-
Summary. This paper describes a 3D numerical model for the
closure of a planar, nonpropped hydraulic fracture under uniform
and layering reservoir conditions. The model sim-ulates
"double-slope" minifracture pressure-decline curves when the
fracture height retracts from high-stress zones during closure.
Applica-tion of the simulation results to mini-fracture analysis is
discussed.
206
3D Numerical Simulation of Hydraulic Fracture Closure With
Application to Minifracture Analysis Hongren Gu, * SPE, BP IntI.
Ltd., and K.H. Leung, SPE, BP Exploration, Europe
Introduction The success of a hydraulic fracture stimu-lation
depends largely on an accurate esti-mate of fluid leakoff during
treatment. The average formation leakoff coefficient can be
determined by analyzing the pressure-decline data from a mini
fracture treatment.
Pressure-decline-analysis methods 1-4 are based on a number of
simplifying assump-tions. The key assumptions are fracture
ge-ometry and a constant fracture area during closure. Despite the
simplifying assump-tions, pressure-decline behavior in many field
observations is consistent with that in-dicated by analysis.
Pressure-decline anal-ysis also has been extended to include
pressure-dependent leakoff5 and leakoff at the interface of two
formations. 6
However, the pressure-decline-analysis theory, which is based on
constant area, does not explain some of the observed phe-nomena
when the fracture is inside a for-mation with stress and
permeability con-trasts. In such cases, the fracture may grow into
the high-stress zones during propagation and shrink back to the
lower-stress zones during closure. Thus, the constant-fracture-area
assumption would be violated. Nolte 2,7 has discussed the effects
of frac-ture-height growth on closure and pressure-decline
analysis.
In this paper, a 3D numerical simulation of fracture closure is
used to study the ef-fects of in-situ stress and leakoff contrasts
on fracture closure and pressure-decline be-havior. The
fracture-closure mechanism is discussed first, and the assumptions
and out-line of a 3D fracture-closure simulator are presented. The
simulation results then are analyzed with the minifracture analysis
tech-nique. The different pressure-decline be-haviors of a
constant-area fracture and a shrinking-height fracture are
demonstrated and explained. A minifracture analysis tech-nique for
shrinking-height fractures and the general principle of deducing
stress contrast from pressure-decline data are discussed.
FractureClosure Mechanism After shut-in during a minifracture
treat-ment, wellbore pressure gradually decreases 'Now at Dowell
Schlumberger.
Copyright 1993 Society of Petroleum Engineers
as the fluid inside the fracture leaks off into the formation.
The fracture is considered closed when the wellbore pressure drops
be-low the minimum horizontal in-situ stress.
When pumping stops, the flow rate inside the fracture reduces
quickly, and the fluid pressure distribution becomes more uniform
because of the reduced friction loss. Ifleak-off is small, the
pressure redistribution may increase the fluid pressure near the
fracture tip, and hence increase the stress-intensity factor. The
fracture may continue to prop-agate. 7-9 If leakoff is high, the
pressure drops quickly and the pressure redistribution may not
increase pressure greatly near the fracture tip. In this case, the
fracture growth after shut-in most likely will be insignificant.
Medlin and Masse's 10 laboratory results showed no fracture growth
after shut-in.
If the fracture is inside a formation with uniform in-situ
stress, fluid pressure inside the fracture is greater than the
minimum in-situ stress over most of the fracture, except for a
small region near the fracture tip. Therefore, the fracture most
likely will close with a constant area until the pressure drops to
near the in-situ stress. Pressure-decline data from some field
observations 11 and laboratory tests 12 agree with the prediction
of the constant -area fracture-closure theory. On the other hand,
numerical simulations with the Perkins-Kern-Nordgren (PKN) model
show decreasing fracture penetration during closure. 7-9 In this
simulation, the fracture penetration in a uniformly stressed pay
zone is assumed to be constant during closure; this assumption is
not verified in this work.
If in-situ stress contrasts exist in the for-mation (Fig. 1),
the fracture may grow into the higher-stress zones during
propagation. After shut-in, the fluid pressure drops and becomes
more uniform. The fluid pressure may drop below the higher in-situ
stress, and the part of the fracture in the high-stress zone most
likely will close earlier than the part in the lower-stress zone.
