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type-curve analysis of pressure data partially falling 
in the pseudoradial flow period because the 
remaining data must follow one of the curves for 
different fracture conductivities. Fig. 14 must be 
used as an auxiliary curve to determine (kfbf) D 
when using Fig. 15. 
Application of Fig. 15 to match pressure data 
provides (PwD)M' (tDr )M' (tlp)M' (t)M' and 
[(kfbf) D]M. Hence, th~ following equations are 
given. 
Reservoir Permeability. For oil, 
DloqBp. k= (PwD) M' ................ (41) h(tlp) M 
102r----------,----------~----------1I----------r_------~ 
.-
til a. 
a. 
E 
<J o 
.D MdLCh Point 
[ot]M c 1 hour 
= 0.35 hour 
t '2 _ 5 hour ebf 
10~----------~----------~----------_t-------
I 
I 
I 
I 
,-- ___ ~_O. 
I 
I o 
I 0 I - -,­
I 
I 
Beginning of Senilog Straight Line 10. 
lo10-1~----~I--~~-----L---L._----i---~------~~~----~~ 3 2 1 101 107 '-_ .!.O-_ I 10-. I 10- _ 1 __ 
- - - - - - - - - - 2 1 102 /). t [hoursl 
10-2 10-1 t DXf (kf bflD 10


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- - - - - - - - - - 2 1 102 /). t [hoursl 
10-2 10-1 t DXf (kf bflD 10 
Fig. 16 -Type-curve matching for Well A. 
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and, for gas, 
(XgqT k= (PwD) M. . ............ (42) 
h[~(p}lM 
Effective WeI/bore Radius. For oil or gas, 
r' =J (3k(t) M w cPJlCt (tDr ~) M· ................. (43) 
By using [(kfbj) DlM in Fig. 14, we obtain 
(r~/xj) Fig. 14; hence, 
r' Xj= [' j ..................... (44) 
r wlXj Fig. 14 
The pressure data falling in the pseudoradial flow 
period also must be analyzed using the semilog 
methods to estimate k and r ~ . 
The discussion in this section clearly indicates (as 
mentioned by Agarwal et af.29) that caution and 
diligence should be exercised when applying the type­
curve matching technique. To perform the type of 
analysis presented in the four cases, it is necessary to 
have prefracture information. 
It should be kept in mind that application of the 
type-curve analysis method to pressure buildup tests 
is appropriate when the producing time is large. 
Examples of Application 
Three examples illustrate the application of several of


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is appropriate when the producing time is large. 
Examples of Application 
Three examples illustrate the application of several of 
the methods and theory previously discussed. 
Well A 
A buildup test was conducted in this fractured well 
producing in a low-permeability reservoir. Table 2 
presents the information and results of the analysis 
for this test. Fig. 16 shows a log-log graph of pressure 
data. matching the type curve given in Fig. 10. Notice 
that the first data points are influenced by well bore 
storage and the rest of the data fall in both the 
bilinear and the transition flow periods. The match­
point results also are presented in Fig. 16. A 
minimum value for (k jb j ) D can be estimated from 
the position of the last data point with respect to the 
type curves; for this case, (kjbj}D min =511". 
The end of well bore storage occurs at ap­
proximately 0.35 hour and the end of bilinear flow is 
at 2.5 hours. We also see that the formation linear


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proximately 0.35 hour and the end of bilinear flow is 
at 2.5 hours. We also see that the formation linear 
flow period was not reached in this test. 
Since the test was not long enough to match a 
curve for a specific value of (kjbj) D' this example 
corresponds to Case 2 in the type-curve-analysis 
section. 
Using the pressure data match and Eq. 38, we 
obtain 
kjbj _ 1,424x7,350x690x6.5 
Xj - 118x109 
=0.3978 md-ftlft. 
SEPTEMBER 1981 TABLE 2 -TEST INFORMATION AND ANALYSIS RESULTS 
FORWELLA 
Reservoir Data 
Production rate q, MscflD 
Producing time tp, hours 
Formation thickness h, ft 
Porosity cJ>, fraction 
Permeability k, md 
Formation temperature T, oR 
Average gas viscosity 1-', cp 
Total compressibility ct, psi-1 
Flowing well bore pressure Pwl, psia 
Analysis Results 
k,b
" md-ft 
x" ft 
(k:~'). md-ftlft 
(klbl)D 
,;.., ft Type-Curve 
Analysis 
148 
373 
0.3978 
15.9 
171.6 7,350 
2,640 
118 
0.1 
0.025 
690 
0.0252 
0.129x 10-3 
1,320 
Bilinear 
Flow Analysis 
154 
368


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Analysis 
148 
373 
0.3978 
15.9 
171.6 7,350 
2,640 
118 
0.1 
0.025 
690 
0.0252 
0.129x 10-3 
1,320 
Bilinear 
Flow Analysis 
154 
368 
0.4185 
-16.71 
-169.3 
From the time-match information and Eq. 39, we can 
calculate 
J 2.637 x 10 4 x 1 
. 0.1 x 0.025 x 1.29 x 10-4 x 0.025 x 3.69 x 10-2 
=373 ft. 
Now, application of Eq. 40 yields 
We also can estimate 
(kjbj}D= kjbf ..!.. 
Xj k 
0.3978 
= 0.025 = 15.9. 
From Fig. 14, r~/xj=0.46. 
:. r~=373x0.46=171.6ft. 
Fi~. 17 shows the bilinear flow graph [~(p) vs. 
llt ~ 1 for this example. Based on the information 
provided by Fig. 16, the correct straight line is 
drawn. The slope of this line is 1.62 X 108 psi2/cp­
hr ~, and at the end of bilinear flow 
llm(p}ebj=2.05x 108 psi2/cp-hr~. Notice that the 
pressure curve after the end of the bilinear flow 
period is concave upward, indicating that 
(kjbj) D > 1.6.


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pressure curve after the end of the bilinear flow 
period is concave upward, indicating that 
(kjbj) D > 1.6. 
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'" 0. 10 ~ 
E • 
<J 
o Cucve concave upwards ... (kfbf) D > 1.6 
1~ V. IllI) (hour I 
Fig. 17 -Bilinear flow graph for Well A. 
TABLE 3 -TEST INFORMATION AND ANALYSIS RESULTS 
FOR WEll B ' 
Reservoir Data 
Production rate q, MscflD 
Producing time tp, hours 
Formation thickness h, ft 
Porosity C/>, fraction 
Permeability k, md 
Formation temperature T, OR 
Average gas viscosity p., cp 
Total compressibility ct, psi-1 
Flowing wellbore pressure Pwt, psia 1,675 
1,800 
85 
0.11 
0.0045 
675 
0.025 
0.152x 10-3 
1,250 
Analysis Results Type·Curve Bilinear 
Analysis Flow Analysis 
ktbt, md·ft 
Xt, ft 
(k:~t), md.ftlft 
(ktbt)D 
r';", ft 
From Eq. 20, 95.3 
;;::692 
::50.1377 
::530.6 
::5325 
(kjbj) Yz = (444.75 X 7,350 X 690)/11.62 X 108 X 118 
·(0.1 x 0.0252 x 1.29 x 10-4 xO.025)1~, 
:. (kjbj) = 154md-ft. 
Using Eqs. 22 and 2, 
(k b ) D "'" 1.38 
j j (0.025X1l8X2.05XI08) 
1,424 X 7,350 x 690 
"'" 16.71. 
1760 109 
0: u 
N~ 
'" 0. 
Ci 108 
E 
<J Slope = 1/4 
II t Ihours)


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(k b ) D "'" 1.38 
j j (0.025X1l8X2.05XI08) 
1,424 X 7,350 x 690 
"'" 16.71. 
1760 109 
0: u 
N~ 
'" 0. 
Ci 108 
E 
<J Slope = 1/4 
II t Ihours) 
Fig. 18 -Log·log graph of pressure data for Well B. 
Hence, 
154 ---- "'" 368 ft. 0.025 x 16.71 
From Fig. 14, r:V1xj = 0.46. 
:.r:V =368 x 0.46 = 169.3 ft. 
We see that the results obtained by using both the 
bilinear flow and the type-curve methods are ap­
proximately the same; this fact increases confidence 
in the analysis performed. 
WellB 
A buildup test was run after fracturing this gas well. 
Information about the test and results of analysis are 
presented in Table 3. A log-log graph of pressure 
data (Fig. 18) indicates that the test was completely 
dominated by bilinear flow (quarter slope), 
corresponding this example to Case 1 in the type­
curve-analysis section. 
The bilinear flow graph (Fig. 19) yields a straight 
line whose slope, m bj' can be used to calculate 
(kjbj) usingEq. 20:


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curve-analysis section. 
The bilinear flow graph (Fig. 19) yields a straight 
line whose slope, m bj' can be used to calculate 
(kjbj) usingEq. 20: 
or (kjbj) Yz = (444.75 x 1,675 x 675)/[0.92 x 108 
x 85 x (0.11 x 0.025 x 1.52x 10-4 
x 0.0045) ~], 
If we assume that the last data point corresponds to 
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0. u 
"''::' U) 
0. 
co 9 
Q 
E 
<J 2 
(.H) last = 18 hours 
mbf = 0.92 x 108 ~, 
cp hrV" 
2 
1;4 1/4 (lit) (hour ) 
Fig. 19 -Bilinear flow graph for Well B. 3 
the end of the bilinear flow period and (k fb f) D ~ 3, 
then from Eq. 31, 
~ lOx2.637xlO-4X(95.3)2 X18 
xf ~ 0.11 x 0.025 x 1.52 x lO -4 x 0.0045 
=692 ft, 
and 
k b 95.3 ~ !S --=0.1377 md-ft/ft. xf 692 
FromEq.6, 
( kfbf) 95.3 
(k fb f) D = kx f !S -0-.0-04-5-x-6-9-2 ' 
From Fig. 14, r~/xf!S0.47; hence, r~ !S0.47 X 692, 
r~ !S325 ft. 
Welle 
After a flowing time of 1,890 hours, a buildup test 
was run on this fractured oil well. Information for 
the test and analysis results are presented in Table 4. 
Fig. 20 shows a log-log graph of the pressure data; 
from this graph we can see that neither a one-fourth 
slope nor a half slope is exhibited by the data. This 
figure also shows that pressure data match the curve 
for (kfbf) D = 27r given in Fig. 15 and the last six 
points fall on the semilog straight line.


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for (kfbf) D = 27r given in Fig. 15 and the last six 
points fall on the semilog straight line. 
From the pressure match obtained in Fig. 20 and 
Eq. 41, we estimate 
k= 141.2 x 220 x 1.2 x 0.8 x 0.34 
49x 100 
=2.07md. 
SEPTEMBER 1981 101 r-----r---,---,---,---,-----, 
Beg~nning of Semllog 
Straight Llnc 
1031-- - -1-- -1--1-- - -
I I I !>latch POl.nt 
I (tlp)1'-! = 100 psi 
(PWO)M '" 0.34 ___ 1 ___ J._ 
I {6t)M = 1 hr 
1 (tDr~)M = 0.19 
1 I ' 
10 '----- -L ___ 1 ____ 1 ___ ..J 
10-1 I 10 102 103 
LIt (hours) 
10-2 L,-----'-:----'--,-----'--~---'7-----'-;;-----.J 
10-3 10-2 10-1 10 1 102 103 
Fig. 20 -Type·curve matching for Well C. 
TABLE 4 -TEST INFORMATION AND ANALYSIS RESULTS 
FORWELLC 
Reservoir Data 
Production rate q, STBID 
Producing time tp, hours 
Formation thickness h, ft 
Porosity </>, fraction 
Viscosity /1-, cp 
Total compressibility ct, psi-1 
Formation volume factor 80, bbllSTB 
Well bore radius r w, ft 
Flowing well bore pressure Pw', pSia 
Analysis Results 
k,md


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Formation volume factor 80, bbllSTB 
Well bore radius r w, ft 
Flowing well bore pressure Pw', pSia 
Analysis Results 
k,md 
(k,btlo 
x" ft 
k,b" md-ft 
r;', ft 
S Type-Curve 
Analysis 
2.07 
211" 
88.7 
1,156 
36.89 
-4.99 220 
1,890 
49 
0.15 
0.8 
17.6x 10-6 
1.2 
0.25 
1,704 
Semi log 
Analysis 
2.28 
30.37 
-4.8 
Using information from the time match in Eq. 43, 
r' _~ 2.637x 10 4 x2.07x 1 
w-0.15xO.8xI7.6xlO 6xO.19 
= 36.9 ft. 
From Fig. 14, r~/xf=0.415; hence, 
36.9 
xf= 0.415 = 88.9 ft. 
The skin factor is estimated by 
r 0.25 s=ln ~ =In-- = -4 99 
r~ 36.9 .. 
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7.--------,-------,--------,---r----, 
6 
5 
2. 4 
'" ~ m = 307 psi/cycle 
-3 
Q. 
<l 
2 
o 
o 
10-1 00 
o 00 
{l>p} : -47 psi // 1hr 
101 
lit Ihours/ 
Fig. 21 -Semi log graph for Well C. 
The fracture conductivity is 
kjbj= {kjbj)nkXj 
=27rx 2.07 x 88.9= 1,156.2 md-ft. 
Fig. 21 is a semilog graph for this example. The 
correct semilog straight line has a slope m = 307 
psi/cycle and (ilp) Ihr = -47 psi. The formation 
permeability can be calculated as 
k = _16_2_._6_qB_fJ. 
mh 
162.6 x 220 x 1.2 x 0.8 = =2.28md. 307x49 
The skin factor is 
S=1.151[{Llp)lhr -log( k 2)+3.2275J 
m 1>fJ.ctrw 
[-47 = 1.151 -307 
2.28 
-log 0.15 xO.8 x 17.6x 10-6 x (0.25)2 
+ 3.2275 J = -4.8. 
Finally, the effective wellbore radius is 
r~ =r we-s =0.25 e4.8 
= 30.37 ft. 
The results provided by both the type-curve 
analysis and semilog analysis methods are in good 
agreement. From these examples it is demonstrated 
that type-curve analysis, when applied properly,


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agreement. From these examples it is demonstrated 
that type-curve analysis, when applied properly, 
provides an excellent diagnostic tool and a technique 
to estimate both reservoir and fracture parameters. 
1762 Conclusions 
Based on the material presented in this work, the 
following remarks are pertinent. 
1. The transient flow behavior of a vertically 
fractured well may exhibit four flow periods: (a) 
fracture linear flow, (b) bilinear flow, (c) formation 
linear flow, and (d) pseudoradial flow. Bilinear flow 
is a new type of flow that has not been considered 
before. 
2. A new technique is presented to analyze data in 
the bilinear flow period. It is shown that, during this 
flow period, a graph of pwj[or m (Pwj)] vs. Vi yields 
a straight line whose slope is inversely proportional 
to h j{kjbj) Yz. 
3. New type curves are now available for pressure 
analysis of fractured wells. The uniqueness problem 
in the analysis is reduced considerably with the use of


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analysis of fractured wells. The uniqueness problem 
in the analysis is reduced considerably with the use of 
these type curves. 
4. Prefracture information about the reservoir is 
necessary to estimate fracture parameters. 
5. The type-curve analysis method must be used 
simultaneously with the specific analysis methods 
(Pwfvs. Vt,Pwjvs.1t, andpwjvs.log t) to produce 
reliable results. 
Acknowledgments 
Part of this work was develo,ped at Stanford U. and 
the Inst. Mexicano del Petroleo. We are grateful to 
many people, especially H.J. Ramey Jr. for his 
encouragement and helpful comments. 
Nomenclature 
b j = fracture width 
B = formation volume factor 
c = compressibility 
C = well bore storage coefficient 
FI ,F2 = correlating parameters for well bore 
storage 
h = formation thickness 
h j = fracture height 
k = permeability 
kjbj = fracture conductivity 
(k jb j) n = dimensionless fracture conductivity 
m = slope of semilog straight line, gas 
pseudopressure


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kjbj = fracture conductivity 
(k jb j) n = dimensionless fracture conductivity 
m = slope of semilog straight line, gas 
pseudopressure 
mbj = slope of straight line for bilinear flow 
P = pressure 
q = well flow rate 
r w = wellbore radius 
r~ = effective well bore radius 
s = skin factor or Laplace space variable 
t = time 
ilt = shut-in time 
T = reservoir temperature 
X,Y = space coordinates 
Xj = fracture half-length 
(X,{3 = unit conversion constants 
r = gamma function 
'Y/ = hydraulic diffusivity 
1> = porosity 
fJ. = fluid viscosity 
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Subscripts 
b= beginning 
bf = bilinear flow 
g gas 
D= dimensionless 
e = end 
f= fracture, flowing 
i = initial 
/f= linear flow 
0 oil 
t total 
xf based on xf 
w wellbore 
References 
1. Matthews, C.S. and Russell, D.G.: Pressure Buildup and 
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California Regional Meeting, Long Beach, April 8-9, 1976. 
6. Raghavan, R.: "Pressure Behavior of Wells Intercepting 
Fractures," Proc., Invitational Well-Testing Symposium, 
Berkeley, CA, Oct. 19-21, 1977.


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6. Raghavan, R.: "Pressure Behavior of Wells Intercepting 
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Bottom-Hole Pressure Build-Up Characteristics," Trans., 
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14. Clark, K.K.: "Transient Pressure Testing of Fractured Water 
Injection WeJls," J. Pet. Tech. (June 1968) 639-643; Trans., 
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15. Millheim, K.K. and Cichowicz, L.: "Testing and Analyzing 
Low-Permeability Fractured Gas WeJls," J. Pet. Tech. (Feb. 
1968) 193-198; Trans., AIME, 243. 
16. Raghavan, R., Cady, G.V., and Ramey, H.J. Jr.: "Well-Test 
Analysis for VerticaJly Fractured Wells," J. Pet. Tech. (Aug. 
1972) 1014-1020; Trans., AIME, 253. 
17. Gringarten, A.C. and Ramey, H.J. Jr.: "Unsteady-State 
Pressure Distributions Created by a WeJl With a Single In­


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17. Gringarten, A.C. and Ramey, H.J. Jr.: "Unsteady-State 
Pressure Distributions Created by a WeJl With a Single In­
finite-Conductivity Vertical Fracture, Soc. Pet. Eng. J. (Aug. 
1974) 347-360; Trans., AIME, 257. 
18. Gringarten, A.C. and Ramey, H.J. Jr.: "Unsteady-State 
Pressure Distributions Created by a Well With a Single 
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19. Gringarten, A.C., Ramey, H.J. Jr., and Raghavan, R.: 
"Applied Pressure Analysis for Fractured WeJls," J. Pet. 
Tech. (July 1975) 887-892; Trans., AIME, 259. 
SEPTEMBER 1981 20. Wattenbarger, R.A. and Ramey, H.J. Jr.: "Well Test In­
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Tech. (May 1969) 625-632; Trans., AIME, 246. 
21. Ramey, H.J. Jr. and Gringarten, A.C.: "Effect of High­
Volume Vertical Fractures on Geothermal Steam WeJl 
Behavior," Proc., Second United Nations Symposium on the


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Volume Vertical Fractures on Geothermal Steam WeJl 
Behavior," Proc., Second United Nations Symposium on the 
Use and Development of Geothermal Energy, San Francisco, 
May 20-29, 1975. 
22. Cinco, H., Ramey, H.J. Jr., and Miller, F.: "Unsteady-State 
Pressure Distribution Created by a Well With an Inclined 
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23. Raghavan, R. and Hadinoto, N.: "Analysis of Pressure Data 
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24. Raghavan, R., Uraiet, A., and Thomas, G.W.: "Vertical 
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Pressure Behavior for a Well with a Finite-Conductivity 
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Pet. Tech. (Oct. 1976) 1169-1178. 
27. Ramey, H.J. Jr., Barker, B., Arihara, N., Mao, M.L., and 
Marques, J.K.: "Pressure Transient Testing of Hydraulically 
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28. Cinco, H. and Samaniego, F.: "Effect of Wellbore Storage 
and Damage on the Transient Pressure Behavior of Vertically 
Fractured WeJls," paper SPE 6752 presented at the SPE 52nd 
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29. Agarwal, R.G., Carter, R.D., and Pollock, C.B.: 
"Evaluation and Prediction of Performance of Low­
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Fracturing," J. Pet. Tech. (March 1979) 362-372; Trans., 
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Permeability Gas WeJls Stimulated by Massive Hydraulic 
Fracturing," J. Pet. Tech. (March 1979) 362-372; Trans., 
AIME,267. 
30. Barker, B.: "Transient Flow to Finite-Conductivity Vertical 
Fractures," PhD dissertation, Stanford U., Palo Alto, CA 
1977. 
31. Barker, B.J. and Ramey, H.J. Jr.: "Transient Flow to Finite­
Conductivity Vertical Fractures," paper SPE 7489 presented 
at the SPE 53rd Annual Technical Conference and Exhibition, 
Houston, Oct. 1-3, 1978. 
32. Scott, J.O.: "A New Method for Determining Flow 
Characteristics of Fractured WeJls -Application to Gas WeJls 
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Conference, Montreal, T-279-186. 
APPENDIX A 
Short-Time Transient Pressure 
Behavior for a Well With 
a Finite-Conductivity Fracture 
Derivation of Solution 
Let us consider the system described in the text of this 
paper. A fractured well produces at a constant flow 
rate from an infinite reservoir. At small time values,


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paper. A fractured well produces at a constant flow 
rate from an infinite reservoir. At small time values, 
the pressure behavior of the system is not affected by 
the tips of the fracture. In addition, the flow in the 
formation is essentially linear and perpendicular to 
the fracture plane. The reservoir and the fracture 
may be treated as two different homogeneous 
regions. 25 
If the flow within the fracture is assumed to be 
linear, the pressure behavior can be described by 
a2pfD 
a 2 + xD 2 apD I 
(kfbf) D aYD YD=O 
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= _1_ ap[D .................. (A-I) 
7JjD atDXj 
for O<xD < 00, tDxj >0. 
Initial Condition. 
PfD =0, tDXj=O, O:s;xD <00. . .......... (A-2) 
Boundary Conditions. 
apfD I = - 7r , tDx >0, 
aXD XD=O (kfbf)D f 
. . . . . . . . . . . . . . (A-3) 
and 
lim PfD =0, tDxj >0, ............... (A-4) 
XD-OO 
where, in oilfield units, 
_ kh[Pi-Pf] 
PjD -141.2 qBJ.I. 
and kh [Pi -p] 
P D = 141.2 qBJ.I. 
x xD=-' xf 
Y YD=-· xf 
The variables not included above are defined in the 
text. P and P f represent the formation pressure and 
the fracture pressure, respectively, x is the flow 
direction in the fracture, and Y is the formation flow 
direction perpendicular to the fracture plane. 
The transient flow in the formation may be 
described by 
a2p ap D_~·O<y <00· ~-!:I ' D ' aYD utDxj 
tDxj>O. . ................... (A-5) 
Initial Condition. 
PD =0; O<YD <00; tDXj=O ............. (A-6) 
Boundary Conditions. 
PD IYD=O =PfD; tDXj>O, .............. (A-7) 
and 
lim PD =0; tDxj>O ................. (A-8)


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Boundary Conditions. 
PD IYD=O =PfD; tDXj>O, .............. (A-7) 
and 
lim PD =0; tDxj>O ................. (A-8) 
YD-OO 
1764 The two partial differential equations are already 
coupled by the boundary conditions. Application of 
the Laplace transformation with respect to. time to 
Eqs. A-I through A-8 and simplification yields 
a2pfD 2 apD I + --
aXD2 (kfbf) D aYD Yd=O 
s _ 
= -PfD; O<xD < 00 ........... (A-9) 
7JjD 
Boundary Conditions . 
apfD I = - 7r , ••••••••••• (A-lO) 
aXD Xd=O (kfbf) D 
lim PjD =0, ..................... (A-ll) 
XD-OO 
and 
a2pD 
aYD2 =SPD; O<YD<oo ............ (A-12) 
Boundary Conditions. 
PD IYD=O =PfD' .................... (A-l3) 
lim PD=O' ..................... . (A-14) 
YD-OO 
where 
PjD (xD' s) =£tDXj [PjD (XD,IDXj)]' 
and 
PD (YD' s) = £tDXj [PD (YD' tDXj) ]. 
Now let us define 
PD (r,s) = £YD [PD (YD,s)]. 
Application of the Laplace transformation, with 
respect tOYD, to Eq. A-12 yields 
2 ~ I ap D I ~ (A 15) r PD-rPD YD=O--a-- =sPD···-
'YD YD=O


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Application of the Laplace transformation, with 
respect tOYD, to Eq. A-12 yields 
2 ~ I ap D I ~ (A 15) r PD-rPD YD=O--a-- =sPD···-
'YD YD=O 
By solving for PD and considering Eq. A-l3, we 
obtain 
_ oPD I rPfD+ -a-
~ 'YD YD=O (A 16) PD= 2 • ...•.•.•.. -r -s 
The inversion of this equation with respect to Y D 
yields 
a-
P D = P jD cosh (...;s Y D) + : D I 
'YD YD=O 
................. (A-17) 
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From Eqs. A-14 and A-17, we can write 
apD I - -- = -PfDYs. . ............. (A-18) 
aYD YD=O 
Next, substitution of Eq. A-18 into Eq. A-9 allows 
us to write a partial differential equation with only 
one dependent variable: 
a2pfD _ S 2Ys_ 
2 - -+ PfD· ........ (A-19) 
aXD 'YlfD (kfbf) D 
The solution of Eq. A-19 with boundary conditions 
given by Eqs. A-tO and A-II can be expressed as 
[ [ S 2Ys] Yz 7rexp -xD -+ 
-_ 'YlfD (kfbf) D ] prD-
p S 2Ys Yz 
(kfbf}DS[-+ b ] 
'YlfD (kf f) D 
....................... (A-20) 
The pressure at the wellbore PwD is calculated at 
xD =0; thus, 
7r 
PwD = S 2Ys Y2·· (A-21) 
(kfbf}D s[-+ ] 
'YlfD (kfbf}D 
Finally, the Laplace inversion of Eq. A-21 with 
respect to 1 DXf produces 
PwD= ~ rDXf 
(kfbf) D 0 
....................... (A-22) 
Unfortunately, Eq. A-22 is too complex to analyze 
both the short-and long-time behavior of the 
solution. Eq. A-21 may be used for this purpose. 
Short-Time Behavior 
The short-time approximation of the solution can be


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solution. Eq. A-21 may be used for this purpose. 
Short-Time Behavior 
The short-time approximation of the solution can be 
obtained from Eq. A-21 by taking the limit as the 
variable S approaches infinity. Thus, 
--~ PwD-(k
fb
f}DS312 ................. (A-23) 
Inversion of this formula produces Eq. 8. 
Long-Time Behavior 
We can obtain the solution for large values of time by 
taking the limit of Eq. A-21 as S approaches zero; 
hence, 
7r 
PwD =::. ..J2(k
fb
f) D s5/4· ............. (A-24) 
Inverting Eq. A-24 produces Eq. 12. This solution 
also may be obtained by considering incompressible 
flow within the fracture. 
SEPTEMBER 1981 Total Formation Flow 
Eqs. A-18 and A-20 can be used to calculate the 
fraction of the well flow rate that is being produced 
from the formation. The formula for this case ap­
pears to be 
q formation = 1 _ exp( 41 DXr 2 ) 
q 7r2CfDf 
( 2..JIDxr ) . erfc . . ........ (A-25) 
7rCfDf 
For small values of 1 Dx' the amount of fluid


