and Valette B. In situ stress determination from hydraulic injection test data. J. Geophys. Res. 89, 11,527-11,537 (1984). 30. Zemanek J., Caldwell R. L., Glenn E. E. Jr, Holcomb S. V., Norton L. J. and Straus A. J. D. The borehole televiewer--a new logging concept for fracture location and other types of borehole inspection. J. Petrol. TechnoL 21, 762-774 (1969). 31. Mardia L. The Statistics qf Orientation Data. Academic Press, London (1972).