Also, high in-situ stress often is related to shale layers, which
have a much lower permeability than the pay-zone rock. Therefore,
the fluid in this part of the fracture flows back to the
more-permeable pay zone to leak off. This also causes the fracture
to shrink back from
March 1993 JPT
-
In-situ stress distribution
-Fracture front
Fig. 1-Fracture under stress contrast.
the high-stress zone. Thus, it is conceiva-ble that the fracture
retracts preferentially from high-stress zones during closure.
Outline of 3D FractureClosure Simulation This simulation of
fracture closure assumes that the fracture has a constant surface
area during shut-in when the in-situ stress and reservoir
conditions are uniform. When in-situ stress and leakoff contrasts
exist, the simulation allows the fracture height to retract from
the high-stress zones. The frac-ture is assumed to have a constant
area once it has shrunk back and become fully con-tained in the
uniformly stressed pay zone. It also is assumed that the leakoff is
con-trolled by filter cake and is pressure-independent. The leakoff
rate is expressed as
qL = CI--J (t-r). . ............... (1) Fracture propagation
during injection is
calculated with a 3D fracture simula-tor. 13,14 Shortly after
shut-in, the fluid pressure redistributes because of short-term
flow transients. The flow equation,
a [ n' b(2n'+I)ln' - ---K'-lIn'----ax 2n' + 1 2(n'+ 1)ln'
Xl (:Y +(:Yr(n'-1)/2n' :J a [ n' b(2n'+ 1)ln'
+- ---K'-lln'----ay 2n'+1 2(n'+I)ln'
Xl (:Y +(:Yr(n'-1)/2n' ::J ab 2C
=-+ , ......... (2) at --J[t-r(x,y)]
and the fracture deformation equation,
JPT March 1993
Well bore
II II
High stress zone
I I Pay zone II II II
High stress zone
+ ~ (~) ~]dX'dY' ay ray'
=p(x,y)-a(x,y), ............... (3) are solved together
iteratively. The bound-ary condition for the flow equation is the
normal flow rate, qn =0, at the wellbore and the fracture front.
The global fluid-volume conservation holds for the balance between
the fracture-volume decrease rate and qL and can be expressed
by
ab I qLdxdy-I -dxdy=O. . ..... (4) A A at f f To perform the
simulation in the time do-
main, a percentage of the current fracture volume is prescribed
as the volume decre-ment. The time needed for fluid in the volume
decrement to leak off is calculated from Eq. 4. After compatible
fracture width and fluid pressure are obtained through iter-ation
for the current timestep, the fracture volume is reduced further
and the compu-tation is carried out for the next timestep.
As a simulation proceeds, the time incre-ment required for a
convergent solution becomes increasingly smaller, probably be-cause
the reduced fracture width causes the discretized flow equation to
become ill-conditioned. At the same time, the pressure distribution
inside the fracture becomes more uniform. At the beginning of
shut-in, the ratio of average excess pressure to well-bore excess
pressure, Fp, is about 0.7. As the simulation proceeds to this
stage, Fp in-creases to about 0.9. The flow effects are considered
less significant when Fp is close to unity. Therefore, the flow
equation is omitted from subsequent calculations. To proceed in the
time domain, the fluid pres-sure is decreased by a prescribed
amount, and the fracture-deformation and volume-conservation
equations are solved step by step until the fracture closes
fully.
For a simulation with stress contrast, the fracture may grow
into the high-stress zones during propagation. After shut-in, when
the width near the fracture tip in the high-stress zones has
reduced to a small prescribed value, that part of the fracture is
considered
"The success of a hydraulic fracture stimulation depends largely
on an accurate estimate of fluid leakoff during treatment."
closed thereafter. The conductivity and leakoff of the closed
fracture are neglected in subsequent calculations because they are
much smaller than those of the fracture that remains open and
because the filter cake probably seals the gap. The fracture height
retracts step by step to the pay zone as the simulation
proceeds.
Simulation Results Uniform In-Situ Stress and Leakoff. Un-der
uniform in-situ stress and leakoff condi-tions, the 3D model
simulates propagation and closure of a penny-shaped fracture. The
fracture is assumed to close with a constant area. Three different
cases of leakoff coeffi-cient, 0.01, 0.005, and 0.0005 ft/min'l>
were considered. For all cases, the plane strain Young's modulus
was 3.2x 106 psi, fluid viscosity was 200 cp, injection rate was 40
bbllmin, and injection time was 10 min.