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q formation = 1 _ exp( 41 DXr 2 ) 
q 7r2CfDf 
( 2..JIDxr ) . erfc . . ........ (A-25) 
7rCfDf 
For small values of 1 Dx' the amount of fluid 
coming from the formation {s negligible; however, at 
large values of 1 Dx' most of the produced fluid 
comes from the formation. This illustrates the 
physical behavior of the system. 
APPENDIXB 
Derivation of Correlation Parameters 
for Bilinear and Linear Flows 
If we consider a fracture of length xf' Eq. A-4 
becomes 
apfD I =0 ....................... (B-1) 
aXD xD=1 
Using Eq. B-1 instead of Eq. A-4 and considering 
incompressible flow within the fracture, the Laplace 
transformation solution for the problem stated in 
Appendix A is 
pWD=(-7r+ E 27rexp[-2n 
n=O 
[~ (kfb~)D2 tJ )/(S(kl'tln 
.[~ (kfb~)D2 t) ............ (B-2) 
or 
PWD (kfbf)D3~( -<+ 00 E 27r eXPt -2n 
n=O 
Hence, 
1 [ S ~] PWD(kfbf}D = (k
fb
f}D2! (k
fb
f}D2 ... (B-4) 
From Laplace inversion tables, 
PWD (kfbf) D =F[tDx/ (kfbf) D2]. . ..... (B-6)


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n=O 
Hence, 
1 [ S ~] PWD(kfbf}D = (k
fb
f}D2! (k
fb
f}D2 ... (B-4) 
From Laplace inversion tables, 
PWD (kfbf) D =F[tDx/ (kfbf) D2]. . ..... (B-6) 
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Hence, a graph of Pw (kfbf)D vs. tDx (kfbf)2 . ld . I D f Yle s a smg e curve. 
For small values of dimensionless time, Eq. B-3 
yields Eq. A-24, corresponding to bilinear flow. 
The long-time approximation of Eq. B-3 yields 
7r P WD ~ 2s312 ' .••••••••••••••••••••••• (B-7) 
which is the Laplace transform of the dimensionless 
pressure for the formation linear flow. 
APPENDIXC 
Derivation of Correlation Parameters 
for Bilinear Flow with 
Wellbore Storage Effects 
When well bore storage effects are considered in the 
flow problem defined in Appendix A, the boundary 
condition at x D = 0 becomes 
apfD I 7r (1-CDf apWD), . (C-l) 
aXD xD=O (kfbf) D atDxf 
where 
C = C Df 27rA.c hx 2' ................... (C-2) 
'I' t f 
and C is the wellbore storage coefficient. 
If flow within the fracture is considered in­
compressible and Eq. C-1 is taken instead of Eq. A-
3, the solution for the wellbore pressure in the 
Laplace space is given by 
7r 
PwD = -V2(kfbf) D s5/4 + 7rCDfS2'


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3, the solution for the wellbore pressure in the 
Laplace space is given by 
7r 
PwD = -V2(kfbf) D s5/4 + 7rCDfS2' 
which can be written as 
1766 ...... (C-3) 4/3 C 4/3 s 5/4 
(kfbf) D - _ ~Y2[ Df ] 
C 5/3 PwD-7r (k b ) 2/3 Df f f D 
CD//3s ~ 
+7r[ (kfbf)D2/3] j" .. (C-4) 
From properties of the Laplace transformation, 
-I _1 [t] L !(as) -~F ~ ................... (C-5) 
Hence, Eq. C-4 can be written as 
(kfbf)D2I3 =F[ (kfbf)D2I3 t ] 
(CDf) 113 PwD (CDf) 4/3 DXf' 
................... (C-6) 
This means that the bilinear flow solutions including 
well bore storage are represented by a single curve 
when graphed in terms of 
(k b ) 213 (k b ) 2/3 
--""f'--Lf'I::;'~;--PWD vs. f f ~/3 tDx' Cw (Cw) f 
SI Metric Conversion Factors 
cp x 1.0* E-03 = Pa·s 
ft x 3.048* E-Ol = m 
psi x 6.894 757 E + 00 = kPa 
·Conversion factor is exact. JPT 
Original manuscript received in Society of Petroleum Engineers office Aug. 
10, 1978. Paper accepted for publication June 4, 1979. Revised manuscript


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10, 1978. Paper accepted for publication June 4, 1979. Revised manuscript 
received July 6,1981. Paper (SPE 7490) first presented at the SPE 53rd Annual 
Technical Conference and Exhibition, held in Houston, Oct. 1-3, 1978. 
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Citation: Gabry, M.A.; Eltaleb, I.;
Soliman, M.Y.; Farouq-Ali, S.M. A
New Technique for Estimating Stress
from Fracture Injection Tests Using
Continuous Wavelet Transform.
Energies 2023 ,16, 764. https://
doi.org/10.3390/en16020764
Academic Editor: Maria Grazia
De Giorgi
Received: 2 December 2022
Revised: 23 December 2022
Accepted: 27 December 2022
Published: 9 January 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
energies
Article
A New Technique for Estimating Stress from Fracture Injection
Tests Using Continuous Wavelet Transform†
Mohamed Adel Gabry *
 , Ibrahim Eltaleb
 , Mohamed Y. Soliman
 and Syed M. Farouq-Ali
Department of Petroleum Engineering, The University of Houston, Houston, TX 77023, USA
*Correspondence: magabry@cougarnet.uh.edu


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Department of Petroleum Engineering, The University of Houston, Houston, TX 77023, USA
*Correspondence: magabry@cougarnet.uh.edu
† Society of Petroleum Engineers Technical Conference and Exhibition, Houston, TX, USA, 3–5 October 2022.
Abstract: The diagnostic fracture injection test (DFIT) is widely used to obtain the fracture closure
pressure, reservoir permeability, and reservoir pressure. Conventional methods for analyzing DFIT
are based on the assumption that a vertical well is drilled in ultra-low permeability reservoirs with
potential multiple closures but fails to consider horizontal wells. There is still signiﬁcant debate about
the rigorousness and validity of these techniques due to the complexity of the hydraulic fracture
opening and closure process and assumptions of conventional fracture detection methods. The
paper introduces a new method for detecting fracture closure pressure using the continuous wavelet


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paper introduces a new method for detecting fracture closure pressure using the continuous wavelet
transform (CWT). The new method aims to decompose the pressure fall-off signal into multiple levels
with different frequencies using the CWT. This “short wavy” function is stretched or compressed and
placed at many positions along the signal to be analyzed. The wavelet then convoluted the signal
yielding a wavelet coefﬁcient value. The signal energy is observed during the fracture closure process
(pressure fall-off) and the fracture closure event is identiﬁed when the signal energy stabilizes to
a minimum level. A predeﬁned simple commercial fracture simulation case with known fracture
closure, ﬂow regime modeling, and actual ﬁeld cases was used to validate the new methodology.
Keywords: continuous wavelet transform; diagnostic fracture injection test; hydraulic fracture;
minimum horizontal stress; geothermal reservoirs
1. Introduction
1.1. Scope of Study


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minimum horizontal stress; geothermal reservoirs
1. Introduction
1.1. Scope of Study
Estimating the minimum horizontal stress is a critical part of building formation
geomechanical models, designing hydraulic fractures, designing drilling schedules, etc.
Hydraulic fracturing design and execution need to calibrate the mechanical Earth model
with a reliable method. This is performed to identify the formation stresses. Classical
methods for obtaining the minimum horizontal stress of the formation relies on the open-
hole well logs (Eaton (1969) [ 1]; Blanton and Olson (1999) [ 2]). These methods can be
categorized into two main techniques. The ﬁrst technique relies on compressional and
shear sonic waves to estimate dynamic geo-mechanical properties. The second technique
uses correlations to calculate the dynamic properties of the rock from conventional well-
logging parameters, then calculates the minimum horizontal stress [3].


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logging parameters, then calculates the minimum horizontal stress [3].
The actual in situ minimum principal stress of the formation is calculated by calibrating
against the closure pressure from diagnostic pumping test [ 4]. That is performed by
pumping a small volume of non-wall building ﬂuids to create a small fracture and then
analyze the pressure decline to know the pressure at which that fracture closes. That
fracture closure pressure should be the average of the minimum principal stress for the
area covered by the created fracture.
As no possible physical observation of the fracture propagation and closure in com-
plicated subsurface formations, there are signiﬁcant debates surrounding the various
methodologies used to determine closure pressure. This is because of several reasons. The
Energies 2023 ,16, 764. https://doi.org/10.3390/en16020764 https://www.mdpi.com/journal/energies


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Energies 2023 ,16, 764 2 of 29
ﬁrst reason is that the hydraulic fracture propagation and closure process are fairly compli-
cated due to the formation stress state, the presence of natural fractures, heterogeneity, and
the anisotropy of the subsurface formations’ properties. Conventional closure detection
methods have pre-set assumptions to simplify the state of the subsurface formation. Some
of them may simplify the leak off to carter leak off, whilst other methods assume the
constant compliance or make certain assumptions regarding the ﬂow regimes or the state
of injection. Many tests may deviate from the assumed pre-set conditions. The detected
closure be confused by the formation pressure response after the fracture closure. The
pressure change propagates in the formation and produces behavior that may be confused
with fracture closure detection.
Although fracture closure is deﬁned as the pressure at which a fracture closes, there


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with fracture closure detection.
Although fracture closure is deﬁned as the pressure at which a fracture closes, there
is an attempt to detect the contact pressure when the two fracture faces ﬁrst come into
contact with one another. Assuming that closure pressure does not really exist, there is an
aperture between fracture faces. Complications regarding how the fracture propagates and
closes cause the interpreted fracture closure pressure debatable. In most cases, the industry
workers try various methods to determine the range of closure pressure. To date, there is no
global technique that can detect closure without pre-assumptions. Each of the conventional
methods detects the closure from a speciﬁc point of view with certain pre-assumptions.
This paper introduces an innovative method to ﬁnd the closure pressure using the wavelet
transform technique as a general technique without any pre-assumptions. It uses an


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transform technique as a general technique without any pre-assumptions. It uses an
independent mathematical microscope to ﬁnd the fracture closure pressure in the pressure
fall-off signal. The new technique is validated using different ways: using simulated
synthetic pressure data from a fracture simulator and using ﬂow regime modeling by
reverse calculating a pressure decay signal assumed to be both before fracture closure and
after fracture closure ﬂow regimes. The main advantage of this technique is that there is no
need to know formation parameters in advance or have a pre-prescribed testing procedure.
Since the beginning of the 1990s, the science and engineering applications of wavelet
transform have increased at a remarkable growth rate. Thousands of peer-reviewed journals
are now interested in wavelet transform applications covering numerous disciplines such
as ﬂuids, medicine, and fractal geometry. However, its use in the oil and gas industry


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as ﬂuids, medicine, and fractal geometry. However, its use in the oil and gas industry
and geothermal energy has been limited to date. A lot of the data recorded in multiple
disciplines are considered as signals to be analyzed to understand the physics-based
models covering the many applications. For example, Germán-Salló et al. (2018) [ 5]
introduced a method for crack detection in manufacturing systems using digital signal
processing methods.
Signal processing is widely used in the medical ﬁeld to detect heart problems by
analyzing electrocardiography (ECG) signals. Research in the medical ﬁeld deploys wavelet
transform to compare the ECG signals of a healthy heart to another ECG signal associated
with a diseased heart. The same methodology is adopted in the industrial application
of signal processing to predict motor failure. For instance, the analysis of the vibration
signal of a motor can help detect bearing failure when the recorded signal is compared to a
functional motor.


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signal of a motor can help detect bearing failure when the recorded signal is compared to a
functional motor.
With the ongoing digitalization of the modern oil and gas industry and geothermal
energy industry, the same technique can be used in the analysis of hydraulic fracturing. In
this case, changes in pressure and rate over time are recorded during the fracture propa-
gation stages, proppant placement, and fracture closure to determine what is happening
thousands of feet below the Earth’s surface during hydraulic fracturing treatment. The
most common methodology used by researchers for engineering wavelets applications is
to try to ﬁnd a correlation between the features detected from the signals and the actual
physical response of the system. The main problem with this methodology in hydraulic
fracturing is the complexity of hydraulic fracturing propagation and closure due to fracture
face roughness or natural fracture intensity and directions and a lot of other factors. This


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face roughness or natural fracture intensity and directions and a lot of other factors. This
may be complex not only because of the heterogeneity of the reservoir, but also because of
the interaction of the hydraulic fracture with natural fractures, ﬁssures, and other geologic


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Energies 2023 ,16, 764 3 of 29
features. Thus, all the complicating factors should be removed from the system to ﬁnd a
signal representing a closure event in a homogeneous reservoir with no geologic complexi-
ties. In order to understand the CWT behavior during fracture closure, a numerical fracture
simulator is used in this study.
The application of wavelet analysis in the oil industry was pioneered twenty years
ago by Soliman et al. (2003) [ 6] as an application for well-test analysis and fracturing. The
wavelet transform technique is a data transform technique that convolves the pressure
and/or temperature data using a short wavy signal called “wavelet”. This convolution can
be performed at discrete points in the discrete wavelet transform (DWT) or continuously
using the CWT. The wavelet transform provides a representation of the pressure signal
by letting the translation and scales of the wavelets vary continuously. This enables the


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by letting the translation and scales of the wavelets vary continuously. This enables the
analyst to ﬁnd the details of the pressure data by observing the wavelet energy spectrum
for the monitored signal (pressure and/or temperature).
As the DWT is computationally cheaper, it was used as the ﬁrst step to detect fracture
closure. The main disadvantage of using the DWT is its requirement of a uniform sampling
rate and the detected closure pressure changes with an increasing level of decomposition.
Soliman, Unal, Siddiqui, Rezaei, and Eltaleb (2019) [7,8] initiated a methodology to detect
the fracture closure pressure using the DWT. Determining the closure pressure in this
methodology is based on analyzing the detail levels to pick the time when the variance
changes. Eltaleb et al. (2020) [ 9] and Eltaleb et al. (2021) [ 10] improved the rigor of the
technique using an energy density plot. The technique was compared to the G-function


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technique using an energy density plot. The technique was compared to the G-function
technique and showed good agreement. The DWT technique involves a multiresolution
wavelet decomposition that breaks the (recorder pressure) signal into high pass (noise) and
low pass (approximation) components at various levels. The plot of energy distribution
in time is constructed using the energy of the high pass component (noise) at different
decomposition levels. These plots provide means of capturing the physical changes in
the fracture system through pressure fall-off. They represented the noise energy in the
recorded pressure at different frequency levels. The closure pressure in this methodology
was determined based on changes in the noise energy of the recorded pressure and where
it drops to a minimum stabilized level.
This paper introduces a closure detection technique similar to that introduced by
Eltaleb et al. (2021) [ 10], however, using the CWT. Using the CWT does not need a


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Eltaleb et al. (2021) [ 10], however, using the CWT. Using the CWT does not need a
uniform sampling rate and captures all signal details. CWT can be used as a method for
detecting the fractal dimension which is one of the key features of complex wavelets that
is sometimes referred to as a “mathematical microscope” to detect the features. Using
CWT was recommended by Fatmir (2020) through personal communication. CWT can
be used as a ﬁlter to detect the signal features. It has the ﬂexibility to be used with or
without a uniform sampling rate. Thus, it could be validated using any type of data. This
new method was applied to both the fracture simulator data and actual ﬁeld data. The
synthetic data were produced using a commercially available fracture simulator developed
by Barree (1983) [ 11] based on fracture propagation and closure simulation principles with
predeﬁned fracture closure. Azlinda et al. (2010) [ 12] showed good agreement between the


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predeﬁned fracture closure. Azlinda et al. (2010) [ 12] showed good agreement between the
used fracture simulator with down-hole events in a stacked ﬂuvial pay system. The CWT
technique showed success with the actual real ﬁeld data conducted on one of the oil and
gas ﬁelds and in another case in a geothermal reservoir.
1.2. History of Closure Pressure Detection Techniques
The basic concept of classical methods of closure event detection from the pressure
decay of the diagnostic fracture injection test is combining the basic concepts of well
testing and hydraulic fracture analysis. After hydraulic fracturing was ﬁrst introduced by
Clark (1949) [ 13], it was recognized that hydraulic fracturing could be used as a method
to estimate the magnitude of minimum principal stress (Hubbert and Willis (1957) [ 14];
Godbey and Hodges (1958) [ 15]; Kehle (1964) [ 16]; Haimson and Fairhurst (1967) [ 17];
Hickman and Zoback (1983) [ 18]). The main idea was to pump ﬂuid at a pressure higher


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Energies 2023 ,16, 764 4 of 29
than the fracturing pressure, to initiate a hydraulic fracture and propagate it through the
formation. The minimum principal stress can be estimated from pressure measurements
during and after injection. Nolte (1979) [ 19] extended the applicability of fracturing tests
by developing analytical techniques that can be used to estimate ﬂuid leak-off parameters.
These techniques can be used to obtain the leak-off rate of complex ﬂuids such as linear gel
or cross-linked gel. These tests are typically called mini-frac tests (McLennan and Roegiers,
(1982) [ 20]) or fracture calibration tests (Mayerhofer and Economides (1993) [ 21]). This is
performed to estimate the efﬁciency of the fracturing ﬂuid after the interpretation of the
minimum horizontal stress. One of the most widely used functions to obtain the closure
pressure is G-function (Nolte (1979) [ 19]). Several techniques have subsequently been


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pressure is G-function (Nolte (1979) [ 19]). Several techniques have subsequently been
published as essential parts of the analysis, including those by Castillo (1987) [ 22] as well
as Barree and Mukherjee (1996) [ 23]. Further modiﬁcations for the analysis were added,
for example, by Mayerhofer and Economides (1993) [ 21] and Mayerhofer et al. (1995) [ 24]
showed how to remove the ﬁlter cake effect and its effect on the ﬂuid leak-off factor from
the analysis. The recent development of the shale gas reservoirs came out with a new test
that uses the non-wall building ﬂuid instead of the cross-linked gel or linear gel. It is called
the “diagnostic fracture injection test” (DFIT). It is routinely performed in unconventional
reservoirs due to the incapability to ﬂow the well to do a pressure transient analysis. The
DFIT is used to obtain the best estimates for stress, permeability, and pore pressure (Craig
and Brown (1999) [25]. The analysis of pressure decay data has two parts.


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and Brown (1999) [25]. The analysis of pressure decay data has two parts.
The ﬁrst part of the DFIT analysis is called “Pre-closure analysis methods” which
are usually based on techniques from Nolte (1979) [ 19] and Mayerhofer and Economides
(1993) [ 21], Mayerhofer et al. (1995) [ 24], Barree and Mukherjee (1996) [ 23], and Barree
et al. (2009) [ 26]. These methods are used to estimate the closure pressure and calibrate
the minimum principal stress. The second part is called ‘Post-closure’ analysis methods
which are usually based on techniques from conventional well testing (Gu et al. (1993) [ 27];
Soliman et al. (2005) [ 28]; Craig and Blasingame (2006) [ 29]) or the ‘linear ﬂow time
function’ technique developed by Nolte et al. (1997) [ 30]. Due the development of uncon-
ventional reservoirs, multiple DFIT interpretation techniques have recently been developed
by Marongiu-Porcu et al. (2011; 2014) [ 31,32], Soliman and Kabir (2012) [ 33], Soliman and


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by Marongiu-Porcu et al. (2011; 2014) [ 31,32], Soliman and Kabir (2012) [ 33], Soliman and
Gamadi (2012) [ 33], Cramer and Nguyen (2013) [ 34], Padmakar (2013) [ 35], Wallace et al.
(2014) [ 36], Meng et al. (2014) [ 37], Craig (2014) [ 38], Zanganeh et al. (2018) [ 39], and
Hawkes et al. (2018) [40].
One of the noticeable techniques is the compliance method that was recently intro-
duced by McClure et al. (2014; 2016) [ 41,42] and Jung et al. (2016) [ 43]. This technique
usually leads to an earlier and higher stress estimate in contrast to the commonly used
tangent method (Barree et al. (2009) [ 26]) which is considered a modiﬁcations to Nolte’s
(1979) [ 19] technique. The technique introduced a new concept which is contact pressure
instead of closure pressure. McClure et al. (2016) [ 42] and Jung et al. (2016) [ 43] per-
formed detailed DFIT simulations considering the effect of a residual aperture fracture


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formed detailed DFIT simulations considering the effect of a residual aperture fracture
after the contact between the fracture walls. Simulations indicate that, in low permeabil-
ity formations, the contacting of the fracture walls causes the magnitude of the pressure
derivative to increase, resulting in a pressure signal that was previously interpreted as
the height recession or closure of transverse fractures (Barree et al. (2009) [ 26]. Wang
and Sharma (2017) [ 44] reproduced these ﬁndings and proposed a modiﬁed approach for
estimating stress called the ‘variable compliance method’. Currently, there are two main
closure detection techniques that are used by the industry, the tangent method, and the
compliance method.
The initial technique of the G function introduced by Nolte (1979) [ 19] relied on several
assumptions for the pumping and shut-in periods with some simpliﬁed concepts of how a
fracture propagates and closes. Nolte (1979) [ 19] provided the G time function to detect


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fracture propagates and closes. Nolte (1979) [ 19] provided the G time function to detect
the changes in the pressure during the shut-in period of the diagnostic pumping. The
second derivative (GdP/dG) is used to detect the closure. The closure deﬁned by this
technique is the pressure at a time when the deviation from the straight line starting from


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zero in GdP/dG occurs. The detailed equations for the G time function were discussed
by Nolte (1979) [ 19]. Castillo (1987) [ 22] added the condition that dP/dG should be with
zero slopes at the same point. The tangent method was introduced to cope with the
complicated fracture closure as a modiﬁcation for the Nolte (1979) [ 19] technique. The
tangent method proposed four types of leak-off in the formation during the closure, normal
leak-off, pressure-dependent leak-off, transverse storage leak-off or height recessions, and
tip extension leak-off. It is taken from a plot of GdP/dG versus G. A straight line is
drawn from the origin to the tangent to the curve. The closure pressure is deﬁned as the
pressure when the GdP/dG starts to deviate downward. The pressure at that point is
taken as the tangent method stress estimate. The tangent method was developed to match
numerical simulations by Barree and Mukherjee (1996) [ 23] and Barree et al. (2009) [ 26].


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numerical simulations by Barree and Mukherjee (1996) [ 23] and Barree et al. (2009) [ 26].
Their work introduced a G-function signature for each type of leak-off. Each type of leak-
off can characterize speciﬁc reservoir properties such as the presence of natural fractures,
the presence of lower permeability streaks or weak barriers surrounding the pay zone.
The tangent method in many cases shows a monotonic increase which makes the closure
pressure too low to be identiﬁed. The tangent method is affected by the formation response
after the fracture closure. It is normally associated with the square root of time and log–log
analysis, as recommended by Baree et al. (2009) [ 26]. The pre-closure ﬂow regime and
post-closure ﬂow regimes should be identiﬁed to support the closure detected using the
tangent method. The second derivative of pressure to the square root of time should show
a local peak at the closure. The relations between the log–log, square root of time, and


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a local peak at the closure. The relations between the log–log, square root of time, and
tangent method were discussed in detail by Baree et al. (2009) [ 26]. The log–log technique
and square root of time method can be combined as the “Holistic Fracture Diagnostics” as
described by Baree et al. (2007) [45].
The compliance method for estimating stress was developed from mathematical
solutions of the fracture closure process (McClure et al. (2016) [ 42]). As per the compliance
method, the very early part of the shut-in is ignored as the pressure drops very rapidly.
This part of the transient is ignored because it is caused by the dissipation of near-wellbore
tortuosity related to the initiation of a transverse hydraulic fracture from a horizontal
well. The straight section of the pressure versus G-time plot can be extrapolated back to
the y-intercept to estimate the pressure in the far-ﬁeld fracture at shut-in (the so-called


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the y-intercept to estimate the pressure in the far-ﬁeld fracture at shut-in (the so-called
effective initial shut-in pressure). After the initial period, the pressure curve settles into
an approximately straight line. The ‘compliance method’ of estimating stress identiﬁes
the contact pressure at the point where the derivative increased from the minimum and
subtracts 75 psi to account for a stress shadow from the residual aperture (McClure et al.
(2019) [ 46]). The 75 psi adjustment is based on a numerical simulation that matches the
ﬁeld DFITs (McClure et al. (2019) [46]).
When the fracture walls contact, the stiffness increases, and the storage coefﬁcient
drops as per Sneddon (1946) [ 47]. This causes the dP/dG to increase when the fracture walls
come into contact. McClure et al. (2016 [ 42], 2019 [ 46]) note that, because of the roughness
of the fracture walls, the ﬂuid pressure may be higher than the minimum principal stress


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of the fracture walls, the ﬂuid pressure may be higher than the minimum principal stress
when the walls come into contact. Thus, it is recommended to estimate the closure pressure
as 75 psi lower than the contact pressure based on numerical simulations. The contact
pressure is estimated by plotting the magnitude of dP/dG and identifying the point where
this derivative reaches a minimum and begins to increase. This is the so-called ‘compliance’
method stress estimate. It is noticeable that the ‘compliance method’ is close to the classical
method by Nolte (1979) [ 19]. However, an upward deﬂection in dP/dG is not always seen
in the shut-in transients.
In a lot of cases, dP/dG continuously decreases during the shut-in. McClure et al.
(2019) [ 46] discussed the possible causes of the lack of an upward deﬂection. That may
be due to the excessive near-wellbore tortuosity as the fracture closes almost immediately


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be due to the excessive near-wellbore tortuosity as the fracture closes almost immediately
after shut-in. The monotonic dP/dG might arise from the continued fracture propagation
after shut-in. One of the key assumptions of G-function is that the fracture does not
propagate after shut-in. This makes the G-function not accurate in many cases. McClure


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Energies 2023 ,16, 764 6 of 29
et al. (2019) [ 46] recommended interpreting tests with a monotonic dP/dG as indicating
‘rapid closure’ in vertical wells where the near wellbore tortuosity is usually minimal and
supplementing dP/dG plots with plots of ‘relative stiffness’ versus pressure. The relative
stiffness is not a normalized value calculated to be proportional to the system stiffness.
In tests where near-wellbore tortuosity is possible, McClure et al. (2019 [ 46]; 2022 [ 48])
recommended declining to provide a stress estimate and labeling the stress uninterpretable.
McClure et al. (2022) [ 48] found that ﬁfty-nine percent of cases showed a clear upward
deﬂection; twenty-ﬁve percent of cases showed an adequate upward deﬂection; and sixteen
percent of cases showed no indication of an upward deﬂection (i.e., a monotonic decrease
in dP/dG).
From the previous discussions, closure detection is still open for research and needs


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in dP/dG).
From the previous discussions, closure detection is still open for research and needs
more studies to understand the complications of fracture closure without pre-assumptions
and fewer data requirements, which is the main advantage of wavelet transform techniques.
2. Methodology
2.1. Wavelet Transform
Wavelet transform is a mathematical transformation applied to signals to obtain
information that is not readily available. This signal can be pressure or any parameter
recorded during ﬁeld operations. In practice, most signals are in the time domain in their
raw format and plotted as a time–amplitude representation of the signal. As a result,
this is not always the best representation of the signal for most signal processing-related
applications; hence, the most valuable features are usually hidden in the signal frequency.
Several transformations may be used to analyze the acquired signals. Fourier trans-


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Several transformations may be used to analyze the acquired signals. Fourier trans-
forms (FTs) are the most widely used signal processing technique that gives signal frequency
information. However, it does not provide timely information on which those frequency
components exist. FT works best in the analysis of stationary signals whose time domain
is periodic; then, the time domain signal can be converted into a combination of sine and
cosine waves to inﬁnity duration (Rioul and Vetidsaerli (1991) [49]).
One modiﬁcation for the Fourier transform is the windowed Fourier transform (WFT).
It is a tool for extracting local-frequency information from the non-stationary signal. The
signal is divided into small stationary segments through window functions whose width
must equal the part of the signal where it is stationary. As discussed by Kaiser (1994) [ 50],
WFT is prone to inaccurate results due to the aliasing of high- and low-frequency compo-


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WFT is prone to inaccurate results due to the aliasing of high- and low-frequency compo-
nents that do not fall within the window function proposed by the technique. Wavelet
transform (WT) has a dynamical frequency spectrum and can represent the signals in
both the time and frequency domains. Figure 1 summarizes the difference between each
signal representation. As the wavelets analysis has a high dynamical frequency and a high
resolution in both frequency and time domains, it is preferred for signals with a wide range
of dominant frequencies and time–frequency localization independence.
Wavelet transform analysis uses small wavelike functions known as wavelets. Math-
ematically, it can be deﬁned as the convolution of the signal with the wavelet function.
Figure 2 shows a few of the most common wavelet types.
Wavelets can be manipulated through translation (i.e., moving the wavelets along
the time axis) and scale (i.e., stretching and squeezing the wavelet at a particular point in


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the time axis) and scale (i.e., stretching and squeezing the wavelet at a particular point in
time). The scale and translation of wavelets govern parameters (b) and (a), respectively, in
Equation (1).
T(a,b)=1√a∫∞
−∞x(t)ψ∗(t−b
a)
dt (1)
The Equation (1) comprised both the signal x(t), which could represent the pressure,
temperature, or any time series data, as well as the ( ψ∗(t−b)/a) mother wavelet function.
The wavelet transform works by superimposing the wavelet function on an arbitrary signal,
as illustrated in Figure 3. Time segments A and B, where both the wavelet and original
signal agree (positive or negative), will result in a large positive transformation value of


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Energies 2023 ,16, 764 7 of 29
the integral given by Equation (1). On the contrary, in regions C, D, and E, when the
wavelet and the signal have opposite signs, the overall transformation results in a negative
contribution to T(a,b).
Figure 1. Comparison between different signal representations.
Figure 2. Shapes of different wavelet transform families.
Figure 3. Mechanism of wavelet transform [51].


