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.

References 1. Nolte, K.G.: "Determination of Fracture Pa-

rameters From Fracturing Pressure Decline," paper SPE 8341 presented at the 1979 SPE Annual Technical Conference and Exhibition, Las Vegas, Sept. 23-26.

2. Nolte, K.G.: "A General Analysis of Frac-turing Pressure Decline With Application to Three Models," SPEFE (Dec. 1986) 571-83; Trans., AIME, 284.

3. Nolte, K.G.: "Principles for Fracture Design Based on Pressure Analysis," SPEPE (Feb. \988) 22-30; Trans., AIME, 285.

4. Martins, J.P. and Harper, T.R.: "Mini-Frac Pressure-Decline Analysis for Fractures Evolving From Long Perforated Intervals and 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)

13869 presented at the 1985 SPEIDOE Sym-posium on Low Permeability Reservoirs, Denver, May 19-22.

5. Castillo, J.L.: "Modified Fracture-Pressure-Decline Analysis Including Pressure-Depend-ent Leakoff, " paper SPE 16417 presented at the 1987 SPEIDOE Symposium on Low Per-meability Reservoirs, Denver, May 18-20.

6. Moschovidis, Z.A.: "Interpretation of Pres-sure Decline for Minifracture Treatments In-itiated at the Interface of Two Formations," SPEPE (Feb. 1990) 45-51; Trans., AIME, 289.

7. Nolte, K.G.: "Fracturing-Pressure Analysis for Nonideal Behavior," JPT (Feb. 1991) 210-18.

8. Kemp, L.F.: "Study of Nordgren's Equation of Hydraulic Fracturing," SPEPE (Aug. 1990) 311-13.

9. Nolte, K.G.: "Fluid Flow Considerations in Hydraulic Fracturing," paper SPE 18537 presented at the 1988 SPE Eastern Regional Meeting in Charleston, WV (Nov.).

10. Medlin, W.L. and Masse, L.: "Laboratory Experiments in Fracture Propagation," SPEJ (June 1984) 256-68.

11. Shlyapobersky, J. et al.: "Field Determina-tion of Fracturing Parameters for Overpres-sure Calibrated Design of Hydraulic Fractur-ing," paper SPE 18195 presented at the 1988 SPE Annual Technical Conference and Ex-hibition, Houston, Oct. 2-5.

JPT March 1993

12. Gu, H.: "Laboratory Fracturing Test Con-firmation of Minifrac Analysis Technique, " unpublished work at BP Research (1990).

13. Gu, H. and Yew, C.H.: "Finite Element So-lution of a Boundary Integral Equation for Mode I Embedded Three-Dimensional Frac-tures," IntI. J. Num. Meth. Eng. (July 1988) 26, 1525-40.

14. Gu, H.: "A Study of Propagation of Hydrau-lically Induced Fractures," PhD dissertation, U. of Texas, Austin (1987).

15. Gu, H. and Leung, K.H.: "Three-Dimen-sional Numerical Simulation of Hydraulic Fracture Closure With Application to Mini-frac Analysis, " paper SPE 20657 presented at the 1990 SPE 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.

255