The computer-generated, wellbore-pres-sure-decline data for the
case of the leakoff coefficient 0.005 ft/min v, are plotted vs.
time, square root of time, and the GL func-tion in Figs. 2 through
4. Fig. 5 shows the p-vs.-Vt plot for leakoff coefficient 0.0005
ft/min v,. The pressure-decline curves were analyzed then with a
minifracture analysis technique. I-4 The penny-shaped fracture
model was used in the analysis. Table 1 shows that the fracture
parameters deduced from the minifracture pressure analysis are in
good agreement with the numerical simu-lation. This demonstrates
that the numerical procedure used in the 3D fracture-closure
simulator is accurate.
InSitu Stress and Leakoff Contrasts. The 3D model then is used
to simulate fracture propagation and closure in a formation with
stress contrasts (Fig. 1). The fluid leakoff is assumed to occur
only in the pay zone, and the 1eakoff area during closure does not
change.
In the simulation examples, stress con-trasts, ~a, of 200, 400,
and 800 psi were used. The pay-zone height was 100 ft. Leak-off
coefficients, C, of 0.005 and 0.01 ftlmin V2 were used. Other data
were the same as those used in the uniform-stress ex-amples.
207
-
300 300
Increasing Fp -- -- eonstant Fp tncreasing FP--+I-eons,ant
Fp
Ol+--------r------_r-------r------~----~~
O+---~----.----.----r---_.----r_--_r--~ 0.2 0.0 0.4 0.6 0.8 1.0 1.2
1.4 1.S 10.5 11.0 11.5 12.0 12.5 13.0 TIME (min) ROOT TIME
(.Imln)
Fig. 2-Excess wellbore pressure vs. time for a penny-shaped
fracture, C = 0.005 ftN mm .
Fig. 3-Excess wellbore pressure vs. square root of time for a
penny-shaped fracture, C = 0.005 ft/.jli1iil.
The wellbore excess pressure after shut-in is plotted vs. the GL
function (Figs. 6 through 9). Two slopes corresponding to pre- and
postfracture height shrinkage can be identified in Figs. 6, 7, and
9. Fig. 10 shows loci tracing the fracture front before and after
height shrinkage for the example with ~a=400 psi and C= 0.005
ft/min '/'. Before the start of shrinkage indicated in the figures,
the width near the tip is still large. The fracture-height
reduction is small even though the simulation allows the fracture
height to reduce. The simulation generates the first slope while
the flow equation is solved fully.
Table 2 compares the results of simulation and pressure-decline
analysis. In the pres-sure-decline analysis, the first slope is
used with an elliptical fracture model, whereas the second slope is
used with a PKN model with modified fracture stiffness. *,15 The
results are discussed in detail later.
Constant Fracture Area Pressure vs. Square Root of Time Plot. As
Fig. 5 shows the p-vs.-.Jt plot has a good linear region for low
leakoff with long closure time, but for high-leakoff cases the
linear region is not obvious (Fig. 3).15 'Personal communication
with J.P. Martins, BP Exploration Co., Glasgow, Scotland
(1988).
300
Increasing Fp -1- Constant Fp
Pressure vs. G Function Plot. The pres-sure-decline data also
can be plotted vs. the G function,S and the slope is
...... (5)
In Eq. 5, the product CFAAfS~ is constant for a constant-area
fracture, and Fp is the only factor causing a deviation from a
linear relationship in the piG plot. When Fp is constant, the
pressure-decline curve has a long linear region even for large
leakoff, as Fig. 4 shows. The nonlinear region just after shut-in
is caused by the increasing Fp and not by fracture extension.
Similarly, a nonlinear region just after shut-in is observed in the
field may not only or necessarily be caused by fracture ex-tension.
s
Therefore, plotting the decline data vs. the G function is a
better way to identify frac-ture-closure pressure and the slope for
cal-culating C. CastilioS first used this plot in pressure-decline
analysis that included pres-sure-dependent leakoff.
Shrinking Fracture Height In BP's Ravenspurn South gas field in
the southern North Sea, the average formation permeabilities are
from 0.5 to 3 md. The
200
mini fracture leakoff coefficients derived from the plGL plots
are 0.003 to 0.007 ft/min v,. Two distinct types of
pressure-de-cline curves have been observed: one similar to that
shown in Fig. 11, which is predomi-nantly linear, and one like the
curve in Fig. 12, which is gently varying but contains two
essentially linear regions.