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When using complex wavelets, the resulting coefﬁcient consists of both real and
imaginary parts. Fourier transforms of complex wavelets are zero for negative frequencies.
Therefore, we can separate the phase and frequency components within the signal using
complex wavelets. For instance, taking Fourier to transform the Mexican hat wavelet
and then performing an inverse Fourier transform after neglecting the zero components
in the Fourier transform will create a complex wavelet. It produces a less oscillatory
transform than in the case of real wavelets because a complex wavelet only responds to
the non-negative frequencies of the particular real signal. Therefore, it outperforms real
wavelets in detecting and tracking instantaneous frequencies (Guerrero, A.P ., and Paredes,
G. E. (2018) [52]). Complex wavelet transform is used in our analysis, which is deﬁned by
Equation (2) where the ﬁrst part is the normalization factor, the second part is the complex


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Equation (2) where the ﬁrst part is the normalization factor, the second part is the complex
sinusoid, and the ﬁnal part is the Gaussian bell curve.
ψ(t)=1
π1/4ei2πf0te−t2/2(2)
To inspect the multi-scale dimensional characteristics of a signal using complex
wavelets, the Wavelet Transform Modulus (WTM) can be calculated using Equation (3).
T(a,b)=√
[Re(T(a,b))]+[Im(T(a,b))] (3)
One of the main features of the signals that can be observed and analyzed to identify
the system properties is the analysis of signal energy. For any signal, the total energy
contained in a signal, x(t), is deﬁned as its integrated squared magnitude as in Equation (4)
with the condition that the signal must contain ﬁnite energy.
E=∫∞
−∞|x(t)|2dt (4)
The relative contribution of the signal energy contained at a speciﬁc scale (a) and
location (b) is given by the two-dimensional wavelet energy density function:
E(a,b)=|T(a,b)|2(5)


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location (b) is given by the two-dimensional wavelet energy density function:
E(a,b)=|T(a,b)|2(5)
The plot E(a)versus time (different location parameters b values) at different dilation
parameter values (scale) (a) can be plotted in a plot called a scalogram. From the scalogram,
the location and scale of dominant energetic features within the signal can be detected from
the scale-dependent wavelet energy spectrum of signal E(a)at a speciﬁc scale. A plot of
E(a,b)as a function of time and frequency is known as a scalogram. Scalograms are usually
plotted with a logarithm and a scale axis. For example, Figure 4 shows a scalogram for The
Nino3 SST monthly climate time data series analyzed using Morlet wavelets (Torrence and
Compo (1998) [53]), as discussed by Addison, P .S. (2016) [51].
The features that are detected at small scales, the compressed wavelet, are the rapidly
changing details (ﬁne features) that cause high frequency in the signal while the features


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changing details (ﬁne features) that cause high frequency in the signal while the features
that are detected at large scales (stretched wavelet) are the slowly changing details (coarse
features) that cause high frequency in the signal.


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Figure 4. Example for the wavelet transform analysis of the monthly climate temperature (Nino3 SST)
time series data: ( a) The Niño3 SST time series data; and ( b) the local wavelet power spectrum using
the wavelet transform analysis of ( a) using the Morlet wavelet (Torrence and Compo (1998) [53]).
2.2. CWT Fracture Closure Detection Technique
In this study, a new innovative technique to ﬁnd a closure pressure using the CWT
technique is introduced. The main idea behind estimating the minimum stress using the
diagnostic fracture injection test is to create a small fracture at a constant rate and then
shut down the pumps and observe the pressure at which the fracture closes. The closure
pressure is the minimum average of the minimum stress for the area covered by the fracture
created by the DFIT. The pressure during the shut-in and the observation period represents
the closure process of the formation. The main goal is to detect and magnify the pressure


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the closure process of the formation. The main goal is to detect and magnify the pressure
changes that reﬂect the fracture’s closure event. The pressure signal during the shut-in
period after DFIT is analyzed using the CWT technique with a complex Morlet wavelet to
obtain the signal energy. The complete workﬂow for the CWT closure detection technique
is shown in Figure 5. The technique starts by getting the coefﬁcient of CWT using the
complex Morlet wavelets at multiple scales (from zero up to a scale of 256) and then getting
the wavelet transform modulus (WTM) using Equation (3) as a scalogram. The scalogram
of signal energy is plotted using a log scale. The scalogram can be averaged to the average
of the log signal energy of all scales. Fracture closure as a dominant feature can be identiﬁed
by the average of all log signal energies at several wavelet scales. The average of the log of
signal energy can be plotted with time. The plot can be used to detect the closure event as


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signal energy can be plotted with time. The plot can be used to detect the closure event as
seen in Figure 5.
Fracture closure can be identiﬁed with the two following features: the fracture closure
starts with a peak in the average of the log signal energy, which represents the fracture walls
coming into contact, then the drop in log signal energy level to a minimum stabilized level,
which represents reaching the equilibrium state. The characteristics of fracture closure
can be seen in Figure 6. When the real ﬁeld data are analyzed, high oscillatory signal
energy may be observed due to the noise of real ﬁeld data. That may be due to several
reasons, and the fracture faces are not smooth faces and the closure does not happen in
subsurface formation instantaneously, or there is noise in the pressure data, or due to that,
the equilibrium state is still disturbed with the DFIT.


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Figure 5. Fracture closure detection workﬂow using the CWT closure detection technique.
Figure 6. Fracture closure pressure using CWT closure detection technique.
2.3. Summary
The new CWT fracture closure detection technique can be conducted as the following
steps: The ﬁrst step is to apply the CWT using a complex Morlet wavelet for the pressure
signal starting from the start of DFIT pumping to obtain the CWT coefﬁcient at different
scales up to (a = 256) as per Equation (1) and then obtain the wavelet transform modulus
(WTM) as per Equation (4) from the complex continuous wavelet coefﬁcient. This can
be followed by the calculation of signal energy using Equation (2). Considering fracture
closure is a dominant feature so it can be detected by averaging the log of signal energy
values at each time point for different scales (up to scale (a) = 256). From the plotted signal


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values at each time point for different scales (up to scale (a) = 256). From the plotted signal
average log energy versus time, the start and the end of the fracture closure event can
be identiﬁed with the following characteristics. The start of the fracture closure which
represents the contact of two fracture faces can be recognized by a peak in average signal
log energy at the start of the fracture closure. The fracture closure can be identiﬁed by the


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Energies 2023 ,16, 764 11 of 29
drop in signal average log energy level to a minimum stabilized level. The start of the
minimum stabilized level characterizes the end of the fracture closure event.
3. Results and Discussion
3.1. Validation Using Fracture Simulation
The numerical simulations in this study were performed using the commercial pla-
nar 3D fracture simulator introduced by Barree (1983). The simulator integrates a ﬁnite
difference formulation for the ﬂuid ﬂow calculation within the fracture with an integral
equation for fracture width. Fluid pressure and fracture width solutions are sequentially
coupled based on the linear-elastic solution of the deformation for an inﬁnite half-space
with a concentrated load. This relies on Poiseuille’s equation along with the continuity
condition. It uses a pressure-dependent leak-off model for the effect of natural fractures.
This simulation is solved using ﬁnite differences with a regular grid. The shear-rate and


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This simulation is solved using ﬁnite differences with a regular grid. The shear-rate and
time-dependent laboratory ﬂuid test data are used to model the ﬂuid rheology. Proppant
transport is coupled with ﬂuid ﬂow and iteratively solved for proppant distribution in
fractures. The simulator is widely used by the oil and gas industry and its fracture geometry
is validated by downhole events in a stacked ﬂuvial pay system as per Nur. Azlinda et al.
(2010) [12].
To test the new CWT closure detection technique using fracture simulation, grids for
the synthetic model are created with ideal properties; the idealized reservoir model has
25 ft. of the sand body between two shale barriers with a sharp stress contrast and constant
values for reservoir characteristics and geo-mechanical parameters. High-stress contrast
between the pay zone and the surrounding shale zones is assumed. It is assumed that
the reservoir is tight (the formation permeability is 5 micro Darcy) and a simulation run


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the reservoir is tight (the formation permeability is 5 micro Darcy) and a simulation run
was conducted with a ﬂuid volume (10,000 gallons of KCL water) of diagnostic fracture
injection test with the typical pumping rate of 8 bbl/min followed by 700 min of leak-off.
This is a standard DFIT procedure that is followed by the oil/gas industry. The KCL
water is commonly used to prevent clay swelling and it is a non-wall building ﬂuid with
viscosity changes with temperatures. The concentration of KCL is dependent on the needed
hydrostatic for the DFIT. The pumping rate is selected to be 8 bbl/min as the maximum
of the range of the DFIT pumping rate (3–8 bbl/MIN). The geomechanical and reservoir
properties are shown in Figure 7. The perforation shot per foot is selected to be six shots per
foot. The DFIT pumping schedule is implemented in the fracture simulator as per Figure 8.
The simulation model showed fracture propagation followed by complete fracture closure


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The simulation model showed fracture propagation followed by complete fracture closure
at 153 min when the average width is equal to zero, as shown in Figure 9. The complete
closure pressure is 5075 psi as per the fracture simulation.


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Figure 7. The mechanical Earth model and reservoir properties for the fracture simulation model
used to test the CWT closure detection technique.
Figure 8. Synthetic model 1 pumping chart.
Figure 9. Average width proﬁle for synthetic model 1.


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The pressure from the fracture stimulation was analyzed with CWT to obtain the
average energy over wavelet scales (up to scale (a) = 256). The average log signal energy
was plotted against the time to detect the fracture closure in Figure 10. The energy signal in
Figure 10 showed a peak at a time of 149 min which happens at a bottom-hole pressure of
5095 psi. That pressure represents the start of the closure. The complete closure is when
the signal average energy drops to a stabilized level. As shown in Figure 10, the CWT
technique showed a closure pressure at 5075 psi.
Figure 10. Fracture closure pressure detection using the CWT technique for synthetic model 1.
To understand how the average log signal energy responds to fracture closure, the
change in the average width proﬁle created by the fracture simulator is plotted as a semi-
log scale against time to magnify the changes in width during the closure. That is to be


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log scale against time to magnify the changes in width during the closure. That is to be
compared with the average log signal energy to understand how the CWT closure detection
technique detected fracture closure. It can be noticed that the difference in the average
width proﬁle reached zero at the same time when the average log signal energy reached
the stabilized level, as seen in Figure 11.
The limitations of the conventional methods are the assumptions behind each method.
For example, the G-function, square root, time, and log–log are based on well-testing con-
cepts, built on a constant pumping rate and shut-in. Therefore, the variable-rate pumping
schedule does not ideally ﬁt the basic concepts of these methodologies. However, the
introduced CWT closure detection is a technique that does not rely on built-in assump-
tions. It depends on the wavelet transform acting as a global mathematical microscope


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tions. It depends on the wavelet transform acting as a global mathematical microscope
to obtain the physical response of the signal. The closure detection technique was tested
for a variable pumping rate case. A pumping schedule with a variable pumping rate was
simulated using the same ideal homogeneous geomechanical parameters used in Figure 7.
The pumping rate started at 4 bbl/min and then increased to 8 bbl/min and ﬁnally to
10 bbl/min, as shown in Figure 12. The simulation model showed fracture propagation
and then complete fracture closure at 184.5 min, where the average width is equal to zero
as shown in Figure 13.


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Figure 11. Comparison between the change in average width proﬁle and CWT closure detection
technique plot.
Figure 12. Synthetic model 2 pumping chart.


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Figure 13. Average width proﬁle for synthetic model 2.
The simulation for synthetic example 2 yielded the same closure pressure of 5075 psi
but at a time of 184.5 min. That is what was expected using as the pumping rate changed
from the synthetic model 1. The pressure from the fracture stimulation was analyzed
with the CWT closure detection technique by plotting the average of the log signal energy
against time. From Figure 14, it is clear that the signal energy showed a peak at 169 min,
representing the bottom hole pressure of 5095 psi, which represents the start of the closure
process. The complete closure pressure is at the moment when the signal average energy
drops to a stabilized level at a time of 184.5 min (closure pressure = 5075 psi). As shown
in Figure 14, the CWT analysis yielded a closure pressure at 5075, the fracture closure is
the same as in the ﬁrst synthetic model since the geomechanical parameters are the same.


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the same as in the ﬁrst synthetic model since the geomechanical parameters are the same.
However, the closure time changed due to a change in the pumping rate and ﬂuid volume.
Figure 14. Signal average log energy using CWT for synthetic model 2.
3.2. Validation Using Flow Regime Modeling
To further test the new technique with the complicated nature of the fracture closure, a
new approach was developed. This approach may be used to validate the CWT technique
using ﬂow regime modeling with a synthetic pressure decay signal that has no noise and
represents the characteristic ﬂow regimes that happen before and after fracture closure. The
ﬂow regime is identiﬁed by the rate of change in pressure with time. A set of ﬂow regimes
were assumed to obtain a synthetic pressure leak-off signal that has a rate of change in
pressure representing those ﬂow regimes, then the wavelet transform technique was tested


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to know how the CWT closure detection technique detects the closure. Various pre-closure
and post-closure ﬂow regimes can be identiﬁed on the log–log ∆t d∆p /d∆t versus ∆t
plot. Table 1 shows the ﬂow regimes that happen pre-closure and post-closure and their
corresponding slopes on the log–log ∆td∆p /d∆t versus ∆t. The basic concept can be
visualized as given in Figure 15. Table 1 the ﬂow regimes that happen pre-closure and
post-closure and their corresponding slopes on the log–log ∆t d∆p /d∆t versus ∆t.
Table 1. The ﬂow regimes that happen pre-closure and post-closure and their corresponding slopes
on the log–log ∆t d∆p /d∆t versus ∆t.
The Slope of the log–log ∆t d∆p/d∆t vs.∆t Flow Regime
Half slope (1/2) Pre-closure Linear ﬂow regime
Quarter slope (1/4) Pre-closure bilinear ﬂow regime
Negative half-slope ( −1/2) Post-closure linear ﬂow regime
Negative three-fourth slope ( −3/4) Post-closure bilinear ﬂow regime


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Negative half-slope ( −1/2) Post-closure linear ﬂow regime
Negative three-fourth slope ( −3/4) Post-closure bilinear ﬂow regime
Negative unit slope ( −1) Post-closure pseudo radial ﬂow regime
Figure 15. Transient ﬂow regimes before and after fracture closure.
Pre-closure and post-closure ﬂow regimes can be identiﬁed using the semi-log pressure
derivative on the log–log plot of ∆p vs.∆t during the shut-in period following the fracture
injection test. A pseudo linear ﬂow period is identiﬁed by parallel (1/2) slope lines on the
log–log ∆p∆t and∆t d∆p/d∆t∆t plot up until fracture closure as seen in Figure 16a,b.
Bilinear ﬂow can be identiﬁed by parallel (1/4) slope lines on log–log ∆pwf versus ∆t and
∆t d∆p/d∆t versus ∆t prior to fracture closure as seen in Figure 16c.
After closure, the pseudo linear formation ﬂow period is identiﬁed by a ( −1/2) slope
of the semi-log derivative of the pressure difference on the same plot, and pseudo radial


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of the semi-log derivative of the pressure difference on the same plot, and pseudo radial
ﬂow is identiﬁed by a ( −1) slope of the semi-log derivative on the log–log plot, as seen in
Figure 16a,b. It is difﬁcult to obtain all the ﬂow regimes in one real ﬁeld case. Thus, the
perfect fracture closure log–log plot can be synthetically created assuming ﬂow regime
slopes and then calculating the pressure decay signal as seen in Figure 17. The slopes that
were assumed are the slopes of the log–log plot of ∆t d∆p/d∆t vs.∆t for each period of
pressure decay signal are shown in Table 2.


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Energies 2023 ,16, 764 17 of 29
Figure 16. (a) Normal leak off log–log plot; ( b) pressure-dependent leak off log–log plot; and ( c) tip
extension log–log plot (Barree et al. (2009) [26]).
Table 2. The slopes of the log–log plot of ∆t d∆P/d∆t vs.∆t for the reverse-calculated pressure
decay signal.
Start, min. End, min.The Slopes of
the log–log Plot
of∆t d∆P/d∆t
vs.∆tPressure at
Start, psi.Pressure at
End, psi.
0 5 Pumping period Pumping period Pumping period
5 15 0.3 12,000 11,692
15 40 0.5 11,692 11,499
40 50 0.25 11,499 11,445
50 200 −0.5 11,445 11,216
200 360 −1 11,216 11,168


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Energies 2023 ,16, 764 18 of 29
The pressure decay signal and its corresponding ﬂow regimes are shown in Figure 17.
The synthetic pressure decay signal can be used to illustrate that the wavelet transform
detects the features within the signal itself and not only the noise associated with it. It can
also be used to investigate the response of each closure detection technique. The synthetic
pressure decay signal was analyzed using each technique as if it was real ﬁeld data.
Figure 17. Synthetic reverse-calculated pressure decay signal.
The application of the conventional technique, the tangent method (Barree et al.
(2009) [ 26]), showed that dP/dG monotonically decreases so it can be either rapid closure
or it is not valid to be analyzed using the compliance method (Mcclure et al. (2022) [ 48]).
However, the effective shut-in pressure is 11,850 psi which is delayed from the pump
shut-down time by a minute and a half. There is no rise in dP/dG. The tangent method


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shut-down time by a minute and a half. There is no rise in dP/dG. The tangent method
using GdP/dG detected the closure pressure at 11,445 psi which represents the end of the
linear pre-closure ﬂow regime. The dP/dG and GdP/dG are shown in Figure 18. The
pre-closure bilinear ﬂow regime is ignored by the tangent method. The application of the
CWT closure detection technique detected the closure at the same point of the change
from the bilinear ﬂow regime to the after-fracture-closure linear ﬂow regime. It accurately
detected the transition period between pre-closure and post-closure, as can be seen in
Figure 19. The fracture closure is usually identiﬁed at the end of the pre-closure linear ﬂow,
but this is not accurate as the pre-closure bilinear ﬂow can be detected in several real ﬁeld
cases. However, that bilinear ﬂow period is very short and cannot be observed in many real
ﬁeld cases. CWT closure detection technique detected the transition between pre-closure


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ﬁeld cases. CWT closure detection technique detected the transition between pre-closure
and post-closure with a clear peak. The change in ﬂow regime between the pre-closure
and post-closure periods can also be identiﬁed with smaller and smoother peaks. The
closure pressure detected using the CWT technique matches the most accurate deﬁnition of
complete fracture closure.
Figure 18. Tangent method (dP/dG and GdP/dG) for the synthetic reverse-calculated pressure
decay signal.
The large peak matches the closure process as seen in Figure 19. This is when the
fracture closes and the ﬂow starts to follow a formation linear ﬂow regime. This includes
the period of bilinear ﬂow where the formation and fracture ﬂow together and each of them


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Energies 2023 ,16, 764 19 of 29
shows a linear ﬂow regime. The multiple peaks that are shown in the reverse-calculated
pressure decay signal are not shown in fracture simulation synthetic data as the fracture
simulation did not include the post-closure reservoir ﬂow regimes. It only simulates
hydraulic fracture propagation and closure.
Figure 19. Closure pressure detection using the CWT technique for the synthetic reverse-calculated
pressure decay signal.
3.3. Application of CWT on Real Field Data
The main challenge for any technique is to be applied to real ﬁeld data. We compared
the closure pressure of a real ﬁeld DFIT case estimated using the CWT technique to the
conventional methods such as the tangent method, log–log, and square root of the time and
compliance methods. This was performed to determine how the CWT closure detection
technique stands up against other techniques used in the industry. Thus, two real ﬁeld


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technique stands up against other techniques used in the industry. Thus, two real ﬁeld
cases, wells X-01 and X-02 were selected for the analysis of the new CWT technique. The
well X-01 was perforated against a conventional oil sandstone reservoir. The well X-02 was
perforated against a deep high temperature (400 degrees Fahrenheit) granite reservoir as
a conventional example for geothermal energy reservoirs. The well X-01 was perforated
through a sandstone formation. One thousand gallons of water were pumped at a rate of
8 bbl/min through a 10 ft perforated interval as a typical DFIT pumping schedule. This
was selected as the well site by the operating company. The pressure fall-off was observed
for 110 min, as shown in Figure 20. The CWT closure detection technique showed that the
closure started at a pressure of 7400 psi (start of closure) and ended at 6600 psi (complete
closure). As shown in Figure 21, the start of closure can be distinguished by increased


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closure). As shown in Figure 21, the start of closure can be distinguished by increased
energy levels followed by a drop in the average of signal log energy. The complete closure
pressure is when the level of energy drops to a lower level. The oscillatory signal energy
with real ﬁeld data may be due to the complexity of fracture closure in addition to the
presence of noise in the pressure signal observed in real ﬁeld data.


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Energies 2023 ,16, 764 20 of 29
Figure 20. Pumping chart for well X-01.
Figure 21. Closure pressure detection using CWT closure detection technique for well X-01.
Application of classical conventional methods, tangent method (Barree et al. (2009) [ 26]),
showed that closure pressure is the same as what the CWT closure detection technique
detected. The closure pressure is 6600 psi as per the tangent method shown in Figure 22.
The log–log technique detected the closure at the same pressure (6600 psi). The closure
pressure is the point at which the fracture linear ﬂow ended (slope of 0.5 at ∆td∆p/d∆t
vs.∆t log–log plot) and the formation linear ﬂow started (slope of −0.5 at∆t d∆p/d∆t vs.
∆t log–log plot) as shown in Figure 23. The square root of the time method detected the
closure at the same point of 6600 psi, as seen in Figure 24. The compliance method (Mcclure
et al. (2022) [ 48]) using dP/dG vs. G-time plot did not show a clear signature for the closure.


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et al. (2022) [ 48]) using dP/dG vs. G-time plot did not show a clear signature for the closure.
There is no rise-up in dP/dG, as shown in Figure 22. However, there is a zero slope period
at a pressure of approximately 7700 psi which can be deﬁned as a closure. According to
McClure et al. (2019) [ 46], the DFIT is invalid due to the high near-wellbore tortuosity or the
fracture may appear to be closed earlier than it should. The closure pressure value picked
by the CWT technique matched the classical closure detection techniques. In addition to
that, CWT showed both the start and end of the fracture closure period. Table 3 lists the
closure pressure detected using each methodology for well X-01. The closure detected using
the CWT technique only used the pressure decline data without any information about the
pumping period or formation properties. In the well X-01 case, the CWT technique was
applied to analyze the classic DFIT in a conventional sandstone formation. However, it


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applied to analyze the classic DFIT in a conventional sandstone formation. However, it
was also tested on a very tight reservoir. The well X-02 was perforated through granite
formation, which is the typical geothermal formation type, and then pumped the DFIT as
per the pumping schedule shown in Figure 25.


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Energies 2023 ,16, 764 21 of 29
Figure 22. Tangentmethod (dP/dG and GdP/dG) for well X-01.
Figure 23. Closure detection using the log–log method for well X-01.


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Energies 2023 ,16, 764 22 of 29
Figure 24. Closure detection using the square root time method for well X-01.
Figure 25. Pumping schedule for well X02.
Table 3. Fracture closure pressure for well X-01.
Method Closure Pressure,psi
Tangent method (G-function) technique 6600 psi
Log–log technique 6600 psi
Square-root time technique 6600 psi
Compliance method Invalid DFIT
CWT fracture closure detection techniqueStart of fracture closure = 7400 psi, complete
fracture closure = 6600 psi
The application of the CWT closure detection technique showed oscillatory signal
energy. That may be due to a noise in the pressure signal recorded during the shut-in period
or the presence of natural fractures in granite formations. The presence of natural fractures
is conﬁrmed with the image log of the well as shown in Figure 26. Natural fractures may
cause noise and aggressive ﬂuctuations in the pressure decay signal. The CWT technique


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cause noise and aggressive ﬂuctuations in the pressure decay signal. The CWT technique
detected the start of fracture closure at the pressure of 5300 psi and complete fracture closure
to be 4700 psi, as shown in Figure 27. The application of the tangent method (Barree et al.
(2009) [ 26]) using GdP/dG showed fracture closure at the same complete fracture closure
that the CWT technique detected (4700 psi), as shown in Figure 28. This was conﬁrmed


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Energies 2023 ,16, 764 23 of 29
by the square root of time, as seen in Figure 29. However, the log–log plot showed the
fracture closure at 4523 psi, as seen in Figure 30. The closure pressure is detected at the
end of before the closure bilinear ﬂow regime. The application of the compliance method
(Mcclure et al. (2022) [ 48]) showed the contact pressure to be 5625 psi as the dP/dG starts
to rise up from the minimum at 5700 psi so the contact pressure is at 75 psi of that pressure,
as seen in Figure 28. The main advantage of the CWT technique is that it illustrates the
conﬂict between the tangent methods and the holistic fracture diagnostic technique and
captured the full image of the fracture closure process. Table 4 showed the comparison
between various techniques for well X-02.
Figure 26. Formation image log for well X-02 indicating the presence of natural fractures in the
granite reservoir.


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Energies 2023 ,16, 764 24 of 29
Figure 27. Closure identiﬁcation using a CWT closure detection technique for well X02.
Figure 28. Tangent method (dP/dG and GdP/dG) for well X-02.
With the application of the CWT closure detection technique on real data cases, fracture
closure can be detected at a lower maximum scale or higher maximum scale than that used
in the study (a = 256) based on the noise of the data and the time of the signal. The selected
maximum scale is iterative until fracture closure is identiﬁed. The signal energy scalogram
can be used as a guide for the maximum value for the maximum scale used. That is because
when the scale (wavelet scale (a)) increases more than required, it can miss the fracture
closure event. The value for the maximum scale (wavelet scale (a)) may depend on several
factors such as the length of the pressure decay signal, the complexity and the duration of
fracture closure, and the level of noise that is in the pressure decay signal.