Similar characteristics also are seen in the simulated
pressure-decline curves. If the reservoir condition is uniform and
the frac-ture has a constant area during closure, the relationship
between p and the GL function is predominantly linear (Fig. 4). If
stress and leakoff contrasts exist and the fracture height shrinks
during closure, the slopes of the curves change like that observed
in the field (Figs. 6, 7, and 9). The departure from a linear
relationship between p and the GL function is explained as
follows.
When the fracture height retracts from the high in-situ stress
zones, the fracture stiff-ness will increase because of the
reduction in fracture height. The leakoff area FAA f is constant if
the leakoff outside the pay zone is negligible. When Fp approaches
a con-stant, Eq. 5 shows that the slope of the curve will be
controlled mainly by stiffness. Therefore, the pressure-decline
curve turns down to reflect the increased stiffness. These
conditions also could exist in the field and
Increasing Fp -1- Constant Fp
O+-----,------r-----r----~----_,--~~ 0.00
O+----,----.-----r----r----.---~----._~~ 0.05 0.10 0.15
Gl 0.20 0.25 0.30
Fig. 4-Excess well bore pressure vs. GL function for
penny-shaped fracture, C = 0.005 ftN min.
208
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 ROOT TIME (lmin)
Fig. 5-Excess well bore pressure vs. square root of time for
penny-shaped fracture, C = 0.0005 ft/v'min .
March 1993 JPT
-
TABLE 1-SIMULATION AND PRESSURE-DECLINE ANALYSIS RESULTS FOR
PENNY-SHAPED FRACTURES
Fracture Leakoff Stiffness Efficiency Radius Coefficient
(psl/fts) (ft.I.J min) (%) ~
Simulation 1.053 7.9 82 d.01* Root time" 1.031 8.5 83 0.01007 GL
function" 1.087 8.5 82 0.01'04 Simulation 0.388 16.8 114 0.005*
Root time 0.425 16.4 112 0.00503 GL function 0.453 16.4 110 0.0052
Simulation a.0587 73_3 218 a.0005* Root time 0.0560 72.0 220
0.00047 G $ function t 0.0548 72.0 218 0.00048
'From prl)$$ure-decline analYSis with p.vs.-Vt plot 'From
prllSsure-decline analysis with p.vs.-G L fl,lnotiOn plot. t From
pressur&-decline analysis with p-vs.-G. function plot. ; Input
for the numericsl simulations.
lead to the pressure-decline behavior shown in Fig. 12. Thus, a
linear relationship be-tween p and the GL function can be
inter-preted as a fracture closing with a constant area; an
increasing slope (in absolute mag-nitude) can be seen as a fracture
closing with a decreasing height, possibly because of stress and
leakoff contrasts.
Identification of Fracture-Height Change. The p-vs.-..fi curve
does not help much in identifying fracture-height change. Fig. 3 is
the simulated pressure-decline curve with constant area, whereas
Fig. 13 is the simulated curve with shrinking height. Com-parison
of these two curves shows no char-acteristic difference.
When plotting the simulated pressure-decline data vs. the GL
function, we can see a distinct difference in the pressure response
caused by height change (Figs. 4 and 7). A straight line indicates
a constant fracture area, and an increasing slope in absolute
magnitude indicates a shrinking height. Therefore, plotting p vs.
the GL function is a useful technique for identifying the
differ-ence in pressure-decline behavior from height change. If the
height change is caused by in-situ stress contrasts, this
identification also implies the existence of stress contrasts.
However, the following situation is an ex-ception.
300
~ S250
~ Zl200
~ ~ 150
~ 100
~ 50
If the stress contrast is very high, the frac-ture does not grow
much into the high-stress zones during propagation. During closure,
the reduction in fracture height and the in-crease in stiffness are
insignificant. Hence, the pressure-decline behavior is similar to
that of a constant-area fracture (Fig. 8). In such cases, the
in-situ stress contrast cannot be detected by the p/GL function
plot.
Pressure-Decline Analysis for Shrinking Height. When a fracture
has a constant area during closure, the slope of the
pressure-decline curve can be identified easily and used in the
pressure-decline analysis to de-termine fracture stiffness and
leakoff coeffi-cient. When a fracture height shrinks during
closure, the slope of the decline curve changes. As Nolte2
discussed, for fracture with height growth, the pressure-decline
analysis can be performed during the latter part of closure instead
of during the initial part. As a numerical experiment, the
simu-lated pressure-decline curves have been ana-lyzed to confirm
Nolte's conclusion and to make use further of the initial slope in
pressure-decline analysis.
The curve has two nearly linear portions, one before and one
after the fracture height shrinks (Figs. 6, 7, and 9). Both slopes
can be used to derive accurate leakoff coeffi-
__ Second slope
End of shrinkage --
350
"In principle, it is possible to deduce some information about
the stress contrast by analyzing the change of fracture stiffness
from the pressure-decline data."
cients, provided that the appropriate frac-. ture model and
fracture height are used in
the analysis. If the fracture height at the wellbore be-
fore shrinkage is known and the half-height is used as the minor
axis of an elliptical frac-ture, good results can be obtained with
the first slope and an elliptical-fracture model. Indeed, the
derived fracture length and leakoff coefficient from the first
slope are very close to the simulation values (Table 2). For the
second slope, the pay-zone height and a PKN model with modified
stiff-ness, *,15
E'E(k) S=--, ................... (6)
7rh 2L
were used. In Eq. 6 E(k) is the elliptical integral of
the second kind, and
k2 = 1-O.25h2/L2.
3D numerical simulations have demon-strated that, for short
fractures, Eq. 6 gives closer fracture stiffness than a
conventional PKN model. For L ~ h, Eq. 6 gives the same fracture
stiffness as a conventional PKN model. The pressure-decline
analysis results based on the second slope and the PKN model with
modified stiffness in Table 2
_Second slope
End of shrinkage-
O+-----.----,-----r----.-----.----.--~., 0.0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 O+------,-------r------r------r------~--
GL
Fig. 6-Excess well bore pressure vs. G L function for
stress-contrast case, 40-=200 pSi, C=0.005 ftl.Jmin.
JPT March 1993
0.0 0.1 0.2 0.3
GL
0.4 0.5
Fig. 7-Excess wellbore pressure vs. GL function for
stress-contrast case, 40-=400 psi, C=0.005 ftNmln.
209
-
!I! ::> Ul Ul II! a.
400
l!! 2 ~ 150 ~ -
Shrinkage
'" '" ~ 7000 M!
~ w ~ 6500
1. For high 1eakoff and short fracture-closure time p-vs.-.ft,
the p-vs.-GL func-tion plot has a more obvious linear region than
the dpl.ft plot. This former plot is eas-ier to use to determine
the slope for 1eakoff coefficient calculation.
2. The p-vs.-GL function plot can be used to identify the
fracture-height change during closure. It has changing slopes, and
two linear regions on the curve often can be identified for a
fracture height that shrinks from zones with moderately high
in-situ stress.
3. Both slopes of the two linear regions on a p-vs.-GL function
plot both can be used to calculate the leakoff coefficient,
provided the correct fracture height and frac-ture model are used
in the pressure-decline analysis. The second slope should be used
with the fracture height after shrinkage, which is likely to be the
pay-zone height in field applications.
4. "It is possible to deduce in-situ stress contrast by
analyzing pressure-decline data. " Further work is required to
develop the concept for field applications.
Nomenclature A f = total fracture surface area,
L2, ft2
-100+----------,---------.----------r---------. o 150 200
6000+-------,-------,-------,-------,-_ 50 100
X(m
Fig. 10-Fracture-front contours before and after shrinkage for
stress-contrast case, .:1a=400 psi, C=0.005 ftNmin.
210
o 0.5 1.0 1.5 2.0 GL
Fig. ii-Field measurement of minifracture pressure deCline,
constant slope.
March 1993 JPT
-
TABLE 2-SIMULATION AND PRESSURE-DECLINE ANALYSIS RESULTS FOR
IN-SITU STRESS AND LEAKOFF-CONTRAST EXAMPLES
Stress Height (ft) Contrast Before After Efficiency Length C ~
Shrinkage Shrinkage (%) ~ (ftN min)
Simulation 200 250 100 30.2 185 O.005t First slope' 250 32.0 167
0.00501 Second slope" 100 32.0 160 0.00507 Simulation 400 176 100
25.8 182 O.005t First slope' 176 26.2 185 0.00499 Second slope" 100
26.2 176 0.00496 Simulation 800 118 100 23.4 190 O.005t Second
slope" 100 22.7 185 0.00497 Simulation 400 151 100 12.0 113 O.01t
First slope' 151 12.8 124 0.0093 Second slope" 100 12.8 101
0.0105
'Pressure-decUne analysis resutt with the lirst slope and the
elUptical-lracture model. "Pressure-decline analysis resutt with
the second slope and the PKN model with modified stiffness. t Input
lor the numerical simulations.