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Energies 2023 ,16, 764 25 of 29
Figure 29. Closure identiﬁcation using the square root of time technique for well X-02.
Figure 30. Closure detection using the log–log analysis technique for well X-02 (closure identified when
the slope of changes from 1/4 (pre-closure bilinear flow) to −1/2 (post-closure formation linear)).
Table 4. Fracture closure pressure for well X-02.
Method Closure Pressure, psi
Tangent method (G-function) technique 4700 psi
Log–log technique 4523 psi
Square-root of time technique 4700 psi
Compliance method 5625 psi
CWT fracture closure detection techniqueStart of fracture closure = 5300 psi, complete
fracture closure = 4700 psi


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Energies 2023 ,16, 764 26 of 29
4. Conclusions
CWT can be used as a general mathematical microscope to analyze the output of
the physical systems (i.e., the diagnostic fracture injection test) to understand the process
of fracture closure. That can be used for multiple purposes in oilﬁeld operations and
geothermal reservoir operations. Time–frequency analysis for a pressure decay signal using
CWT reliably detects the closure pressure. The following conclusions can be concluded
from previous discussions:
1. The average of the log signal energy versus time calculated using CWT characterizes
the closure event as a local increase in the signal log energy followed by a drop in the
signal log energy to a minimum stabilized level.
2. The proposed technique has shown good agreement with synthetic fracture simula-
tor data.
3. The application of the CWT technique matched the classical closure detection tech-
niques in real ﬁeld cases with no pre-assumptions and less available data.


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niques in real ﬁeld cases with no pre-assumptions and less available data.
4. The pressure decay signal reﬂects the physical process of fracture closure and the
average signal energy using CWT is just a mathematical independent magniﬁcation
for the pressure decay.
5. The start of the fracture closure can be identiﬁed with the CWT closure detection
technique at the local peak of average signal energy. Fracture closure ends when the
average signal energy reaches the minimum stabilized level. Those characteristics
matched numerical simulation, ﬂow regime modeling, and real ﬁeld cases. The
technique is consistent, reliable, and can efﬁciently detect fracture closure.
6. The CWT closure detection technique can be used to analyze the real ﬁeld data and
can detect a reliable closure pressure compared to current methods.
7. The CWT closure detection technique can detect fracture closure regardless of the noise


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7. The CWT closure detection technique can detect fracture closure regardless of the noise
that may accompany the real ﬁeld pressure signal, it showed accurate fracture closure
in synthetic data when there was no noise included in the pressure decay signal.
8. The new technique has no built-in assumptions and works on the pressure decay
signal during the shut-in period.
9. Using CWT to detect the closure can be generalized to detect the features from oilﬁeld
signals such as rate, torque, and drilling vibrations. Another perspective of the CWT
closure detection technique is using the CWT wavelet transform to investigate the
natural fracture in the subsurface formations.
5. Patents
This study was ﬁled under U.S. Patent Application No. 63/412,269 on 30 Septem-
ber 2022, and entitled “SYSTEMS AND METHODS FOR MONITORING SUBSURFACE
EVENTS USING CONTINUOUS WAVELET TRANSFORMS.”
Author Contributions: Conceptualization, M.A.G. and I.E.; methodology, M.A.G.; software, I.E.;


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EVENTS USING CONTINUOUS WAVELET TRANSFORMS.”
Author Contributions: Conceptualization, M.A.G. and I.E.; methodology, M.A.G.; software, I.E.;
validation, I.E.; supervision, M.Y.S. and S.M.F.-A.; writing—original draft, M.A.G. and I.E.; writing—
review and editing, M.Y.S. and S.M.F.-A. All authors have read and agreed to the published version
of the manuscript.
Funding: This research received funding from HESS Corporation, Shell PLC, and Halliburton Energy
Service, members of the hydraulic fracturing consortium at the University of Houston.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data presented in this study are available on request from the
corresponding author.
Acknowledgments: We thank HESS Corporation, Shell PLC, and Halliburton Energy Service, mem-
bers of the hydraulic fracturing consortium at the University of Houston, for sponsoring this work.


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bers of the hydraulic fracturing consortium at the University of Houston, for sponsoring this work.
Conﬂicts of Interest: The authors declare no conﬂict of interest.


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Energies 2023 ,16, 764 27 of 29
Abbreviations
The following abbreviations are used in this manuscript:
∆t Shut in time
a Dilation parameter or scale
b Transportation parameter
CWT Continuous wavelet transform
DFIT Diagnostic fracture injection test
DWT Discrete wavelet transform
FT Fourier transform
G G-time function
P Pressure
t Time
T Wavelet transform modulus
WFT Windowed Fourier transform
WT Wavelet transform
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people or property resulting from any ideas, methods, instructions or products referred to in the content.


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Citation: Gabry, M.A.; Eltaleb, I.;
Soliman, M.Y.; Farouq-Ali, S.M.
Validation of Estimating Stress from
Fracture Injection Tests Using
Continuous Wavelet Transform with
Experimental Data. Energies 2023 ,16,
2807. https://doi.org/10.3390/
en16062807
Academic Editor: Rafael J. Bergillos
Received: 10 February 2023
Revised: 6 March 2023
Accepted: 8 March 2023
Published: 17 March 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
energies
Article
V alidation of Estimating Stress from Fracture Injection T ests
Using Continuous Wavelet T ransform with Experimental Data†
Mohamed Adel Gabry *
 , Ibrahim Eltaleb
 , Mohamed Y. Soliman
 , S. M. Farouq-Ali
Department of Petroleum Engineering, The University of Houston, Houston, TX 77023, USA
*Correspondence: magabry@cougarnet.uh.edu


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, S. M. Farouq-Ali
Department of Petroleum Engineering, The University of Houston, Houston, TX 77023, USA
*Correspondence: magabry@cougarnet.uh.edu
† In Proceedings of the Society of Petroleum Engineers Hydraulic Fracture Technology Conference,
The Woodlands, TX, USA, 31 January–2 February 2023.
Abstract: The article discusses the new technique for fracture closure pressure detection using
continuous wavelet transform (CWT). The study focuses on calibrating the CWT technique and
comparing it with different techniques for closure detection. According to the article, traditional
methods for identifying the closure of hydraulic fracturing operations are based on assumptions
that can conﬂict with one another, resulting in greatly varying approximations of closure pressure
and duration. To address this issue, the article employs a set of diagnostic fracture injection tests
that utilize the Step-Rate Injection Method for Fracture In-Situ Properties tool (SIMFIP). By directly


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that utilize the Step-Rate Injection Method for Fracture In-Situ Properties tool (SIMFIP). By directly
observing wellbore deformation, the SIMFIP tool determines the minimum principal stress, while
strain gauges monitor the opening and closing of fractures during multiple tests. The publicly
accessible data are used to evaluate the accuracy of the new closure detection technique using CWT.
The ﬁndings indicate that the CWT method aligns with measurements of deformation and can identify
the impact of intricate closure events and pre-existing natural fractures. In conclusion, the article
suggests that the CWT technique shows great potential as an alternative to traditional approaches for
detecting closure pressure.
Keywords: strain measurements; fracture closure; geothermal reservoirs; hydraulic fracture; DFIT
1. Introduction
Using the continuous wavelet transform (CWT) as a fracture closure detection tech-


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1. Introduction
Using the continuous wavelet transform (CWT) as a fracture closure detection tech-
nique and its validation using fracture simulation and ﬂow regime modeling with the
discussion of its applications on real ﬁeld cases was introduced by Gabry et al. (2023) [ 1].
In this paper, the CWT fracture closure detection technique was validated using ﬁeld
experimental data. The fracture closure was identiﬁed by measuring the rock deformation
using strain gauges while observing the pressure decline during the shut-in period. The
pressure decline data during the shut-in period were analyzed using the various existing
techniques such as the log–log technique, square root of time technique, and tangent method
that were discussed in detail by Baree et al. (2009) [ 2]. The log–log technique and square
root of time method can be combined with G function technique as the “Holistic Fracture
Diagnostics” as described by Baree et al. (2007) [ 3]. The data are analyzed also using the


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Diagnostics” as described by Baree et al. (2007) [ 3]. The data are analyzed also using the
compliance method (McClure et al. (2016) [ 4]). The analyses were carried out while strain
gauge data provided an actual change in fracture width and finally closure. This would
provide a deep understanding of the accuracy and validity of the various fracture closure de-
tection techniques. Dutler et al. (2020) [ 5] used strain measurements to compare the various
fracture closure detection techniques and showed that the tangent method gave the lowest
minimum principal stress among the various conventional techniques. However, there
are some issues with Dutler et al. [ 5] work. The main issue is that the injection protocol
does not follow the main assumptions for G-function which assume a constant injection
rate for a fairly long period. The test’s injection period was analyzed and performed under
Energies 2023 ,16, 2807. https://doi.org/10.3390/en16062807 https://www.mdpi.com/journal/energies


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Energies 2023 ,16, 2807 2 of 21
constant pressure instead. The methodology of the closure pressure detection from the
measured strain data by Dutler et al. (2020) [ 5] did not follow the x-intercept rule intro-
duced by Gulrajani and Nolte (2001) [ 6]. The EGS Collab project presented an occasion to
validate reservoir modeling, while concurrently analyzing geophysical and other fracture
characterization data, with the overarching objective of comprehending the fundamental
correlation among stress, seismicity, and permeability enhancement. Strain gauges were
utilized in experiments to determine rock deformation and measure wellbore displacement
during fracture propagation and closure. Observing the three-dimensional displacements
of a hydraulic fracture during water injections at various depths in the crystalline rock of
the Sanford Underground Research Facility (USA) was undertaken in order to characterize


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the Sanford Underground Research Facility (USA) was undertaken in order to characterize
the stimulation mode of a fracture. EGS Collab has undertaken a sequence of experiments
involving the injection of fractures to determine the lowest principal stress through direct
observation of rock deformations, using the SIMFIP tool (Step-Rate Injection Method for
Fracture In-Situ Properties) as described by Guglielmi et al. (2020) [ 7], and (2022) [ 8]. The
tool monitors the formation deformation as a fracture opens and closes during multiple
tests. Guglielmi et al. (2022) [ 8] compared the measurements for four extended shut-in tests
performed in two different wellbores at the Sanford Underground Research Facility as part
of the EGS Collab project against several fracture closure detection techniques. The tool is
used by Eltaleb (2023) [9] to validate the new Pump-In Flowback Model.
2. Material and Methods
2.1. Cwt Fracture Closure Detection Technique


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used by Eltaleb (2023) [9] to validate the new Pump-In Flowback Model.
2. Material and Methods
2.1. Cwt Fracture Closure Detection Technique
As discussed by Gabry et al. (2023) [ 1], the new fracture closure detection technique is
a way to ﬁnd fracture closure pressure using wavelet transform. The pressure signal during
the shut-in period after DFIT is analyzed using CWT with a complex Morlet wavelet.
The CWT analysis is applied to obtain signal average log energy at different scales. As
a dominant feature fracture closure can be identiﬁed from the average of all signal energies
at several wavelet scales. Figure 1 shows the workﬂow to detect fracture closure using the
CWT technique. Fracture closure is identiﬁed by the two following features; the fracture
closure starts with a peak in signal average energy that drops into a stabilized level. Still,
there is a ﬂuctuation in signal average log energy due to the noise of real ﬁeld data. Fracture


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there is a ﬂuctuation in signal average log energy due to the noise of real ﬁeld data. Fracture
faces are not smooth and the fracture closure does not happen instantaneously, it takes
up to several minutes to occur. The characteristics of fracture closure in the CWT fracture
closure detection technique can be seen in Figure 2.
Figure 1. CWT fracture closure detection technique workﬂow (Gabry et. al. (2023) [1]).


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Energies 2023 ,16, 2807 3 of 21
Figure 2. Closure pressure using CWT fracture closure detection technique(Gabry et al. (2023) [1]).
2.2. Validation Using Experimental Data from EGS Collab Project
The data published by the EGS Collab team were used to validate the newly introduced
CWT fracture closure detection technique against the SIMFIP tool. The SIMFIP tool is based
on a double-packer hydro fracturing probe customized with a 6-component displacement
Fiber Bragg grating sensor. The sensor is integrated with the injection interval sealed between
the two packers, as seen in Figure 3a full description of the procedure used in the testing was
described by Guglielmi (2013) [ 10]. The injection interval is 2.41 m long. The displacement
sensor is set in the center of the interval. It is a 0.24 m long and 0.1 m diameter pre-calibrated
aluminum cage connected to two-0.58 m long elements that allow clamping of both ends


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aluminum cage connected to two-0.58 m long elements that allow clamping of both ends
of the cage on the borehole wall (clamps correspond to points A and B in Figure 3). When
the tool is clamped, the cage is disconnected from the straddle packer system allowing it
to measure the relative borehole wall deformation/displacement between the two anchors
which are 1.4 m apart. When a fracture is propagating or closing, the SIMFIP sensor measures
the three-dimensional displacement of the wellbore. The sensitivity of the SIMFIP tool is at
the micron level. Guglielmi et al. (2022) [8] give further details of the tool.
The tests conducted by EGS Collab are short rate-controlled pumping periods followed
by a long shut-in period. The shut-in period was planned to be at least a few hours. The
SIMFIP measurements were recorded and synchronized during the testing cycle, the
injection is designed not to cause a fracture intersection and not to be drained into a


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injection is designed not to cause a fracture intersection and not to be drained into a
neighboring borehole. Guglielmi et al. (2022) [ 8] published an analysis of four injection
sequences that were conducted using this tool. The measurements were taken in 2018
and 2019 during hydrofracturing experiments in boreholes drilled from drifts at the mine
4890 ft and 4101 ft below ground surface (BGS), respectively.


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Energies 2023 ,16, 2807 4 of 21
Figure 3. SIMFIP probe. ( a) Design of the probe; ( b) Schematic concept of the borehole three-
dimensional relative displacement between anchors A and B; ( c) Example of a SIMFIP displacement
signal captured across the activated hydrofracture during E1-I 164 Test 2. The red segment is the
displacement during injection and growth of the fracture. The orange segment is the displacement
during shut-in. (Guglielmi et al. (2022) [8]).
2.3. Theoretical Basis of Using SIMFIP Tool
Gulrajani and Nolte (2001) [ 6] explained how to estimate the magnitude of the closure
pressure from deformation measurements; when the fracture is open but not propagating,
there is an approximately linear relationship between deformation and pressure. The
rock displacement is plotted versus pressure as per Figure 4. It is based on Equation (1)
(Sneddon (1946) [11]; Gulrajani and Nolte (2001) [6]).
W=P−Shmin
Sf(1)


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(Sneddon (1946) [11]; Gulrajani and Nolte (2001) [6]).
W=P−Shmin
Sf(1)
where Sfis the fracture stiffness, Wis the average fracture width, and Shmin is the minimum
principal stress. The extrapolation line of the ‘open fracture’ curve back to zero rock
displacement response indicates the minimum principal stress as shown in Figure 4. When
an inclined fracture that is aligned with the minimum horizontal stress ( Shmin) is reopened,
the normal stress acting on the fracture will exceed Shmin, potentially leading to irreversible
shear. This scenario becomes more likely if the fracture walls contain asperities that come
into contact during the closure, causing a non-linear increase in stiffness, as outlined by
Barton et al. (1985) [12]).


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Energies 2023 ,16, 2807 5 of 21
Figure 4. Schematic of hydraulic fracture reopening/closure. (Gulrajani and Nolte (2001) [6]).
The asperities contact at a critical fracture width is Wc. If the fracture were perfectly
smooth, then Wcwould equal zero, and the straight line in Figure 4 would extrapolate to
zero on the y-axis in Figure 4. In other words, the existence of nonzero width at fracture
closure Wcindicates the existence of the rough fracture face. A typical SIMFIP test consists
of several injection cycles. During the ﬁrst cycle, pressure increases gradually up to a
maximum value considered below the fracturing opening pressure. Then pressure is
stepped down to the initial value. This cycle is used for quality control and to characterize
the hydro-mechanical response of the injection chamber that integrates the formation
mechanical response, the effect of the mechanical coupling of both SIMFIP clamps, packers,


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mechanical response, the effect of the mechanical coupling of both SIMFIP clamps, packers,
and the probe stiffness. The main test cycles are then conducted. These tests are the
classical DFITs. The SIMFIP displacements data are affected by the injection pressure to
the orientation and activation mode of the created fracture. Therefore, several stimulation
cycles may be necessary to get a clear understanding of displacement orientation and of
the local borehole tortuosity effects. The borehole displacements are also affected by the
zone pressure. The displacements should be reversible when pressure is stepped back
to the initial to be used to detect fracture closure. The fracture closure should be at zero
fracture width.
2.4. Experiments’ Location
The injection tests were conducted at two boreholes, the E1-I and TV4100. The E1-I
borehole on the 1490 m (4891 ft) level was 69 m (226 ft) long and slightly dipping from


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borehole on the 1490 m (4891 ft) level was 69 m (226 ft) long and slightly dipping from
horizontal as shown in Figure 5. The E1-I test was the stimulation of a zone at a measured
depth of 164 ft., which is called the E1-I 164 Tests. TV4100 was approximately 1249 m
(4100 ft) long and vertical. The TV4100 tests were performed at measured depths between
10 m (32.8 ft) and 44.98 m (147.6 ft), and these are referred to by test numbers TV4100 Test
4 and TV4100 Test 7. The host rock for the tests was a metamorphic rock of the Sanford


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Energies 2023 ,16, 2807 6 of 21
Underground Research Facility (SURF) in South Dakota, USA (Kneafsey et al. (2019) [ 13],
(2020) [14]; Guglielmi et al. (2021) [7]; Guglielmi et al. (2022) [8]).
2.4.1. E1-I 164 Interval
In the E1-I 164 borehole which is located 4891 ft below the ground surface, the vertical
stress magnitude is estimated to be around 41.8 MPa (6062 psi) based on the density
of overlying rocks. The maximum horizontal stress, SHmax , is estimated to be 34 MPa
(4931 psi), while the minimum horizontal stress, Shmin, is 21.7 MPa (3147 psi) based on
Kneafsey et al. (2020) [ 14], Singh et al. (2019) [ 15], and Wang et al. (2017) [ 16]. The borehole
is drilled subparallel to the estimated Shmin. The type of host rock is a metamorphic rock
(Poorman Formation). The 3D view of the injection borehole E1-I and monitoring wellbore
E1-OT for E1-I 164 tests are shown in Figure 5.
Figure 5. (a) Numerical model setting with the calculated fracture at the end of the injection for E1-I


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Figure 5. (a) Numerical model setting with the calculated fracture at the end of the injection for E1-I
164 Test ( b) 3D view of the injection borehole E1-I and monitoring wellbore E1-OT (Guglielmi et al.
(2020) [7]).
The E1-I 164 tests consisted of three propagation and shut-in steps, each step extending
the fracture length until it intersected an observation borehole located 10 m (32.8 ft) away
from the injection well. Using basic modeling, a mixed mode of fracture growth best
matched the measurements in accordance with the normal stress regime (Guglielmi et al.
(2021) [ 17]; Schoenball et al. (2020) [ 18]; Fu et al. (2021) [ 19]). Table 1 shows the injection
protocol conducted by EGS Collab team during E1-I 164 tests as reported by Guglielmi et al.
(2022) [8]. The values of E1-I 164 Test 3 are calculated from the publicly published data.
Table 1. Injection duration, total injection volume, average injection rate, and shut-in duration for


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Table 1. Injection duration, total injection volume, average injection rate, and shut-in duration for
E1-I 164 tests (Guglielmi et al. (2022) [8]).
NameInjection Duration
(min:sec)Injection
Volume, LAvg. Injection
Rate, L/minShut-In
Duration, h
E1-I 164 Test 1 10:00 2.1 0.21 2
E1-I 164 Test 2 63:00 23 0.37 15
E1-I 164 Test 3 20:00 75 5 15
Due to the early loss of SIMFIP tool recording and the lack of available test calibration
information, it is not feasible to estimate stress using the SIMFIP measurements from the
E1-I 164 test 1 shut-in period. Additionally, the measurements failed to reach an asymptotic
minimum width.


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Energies 2023 ,16, 2807 7 of 21
During E1-I 164 Test 2, the zone pressure was initially maintained at 13.8 MPa (2001 psi)
for approximately 5 min, after which it was raised to 20.7 MPa (3002 psi) for around 1 min.
Subsequently, the protocol was changed from pressure-controlled to ﬂow rate-controlled,
and the ﬂow rate was progressively increased in increments until it reached a maximum of
0.4 L/min, where it was maintained for 57 min (Guglielmi et al., 2022) [ 8]). The pressure
increased to 26.8 MPa (3887 psi) and then decreased to a relatively constant value of
26.7 MPa (3872 psi) just before the pump was stopped, and the zone was left shut in for 15 h
as seen in Figure 6. The SIMFIP tool readings may be seen in Figure 7. The application of
the x-axis intercept conﬁrms the closure stress to be 21.4 MPa (3103 psi) as seen in Figure 7.
The displacement lines representing fracture reopening (injection) and fracture closure


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The displacement lines representing fracture reopening (injection) and fracture closure
(shut-in) in Figure 7 are almost identical. In E1-I 164 Test 3 is somewhat more complex
because the injection started with one pump at 0.1 L/min for 83 min before the pump was
changed to a larger one. The SIMFIP tool showed the closure to a negative displacement
value (−50 microns) as given in Figure 7. The test procedure may be the reason behind this
obviously wrong value.
Figure 6. E1−I 164 Test 2 injection protocol with SIMFIP displacement normal and tangential to the
activated fracture during E1-I 164 Test 2. (Guglielmi et al. (2022) [8]).
Figure 7. Rock displacement normal to the activated fracture versus zone pressure measured using
SIMFIP tool for the E1-I 164 tests (Guglielmi et al. (2022) [8]).
2.4.2. Tv4100 Interval
Eight SIMFIP tests in TV4100 intervals were conducted using the same SIMFIP tool
probe to monitor fracture displacement response at different depths along the 164 ft (50 m)


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Energies 2023 ,16, 2807 8 of 21
deep borehole to build a stress proﬁle and to characterize the stress variability along with
this proﬁle. The complete data sets (injection protocol with SIMFIP tool readings) for tests 4
and 7 and SIMFIP tool measurements were publicly published and discussed by Guglielmi
et al. (2022) [ 8]. Table 2 shows the Injection duration, total injection volume, average
injection rate, and shut-in duration for TV4100 tests.
Table 2. Injection duration, total injection volume, average injection rate, and shut-in duration for
TV4100 tests (Guglielmi et al. (2022) [8]).
NameInjection Duration
(min:sec)Injection
Volume (L)Avg. Injection
Rate (L/min)Shut-In
Duration (h)
TV4100 Test 4 10:29 20.5 1.08 7
TV4100 Test 7 12:21 45.2 2.15 16
Test 4 had a constant 1.2 L/min injection rate for 10 min and 29 s followed by a
7-h-long shut-in as shown in Figure 8. Guglielmi et al. (2022) [ 8] reported that the SIMFIP


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7-h-long shut-in as shown in Figure 8. Guglielmi et al. (2022) [ 8] reported that the SIMFIP
measurements indicate reactivation of a cemented natural fracture set between the two
SIMFIP anchors with the approximate orientation of 236 °/54°dip direction/dip angle.
SIMFIP tool showed a more complex displacement response as compared to previous
tests. Figure 8 shows high tangential displacement readings and a smooth normal closure
displacement during shut-in. It is noticeable that at 2 min and 21 s after injection, the
fracture opened with a limited normal displacement of about 5 microns. Using the SIMFIP
tool it appears that fracture closure in this test is affected by complex hydro-mechanical
effects as seen in Figure 9. The normal displacement drops to near zero during the shut-in
period at 18.6 MPa (2700 psi) and then climbs back up.
Figure 8. SIMFIP displacement normal and tangential to the activated fracture during TV4100 Test 4.


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Figure 8. SIMFIP displacement normal and tangential to the activated fracture during TV4100 Test 4.
Displacements are plotted with injection pressure and injection ﬂow rate (Guglielmi et al. (2022 [ 8]).
In TV4100 Test 7, a 12:21 min, 2.8 L/min injection test was run, followed by a 16-h
shut-in period as seen in Figure 10. There is a smooth variation of fracture opening and
shearing during the shut-in period. This may indicate that one or more natural fracture
segments were reactivated between the SIMFIP anchors. The fracture opens to a maximum
of 43 microns as seen in Figure 11.


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Energies 2023 ,16, 2807 9 of 21
Energies 2023 ,1, 0 9 of 22
Figure 9. Rock displacement normal to the activated fracture versus zone pressure measured using
SIMFIP tool for the TV4100 test 4 (Guglielmi et al. (2022) [8]).
Figure 10. Rock displacement normal and tangential to the activated fracture using SIMFIP tool
during TV4100 test 7. Rock displacements are plotted with injection pressure and injection ﬂow rate
(Guglielmi et al. (2022) [8]).
Figure 9. Rock displacement normal to the activated fracture versus zone pressure measured using
SIMFIP tool for the TV4100 test 4 (Guglielmi et al. (2022) [8]).
Figure 10. Rock displacement normal and tangential to the activated fracture using SIMFIP tool
during TV4100 test 7. Rock displacements are plotted with injection pressure and injection ﬂow rate
(Guglielmi et al. (2022) [8]).
Energies 2023 ,1, 0 9 of 22
Energies 2023 ,1, 0 9 of 22
Figure 9. Rock displacement normal to the activated fracture versus zone pressure measured using


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Energies 2023 ,1, 0 9 of 22
Figure 9. Rock displacement normal to the activated fracture versus zone pressure measured using
SIMFIP tool for the TV4100 test 4 (Guglielmi et al. (2022) [8]).
Figure 10. Rock displacement normal and tangential to the activated fracture using SIMFIP tool
during TV4100 test 7. Rock displacements are plotted with injection pressure and injection ﬂow rate
(Guglielmi et al. (2022) [8]).
Figure 9. Rock displacement normal to the activated fracture versus zone pressure measured using
SIMFIP tool for the TV4100 test 4 (Guglielmi et al. (2022) [8]).
Figure 10. Rock displacement normal and tangential to the activated fracture using SIMFIP tool
during TV4100 test 7. Rock displacements are plotted with injection pressure and injection ﬂow rate
(Guglielmi et al. (2022) [8]).
Figure 11. Rock displacement normal to the activated fracture versus zone pressure measured using
SIMFIP tool for the TV4100 test 7 (Guglielmi et al. (2022) [8]).


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SIMFIP tool for the TV4100 test 7 (Guglielmi et al. (2022) [8]).
Figure 11. Rock displacement normal to the activated fracture versus zone pressure measured using
SIMFIP tool for the TV4100 test 7 (Guglielmi et al. (2022) [8]).


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Energies 2023 ,16, 2807 10 of 21
Estimating the minimum stress from the x-intercept in test TV4100 is somewhat
challenging because the observed data do not usually yield a smooth, ideal straight line, as
is seen in the E1-I 164 tests. The EGS Collab team reported the presence of apparently ﬁve
cemented natural fractures in the TV4100 test 4 intervals and ten cemented natural fractures
in the TV4100 test 7 intervals. Some natural fractures contained a several-centimeter thick
in-ﬁlling cement and a centimeter-thick damage zone. Acoustic logs before and after both
tests were compared and the EGS collab team concluded that there were no new fractures.
The stronger acoustic contrast indicates that the fractures had been reactivated. EGS Collab
team conducted a pump-in ﬂow-back test on TV4100 interval as the second cycle for test 7.
Using the analysis technique introduced by Plahn et al. (1997), the closure stress may be


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Using the analysis technique introduced by Plahn et al. (1997), the closure stress may be
estimated from a plot of pressure versus cumulative ﬂow-back volume. A straight line is
drawn through the linear periods on the plot before and after closure, and the intercept
is taken as the closure as per Guglielmi et al. (2022) [ 8]. The closure estimate of around
18.6 MPa (2700 psi) as seen in Figure 12.
Figure 12. Pressure versus cumulative ﬂow-back volume for the TV4100 test 7-Cycle 2 (Pump-in
ﬂow-back test) (Guglielmi et al. (2022) [8]).
3. Results
The ﬁnal comparison between the compliance, tangent, log–log, and square root of
time with the SIMFIP tool reading is shown in Table 3. The new CWT fracture closure
detection technique showed the most accurate fracture closure pressure compared with the
physical measurement using the SIMFIP tool in the four tests published by Guglielmi et al.
(2022) [8].
Table 3. Comparison of the closure events period of the four EGS Collab tests.


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(2022) [8].
Table 3. Comparison of the closure events period of the four EGS Collab tests.
Fracture Closure Pressure
Detection TechniqueE1-I 164 Test 2 E1-I 164 Test 2 TV4100 Test 4 TV4100 Test 7
Using SIMFIP 21.37 MPa (3100 psi) 21.37 MPa (3100 psi)18.69 MPa
(2712 psi)–19.2 MPa
(2785 psi)18.6 MPa (2700 psi)
Using CWT fracture closure
detection TechniqueStart of closure =
21.59 MPa (3132 psi)Start of closure =
20.68 MPa (3000 psi)Start of closure =
18.89 MPa (2740 psi)Start of closure =
19.02 MPa (2760 psi)
Complete closure =
21.39 MPa (3103 psi)Complete closure =
20.34 MPa (2950 psi)Complete closure =
18.6 MPa (2700 psi)Complete closure =
18.6 MPa (2700 psi)
Using Compliance MethodRapid closure
20.68 MPa
(3000 psi)–24.13 MPa
(3500 psi)18.79 MPa (2725 psi) 17.06 MPa (2475 psi) 19.31 MPa (2800 psi)


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Energies 2023 ,16, 2807 11 of 21
Table 3. Cont.
Fracture Closure Pressure
Detection TechniqueE1-I 164 Test 2 E1-I 164 Test 2 TV4100 Test 4 TV4100 Test 7
Fracture closure pressure
using Tangent MethodLess than 15.17 MPa
(2200 psi)Less than 15.17 MPa
(2200 psi)Less than 15.17 MPa
(2200 psi)18.61 MPa (2700 psi)
Using Nolte Technique 23.44 MPa (3400 psi) No closure signature 20.06 MPa (2910 psi) 21.90 MPa (3177 psi)
Using log–log method No closure signature No closure signature No closure signature No closure signature
Using the square root of
time methodNo closure signature No closure signature No closure signature 18.6 MPa (2700 psi)
4. Discussion
4.1. Analysis of E1-I 164 Test 2
The application of the conventional G-function methods to E1-I 164 Test 2 is shown in
Figure 13. Following the initial Nolte (1979) [ 20] technique detected the fracture closure at
23.44 MPa (3400 psi) where GdP/dG deviated from linear. The GdP/dG curve continues in-


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23.44 MPa (3400 psi) where GdP/dG deviated from linear. The GdP/dG curve continues in-
creasing and does not reach a peak by the end of the test. The dP/dG curve showed a mono-
tonic decrease reaching a minimum pressure of 15.17 MPa (2200 psi). The interpretation
using compliance (McClure et al. (2019) [ 21]) that is discussed by Guglielmi et al. (2022) [8]
implies a wide range of stress estimates from 20.68 MPa (3000 psi) to 24.13 MPa (3500 psi).
In essence, there was no agreement between the three methods. Using the classical square
root of time given in Figure 14 did not show any closure event signature. The log–log
analysis Figure 15 showed a pre-closure linear ﬂow regime (slope of 1). The curve con-
tinued to increase at a half slope line till the end of the test indicating no closure. The
SIMFIP tool detected closure at 21.37 MPa (3100 psi). There was no agreement among the
conventional techniques, however, the compliance technique gave the closest value to that


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conventional techniques, however, the compliance technique gave the closest value to that
of the SIMFIP . These results indicate that conventional techniques may fail in analyzing
complex or non-ideal tests.
Figure 13. Tangent method (dP/dG & GdP/dG) and relative stiffness plots for E1-I 164 Test 2
(Guglielmi et al. (2022) [8]).


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Energies 2023 ,16, 2807 12 of 21
Figure 14. Square root of time analysis for E1-I 164 Test 2.
Figure 15. Log–log analysis for E1-I 164 Test 2.
Application of CWT fracture closure detection technique to the SIMFIP tool observed
data is given in Figure 16 as signal average log energy of pressure versus time. Local
peaks are clear at the pressure of 21.59 MPa (3132 psi) indicating initiation of closure and
again 3 min later at 21.4 MPa (3103 psi) with a stabilized energy level over a long period
indicating complete fracture closure. These results compare very well with the SIMFIP tool
measurements reported in Figure 7. These clearly show that CWT techniques work very
reliably regardless of the level of complexity of the test.


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Energies 2023 ,16, 2807 13 of 21
Figure 16. Closure pressure detection using CWT technique for E1-I 164 Test 2.
4.2. Analysis of E1-I 164 Test 3
E1-I 164 Test 3 is even more complex than the previous case as it started with a low
pumping rate for a long period of time (almost 82 min of injection rate of 0.1 L/min) then
the pumping rate increased to 5 L/min for 20 min, followed by 15 h of the shut-in period
as seen in Figure 17. The SIMFIP tool signature for E1-I 164 Test 3 was similar to E1-I 164
Test 2, however, the rock displacement line during shut-in dropped back to a large negative
displacement value of −50 microns at 19 MPa (2755 psi) as seen in Figure 7. The shut-in
line intersection with zero displacements at 21.37 MPa (3100 psi). It conﬁrmed the same
values of fracture closure pressure of 21.4 MPa (3100 psi).
Figure 17. Injection pressure and injection ﬂowrate for EI-164 test 3.


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Energies 2023 ,16, 2807 14 of 21
The application of conventional methods using the compliance method on this test
showed that the transient reaches a minimum dP/dG at a value yielding a contact pressure
of 19.30 MPa (2800 psi) and a stress estimate of 18.78 MPa (2725 psi) as given in Figure 18.
GdP/dG monotonically increases throughout the test implying that the fracture never
closed. This implies that the minimum stress is lower than the ﬁnal pressure value of
15.17 MPa (2200 psi). The compliance method stress estimate is 2.07–2.59 MPa (300–375 psi)
lower than the SIMFIP estimates, i.e., yields about a 10% error. The application of conven-
tional log–log and the square root of time did not show any fracture closure signature as
seen in Figure 19. The compliance method is the only fracture closure detection technique
that showed an almost reasonable stress estimate.
Figure 18. Tangent method plots (dP/dG & GdP/dG) for E1-I 164 test 3.


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that showed an almost reasonable stress estimate.
Figure 18. Tangent method plots (dP/dG & GdP/dG) for E1-I 164 test 3.
Figure 19. Square root of time analysis and log–log analysis for E1-I 164 Test 3.
The application of the CWT fracture closure detection technique is shown in Figure 20.
The signal average log energy showed a local peak at the pressure of 20.68 MPa (3000 psi)
which indicates the start of fracture closure. That peak is followed by a drop in the level
of signal average energy to a long and stable value which indicates complete closure at
the pressure of 20.34 MPa (2950 psi). This value agrees very well with the value of closure
pressure that is observed by the SIMFIP tool as per Figure 7. The conventional techniques
detected the closure 1.034 MPa (150 psi) below the SIMFIP tool measurement. However,
the closure pressure detected using the CWT technique matches the start of stabilization of


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the closure pressure detected using the CWT technique matches the start of stabilization of
the shut-in pressure line in negative displacement measurements. The difference between
closure pressure values detected by CWT and the one detected by the SIMFIP tool is less
than 5%. The time of closure event between the start of fracture closure at 20.68 MPa
(3000 psi) and complete closure at 20.34 MPa (2950 psi) is 20 min.


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Energies 2023 ,16, 2807 15 of 21
Figure 20. Closure pressure detection using CWT fracture closure detection technique for E1-I 164
test 3.
4.3. Analysis of TV4100 Test 4
The application of conventional methods using the compliance method on TV4100
interval test 4 showed that the transient reaches a minimum dP/dG and rises back up,
yielding a contact pressure of 17.58 MPa (2550 psi) and a stress estimate of 17.06 MPa
(2475 psi) as seen in Figure 21. The relative stiffness plot shows an upward deﬂection
at approximately 17.24 MPa (2500 psi) as seen in Figure 21. The GdP/dG curve is still
bending upwards at the end of the transient. Thus, the tangent method interpretation
indicates that the fracture never closed as GdP/dG monotonically increases. This would
also indicate that the stress should be lower than the ﬁnal pressure of 15.17 MPa (2200 psi).
The compliance method stress estimate is 1.6–2.14 MPa (237–310 psi) (approximately 10%


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The compliance method stress estimate is 1.6–2.14 MPa (237–310 psi) (approximately 10%
error) lower than the estimates from SIMFIP . The tangent method stress estimates are
even lower. However, by applying the initial technique by Nolte (1979) [ 20] the closure
pressure is detected at 2910 psi where the GdP/dG deviated from linear. The application
of conventional log–log and the square root of time did not show any fracture closure
signature as seen in Figure 22. As a summary of TV4100 test 4, the fracture closure pressure
is less than 15.17 MPa (2200 psi) using the tangent method such as the log–log analysis and
the square root of time analysis, it is 17.06 MPa (2475 psi) using the compliance method, and
20.06 MPa (2910 psi) using Nolte (1979) [ 20] technique. The closure pressure detected using
the SIMFIP tool ranges from 19.20 MPa (2785 psi) to 18.69 MPa (2712 psi). Figure 23 shows
the application of the continuous wavelet transform (CWT) technique to data of TV4100


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the application of the continuous wavelet transform (CWT) technique to data of TV4100
test 4. It shows the average log signal energy using CWT of pressure versus time during
the shut-in period. Several observations may be made from Figure 23. It shows a small
peak at pressure 18.89 MPa (2740 psi) and a decrease of the signal average wavelet energy
level starting from that peak till complete closure after 28 min at 18.61 MPa (2700 psi). This
value matches the SIMFIP tool readings.


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Energies 2023 ,16, 2807 16 of 21
Figure 21. Tangent method plots (dP/dG & GdP/dG) and relative stiffness plots for TV4100 test
4(Guglielmi et al. (2022) [8]).
Figure 22. The log–log analysis and square root of time analysis plots for TV4100 test 4.
Figure 23. Closure pressure detection using CWT for TV4100 interval test 4.


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Energies 2023 ,16, 2807 17 of 21
4.4. Analysis of TV4100 Test 7
The application of conventional methods on TV4100 interval test 7 indicates that
dP/dG plot may be considered as an ‘adequate’ indication of closure (McClure et al.,
(2022) [ 22]) as seen in Figure 24. There is an upward deﬂection in dP/dG, but it never
reaches a minimum. The upward deﬂection occurs at approximately 19.82 MPa (2875 psi),
suggesting a stress estimate of 19.31 MPa (2800 psi) as per Guglielmi et al. (2022) [ 8].
On the other hand, the relative stiffness plot, Figure 24, does not show a clear upward
inﬂection, the curve stayed ﬂat in spite of the increase in pressure. The tangent method
closure estimate is around 18.61 MPa (2700 psi). The application of the initial technique by
Nolte (1979) [ 20] detected the closure at 21.90 MPa (3177 psi) at the deviation of GdP/dG
from the linear. The application of square root of time showed a good signature for fracture


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from the linear. The application of square root of time showed a good signature for fracture
closure at the pressure of 18.61 MPa (2700 psi), however, conventional log–log does not
show the fracture closure behavior as seen in Figure 25. As a summary of TV4100 test 7, the
fracture closure is 18.61 MPa (2700 psi) using the tangent method and 19.31 MPa (2800 psi)
using the compliance method, and the square root of time analysis showed the closure at
18.61 MPa (2700 psi). All these values are close to the closure pressure detected using the
SIMFIP tool (18.61 MPa (2700 psi)).
Figure 24. Tangent method plots (dP/dG & GdP/dG) and relative stiffness plots for TV4100 test 7
(Guglielmi et al. (2022) [8]).
Figure 25. The log–log analysis and the square root of time analysis plots for TV4100 test 7.


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Energies 2023 ,16, 2807 18 of 21
Figure 26 illustrates the application of the continuous wavelet transform (CWT) tech-
nique to TV4100 test 7. It may be noticed that the signal average energy showed a small
peak at pressure 19.03 MPa (2760 psi) and may also be noticed that the drop of the signal
average energy level may be identiﬁed after 28 min at 18.61 MPa (2700 psi). This agreed
very well with the SIMFIP tool reading of closure pressure (18.61 MPa (2700 psi)). That
complex fracture closure observed by the SIMFIP tool matches the gradual decline of the
signal average energy as seen in Figure 26. It is interesting to observe that the presence of
natural fracture dampens the wavelet signal average log energy.
Figure 26. Closure pressure detection using CWT for TV4100 interval Test 7.
The number of natural fractures reported by The EGS Collab Team (Guglielmi et al.,
(2022) [ 8]) can be compared with the time of fracture closure based on the new CWT


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(2022) [ 8]) can be compared with the time of fracture closure based on the new CWT
fracture closure detection technique as per Table 4. Plotting these two parameters yields
a linear relationship as given in Figure 27. This indicates that the new CWT closure
technique may be used as a deep investigative tool to understand the complexity of the
fracture closure process. More deformation/strain measurement experiments are needed
to further investigate the relationship between the natural fracture intensity and types and
its response on the signal average energy plot using the CWT technique.
Table 4. Comparison of the closure events period of the four EGS Collab tests.
TestPressure at Start of
Closure Event,
MPa (psi)Pressure at Start of
Closure Event,
MPa (psi)Period of Closure
Event, minNumber of Reported
Natural Fractures
E1-I 164 Test 2 21.59 (3132) 21.39 (3103) 3 0
E1-I 164 Test 3 20.68 (3000) 20.34 (2950) 20 Not Reported
TV4100 Test 4 18.89 (2740) 18.61 (2700) 28 5


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E1-I 164 Test 2 21.59 (3132) 21.39 (3103) 3 0
E1-I 164 Test 3 20.68 (3000) 20.34 (2950) 20 Not Reported
TV4100 Test 4 18.89 (2740) 18.61 (2700) 28 5
TV4100 Test 7 19.03 (2760) 18.61 (2700) 42 10


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Energies 2023 ,16, 2807 19 of 21
Figure 27. The relationship between fracture closure periods vs. number of natural fractures reported
in EGS Collab experiments.
5. Conclusions
The new CWT closure detection technique showed an accurate match with the closure
pressure detected by the measurement of the rock displacement with the strain gauges. The
following conclusions can be concluded:
1. The signal average energy using the CWT closure detection technique is a reliable
mathematical independent magniﬁcation for the pressure decay. This has been con-
ﬁrmed using the SIMFIP tool.
2. The presence of natural fracture dampens the peak of the indicating the start of closure
as it makes the fracture closure a more gradual process causing the average signal
energy to drop in a more gradual manner.
3. A linear relationship between the closure time can be detected by the new CWT
technique and the number of natural fractures observed.
6. Patents


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3. A linear relationship between the closure time can be detected by the new CWT
technique and the number of natural fractures observed.
6. Patents
This study was ﬁled as U.S. Patent Application No. 63/412,269 on 30 September 2022,
and entitled “SYSTEMS AND METHODS FOR MONITORING SUBSURFACE EVENTS
USING CONTINUOUS WAVELET TRANSFORMS”.
Author Contributions: Conceptualization, M.A.G. and I.E.; methodology, M.A.G.; software, I.E.;
validation, I.E.; Supervision, M.Y.S. and S.M.F.-A.; Writing—original draft, M.A.G.; Writing—review
and editing, M.Y.S. and S.M.F.-A. All authors have read and agreed to the published version of
the manuscript.
Funding: This research received funding from HESS Corporation and Halliburton Energy Service,
members of the hydraulic fracturing consortium at the University of Houston.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data that support the ﬁndings of this study are openly available on


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Informed Consent Statement: Not applicable.
Data Availability Statement: The data that support the ﬁndings of this study are openly available on
1 May 2022 in (https://gdr.openei.org/submissions/1250) and (https://gdr.openei.org/submissions/
1229). Derived data supporting the ﬁndings of this study are available from the corresponding author
[M.A.G.] on request.
Acknowledgments: We thank HESS Corporation, and Halliburton Energy Service, members of the
hydraulic fracturing consortium at the University of Houston, for supporting this work. We thank
also The EGS Collab team for conducting that remarkable experiments and publishing the results for
public use.


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Energies 2023 ,16, 2807 20 of 21
Conﬂicts of Interest: The authors declare no conﬂict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
∆t Shut-in time
CWT Continuous wavelet transform
DFIT Diagnostic fracture injection test
G G-time function
P Pressure
SIMFIP Step-Rate Injection Method for Fracture In-Situ Properties.
t Time
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SPE 8341 
DETERMINATION OF FRACTURE PARAMETER~ 
FROM FRACTURING PRESSURE DECLINE 
by Kenneth G. Nolte, Member SPE-AIME, 
AMOCO Production Co. 
g. Copynght t979. Amencan Institute of Mming. Metallurgical. and Petroleum Engineers. Inc. SPE 
Society of Petroleun Engineers of AIM E 
Th1s paper was presented at the 54th Annual Fall Techn1ca/ Conference and Exhibition of the Society of Petroleum Engineers of A/ME. held in Las Vegas. Nevada. September 23·26. 1979. The matenal is sub,ect to correct1on by the 
author. Permiss1on to copy IS restncted to an abstract of not more than 300 words. Wnte 6200 N. Central Expy .. Dallas. Texas 75206. 
ABSTRACT 
This paper presents analyses which permit some of The basic assumptions for the applicability of the 
analyses are that the fracture: 
the parameters that quantify a fracture and the frac- 1. has essentially constant height. 
turing process to be estimated from the pressure decline


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the parameters that quantify a fracture and the frac- 1. has essentially constant height. 
turing process to be estimated from the pressure decline 
following fracturing. The primary assumptions are 2. propagates through a quasi-elastic formation with 
negligible slip of bedding planes. consistent with those of current practice--a vertical 
fracture of essentially constant height, propagating 
through a quasi-elastic formation, with continuous 3. was created by a constant injection rate of a 
power-law fluid into two symmetric wings. displacements (i.e., no slip) at the planes bounding 
the top and bottom of the fracture. The parameters 
which can be quantified from the pressure decline are 4. propagates continuously during pumping and prop­
agation stops when pumping stops. the fluid loss coefficient, the fracture length and 
width, fluid efficiency, and time for the fracture to 
close. 5. closes freely without significant interference 
from proppant. 
INTRODUCTION


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width, fluid efficiency, and time for the fracture to 
close. 5. closes freely without significant interference 
from proppant. 
INTRODUCTION 
At the present time there is no direct and simple 
procedure for evaluating the basic parameters achieved 
during a fracture treatment, such as the length, width 
and fluid efficiency at the time the treatment ends. 
There exist fracture models which permit estimation of 
fracture length and width based on assumed heights, 
fluid loss coefficient, fluid viscosity, and formation 
modulus. However, at the end of the treatment there is 
no way of knowing if the assumed parameters were cor­
rect, with the possible exception of fracture height 
near the wellbore. Techniques for defining the frac­
ture geometry from production performance do not sat­
isfy this need completely because of the time delay 
required for the data, and the inferred lengths and 
widths (fracture conductivity) are average values over


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required for the data, and the inferred lengths and 
widths (fracture conductivity) are average values over 
sections which significantly contribute to production 
and are at best lower bounds on the actual dimensions 
created by the fracturing fluid. As the volume and 
unit cost of fluids used for a typical fracture treat­
ment continue to increase, so does the need for better 
definition of the fracture geometry created by a par­
ticular treatment in a particular zone. 
SUMMARY OF DERIVATIONS 
Ba$ed on the concept of a vertical fracture as 
presented by Perkins and Kern2 and as extended by 
Nordgren,4 relationships based on the fracturing pres­
sure decline are derived in this paper. 
References and illustrations at end of paper. Since actual hydraulic fractures will deviate by 
varying degrees at various times from these idealized 
assumptions, the utility of the analyses presented for 
any application will depend on the degree of deviation


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assumptions, the utility of the analyses presented for 
any application will depend on the degree of deviation 
and the sensitivity to the deviation. The ultimate 
utility will depend on the ability to provide realistic 
engineering correlations and predictions. As discussed 
in the "Discussion of Applications" section, the most 
likely deviations from assumptions 1, 4, and 5 above, 
would produce estimates erring on the conservative side 
for the fluid loss coefficient and length. 
In the following, the derived relationships are 
summarized with equation numbers corresponding to those 
in later sections of the paper. The relationships are 
expressed in terms of the dimensionless shut-in time, 
o = ~t/t , where ~t is the time since pumping stopped 
(shut-in~ and t is the pump time prior to shut-in. . 0 . 
The rate of pressure decline at time o is 
4CH E' dP(8) p f(8) ~= TIH2S It 
s 0 (9) 
with f(o) defined by the bounds


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The rate of pressure decline at time o is 
4CH E' dP(8) p f(8) ~= TIH2S It 
s 0 (9) 
with f(o) defined by the bounds 
-1 -1/2 2 (11+8 -18) > f (8) > sin [ (1+8) ] (3) Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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2 DETERMINATION OF FRACTURE PARAMETERS FROM FRACTURING PRESSURE DECLINE S"'?E 8341 
shown in Fig. 1. The symbols in the above and follow­
ing equations are defined at the end of the paper. 8 
is deftned in Eq. 8 and shown graphically in Fig. 2. s 
The difference in the decline pressure between 
times o
0 and o is given by 
(15) 
and G(o,o
0) is defined in Eq. 16 and shown on Fig. 3. 
The above equation can be used with actual decline 
pressures in a curve matching technique (Applications 
Section) which defines a match pressure 
P* = (17) 
that determines the value of the variable group on the 
right-hand side for the given application, from which 
one of the variables can be defined in terms of the 
remaining variables. It is noted that these equations 
are independent of the fracture length. 
The volume of fluid lost to the formation during 
pumping is given in terms of the bounds 
nCR L ft > V > ~ CR L /t p o L 3 p o (22) 
The length of the fracture can be determined by 
L =


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pumping is given in terms of the bounds 
nCR L ft > V > ~ CR L /t p o L 3 p o (22) 
The length of the fracture can be determined by 
L = 
nCR /t (l+p) p 0 
with the variable 
or, 28 P/t = c~) P o P f u 7f"i3 dP/dt s 
~p P = G(o,oo) 28 -6-P~(o-,-o--) 
s 0 (25) 
(26) 
(27) 
in which P is the wellbore pressure, above closure 
pressure, at shut-in and dP/dt is evaluated at o. 
Figure 4 illustrates the pressure terms in the above 
expressions. 
The fluid efficiency during the treatment is given 
by the simple ratio 
eff = _2_ __ l+p 
with p defined above. 
Combining (1) and (7) gives 
Qt
0(1-eff) 
L = 
nCR /t p 0 (28) 
(31) for the fracture length in terms of the fluid efficiency 
and fluid loss coefficient. 
The average fracture width, over the height and 
length at the end of pumping is given by 
R 
w = TIC _E_ ft p R o (37) 
The dimensionless shut-in time for the fractur~ to 
close can be found by either of the expressions 
--1 wH ] (41) 0 = g [ 
2CH It p 0 
0 -1 [~] = g 2


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close can be found by either of the expressions 
--1 wH ] (41) 0 = g [ 
2CH It p 0 
0 -1 [~] = g 2 
where g-l is the inverse function for g(o) given in 
Fig. 1 and defined in Eq. 13. 
In the remaining sections of the paper, the above 
expressions are derived and discussed, with example 
applications given in a later section. 
DERIVATIONS 
Background: There are two fundamentally different 
concepts for the geometry of a constant-height vertical­
fracture which lead to very different results. One 
concept1 is that the fracture width is constant across 
the height of the fracture. This requires the assump­
tion that the formation bed being fractured is inde­
pendent of the beds above and below--that is, the beds 
can slip freely of one another at their boundaries. 
This assumption leads to the conclusion that the fluid 
pressure required to extend the fracture decreases with 
time. 
The other concept, utilized by Perkins and Kern,2 
assumes that there is no, or negligible, slip of bound­


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time. 
The other concept, utilized by Perkins and Kern,2 
assumes that there is no, or negligible, slip of bound­
aries along the horizontal planes which confine the 
fracture height. This assumption leads to the conclu­
sion that the fluid pressure required ~o extend the 
fracture increases with time. 
Since the two concepts fundamentally differ only 
in the assumption of slip along bedding planes, the 
applicable concept for a given fracture application 
depends on the potential for bedding plane slip. 
Bedding plane slip is more likely at shallower depths 
than at deeper depths due to the difference in over­
burden stress on the planes and the resulting friction 
which must be overcome for slip to occur. 
Measurements3 of bottom-hole pressures during 
massive-hydraulic-fracturing in three different areas 
of North America, at depths greater than 7900 ft indi­
cate that for these conditions the fracturing pressure 
increases with time in a manner consistent with the


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cate that for these conditions the fracturing pressure 
increases with time in a manner consistent with the 
concept presented by Perkins and Kern2 and as extended 
by Nordgren.4 As a result of these measurements indi­
cating the validity of the concepts presented by 
Perkins and Kern, the assumptions used in this paper 
will be consistent with this reference and the refine­
ments presented by Nordgren. Nordgren presents an 
excellent review and critique of these assumptions 
which will not be repeated in this paper. Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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K. G. NOLTE 
Fluid Loss Coefficient: The basic equation used in 
this paper is the continuity equation for flow down a 
fracture, and as presented by Nordgren4 is 
_ aQ(z,t) =A( ) + aA(z,t) (l) az z,t at. 
which states that the gradient of the flow rate is 
equal to the rate fluid is lost to the formation, per 
unit length, plus the time rate of fluid storage due to 
cross-sectional area change. The assumption for fluid 
loss, as critiqued in Ref. 4, that 
2CH p (2) 
will be used in the following. C is the fluid loss 
coefficient, H is the height over the fracture where 
fluid loss occRrs (e.g., sand section), and Tis the 
time the fracture was created at point z. Also the 
assumption will be used that negligible slip occurs 
across bedding planes and that the formation responds 
quasi-elastic. These assumptions yield2 
A = !. wH 4 7TH2 
2E' p (3) 
where H is the fractured height, w is the maximum 
fracture width at z, p is the pressure at z, above the


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A = !. wH 4 7TH2 
2E' p (3) 
where H is the fractured height, w is the maximum 
fracture width at z, p is the pressure at z, above the 
in-situ closure pressure of the formation, and E' is 
the effective plane-strain modulus across the fracture 
height. The term "effective" refers to a constant 
modulus which gives the same average width across the 
height as for an actual case in which the fracture may 
cross several beds with different in-situ pressures and 
elastic properties. 
Combining Eqs. 1, 2, and 3 yields 
-~ = 3 + !. H2 l£ 
az It-T 2 E' at (4) 
assuming that H and E' do not change significantly with 
time. 
Integrating (4) over the length of the fracture 
yields 
L 
-Q(L) + Q(O) 2CH f dz 
p It-T (z) 0 
7T H2 L 
J lE.d (5) + 2E' at z 
0 2CH L 7T H2 lE_ 0 ~f(t) + 2 ElL rt at (6) 
0 
where 
It L 
f (t) 0 J dz 
L lt-T(z) 0 (7) 
L 
1 J p dz s p p =-L s (8) 
0 
t in (6) is the injection time prior to shut-in and P 
iR (8) is the wellbore pressure. S is defined as the


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L 
1 J p dz s p p =-L s (8) 
0 
t in (6) is the injection time prior to shut-in and P 
iR (8) is the wellbore pressure. S is defined as the 
ratio of the average pressure to th~ wellbore pressure 
and is given by Eq. 13 of Appendix A and shown in 
Fig. 2. f(t) is the average over the length of the 
inverse-root of the fracture's age. Combining (6), 
(7), and (8) gives 
dP = 
dt CH E' 4 p f (t) 
7T H2S It s 0 (9) 
The pressure decline function, f, in (7) can be 
evaluated by an upper and lower bound. These bounds 
come from Ref. 4 for the time of fracture creation, 
T(z), and require the assumptions that the height, 
injection rate and the injected fluid are constant, and 
that the fracture propagates continously during pumping 
(i.e., no flow restrictions such as a screen-out). The 
lower bound on the rate of fracture extension is for 
the fluid-loss dominated case and assumes that the 
first term on the right-hand side of (1) dominates the


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the fluid-loss dominated case and assumes that the 
first term on the right-hand side of (1) dominates the 
second term to the extent that the second term, for 
fluid storage, can be neglected. As shown in Ref. 4, 
this assumption yields 
z(t) = L (t/t )1/2 
0 (10) 
or 
t(z) = t (z/1)2 
0 
where z is the length at time t. The upper bound on 
the rate of fracture extension is for the minimal fluid 
loss case and assumes the A term of .(1) can be neglect­
ted. For this case4 the equivalent exponent of time in 
(10) would be 4/5 for a Newtonian fluid. However, an 
equivalent result for a power-law fluid can be obtained. 
Integrating (5), with respect to time and neglecting A, 
yields Q~ to be proportional to pL; and (9) of Appendix 
A gives pL to be proportional to L raised to the 
l+l/(2n+2) power, with n being the power-law fluid 
for which C, H , H, and E' are assumed to be effective- exponent. Combining these two proportionalities yields 
ly constant do~ the fracture.