b = fracture width, L, ft C = leakoff coefficient, Lit y, ,
ft/min \I, e = fluid efficiency at end of
injection E = (k)=elliptical integral of the
second kind E' = plane strain Young's modulus,
m/Lt2 , psi FA = ratio of leakoff area to total
fracture area Fp = ratio of average excess
pressure to well bore excess pressure
G = shear modulus of formation, m/Lt2, psi
GL = Nolte's leakoff function,2 fluid-loss dominated
Gs = Nolte's leakoff function, 2 fracture storage dominated
h = fracture height, L, ft k = (1-0.25 h2 /0) y,
K' = power-law fluid consistency, Ibf-sec n '/ft2
L = half-fracture length, L, ft n' = power-law fluid
exponent
p = fluid pressure, m/Lt2, psi ji = average excess pressure,
m/Lt2, psi qL = leakoff rate per unit area, Lit,
ft/min qn = normal flow rate per unit
area, Lit, ft/min r = [( x-x')2 +(y_y')2] y" L S = fracture
stiffness=djildV,
miL 4t2, psi/ft3 t = time or elapsed time after
shut-in, t, minutes tD = dimensionless shut-in time,
tlto to = injection time, t, minutes V = fracture volume, L3,
ft3
x,y = coordinates on fracture surface, L, ft
x ',y' = integration variables, L, ft p. = Poisson's ratio a =
in-situ stress, m/Lt2 , psi
da = in-situ stress contrast, m/Lt2, psi
7 = leakoff beginning time, t, minutes
350
";; 300 :; II! ::J 250 (/) (/) w a: n. 200 II! _ Second Slope
0
~ 150 _ Fracture Closure ..J
~ ~ 100 w () x 50 w
a 0.0 ~+------------r-----------.----------~
o 0.5 GL
1.0 1.5
0.5
"It is possible to deduce in-situ stress contrast by analyzing
pressure-decline data."
Acknowledgments We thank the management of BP Explora-tion and
BP Research for permission to pub-lish this paper . We also thank J
. P. Martins, A.H. Carr, and M.R. Jackson ofBP Explo-ration and N.
C. Last of BP Research for useful discussions and appreciate the
as-sistance from C.H. Yew at the U. of Texas at Austin.
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Unaffected by Confining Strata, " paper SPE
(To Page 255)
1.0 1.5 2.0 ROOT TIME (,Imin)
Fig. 12-Field measurement of minifracture pressure decline,
double slope.
Fig. 13-Excess wellbore pressure vs. square root of time for
stress-contrast case, da=400 psi, C=O.005 ftNmin.
JPT March 1993 211
-
3D Numerical Simulation of Hydraulic Fracture Closure With
Application to Minifracture Analysis (From Page 211)
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JPT March 1993
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Annual Technical Conference and Exhibition, New Orleans, Sept.
23-26.
Sl Metric Conversion Factors bbl x \.589873 E-01 = m' cp x \.0'
E+OO = mPas ft x 3.048* E-Ol = m
ft' x 2.831 685 E-02 = m' psi x 6.894757 E+OO = kPa
'Conversion factor is exact.
Provenance Original SPE manuscript, Three-Dimen-sional Numerical
Simulation of Hydraulic Fracture Closure With Application to
Minifrac Analysis, received for review Sept. 2, 1990. Revised
manuscript received March 12, 1992. Paper accepted for publi-cation
July 23, 1992. Paper (SPE 20657) first presented at the 1990 SPE
Annual Technical Conference and Exhibition held in New Orleans,
Sept. 23-26.
JPT
Authors
Gu Leung
Hongran Gu, a senior development en-gineer at Dowell
Schlumberger In Tul sa, previOUsly worked at the BP Sunbury
Research Centre in hydrauHc fracturing simulation, anatysls, and
design. He holds an MS degree from Xian Jlaotong U. and a PhD
degree from the U. of Texas, both In engineering mechanics. K. Hong
Leung Is a well-test analyst at BP Exploration, Europe. He has 5
years' experience in developing fracture sima ulators,
fracture-treatment deSign, and fractured well-test analysis. He
holds MS and PhD degrees In civil engineer-Ing from the U. of
Wales.
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