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for which C, H , H, and E' are assumed to be effective- exponent. Combining these two proportionalities yields 
ly constant do~ the fracture. 
Further assuming that the fracture is shut-in 
(i.e., Q(O) = 0) and free extension of the tip after 
shut-in has ceased (i.e., Q(L) = 0), and replacing the 
above integrals by their average values over the length 
(5) can be expressed as 2n+2 
z(t) = L (t/t )2n+3 
0 (11) 
Since the exponent in (11) is less than unity for 
any positive value of n, a more conservative upper 
bound than (11) is 3 Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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4 DETERMINATION OF FRACTURE PARAMETERS FROM FRACTURING PRESSURE DECLINE SPE 8341 
z (t) 
or 
-c (z) L t/t 
0 
t z/L 
0 (12) 
as shown in Figure 5. This figure also shows that (10) 
and (12) give, respectively, lower and upper bounds for 
the actual value of -c(z). These bounds can be used for 
-c(z) in (7) to bound the value of the integral which 
defines f(t). It can be shown that using (10) and 
(11), respectively to evaluate (7) gives 
2 (ll+~t/t -/trtTt) 0 0 (13) 
. -1 -1/2 sin [(1-~t/t) ] 
0 (14) 
where f1 results from (12) and f2 from (10). ~t/t is 
the dimensionless shut-in time in terms of the pum~ 
time, t , and the time since shut-in, D.t. The total 
time siRce pumping began is t = t + ~t. In the fol­
lowing ~t/t will be denoted simp~y as 8. The above 
bounds for ~ are shown in Fig. 1. The bounds are 
surprisingly close and differ by less than 10% for 
shut-in times greater than one-quarter the pumping 
time, i.e., D.t/t > 0.25. As a result, either of the


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shut-in times greater than one-quarter the pumping 
time, i.e., D.t/t > 0.25. As a result, either of the 
two bounds can bg used without compromising the accu­
racy of (9) due to the larger uncertainty in quantify­
ing the other parameters in (9). 
The pressure difference between two shut-in times 
can be found by integrating (9), using (13), between 
the two times, i.e., from 80 too. This can be shown 
to give 
for which 
0 = D.t/t 
0 
D.P(o
0,o) = P(o
0) -P(o) 
4 C(o,o0) = TI" [g(o) -g(o
0)] 
g(o) = i [(l+o)3/2- o3/2-1] 3 (15) 
(16) 
Graphs of G(o,o ) and g(o) are shown in Figs. 3 
and 1, respectively.0 Eq. (15), in terms of pressure 
differences, is more readily applied to field data than 
(9). In particular, (15) can be used in terms of curve 
matching to determine one of the four variables; C, H , 
E', or H, if the other three can be quantified from p 
other sources. This is undertaken by plotting D.P(o
0,o) from field data, for the 8 's corresponding to those in


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other sources. This is undertaken by plotting D.P(o
0,o) from field data, for the 8 's corresponding to those in 
Fig. 3, versus the actual ~hut-in time ~t. The field 
data is plotted on the same log-log scale as Fig. 3, 
and then Fig. 3 is overlain the field data with the 
vertical axis of Fig. 3, for 8 = 1, aligning with t , 
the pump time, of the field data. Then the val~e o~ ~p 
from the field data which corresponds to G(o,o
0)=1 of 
Fig. 3, is the match pressure and is equal to 
* p (17) 
The curve-matching procedure is demonstrated in 
the Applications Section. From this curve-matching 
procedure, the determined value of P* is equal to the 
variable group on the right-hand side of (17). If the 
value of H can be inferred from well logs, E' from the 
mechanicalpproperty tests on cores, and H from post­
fracture temperature logs or other means, this pro­
cedure would permit determination of the fluid-loss 
coefficient. 
c (18)


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fracture temperature logs or other means, this pro­
cedure would permit determination of the fluid-loss 
coefficient. 
c (18) 
It is important to note that this expression for deter~ 
mining the fluid-loss coefficient is independent of the 
fracture length or the constant injection rate while 
pumping. 
Before leaving this section, the variation of 
wellbore pressure with respect to time during pumping 
will be recorded for a power-law fluid. The relation­
ships for a Newtonian fluid were given in Ref. 4. 
Combining (11) of Appendix A with the bounds in (10) 
and (11) yields the following proportionalities for the 
exponent of time, 
1 
P(t) a t2(2n+2) high fluid loss (19) 
1 
P(t) a t2n+3 low fluid loss (20) 
for the bounds on the rate of pressure increase. 
Fracture Length and Fluid Efficiency: Approximations 
of the fracture length and fluid efficiency can be 
obtained from relationships derived in the previous 
sections. Integrating (5) with respect to time over


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obtained from relationships derived in the previous 
sections. Integrating (5) with respect to time over 
the time of injection (i.e., 0 tot) and using (8) 
yields 0 
Qt 
0 t z ( t) 
2CHP J J 
0 0 
2 + .!_ H L S p 2 E' p dz dt 
lt--c (z) 
(21) Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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., SPE 8J41 K. G. NOLTE 
In (21), the terms can be identified as the fluid 
~olume injected which is equal to the sum of the fluid 
~olume lost to the formation and the fluid volume 
occupying the fracture. Utilizing the bounds on frac­
ture growth rate and the corresponding bounds for T(z) 
in (10) and (12), the integral term in (1), whi.ch is 11Discussion of Applications11), then (25) would depend 
only on the fluid-loss coefficient, the ISIP and the 
rate of pressure decline at some time after shut-in. 
the volume of fluid lost, v1, can be shown to be bounded 
by The pressure decline ratio, pin (26), can be 
expressed also in terms of the decline pressure differ­
ence defined in (15). It can be shown that combining 
(9), (15), and (26) gives 
rrCH L /t > V > ~ CH L /t p o L 3 p o (22) 
The upper bound for V in (22), which results from 
(10), will be used in the rollowing in order to provide 
more conservative estimates on the fracture length and


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(10), will be used in the rollowing in order to provide 
more conservative estimates on the fracture length and 
fluid efficiency than would result from the lower 
bound. As a result (21) can be expressed as 
Qt 
0 (23) 
In (23) L represents the total length of both 
wings of the fracture since Q represents the total 
injection rate into both wings. Also, the subscript, 
p, on S indicates that the appropriate condition to 
evaluate S is for pumping, which is given by Eq. 13 of 
App. A and shown on Fig. 2. Solving for L from (23) 
yields 
L Q t 
0 (24) p (27) 
where P is the ISIP pressure above closure pressure and 
6P(8 ,8) is the difference in the decline pressure 
betwgen times 8 and o. The pressure terms in (26) and 
(27) are illust~ated in Fig. 4. 
The fluid efficiency, defined as the ratio of the 
volume of fluid (i.e., slurry) in the fracture at shut­
in to the volume pumped, can be obtained in a similar 
manner as (25) by the previous identification of the


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in to the volume pumped, can be obtained in a similar 
manner as (25) by the previous identification of the 
terms in (21). However, more directly, noting that the 
term in parentheses of the denominator of (25) relates 
to the total volume pumped and that the last term in 
the parentheses is the proportion left in the fracture 
(denoted as p) it follows that the efficiency is 
eff __ P_ = __ 1_-:-
1 + p 1 + 1/p 
Rearranging (28) yields 
eff 
P = 1 -eff (28) 
(29) 
The dependence of L on fracture height and forma- and 
tion modulus can be eliminated by using (9). Combining 
this equation with the last term in the denominator of 1 + p 1 (30) (24) yields 
L 
or 
L 
p 2CH f (8) S P 
rr[CH /t + p p ] 
p 0 
1T s It dP I d t s 0 
Q t 
0 
rrCH It (1 +p) p 0 
S P/t 
2f(8) ~ dP/~t 
s (25) 
(26) 
in which f(8) can be selected as (14) to be consistent 
with the bound used in (23), and is evaluated for some 
selected shut-in time, 8; dP/dt is the wellbore pres­


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with the bound used in (23), and is evaluated for some 
selected shut-in time, 8; dP/dt is the wellbore pres­
sure decline rate at 8; and P is the instantaneous 
shut-in pressure (ISIP) above the formation closure 
pressure. Assuming that the height over which fluid 
loss will occur, H , can be selected from well logs 
(e.g., net sectionpof sand) and the closure pressure 
can be determined (e.g., as discussed in the section 1 -eff 
Combining (25) and (30) yields 
Qt
0(1 -eff) 
L (31) 
rrCH It p 0 
for the fracture length in terms of the fluid effi­
ciency and loss coefficient. 
Fracture Width and Closure Time: From the expressions 
in previous sections, the average and maximum fracture 
widths, and the time for the fracture to close can be 
found. The volume of a fracture is 
(32) 
where w is the average width over the fracture length 
and height. Using the definition of S from Appendix A 
as the ratio of the average pressure over the fracture


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and height. Using the definition of S from Appendix A 
as the ratio of the average pressure over the fracture 
to the pressure at the wellbore (pressures referenced 
to the closure pressure) it follows that S also relates 
the ratio of average width to maximum width. In addi­
tion, the average width, w, over the height at a given 
fracture location is rr/4 times the maximum width.2 5 Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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6 DETERMINATION OF FRACTURE PARAMETERS FROM FRACTURING PRESSURE DECLINE SPE 8341 
As a result, the average width, w, is related to the 
maximum width at the wellbore, W, by 
7T ss W: shut-in w = 4 (33) 
7T 
sP W: pumping w = 4 (34) 
During the period of shut-in, w is essentially constant 
and (33) and (34) relate the change of W during this 
short period before to after shut-in. From the defini­
tion of fluid efficiency 
(35) 
Combining (32) and (35), and substituting into (25) 
gives 
w = TICH it (1 + p) (eff)/H (36) p 0 
and substituting (28) yields, at the end of pumping, 
w = 7TC1 It p 
0 
C' = CH /H p (37) 
in which C' is the average fluid loss coefficient with 
respect to the total fracture height, and p is the 
decline pressure ratio given in. (26) or (27). 
From (34), the maximum width at the wellbore, at 
the end of pumping, is 
4 w w = -;s 
p 
with w given by (37). (38) 
The time for a fracture to close after shut-in is 
given in Appendix B and is found by integrating (9)


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4 w w = -;s 
p 
with w given by (37). (38) 
The time for a fracture to close after shut-in is 
given in Appendix B and is found by integrating (9) 
from shut-in to the time to close using f1 (~t/t ) of 
(13). As a result, the following express~on is0deter­
mined in Appendix B, 
g (!J.t/t ) = ---'w-'----
0 2C' /t (39) 
0 
where the time to close, !J.t, is the value of g(!J.t/t ), 
defined in (16), which equals the right-hand side o~ 
(39). This expression assumes that the average width 
and average fluid loss coefficient can be estimated 
through the fracture design process. 
Another expression for closure time can be obtained 
from pressure decline data shortly after the fracture 
treatment which would be useful for estimating the 
additional time for the fracture to close. This expres­
sion is obtained by combining (37) and (39). 
g(!J.t/t ) = _:~!£_ 
0 2 (40) This expression depends only on the pressure decline 
ratio given in (26) or (27).


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g(!J.t/t ) = _:~!£_ 
0 2 (40) This expression depends only on the pressure decline 
ratio given in (26) or (27). 
Using symbolic notation, in terms of the inverse 
function of g, i.e., g-1, (39) and (40) can be rewritten 
for the dimensionless closure time as 
t:.t/t 
0 
t:.t/t 
0 
APPLICATIONS g-1 [ w ] 
2C'/t 
0 (41) 
In this section the expressions derived previously 
will be applied to fracturing pressure decline data for 
treatments of three wells. More specific information 
on the treatments is given in Table I. 
Case 1: This was a calibration treatment of 500 
bbls in the Muddy J sand of the Denver Basin. The 
calibration treatment was pumped without proppant to 
determine information on the tendency of fracture 
height to increase with respect to volume and pressure, 
and on fluid-loss behavior of the zone. The treatment 
consisted of polymer-emulsion fluid consisting of two­
thirds condensate and one-third water with 50 lbs of 
polymer per 1000 gals.


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consisted of polymer-emulsion fluid consisting of two­
thirds condensate and one-third water with 50 lbs of 
polymer per 1000 gals. 
Figure 6 shows the SP log from which a fluid-loss 
height of 32 feet over the height of the fracture was 
estimated, and the post-fracture temperature log from 
which a fracture height of 60 ft. was estimated. 
Figure 7 shows a "pump-in/flow-back" determination of 
closure pressure (discussed in next section) and the 
surface pressure decline, after shut-in. The left-hand 
side of Fig. 8 shows a pressure difference plot con­
structed from the decline data and compatible with 
Eq. 15 and Fig. 3. The right-hand side of Figure 8 
shows a "curve match" of the master curves, Fig. 3, 
with the field data from which a P~\' value of 350 psi 
was.obtained. The master curves are shown as dashed 
lines in Fig. 8. A discussion of the "curve match" 
interpretation will be given in a later section. Using 
the data from Table 1 and Eq. 18, the fluid-loss


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interpretation will be given in a later section. Using 
the data from Table 1 and Eq. 18, the fluid-loss 
coefficient is estimated to be 
c 
E'H /t p 0 .00053 ft/lmin 
From Fig. 7, the instantaneous shut-in pressure above 
closure pressure and the pressure decline difference 
between o=.S and 1.0 can be obtained as 800 and 210 
psi, respectively. From Fig. 3, the value of 
G(.S,l.)=O.S9, and from Fig. 2 the S /S ratio is 
obtained as .88 for estimated valuespofsn=.75 and 
a=l.O. Using the above data and Eq. 27, the pressure 
decline ratio is 
p 0.98 
From Eq. 28, the fluid efficiency is 
eff = 0.49 Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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0 SPE 8341 K. G. NOLTE 
From the data above and Table I, the fracture length 
(tip to tip) can be estimated from Eq. 25 as 
L = 2700 ft 
For Eq. 25, the injected volume, Qt , must be converted 
to ft.3 fo~ compatibility of units Rsed above and in 
Table I. Eq. 37 can be used to estimate the average 
width across the height and length from the above data, 
and yields 
w = 0.0094 ft 0.11 inch can be estimated. Forming a ratio of P<'~, Hz, and 
~ for the two cases gives 
0 
(25'300 =.; 250150 = •63 
H1 = .sox H2 = .80 x 150 120 ft 
for the fracture height at the time of the pad shut-in 
The maximum width can be found from w and Eq. 34 and is and indicates the fracture height changed very little 
W = 0.24 inch 
The dimensionless time for the fracture to close is 
given by Eq. 41 and for p=0.98 is 
-1 0 = g (1.54) = 1.6 
6t = ~ t = 1.6 X 100 
0 160 min. 
where Fig. 1 can be used to evaluate g-1• The actual 
closure time from Fig. 7 was 150 min. This is the only


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6t = ~ t = 1.6 X 100 
0 160 min. 
where Fig. 1 can be used to evaluate g-1• The actual 
closure time from Fig. 7 was 150 min. This is the only 
parameter of those determined above which can be check­
ed with data. 
Case 2: This treatment consisted of 700,000 lbs 
of 20/40 sand and 300,000 gals of a cross-linked, 
water-base fluid with 40 to 60 lb/1000 gals of polymer 
and a 5% hydrocarbon phase to reduce leakoff. The 
treatment was stopped after 30,000 gals for about 
80 min. The data used in the following calculations 
neglects this first part of the treatment. The zone 
treated was the same as for Case 1. A fluid-loss 
height of 70 ft and fracture height of 150 ft were 
estimated from the appropriate logs for the treatment. 
The surface pressure decline data is shown in Fig. 9 as 
is the pressure-difference "curve match". The master 
curves are shown as dashed lines. Using the same 
equations as in Case 1, the following parameters are 
estimated for this treatment. 
P* = 300 psi


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curves are shown as dashed lines. Using the same 
equations as in Case 1, the following parameters are 
estimated for this treatment. 
P* = 300 psi 
C = .00076 ft/lmin 
p = 2.4 
eff = 0.59 
L = 4600 ft 
w 0.43 in 
W = 0.86 in 
0 = 5.5; 6t = oto 
6t = 5.5 x 250 = 1380 min = 23 hrs 
The well was flowed back before closure and, therefore, 
the 23-hour closure estimate cannot be compared with 
actual data for this case. 
As stated above, this treatment was stopped during 
the pad, i.e., shut-in, after 30,000 gals and a pump 
time of 25 minutes. An analysis of the decline data 
(not given here) after the pad shut-in gave a P* value 
of 150 psi. Using Eq. 17, and assuming that all param­
eters, except the pump time, t , and the height are the 
same for the pad shut-in and tRe shut-in after the 
treatment, the height at the time of the pad shut-in during the major portion of the treatment. 
Case 3: the final example uses data on pressure 
decline reported by Novotny.5 The reported data is


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Case 3: the final example uses data on pressure 
decline reported by Novotny.5 The reported data is 
summarized in Table 1. This treatment was 400 bbl of a 
polymer-emulsion fluid. From the decline data in this 
reference, a rate of decline of dP/dt=4.5 psi/min is 
inferred at a shut-in time of 40 minutes (i.e., o=l for 
a 40-minute pump time). Equation 9 will be used to 
estimate the fluid-loss coefficient with a value of 
f(o=l)=0.78 from Fig. 1 and a B value from Fig. 2 of 
0.67 (assuming values of a=.75 ~nd n=.75). This equa­
tion gives 
c nH2B s rt:" dP 
4H Ef(o) dt = 0.00035 
p 
From the referenced decline data, an inferred value of 
instantaneous shut-in pressure above closure pressure 
of P=780 psi is obtained, and from Fig. 2 a B /Bs 
value of 0.88. Using the above data, the pregsure 
decline ratio is 
2B P/t 
P = f ( 0 ) n B P dP I~ t = 1. 8 7 
s 
The fluid efficiency, from Eq. 28 is 
eff _P_ = .65 
1 + p 
and estimating the length from Eq. 31 gives 
L Qt
0(1-eff)


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s 
The fluid efficiency, from Eq. 28 is 
eff _P_ = .65 
1 + p 
and estimating the length from Eq. 31 gives 
L Qt
0(1-eff) 
nCH It p 0 5600 ft 
The average width, over the height and length, and 
maximum width at the wellbore, obtained from Eqs. 37 
and 38, are 
w .0043 ft = 0.052 inch 
w 0.11 inch 7 Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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8 DETERMINATION OF FRACTURE PARAMETERS FROM FRACTURING PRESSURE DECLINE SPE 8341 
The time for the fracture to close is obtained from 
Eq. 36 and Fig. 1 as 
6t = at = 4.0 X 40 = 160 min. 
0 
The estimated closure time of 160 min. and fluid effi~ 
ciency are in reasonable agreement with that obtained 
from the referenced data. However, the above length is 
significantly greater and the above width is signifi­
cantly less than those esimated in the cited reference. 
These values can be brought in closer agreement by 
reducing the modulus for the referenced data from 107 
to 6 x 106 psi, (i.e., similar to the value for the 
previous cases). For the reduced modulus, the above 
parameters which would change and their resulting 
values are listed below 
c = .00055 ft/lmin 
L = 3400 ft -w = .087 in 
w = 0.18 in 
For the reduced modulus, the above value of W is in 
reasonable agreement to the referenced value of 0.21. 
However, the referenced value of L=852 ft. is still


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reasonable agreement to the referenced value of 0.21. 
However, the referenced value of L=852 ft. is still 
signi~icantly different than the above value. It is 
noted that the referenced values of fluid pumped, 
efficiency, and wellbore width are not consistent with 
the referenced value of height. 
DISCUSSION OF APPLICATIONS 
The "curve match" for Case 1, shown in Fig. 8, 
indicates that the data curve for 8=.25 is larger than 
the theoretical pressure difference curve for 8=.25. 
This indicates that the average rate of pressure 
decline between 8=.25 and subsequent times is greater 
than predicted by fluid loss to the formation alone. 
An expianation is that the fracture is still extending 
after time 8=.25. This explanation is supported by a 
numerical simulator for fractures (similar to that 
discussed in Ref. 4) which has been extended to include 
fracture closure as well as extension. The continued 
extension for some time after shut-in is contrary to


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fracture closure as well as extension. The continued 
extension for some time after shut-in is contrary to 
the assumption used to derive the expressions in this 
paper. However, the assumption of extension stopping 
at shut-in provides conservative estimates in that the 
determined fluid-loss coefficient will be greater and 
the length less than in reality. This follows because 
continued extension creates larger rates of pressure 
decline (proportional to estimated value of C) than for 
no extension, because the increased length reduces the 
average fracture width (for a fixed volume of fluid) 
and thus reduces the pressure. Also, the additional 
extension creates new fracture area (i.e., fluid-loss 
area) not accounted for by the analysis which assumes 
no extension after shut-in. 
Other possible deviations from the assumptions, 
which would lead to conservative estimates of the 
fluid-loss coefficient and length, are interference


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which would lead to conservative estimates of the 
fluid-loss coefficient and length, are interference 
from proppant and variable fracture height. If the 
fracture height is not essentially constant, it will 
tend to decrease away from the wellbore. This follows 
because the pressure decreases toward the tip as a 
result of the pressure gradient required to maintain 
fluid flow; and since an increase in fracture height 
most likely results from an increase in pressure, the 
fracture height will tend to decrease toward the tip. Therefore, the average fracture height will most likely 
be equal to or less than that determined at the well­
bore. The effect of proppant on fracture closure is to 
make the system effectively stiffer which is equivalent 
to increasing the formation modulus. Either a stiffer 
fracture system than would actually occur without 
proppant, or a smaller than estimated average-fracture­
height would increase the pressure decline rate and


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proppant, or a smaller than estimated average-fracture­
height would increase the pressure decline rate and 
lead to estimating a larger than actual fluid-loss 
coefficient. Since the effect of proppant is to 
increase the decline rate, an indication of the effect 
of proppant on the match curves would be for the data 
values to deviate toward larger values than indicated 
by the theoretical curves. 
The match curves in Fig. 8 also show that all the 
data curves, except 8=.25, tail off to lower values 
than the theoretical curves for times greater than 125 
minutes. This indicates that the rate of fluid loss 
after 125 minutes is less than predicted. An inter­
pretation of this observation is that for times greater 
than 125 minutes the fracture walls are closing to­
gether from the tip toward the wellbore, since there 
was no proppant in this case. The closing of the 
unpropped fracture reduces the area for fluid loss in 
the relatively high loss area of the tip and conse­


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unpropped fracture reduces the area for fluid loss in 
the relatively high loss area of the tip and conse­
quently reduces the net rate of loss and pressure 
decline. 
The match curves for Case 2 also indicate fracture 
extension after 8=.25, or 63 minutes, but none after 
o~0.5, or 125 minutes. The curves for this case do not 
tail-off to the same degree as for Case 1. 
For Cases 1 and 2, a fracture closure pressure is 
required to determine the net fracture pressure above 
closure, denoted by P. The closure pressures were 
determined for these cases by a procedure prior to the 
treatments which consists of creating a fracture with a 
relatively small amount of fluid and flowing the fluid 
back out of the fracture at a controlled rate through a 
choke. The decline curve for the flow-back stage 
(e.g., Fig. 7) shows a recognizable dogleg of an in­
crease rate of decline at the closure pressure due to 
the restriction to flow out of the fracture introduced


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crease rate of decline at the closure pressure due to 
the restriction to flow out of the fracture introduced 
as the fracture closes at the wellbore. 
SUMMARY OF APPLICATIONS 
1. The example cases given in this paper illustrate 
that the parameters which describe the geometry of 
a fracture can be obtained simply and directly 
from the pressure decline data after a treatment. 
2. Using the parameters determined from an actual or 
calibration treatment (e.g., Case 1) improved 
designs can be achieved for future treatment in a 
given pay zone. 
3. In particular, the"fluid-loss coefficient should 
be similar for all treatments of a given zone for 
similar fluids. As a result, using a "calibrated" 
fluid-loss coefficient and using decline data from 
subsequent treatments, the fracture height or the 
formation modulus, and the fracture length can be 
estimated for these treatments. 
4. The decline pressure ratio, p, obtained from 
decline data only, provides estimates of the fluid


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estimated for these treatments. 
4. The decline pressure ratio, p, obtained from 
decline data only, provides estimates of the fluid 
efficiency and fracture closure time. 
5. The only direct check on the accuracy of the 
estimates provided by this analysis is the time 
for the fracture to close. In the two cases where Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE 8341 K. G. NOLTE 9 
closure time data exis.ted, the predicted time Wq1;1. 
in good agreement with the data. 
NOMENCLATURE 
A = cross-sectional area of fracture 
C = fluid-loss coefficient 
E' = plane strain modulus 
= E/(l-v2) 
E =Young's Modulus 
eff = fluid efficiency 
f(o) =dimensionless pressure decline function, Eq. 7. 
G(o,o
0) = dimensionless pressure 
difference function, Eq. 16. 
g(o) = avg. decline rate function, Eq. 16. 
H = fracture height 
H = fluid-loss height p 
L = fracture length 
P = fracture pressure, above closure pressure, 
at wellbore and at shut-in 
P* =match pressure, Eq. 17. 
6P(o
0,o) = pressure decline difference 
dP/dt = pressure decline rate 
p = pressure at some point in fracture. 
p = average pressure over length of fracture. 
Q = pump rate. 
t = time variable. 
t
0 = pump time. 
6t = time since shut-in. 
w = maximum fracture width 
w = maximum fracture width 
in the fracture. 
-at 
at 
w = average width over height well bore. 
some point


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w = maximum fracture width 
w = maximum fracture width 
in the fracture. 
-at 
at 
w = average width over height well bore. 
some point 
and length. 
z = distance v~riable down fracture. 
a = denotes proportionality 
S = ratio of average and wellbore pressure. 
sp = value of s while pumping. 
Ss = value of S while shut-in. 
o = dimensionless shut~in time = 6t/t 
0 
8
0 = reference shut-in time for pressure 
differences. 
A = fluid loss per unit fracture length. 
v =Poisson's ratio. 
T = time of local fracture creation. 
!REFERENCES 
1. Khristianovic, S. A. and Zheltov, Y. P., "Forma­
tion of Vertical Fractures by Means of Highly 
Viscous Liquid," Proc. Fourth World Pet. Gong., 
·Rome (1955). 2. 
3. 
4. 
5. 
6. Perkins, T. K., and Kern, L. R., "Widths of 
Hydraulic Fractures," J. Pet. Tech. (Sept., 1961). 
Nolte, K. G., and Smith, M. B., "Interpretation of 
Fracturing Pressures," 54th Annual Tech. Con£. and 
Exhib., Las Vegas (Sept., 1979). 
Nordgren, R. P., "Propagation of a Vertical


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Fracturing Pressures," 54th Annual Tech. Con£. and 
Exhib., Las Vegas (Sept., 1979). 
Nordgren, R. P., "Propagation of a Vertical 
Hydraulic Fracture, ll Soc. Pet. Eng. Jour. (August, 
1972). 
Novotny, E. J., "Proppant Transport," 52nd Annual 
Tech. Conf. and Exhib., Denver (October, 1977). 
Dobkins, T. A., "Methods to Better Determine 
Hydraulic Fracture Height," 54th Annual Tech. 
Con£. and Exhib., Las Vegas (Sept., 1979). 
APPENDIX A 
Fluid Flow in a Vertical Fracture 
In the following, the power-law flow model will be 
a.ssumed to represent the flow behavior of a fracturing 
fluid over the range of conditions generally encount­
ered in fracturing applications. The parameters for 
this model, as used ·in the following, are 
Shear stress = k(shear rate)n 
For this assumption, the pressure gradient2 is 
~ 
dz n 
=2k~ 2n+l w 
m = (2n+l)/n (1) 
for flow of q (volumetric flow rate per unit height) 
between walls separated by the width w. The total flow


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~ 
dz n 
=2k~ 2n+l w 
m = (2n+l)/n (1) 
for flow of q (volumetric flow rate per unit height) 
between walls separated by the width w. The total flow 
rate, Q, down both wings of the fracture is twice the 
rate down either wing and is found by rearranging (1) 
and integrating over the cross section. 
H/2 1/n Jq 1 Cdp L) Q = 2 dy = -m dz 2k 
-H/2 
H/2 
X J wm dy 
-H/2 
or 
1/n 
Q = cL dp) Hifl cp (n) 
2k dz m (2) 
where 
1/2 m 
<j>(n) = J c~r d (Y) w H (3) 
-1/2 
m = (2n+l)/n Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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10. DETERMINATION OF FRACT'ORE PARAMETERS FROM FRACTURING PRESSURE DECLINE SPE 8341 
and W is the maximum cross-sectional width. In (2) the Also from (8), the pressure at the wellbore is 
assumptions were made that k and n are constant across 
the section and that dp/dz is also constant across the 
section, i.e., one-dimensional flow. Rearranging (2) 
yields the proportionality 
dp = Mk~t ~ 
dz HnW2n+l a w2n+l 
n 
M = 2 (<P (:)) m (2n+l) /n (4) 1 
p (11) 
Forming the ratio of p, from (9), to the pressure at 
the wellbore, P from (11), gives 
p/P = ___ 1 ___ = 2n+2 = 8 (e+l) 2n+bn+a+3 
p = 8P (12) 
for the pressure gradient down a constant height frac- Without a detailed discussion, it has been found from 
ture. If the fracture responds elastically to internal a numerical fracture simulator (similar to that 
pressure discussed in Ref. 4) that the appropriate approxima­
tions for flow rate in (7) or (12) are 
p sw (5) 
where Sis the fracture stiffness (i.e., S = E'/2H for


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tions for flow rate in (7) or (12) are 
p sw (5) 
where Sis the fracture stiffness (i.e., S = E'/2H for 
a constant-height fracture in a homogeneous, infinite, 
elastic medium), (4) and (5) yield 
p2n+l dp a kQn dz (6) 
for which the fracture stiffness is assumed to be 
constant. For the assumption of a homogeneous elastic 
medium, the fracture cross section is elliptical2 with 
a minor axis of W and a major axis of H. For this case 
a numerical integration for <P(n) and M defined in (3) 
and (4) is possible and is shown graphically in 
Fig. 10. 
Assuming k and Q have the exponential relation­
ships 
k k (2z/L) a 
0 (7) 
Q Q (2z/L)b 
0 
along the fracture, and integrating (6) from the tip 
(p=O, z=O) to some distance z toward the wellbore 
yields 
-(8) 
for which n was assumed effectively to be constant. 
The average value of p, along the length L of the 
fracture, from (8) is 
L 
p = f J p (z) dz 
0 
(e+l) 
e = (a+bn+l)/(2n+2) (9) 
(10) b = 0 shut-in 
ss 2n+2 
2n+3+a 
(13)


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fracture, from (8) is 
L 
p = f J p (z) dz 
0 
(e+l) 
e = (a+bn+l)/(2n+2) (9) 
(10) b = 0 shut-in 
ss 2n+2 
2n+3+a 
(13) 
b = 1 pumping 
sP 2n+2 
3n+3+a 
implying a nearly constant flow rate, greater than 
zero, down the fracture for shut-in even though the 
flow rate is zero at the wellbore and fracture tip; 
and a linearly varying flow rate while pumping of Q 
at the wellbore to essentially zero at the fracture0 
tip. From (7) the variation of k, or equivalently 
the viscosity, can be represented as 
a = 0 constant viscosity 
a = 1 medium degradation (14) 
a = 2 strong degradation 
where "a" reflects the "order of magnitude" decrease 
in fluid viscosity down the fracture due to tempera­
ture, shear and time degradation of fluid system. 
The ratio of 8 for pumping, 8 , to that for 
shut-in, 8 , is found from (13) asp s 
8 _E.. = 2n+3+a = 1 _ n 
8s 3n+3+a 3n+3+a (15) 
For the sake of completeness, the actual equality for 
(11) is 
1 
{ s2n+l k Qn 1} 2n+2 
p(L) M' o 0 (16)


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8s 3n+3+a 3n+3+a (15) 
For the sake of completeness, the actual equality for 
(11) is 
1 
{ s2n+l k Qn 1} 2n+2 
p(L) M' o 0 (16) 
(a+bn+l) Hn 
where 
1 
M' [ (2n+2)M] 2n+Z Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE 8341 K. G. NOLTE 
with M defined in (4). By using the relationship for 
M in terms of n in Fig. 10, it can be shown, surpris­
ingly, that M' in (16) is essentially independent of 
nand is equal to 2.53 ± 0.5% over the range of n=0.4 
to 1.0. In (16), Q is the total flow rate into both 
fracture wings, L i~ the length of a fracture wing, 
and Sis defined by (5). 
APPENDIX B 
Time for Fracture to Close 
The time for a fracture to close can be found 
from Eq. 9 of the text. Rewriting this equation· 
results in 
2H dP 8CHP f(t) 
E' dt = 
TIH S /t s 0 (1) 
and integrating over the shut-in time 6t (time for 
pressure to go to zero) gives 
2H p 
E' (2) 
where P is the pressure at shut-in. Noting that the left-hand side of (2) is the maximum width W at the 
wellbore,2 and rewriting yields 
0 
C' 6t/t 
0 f t CL.lt/t ) d6t = ci ws ) -1-­
o to s 2C'/t 
0 (3) 
The integral can be shown to be equal to g(6t/t ) 
given in (16) of the text, if (13) of the text ~s


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o to s 2C'/t 
0 (3) 
The integral can be shown to be equal to g(6t/t ) 
given in (16) of the text, if (13) of the text ~s 
used for f. C' is the average fluid-loss coefficient 
over the fracture height, and using (33), the term in 
parentheses on the right-hand side can b~ expressed 
in terms of the average fracture width, w, at shut­
in. Therefore, (3) can be rewritten as 
g(6t/t ) 
0 = __ w __ 
2C'/t 
0 (4) 
Table 1 
Data for Example Well Analysis 
Case 1 Case 2 Case 3 (Ref. 5) 
Treatment Vol. (bbl) 500 6,400 400 
Treatment Rate (BPM) 5 30 10 
Treatment Time (min.) 100 250 40 
Fluid Type Poly. -Em. Cross-Lk. Poly. -Em. 
Reservoir Depth (ft.) ?,900 8,000 10,000 
Temperature (°F) 265 265 200 
Permeability, (md) .01 .01 1.0 
Fracture Height (ft.) 60 150 60 
Fluid-Loss Height (ft.) 32 70 20 
Formation Modulus (106 psi) 5 6 10 
Time to Close (min.) 150 130


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Fracture Height (ft.) 60 150 60 
Fluid-Loss Height (ft.) 32 70 20 
Formation Modulus (106 psi) 5 6 10 
Time to Close (min.) 150 130 
Proppant No Yes Yes 11 Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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4 
3 
§: 2 = 
1 
0 0.25 0.5 0. 75 1.0 1.5 2.0 3.0 4.0 6.0 
1.5 
o = ~t/t0, Dimensionless Shut-in Time 
Fig. 1 -Dimensionless time function versus rlimensionless time. 
0 2 0 2 0 21.0 
/3p f3s 0.9 0.9 
(Pumping) (Shut-in) 
0.8 0.8 
co.. 
0 0.7 0.7 0.7 
G) 
f3r/f3s :I 'i > 0.6 0.6 0.6 
0.5 0.5 0.5 
0.4 0.4 
0 2 0 2 0 2 
Viscosity Exponent, a 
Fig. 2 -Variation of S with viscosity exponent, Eq. 7 -Appendix A. Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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-0 "0 .. "0 
(!} 
.. c 
0 ... 
(.J c 
::::J 1 LL. 
Q) .8 (.J c .6 Q) .... 
Q) = c .4 
Q) .... 
::::J en en .2 Q) .... 
Cl. 0.25 0.5 1.0 1.5 2.0 3.0 5.0 
0 -o-
0.25 
0.5 
0.75 
1.0 
0.25 0.5 1.0 1.5 2.0 3.0 5.0 
Dimensionless Shut-in Time, o -~t/t0 
Fig. 3 -Master curves for pressure differences. 
Pressure Decline Ratio, p, Eqs.26 & 27 
/ / 
/ 
/ 
/(Slope / 
p P/t0 P Proportional to --or -
dP/dt ~P 
-------T 
~P (8o, o) 
-~ / 1 / = Plt0 
z ~-~ Fracture _ _ _ _ _ :\ _ dP I Slope ~Cit Closure Pressure I 
~t 0 - = 8=0 0.5 • t0 ~~----L---~1----L---~J 
~t == 0 0.5t0 t0 I I I ~ 1.0 
0 t0 1 .5 t0 2t0 t, time 
1~•--Pumping------+--Shut-in ---
Fig. 4 -Illustration of pressure terms for decline analysis. Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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..J 
........ 
N 1.0 
/% / / /Upper bound, Eq.12 
/...,._, z/L = t/t0 / 
/ 't (z) 
~------~A~-------r Lower Upper' 
bound Actual bound 
log t/t0 1.0 
Fig. 5 -Bounds for rate of fracture extension and time 
of fracture creation. 
SP-Log 
Fluid Loss 7850 
..... -..c: ..... c. 
Q) c Post-Fracture 
Temperature Log 
7950 *See Ref.6 
Fig. 6 -Input data for analysis of case 1. Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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Pressure Decline After Fracturing-
1800 
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50 75 100 ~ 25 150 
Time Since Shut-In, min. 
Fig. 7 -Pressure decline for case 1. 
Pressure Differences for Data Match of Data and Master Curves 
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I I 1 
100 c I I I 50 ......_--'---!..----L---'-----'----....1 
50 75 100 150 200 50 75 100 150 200 50 
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Fig. 8 -Match of pressure differences for case 1. Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/79SPE/All-79SPE/SPE-8341-MS/2057509/spe-8341-ms.pdf/1 by University of Houston user on 27 October 2023


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Match of Data and Master Curves 
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Fig. 9 -Decline data and pressure difference match for case 2. 
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drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE_187038_(Craig)_Frac_Closure_Stress_Field_y_Lab_Experiments_Frac_Closure.pdf


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SPE-187038-MS
Fracture Closure Stress: Reexamining Field and Laboratory Experiments of
Fracture Closure Using Modern Interpretation Methodologies
D. P. Craig, DFITpro.com; R. D. Barree, Barree & Assocs. LLC; N. R. Warpinski, Retired and T. A. Blasingame,
Texas A&M University
Copyright 2017, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 9-11 October 2017.
This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents
of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect
any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written


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consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may
not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract
During the 1990s, field and laboratory experiments measured hydraulic fracture creation, propagation,
and closure, and the archived data represent the finest collection of measurements that can be used to
evaluate fracture models and fracture closure interpretation methodologies. None of the current fracture
closure interpretation methods, including G-function derivative analysis, log-log storage diagnostics, and
the changing-compliance method have been evaluated versus the field and laboratory measured data.
Recent papers have proposed fracture closure pressure interpretations that differ from established
methodologies, and under some circumstances, will result in a closure pressure that is higher than


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methodologies, and under some circumstances, will result in a closure pressure that is higher than
traditionally accepted. Thus, it seems an opportune time to reexamine the field and laboratory fracture
closure data using interpretation methodologies developed over the last twenty years.
Additional issues cloud closure pressure interpretations, including different definitions of fracture closure
used in numerous publications, like mechanical fracture closure, hydraulic fracture closure, progressive
fracture closure, and complete fracture closure. Evidence from downhole tiltmeters and finely-instrumented
laboratory experiments of fracture propagation and closure all demonstrate that residual width is retained
after closure. Consequently, closure is somewhat of a misnomer, and if a "closed" fracture remains open,
the relationship between what we interpret as fracture closure and the minimum horizontal stress needs to
be clearly defined based on measurements as opposed to simulation.


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be clearly defined based on measurements as opposed to simulation.
Based on field tiltmeter deformation and pressure measurements in hard rock formations, we find that
G-function derivative analysis and the log-log storage diagnostic plot interpretations together provide a
fracture closure pressure that is consistent with the minimum horizontal stress identified using tiltmeter-
measured rock deformation. Additionally, the closure pressure interpretations, and corresponding minimum
horizontal stress, are invariant over multiple injection/falloff sequences of varying volume and time.
Field experiments exhibiting variable-storage/changing-compliance signatures were also observed, and the
changing-compliance method interpretations of fracture closure pressure are inconsistent with tiltmeter-
measured rock deformation. Finally, we find the fracture re-opening pressure identified using tiltmeter


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measured rock deformation. Finally, we find the fracture re-opening pressure identified using tiltmeter
deformation and the fracture closure pressure interpreted using pressure falloff data are essentially equal.


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2 SPE-187038-MS
Based on laboratory measurements of fracture closure and pressure, we find that G-function derivative
analysis and the log-log storage diagnostic plot together provide a fracture closure interpretation generally
consistent with measured fracture closure, but despite attempts to define an objective closure identification
methodology, fracture closure signatures are often non-distinct and interpretations are subjective. In soft
rock reservoirs, like unconsolidated sand, the fracture closure pressure interpretation does not correspond
to the minimum horizontal stress, but in hard rock reservoirs, the fracture closure pressure identified using
G-function derivative analysis and log-log storage diagnostic interpretations are approximately equal to the
imposed minimum horizontal stress.
Introduction
Hydraulic fracture closure has been studied for decades by researchers inside and outside of the petroleum


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Introduction
Hydraulic fracture closure has been studied for decades by researchers inside and outside of the petroleum
industry. Even after decades of study, opinions vary widely with respect to the fracture closure process,
the identification of fracture closure pressure, and the relationship between hydraulic fracture closure
pressure and the minimum horizontal stress. Hydraulic fracture closure pressure is typically inferred from
pressure data during fracture initiation, re-opening, or falloff. While we recognize small-volume stress tests,
fracture step-rate tests, fracture-injection/flow back tests, and fracture pulse tests are all used to infer closure
pressure, we focus our study on fracture closure interpretations from the analysis of pressure and time data
during a fracture-injection/falloff test (DFIT).
Numerous models and methods exist for identifying fracture closure pressure from pressure versus time


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Numerous models and methods exist for identifying fracture closure pressure from pressure versus time
data recorded during injection/falloff sequences. In geophysics and rock engineering, Hayashi and Haimson
(1991) provided a summary of commonly-used methods outside of petroleum engineering, and the authors
described a fracture closure process based on observations of laboratory fracture closure experiments.
Hayashi and Haimson suggested a 3-stage fracture closure process during the shut-in period following a
fracture inducing or dilating injection. Stage 1 of the process begins at the end of fracture-tip extension after
shut-in and as aperture along the entire fracture half-length decreases. According to Hayashi and Haimson,
fracture closure begins at the fracture tip, which defines Stage 2 of the closure process, and continues as the
open width and fracture length decrease from the fracture tip back to the wellbore. Stage 3 of the process is


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open width and fracture length decrease from the fracture tip back to the wellbore. Stage 3 of the process is
defined by "complete" closure of the hydraulic fracture, where complete closure allows for the possibility of
residual fracture width. Hayashi and Haimson define the pressure observed at fracture-tip closure (Stage 2)
as the minimum horizontal stress. Although Hayashi and Haimson based their model on observed field and
laboratory closure data, they did not present fracture dimensions during closure to validate their proposed
fracture closure process.
Several derivative works have followed from the work of Hayashi and Haimson (1991), notably Raaen,
Skomedal, Kjørholt, Markstead, and Økland (2001) who recognized that a change in fracture compliance
should be observed as fracture closure begins at the fracture tip, and suggested a fracture-injection/flow
back test and interpretation methodology to identify closure pressure from a change in fracture compliance.


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back test and interpretation methodology to identify closure pressure from a change in fracture compliance.
McClure, Jung, Cramer, and Sharma (2015) extended the work of Raaen et al., and using simulations
of synthetic cases and interpretations of field data, described a changing fracture-compliance method for
identifying fracture closure from pressure falloff data in the absence of flow back. Recently, Zanganeh,
Clarkson, and Hawkes (2017) have suggested a fracture closure process and closure pressure identification
analogous to that of Hayashi and Haimson (1991). Zanganeh, Clarkson, and Hawkes also suggest that the
fracture closure pressure corresponding to the minimum horizontal stress is observed at the start of tip
closure, and base their conclusions on finite-element modeling of synthetic cases and interpretations of
field data.
Classical fracture closure pressure interpretation methods include plotting the shut-in pressure and


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field data.
Classical fracture closure pressure interpretation methods include plotting the shut-in pressure and
pressure derivative versus the square root of time, which have been described by Middlebrook, Aud,
Harkrider, and Hansen (1997). Perhaps the best known fracture closure pressure interpretation methodology


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SPE-187038-MS 3
is G-function analysis, which was initially presented by Nolte (1979), and expanded to include G-function
derivative analysis by Castillo (1987) and Barree and Mukherjee (1996). A "holistic" approach to fracture
closure interpretation that included a square-root of time plot, G-function derivative analysis, and a log-log
storage diagnostic plot was also presented by Barree, Barree, and Craig (2009). We assume for the purposes
of this research that the de facto standard for closure pressure interpretation from fracture-injection/falloff
data is G-function derivative analysis combined with the log-log storage diagnostic plot.
The G-function is defined as follows
(1)
where the dimensionless loss-volume function at the end of pumping is defined by Valkó and Economides
(1999) as
(2)
and αN is Nolte's power-model growth exponent (½ ≤ αN ≤ 1) and Γ is the Euler gamma function. The
dimensionless loss-volume function is defined at any time after the end of pumping and written as


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dimensionless loss-volume function is defined at any time after the end of pumping and written as
(3)
where F is the hypergeometric function, dimensionless time is defined as ΔtD = (t – te)/te, and te is the time
at the end of pumping.
Closure Interpretation and Leakoff Types
G-function derivative analysis requires a Cartesian plot of three curves: bottomhole pressure versus the G-
function; the derivative, dp w/dG, versus the G-function, and the derivative, Gdp w/dG, versus the G-function.
G-function derivative analysis can be used to identify hydraulic fracture closure and has been classically
used to identify several non-ideal leakoff types, including pressure-dependent leakoff, fracture-height
recession/transverse or variable storage, and fracture tip extension. Examples of the leakoff types commonly
observed have been provided by Barree, Barree, and Craig (2009) and Craig, Eberhard, and Barree (2000).


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observed have been provided by Barree, Barree, and Craig (2009) and Craig, Eberhard, and Barree (2000).
Fig. 1 contains graphs of the four common leakoff types and has been updated from that of Craig, Eberhard,
and Barree to include variable storage/changing compliance. A brief review of the classical leakoff types
is as follows.
Normal leakoff  behavior occurs when fracture area is constant during shut-in and leakoff is through
a homogeneous rock matrix. With G-function derivative analysis, normal leakoff is indicated by a
constant derivative and when the derivative Gdp w/dG lies on a straight line extending from the origin.
Fracture closure is identified when the Gdp w/dG data deviate downward from the straight line.
Pressure-dependent leakoff  from dilated fractures/fissures is indicated by a characteristic "hump" in
the Gdp w/dG derivative that lies above an extrapolated straight line from the origin through the normal


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the Gdp w/dG derivative that lies above an extrapolated straight line from the origin through the normal
leakoff data. The fissure opening pressure is identified at the end of the hump when the Gdp w/dG
derivative data begin to follow the extrapolated straight line. A period of normal leakoff behavior
is generally observed before fracture closure is identified when the Gdp w/dG derivative data deviate
downward from the extrapolated straight line.
Fracture height recession or transverse storage or variable storage or changing compliance  during
shut-in are indicated by G-function derivative analysis when the Gdp w/dG derivative data fall below
a straight line extrapolated through the normal leakoff data. Traditionally, hydraulic fracture closure


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4 SPE-187038-MS
is identified when the Gdp w/dG derivative data deviate downward from the straight line; however,
McClure, Jung, Cramer, and Sharma (2015) argue that fracture closure should be identified as the
pressure corresponding to the beginning of variable storage or changing compliance. Storage and
compliance are related to system stiffness by a reciprocal relationship. For example, Craig and
Blasingame (2005, 2006) and Craig (2014) define storage during fracture propagation and closure as
(4)
where C is the storage coefficient, cw is the wellbore-fluid compressibility, cf is fracture-fluid
compressibility, Vw is the volume of the wellbore, Vf is the volume of one wing of the fracture, Af is
the area of one wing of the fracture, and Sf is the fracture stiffness, which can be written in a general
form for 2D-analytical expressions of fracture stiffness as
(5)
where c is constant, E' is the plane-strain modulus, and Lc is a characteristic length. McClure, Jung,


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(5)
where c is constant, E' is the plane-strain modulus, and Lc is a characteristic length. McClure, Jung,
Cramer, and Sharma (2015) frame their discussion of closure around system stiffness, which is the
reciprocal of storage, that is, using the definitions of storage in Eq. 4, system stiffness is defined as
(6)
where Seff is a system stiffness. In another recent work, van den Hoek (2016) frames his discussion
of fracture closure in terms of total compliance, which using the definitions in Eq. 4, he defines as
(7)
where Ct is a system compliance and Cf is a time-dependent fracture compliance, which van den
Hoek defined as
(8)
where h is the fracture height, L is the fracture half length, and E(m) is complete elliptical integral. van
den Hoek (2016) provided an interesting modification in that he allowed a time-dependent fracture
compliance during the falloff. In other words, van den Hoek's solution, which was formulated based


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compliance during the falloff. In other words, van den Hoek's solution, which was formulated based
on laboratory experiments of fracture closure, explicitly allows fracture height and or length recession
before hydraulic fracture closure is observed.
It should be clear that discussions of variable storage, effective stiffness, and changing total
compliance all refer to the same phenomena during a fracture-injection/falloff sequence. In Fig.
1, the graph labeled "Fracture Height Recession/Transverse or Variable Storage" illustrates the
Gdp w/dG derivative response during the closure process with variable storage, compliance, or


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SPE-187038-MS 5
stiffness. Additionally, the figure illustrates the different interpretations of fracture closure based
on the changing fracture compliance method or the classical Gdp w/dG derivative interpretation—a
difference of about 500 psi between the two interpretations in the Fig. 1 example.
Fracture tip extension , which occurs when a fracture continues to grow after injection is stopped, is
indicated when the Gdp w/dG derivative data lie along a straight line that extrapolates above the origin.
Figure 1—Classical G-function derivative analysis qualitative interpretations.
As noted, Fig. 1 shows the traditional interpretation of leakoff types during a falloff, but other effects
can result in a non-linear Gdp w/dG derivative response. Barree and Miskimins (2016) summarized several
additional possibilities to explain Gdp w/dG derivative deviation from a ‘normal’ leakoff signature, and


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additional possibilities to explain Gdp w/dG derivative deviation from a ‘normal’ leakoff signature, and
in some cases, the root cause of the Gdp w/dG derivative deviation from normal leakoff may be either
unknown or ambiguous. Additionally, combinations of the above leakoff-types can be observed in some data
sets. Fracture-injection/falloff tests conducted at the toe of horizontal wells predominately exhibit variable
storage or changing compliance during the falloff, so the correct fracture closure interpretation is important
for determining a representative minimum horizontal stress.
The holistic analysis method (Barree, Barree, and Craig 2009) also requires a log-log plot of the derivative
td[p e – pw(t)]/dt plotted versus time, t, which we refer to here as the log-log storage diagnostic. Craig (2014)
has proved that the derivative td[p e – pw(t)]/dt is related to a constant-rate pressure transient solution as
follows.
(9)


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6 SPE-187038-MS
where pi is the initial reservoir pressure, pwsD is the slug-test pressure difference, and dimensionless pressure,
dimensionless time, and the dimensionless storage coefficient are defined as
(10)
(11)
(12)
The fracture closure process is a storage phenomenon, so storage consists of both wellbore and
fracture storage components. In general, an observed deviation of the Gdp w/dG derivative off of the
extrapolated straight line, for example, pressure-dependent leakoff, will result in an observed deviation of
the td[p e – pw(t)]/dt derivative on the log-log storage diagnostic plot, which signals a change in fracture
storage. Additionally, the deviation of the Gdp w/dG derivative identified as fracture closure coincides with
dissipation of storage distortion of the pressure falloff data identified by the log-log storage diagnostic in
soft and hard rock types.
The above discussion of the fracture closure process and fracture closure identification are all based


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soft and hard rock types.
The above discussion of the fracture closure process and fracture closure identification are all based
on analytical or numerical models, and although the models are based on fundamental physics and sound
principles, their validation is often based on interpretation, which can be ambiguous. Both laboratory
and field experiments of hydraulic fracture propagation and closure have been completed over the last
two decades; however, the results of fracture closure models have not been routinely tested against the
experimental observation.
Analysis and Interpretation of Data
We examined both field and laboratory data with the intent of validating fracture closure interpretation
models and methodologies with known or carefully measured results. The field data include virtually all
the time, rate, pressure, and deformation measurements from the M-Site project that has been summarized


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the time, rate, pressure, and deformation measurements from the M-Site project that has been summarized
by Peterson, et al (1996) and Warpinski, et al (1998). We recognize that we are not the first to analyze
all, or portions, of the M-Site field measurements. The original interpretations relied on fracture closure
identification methods that preceded the development of G-function derivative analysis and the log-log
storage diagnostics; consequently, the original interpretations were often inconsistent between data sets and
analysts. Subsequent analysis of the M-Site field data, for example, the study presented by Gulrajani, Nolte,
and Romero (2001), also noted inconsistent interpretations of the multiple data sets, which the authors
rationalized was caused by repeated injections of increasing volume connecting higher-stress reservoir rock
to the wellbore.
Our intent is to provide a consistent interpretation of all relevant time, rate, pressure, and deformation


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to the wellbore.
Our intent is to provide a consistent interpretation of all relevant time, rate, pressure, and deformation
data recorded at the M-Site. Additionally, our objective is to use the downhole tiltmeter fracture re-opening
and fracture closing deformation measurements to define the minimum horizontal stress. With the minimum
horizontal stress defined independently, we then compare fracture closure pressure interpretations from the
different methodologies to validate a closure model.
Many laboratory experiments have also been completed of hydraulic fracture creation, propagation, and
closure. We use publicly-available laboratory data, where the minimum horizontal stress is experimentally
imposed, to compare fracture closure pressure interpretations with the known solution to validate a closure
model. Additionally, some publicly available data includes measurements of fracture dimensions that were


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model. Additionally, some publicly available data includes measurements of fracture dimensions that were
recorded during closure, so the effect of changing storage (or changing compliance) can be evaluated with
respect to the fracture closure interpretations. Finally, we include a never-published laboratory experiment


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SPE-187038-MS 7
where the minimum horizontal stress is experimentally imposed, and a blind test is used to compare the
interpreted fracture closure pressure to the known minimum horizontal stress.
Field Experiments of Fracture Closure
Perhaps the most comprehensive, best-documented, and publicly available field data set of fracture closure
was recorded at the M-Site in the mid-1990s. While the scope of the M-Site project was much more
comprehensive than simply fracture closure determination, two separate hard rock, tight-gas sandstone
formations were tested during the field experiments, and the results provide insight into the fracture
closure process and fracture closure pressure determination. Experiments in both sands included multiple
injections of varying injection volume, injection rate, falloff time, flow back periods, time between
injections, and observations of hydraulic fracture closure. Many injections were monitored by downhole


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injections, and observations of hydraulic fracture closure. Many injections were monitored by downhole
tiltmeters cemented in place in an offset wellbore along with microseismic imaging of fracture propagation
from two arrays in adjacent wellbores. Fig. 2 shows a plan view of the surface location of the M-Site
wells. The Monitor Well shown in Fig. 2 held six tiltmeters and a microseismic array cemented in place
while a wireline-run microseismic array monitored events from MWX-3. Fig. 3 contains the location and
orientation of the tiltmeters in the Monitor Well along with the location of the triaxial seismic receivers
with respect to the B-Sand and C-Sand in the MWX-2 treatment well. Within Fig. 3, the measurement-
axis orientation is shown for each tiltmeter, and for all tiltmeters the y-axis is essentially normal to the
primary fracture orientation in the treatment well, which is shown in Fig. 2. With the tiltmeter y-axes


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primary fracture orientation in the treatment well, which is shown in Fig. 2. With the tiltmeter y-axes
essentially normal to the primary fracture plane, the tiltmeters have proven to be extremely sensitive
to the rock deformation resulting from small fracture-width changes in the treatment well. Additionally,
the deformation and tiltmeter response is virtually immediate with respect to rate or pressure changes in
the treatment well once the treating pressure exceeds the minimum horizontal stress. Additional M-Site
wellbore configuration, experimental measurements, and experimental procedures have been exhaustively
documented by numerous authors, including Warpinski et al (1995, 1996, 1997, 1998), Peterson, et al
(1996), and Branagan, et al (1996a, 1996b, 1997a, 1997b).
Figure 2—Plan view of M-Site, including locations of treatment well
(MWX-2) and Monitor Well (downhole tiltmeters). After Branagan, et al. 1996.


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8 SPE-187038-MS
Figure 3—Tiltmeter placement within Monitor Well, and tiltmeter axis orientation. After Branagan et al. (1996).
For the purposes of this paper, the pressure, rate, and time data recorded during the injection/falloff
sequences are interpreted using G-function derivative analysis and the log-log storage diagnostic plot
to infer hydraulic fracture closure, and where relevant tiltmeter measurements are available, closure-
stress interpretations are compared with tiltmeter-measured rock deformation. As noted, the downhole
tiltmeters are extremely sensitive to deformation created by an opening or closing fracture; thus, the tiltmeter
measurements provide the best direct indication of hydraulic fracture width changes in field experiments.
Injection/falloff sequences were completed in both the B-sand and the C-sand. In the B-sand, a total of
10 injection/falloff, or injection/falloff/flowback, sequences were recorded over a period of nine months


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10 injection/falloff, or injection/falloff/flowback, sequences were recorded over a period of nine months
with different fluid types, volumes, injection rates, falloff period, and flowback period. The first three
injection/falloff sequences were three small volume and low rate classical stress tests to establish an
estimated minimum horizontal stress. In the C-sand, a total of 8 injection/falloff, or injection/falloff/
flowback, sequences were recorded with different fluid types, volumes, injection rates, falloff period, and
flowback period. No small-volume stress tests were pumped in the C-sand in the MWX-2 wellbore.
Fracture re-opening experiments were pumped in both the B-Sand (Injection 4B) and the C-Sand
(Injection 4C). Prior injections in both sands had created and propagated a hydraulic fracture, so the re-
opening experiments were pumped at very low rates to observe the dilation of an existing fracture, which


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opening experiments were pumped at very low rates to observe the dilation of an existing fracture, which
would be indicated by the rock deformation measured by the tiltmeters. As previously noted, an existing
fracture should dilate (open) as the pressure within the fracture exceeds the minimum horizontal stress.
Conversely, an existing fracture should contract (close) during a falloff to a residual-width. In hard rock (E
> 1.8 × 106 psi [12.5 GPa]), the pressure observed at fracture closure (residual width) corresponds to the
minimum horizontal stress. Thus, in both the B-Sand ( E = 4.57 × 106 psi [31.5 GPa]) and C-Sand (E = 5.25


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SPE-187038-MS 9
× 106 psi [36.2 GPa]) fracture re-opening and fracture closure should occur at the same pressure, which is
equivalent to the minimum horizontal stress (within experimental error). In soft rock (E < 7.25 × 105 psi [5.0
GPa]), which includes unconsolidated sand formations, fracture closure to a residual width can be observed
at a pressure below the minimum horizontal stress (Dong and de Pater 2007).
B-Sand Stress Tests and Fracture Closure Experiments
B-Sand experiments at the M-Site began with three stress tests in November 1994. Fig. 4 shows graphs
for each of the three stress tests, which are labeled within each graph "Stress Test #1," "Stress Test #2,"
and "Stress Test #3." For each injection/falloff sequence, Fig. 4 shows graphs of bottomhole pressure and
injection rate versus time; bottomhole pressure, the derivative of bottomhole pressure with respect to the G-


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injection rate versus time; bottomhole pressure, the derivative of bottomhole pressure with respect to the G-
function, dp w/dG, and the product of the G-function and the derivative of bottomhole pressure with respect
to the G-function, Gdp w/dG, plotted versus the G-function during the shut-in period; and a log-log storage
diagnostic graph of the fracture-pressure difference, pe – pw(t), and the product of time and the derivative of
the fracture-pressure difference with respect to time, td[p e – pw(t)]/dt, plotted versus time since the beginning
of the injection.
Figure 4—MWX-2 stress tests recorded in November 1994.


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10 SPE-187038-MS
Stress Test #1 was a very small injection of 41-gal of water pumped at injection rates of less than 20
gal/min. For Stress Test #1 the 1994 interpretation of closure of (p c)1994 = 3,020 psi is shown along with
a new interpretation of the fissure-opening pressure, (p fis)2017 = 2,941 psi, and fracture closure pressure,
(pc)2017 = 2,886 psi using G-function derivative analysis. The log-log storage diagnostic graph shows that
the fracture closure time interpreted using G-function derivative analysis corresponds to the dissipation of
storage distortion of the pressure falloff data.
Stress Test #2 was a slightly larger injection of 47-gal of water pumped at injection rates of less than 20
gal/min. For Stress Test #2, the new interpretation includes a fissure opening pressure of ( pfis)2017 = 2,915
psi, and hydraulic fracture closure pressure of (p c)2017 = 2,893 psi.
Stress Test #3 was again a slightly larger injection of 51-gal of water pumped at injection rates of less


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Stress Test #3 was again a slightly larger injection of 51-gal of water pumped at injection rates of less
than 20-gal/min, but the test included a step-down in injection rates. G-function derivative analysis shows a
fissure-opening pressure, (p fis)2017 = 2,960 psi, but the final recorded pressure during the falloff was ( pw)final
= 2,938 psi, and G-function derivative analysis confirms that hydraulic fracture closure was not observed,
which is indicated by the increasing Gdp w/dG derivative through the end of the recorded pressure data. The
log-log storage diagnostic also demonstrates that the end of storage distortion was not observed during the
recorded falloff, which is also indicated by the increasing derivative, td[p e – p w(t)]/dt.
After a 19-bbl borate crosslinked-gel injection (Injection 1C) was completed, the fracture diagnostics
continued with Injection 2B, Injection 3B, and Injection 4B, and the fracture closure diagnostic graphs are


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continued with Injection 2B, Injection 3B, and Injection 4B, and the fracture closure diagnostic graphs are
shown in Fig. 5 with each injection/falloff sequence labeled "Injection 2B," "Injection 3B," and "Injection
4B," respectively. Injection 2B consisted of 27.1 bbl of 2% KCl-treated water injected in step rates up
to 3.3 bbl/min. There are several important features of Injection 2B to note. First, the step-rate injection
can theoretically be used to interpret fracture closure pressure from a graph of bottomhole pressure versus
injection rate; however, close examination of the bottomhole pressure and injection rate versus time graph
for Injection 2B clearly shows the pressure response was not smooth or constant at the lowest injection rate
of 0.5-bbl/min; consequently, an interpretation of the fracture closure stress from the step-rate pressure data,
like that presented by Gulrajani, Nolte, and Romero (2001), is highly questionable.


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SPE-187038-MS 11
Figure 5—MWX-2 Injections 2B (27 bbl), 3B (104 bbl), 4B (215 bbl) April 1995.
Second, the G-function derivative analysis graph clearly shows the "sudden" appearance of variable
storage or changing compliance during the falloff, which was not observed in any of the stress tests recorded
before the crosslinked-gel Injection 1B. The fracture-compliance method interpretation suggests a fracture
closure pressure of (p c)ΔCbc = 3,208 psi, where the subscript ‘ΔCbc’ denotes variable storage/compliance
before closure. Since the final recorded pressure before beginning flow back was ( pw)final = 2,893 psi, fracture
closure was not observed during the falloff based G-function derivative analysis. Additionally, the log-
log storage diagnostic clearly shows storage distortion is observed through the end of the recorded falloff
data; thus, hydraulic fracture closure was also not observed during the falloff based on the log-log storage
diagnostic.


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data; thus, hydraulic fracture closure was also not observed during the falloff based on the log-log storage
diagnostic.
Injection 3B was a 104-bbl injection of 2% KCl-treated water, which except for a brief shut-down early
in the test, was injected at a constant rate. The bottomhole pressure and injection rate versus time plot shows
the final recorded pressure during the falloff was (p w)final = 2,972 psi. The G-function derivative analysis
graph once again suggests variable storage/changing compliance was observed during the falloff. Using the
fracture-compliance method, the fracture closure is (p c)ΔCbc = 3,308 psi. Fracture closure was not observed


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12 SPE-187038-MS
based on G-function derivative analysis as indicated by the increasing derivative, Gdp w/dG, and based on the
log-log storage diagnostic plot, which shows an increasing derivative, td[p e – pw(t)]/dt, and corresponding
storage distortion of the pressure falloff.
Injection 4B was a 214.7 bbl injection of 2% KCl-treated water with a brief shut-down during the
injection, but with a hydraulic fracture re-opening experiment included at the beginning of the injection.
As was observed in Injection 2B and Injection 3B, the G-function derivative analysis plot shows a strong
indication of variable storage/changing compliance, and the fracture-compliance method closure pressure is
(pc)ΔCbc = 3,330 psi. Conversely, fracture closure was not observed based on G-function derivative analysis
and the log-log storage diagnostic plots, which both show increasing derivatives, Gdp w/dG and td[p e – pw(t)]/
dt, respectively, through the end of the falloff data.


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dt, respectively, through the end of the falloff data.
Although the low-volume stress tests should have established a fracture closure pressure corresponding
to the minimum horizontal stress, the test analyses remain pressure interpretations of fracture closure and
are not a direct observation of closure. The tiltmeters, however, are a measurement of rock deformation
as a function of increasing hydraulic fracture width. Consequently, a re-opening experiment where 2%
KCl-treated water is slowly injected into an existing fracture, and the pressure gradually increases, should
indicate measurable rock deformation as the pressure within the fracture exceeds the minimum horizontal
stress (closure pressure) and the hydraulic fracture dilates.
Fig. 6 contains plots of bottomhole pressure and y-axis deformation of the tiltmeters at 4,628-ft (below
the B-Sand), 4,558-ft (within the B-Sand), 4,487-ft (above the B-Sand). In each figure, the tiltmeter


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the B-Sand), 4,558-ft (within the B-Sand), 4,487-ft (above the B-Sand). In each figure, the tiltmeter
measurements are essentially constant before the injection begins and as pressure starts to increase. As
bottomhole pressure exceeds approximately 2,800 psi, deformation is observed in every tiltmeter. The
tiltmeter measurements suggest existing-fracture dilation begins at approximately (p c)ReOpen  ≈ 2,800 psi,
which may be the fracture opening pressure corresponding to the minimum horizontal stress. Each figure
shows a vertical red dashed line that corresponds to the fracture closure pressure interpreted from the stress-
test data, 2,855 psi ≤ (p c)2017 ≤ 2,893 psi and the black dashed line corresponds to the fracture closure stress
interpreted using the fracture-compliance method, which is 3,208 psi ≤ (p c)ΔCbc ≤ 3,330 psi.
Figure 6—MWX-2 B-Sand Injection 4B re-opening experiment tiltmeters identifying deformation and fracture opening.


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Figure 6—MWX-2 B-Sand Injection 4B re-opening experiment tiltmeters identifying deformation and fracture opening.
Unfortunately, tiltmeter re-opening experiments also have uncertainty, for example, if the re-opening
injection rate is too high, deformation may be observed before reopening. The tiltmeter measurements
are, however, consistent with the fracture closure pressure interpreted from G-function derivative analysis
and log-log storage diagnostic. Additionally, the tiltmeter measurements seem to validate that the fracture
closure pressure interpretations based on identifying the dissipation of storage distortion correspond to the
minimum horizontal stress. Conversely, the tiltmeter measurements are inconsistent with the findings of the
fracture-compliance method for closure identification, that is, the fracture-compliance method interpreted
closure pressure may not correspond to the minimum horizontal stress. Fig. 6 does show a significant


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SPE-187038-MS 13
increase in deformation as the pressure exceeds approximately 3,100 psi, but the initial deformation is
observed as bottomhole pressure increases above approximately (p c)ReOpen  ≈ 2,800 psi.
Injections 5B and 6B were both 400-bbl 40-lb/1,000-gal linear gel injections, and the pressure falloff did
not extend through hydraulic fracture closure for either injection/falloff sequence. Injection 7B consisted
of the final KCl-treated water injection/falloff in the B-Sand, and the falloff did extend through hydraulic
fracture closure. Fig. 7 shows the Injection 7B diagnostic plots, and the bottomhole pressure and injection
rate versus time graph show that the injection consisted of 30.5-bbl of 2% KCl treated water at injection
rates up to 19 bbl/min. Unlike Injection 2B, Injection 3B, and Injection 4B, but like the three stress tests, the
leakoff-off type identified using G-function derivative analysis is pressure-dependent leakoff with a fissure-


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leakoff-off type identified using G-function derivative analysis is pressure-dependent leakoff with a fissure-
opening pressure of (p fis)2017 = 2,893 psi. G-function derivative analysis shows no indication of variable
storage/changing compliance, but it does indicate fracture closure was observed at ( pc)2017 = 2,855 psi, which
is consistent with the three stress tests and validated by the tiltmeter measurements from the Injection 4B re-
opening experiment. The log-log storage diagnostic also shows storage distortion dissipates as the fracture
closes, which is indicated by the decreasing derivative td[p e – p w(t)]/dt, after closure.
Figure 7—MWX-2 B-Sand Injection 7B -- 31 bbl KCl water prefrac DFIT August 1995.
The tiltmeter measurements during the re-opening experiment provided a clear indication of hydraulic
fracture re-opening and dilation during the injection, but the measurements are not as definitive during


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fracture re-opening and dilation during the injection, but the measurements are not as definitive during
closure. For example, Fig. 8 contains a graph of bottomhole pressure and tiltmeter deformation recorded
during the pressure falloff for the tiltmeter located at 4,487-ft (above the B-Sand). Notice that as bottomhole
pressure falls below the re-opening pressure and falloff-interpreted fracture closure pressure, the tiltmeter
measurements continue to show decreasing deformation, which suggests fracture width continues to
decrease. Examination of the tiltmeter measurements also shows that the data are relatively noisy; however,
the shape of the tiltmeter-measured deformation curve is similar to the pressure falloff curve, and the
tiltmeter-measured deformation might follow relationships known to exist for fracture pressure falloff data.


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14 SPE-187038-MS
Figure 8—MWX-2 B-Sand Injection 7B bottomhole pressure and tiltmeter (4,487 ft) falloff during shut-in.
As previously noted, Craig (2014) demonstrated that the derivative, td[p e – pw(t)]/dt, is related to a general
pressure transient solution as follows.
(13)
Generally, an analytical solution for a fractured well is used to generate the pressure transient derivative,
and the observed DFIT data are matched to an appropriate solution. The method is general, and any
appropriate transient solution, including flow from an infinite- or finite-conductivity fracture, dual porosity
reservoir, anisotropic reservoir, multiple producing fractures, etc., can be used in the matching process. For
the purposes of the current paper, it's important to note the behavior of the pressure-transient derivative,
which is shown in Fig. 9. In Fig. 9, the increasing derivative is indicative of wellbore and fracture storage


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which is shown in Fig. 9. In Fig. 9, the increasing derivative is indicative of wellbore and fracture storage
distortion. Wellbore and fracture storage distortion dissipates as the curve reaches the apex and begins to
fall. A closing fracture is a type of storage ( Craig and Blasingame 2005, 2006), and in most cases, fracture
storage affects the falloff data long after wellbore storage has dissipated. Thus, when the derivative, td[p e
– p w(t)]/dt, is increasing, the fracture is still closing.
Figure 9—Pressure transient solution derivative versus dimensionless time.


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SPE-187038-MS 15
In terms of the tiltmeter response, tiltmeter measurements peaked at the end of pumping, that is, the
hydraulic fracture maximum width was observed at the end of pumping and pressure pe. Since tiltmeter
deformation is proportional to the fracture width, and the fracture width changes over time are a function
of pressure, it's anticipated that the tiltmeter deformation-difference derivative, td[(u y)e – uy (t)]/dt, curve
should mirror the fracture pressure-difference derivative curve. In other words, we postulate that in a
homogeneous reservoir with normal leakoff, the tiltmeter deformation-difference derivative, td[(u y)e – uy(t)]/
dt, will increase through hydraulic fracture closure. Similarly, if pressure-dependent leakoff is observed,
the tiltmeter deformation-difference derivative should show a positive "hump," like the fracture pressure-
difference derivative.


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the tiltmeter deformation-difference derivative should show a positive "hump," like the fracture pressure-
difference derivative.
One issue, however, in attempting derivative analysis of tiltmeter data is the noise noted in Fig. 8. If finite-
difference differentiation of the noisy tiltmeter data is attempted, the noise will be amplified by the derivative
calculation, and the derivative may be of limited value. An alternative technique for calculating a "noiseless"
underlying derivative of noisy experimental data was developed by Chartrand (2011) and is called total
variation regularization. Fig. 10 illustrates the derivative calculation of the tiltmeter measurements (4,487
ft) using both "smoothed" central-difference calculations and total variation regularization. With a smooth
derivative calculation, the tiltmeter deformation-difference derivative, td[(u y)e – uy (t)]/dt, should show
trends similar to the fracture pressure-difference derivative, td[p e – p w(t)]/dt.


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trends similar to the fracture pressure-difference derivative, td[p e – p w(t)]/dt.
Figure 10—MWX-2 B-Sand Injection 7B tiltmeter du y/dt derivative
calculated with central-difference and total variation regularization.
Fig. 11 shows a log-log graph of the tiltmeter deformation-difference derivative versus time since the
beginning of pumping for the tiltmeters at 4,487-ft (above B-Sand), 4,558-ft (within B-Sand), and 4,628-
ft (below B-Sand). All three tiltmeter derivative curves have a similar behavior that includes a pressure-
dependent leakoff "hump" followed by an increasing derivative until fracture closure is observed. Fig. 11
also contains a log-log plot of both the fracture pressure-difference derivative and the tiltmeter deformation-
difference derivative (4,558-ft within B-Sand), and shows how both exhibit a pressure dependent hump and
an increasing derivative before hydraulic fracture closure is interpreted. Figs. 10 and 11 demonstrate that


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an increasing derivative before hydraulic fracture closure is interpreted. Figs. 10 and 11 demonstrate that
a characteristic tiltmeter deformation-derivative signature was observed and is consistent with pressure-
inferred hydraulic fracture closure. While the tiltmeter deformation-derivative curves may not provide an
easily-identified and definitive closure event, the tiltmeter deformation-difference derivative interpretation
along with the analog fracture pressure-difference derivative methodology provide convincing evidence of
hydraulic fracture closure, and the closure pressure is consistent with the fracture re-opening and pressure-


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16 SPE-187038-MS
falloff inferred fracture closure pressure. To our knowledge, a relationship between the deformation-
difference derivative and the pressure-difference derivative has not been studied. The trends observed in the
tiltmeter and pressure data suggest additional research is warranted to mathematically define a relationship
if one exists.
Figure 11—Comparison of tiltmeter deformation-difference derivatives, td[(u y)e-uy(t)]/dt, and comparison of
tiltmeter deformation-difference derivative (4,558 ft) with observed fracture pressure-difference derivative, tdΔp/dt.
Fracture Re-Opening Experiment – C-Sand
The C-Sand is located between 4,303-ft and 4,390-ft in the MWX-2 wellbore, and the experiments of interest
for the present paper are Injection 3C and Injection 4C, which includes a fracture re-opening experiment.
Injection 1C, like Injection 1B in the B-Sand, was a 95-bbl injection of 40-lb/1,000-gal borate-crosslinked


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Injection 1C, like Injection 1B in the B-Sand, was a 95-bbl injection of 40-lb/1,000-gal borate-crosslinked
gel, and Injection 2C was a 40-lb/1,000-gal linear gel injection. Injection 3C was a 247-bbl injection of
40-lb/1,000-gal linear gel, with a significant falloff period; however, fracture closure cannot be interpreted
from the falloff data using G-function derivative analysis and the log-log storage diagnostic plot.
Fig. 12 contains a graph of bottomhole pressure and injection rate versus time, the G-function derivative
analysis plot, and the log-log storage diagnostic graph. The graph of bottomhole pressure and injection rate
versus time shows a maximum injection rate of 19-bbl/min and a falloff that was recorded through a final
bottomhole pressure of (p w)final = 2,956 psi. G-function derivative analysis suggests fracture closure was not
observed through the end of the falloff, and the log-log storage diagnostic also shows an increasing fracture


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observed through the end of the falloff, and the log-log storage diagnostic also shows an increasing fracture
pressure-difference derivative through the end of the falloff, which confirms storage was still distorting the
pressure falloff and suggests the fracture was still closing.


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SPE-187038-MS 17
Figure 12—MWX-2 C-Sand Injection 3C DFIT Analysis.
A fracture re-opening experiment was also conducted in the C-Sand, and Fig. 13 contains a graph of
bottomhole pressure and injection rate during the re-opening. The injection rates were kept lower during
re-opening as compared to the B-Sand re-opening experiment; thus, the tiltmeter re-opening deformation
measurements may provide a more definitive fracture re-opening pressure. As shown in Fig. 13, the injection
rates were maintained at generally less than 10-gal/min until fracture opening was observed. Fig. 14 contains
graphs of bottomhole pressure and tiltmeter deformation for the tiltmeters at 4,418-ft (below the C-Sand),
4,348-ft (within the C-Sand), and 4,273-ft (above the C-Sand). Even with the noise in the tiltmeter-recorded
deformation, a fracture re-opening pressure of approximately (p c)ReOpen  ≈ 2,900 psi was observed. Pressure


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deformation, a fracture re-opening pressure of approximately (p c)ReOpen  ≈ 2,900 psi was observed. Pressure
falloff analysis of the entire Injection 4C injection/falloff sequence is shown in Fig. 15. The final recorded
pressure during the falloff was (p w)final = 3,108 psi, which is clearly above the hydraulic fracture re-opening
stress, and both G-function derivative analysis and the log-log storage diagnostic confirm that hydraulic
fracture closure was not observed during the falloff, which is indicated by the increasing derivatives, Gdp w/
dG and td[p e – p w(t)]/dt, respectively, through the end of the falloff data.
Figure 13—MWX-2 C-Sand Injection 4C Reopening Experiment.


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18 SPE-187038-MS
Figure 14—MWX-2 C-Sand Injection 4C Reopening Experiment.
Figure 15—MWX-2 C-Sand Injection 4C DFIT Analysis
Summarizing the findings from both the B-Sand and C-Sand injection/falloff and fracture re-opening
experiments, we observed the following.
•Fracture re-opening experiments using bottomhole pressure and tiltmeter deformation
measurements suggest a B-Sand fracture re-opening pressure of (p c)ReOpen  ≈ 2,800 psi and a C-Sand
re-opening pressure of (p c)ReOpen  ≈ 2,900 psi.
•Fracture closure pressure interpretations using G-function derivative analysis and the log-log
storage diagnostic suggest a fracture closure pressure approximately equivalent to the fracture re-
opening pressure. In the B-Sand, the falloff-interpreted closure pressure was 2,855 psi ≤ ( pc)2017 ≤
2,893 psi. However, the C-Sand injection/falloff sequences did not allow bottomhole pressure to
decline below pw(t) = 2,956 psi, but we can confidently state that the falloff analysis in the C-Sand


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decline below pw(t) = 2,956 psi, but we can confidently state that the falloff analysis in the C-Sand
suggests a fracture closure pressure of (p c)C-Sand < 2,956 psi.
•Fracture closure pressure when interpreted using G-function derivative analysis and the log-
log storage diagnostic was invariant over multiple injection/falloff sequences, including different
fluids, fluid volumes, and rates, and invariant over time.
•The fracture-compliance method interpretations of the G-function derivative, Gdp w/dG, in the B-
Sand injections suggested a fracture closure pressure of 3,208 psi ≤ ( pc)ΔCbc ≤ 3,330 psi, which
are inconsistent with the small-volume stress tests, holistic G-function derivative analysis, and the
fracture re-opening experiments.
•Tiltmeter measurements during a pressure falloff do not exhibit an easily-identifiable and definitive
closure event.


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SPE-187038-MS 19
•Tiltmeter deformation-difference derivative curves are generally similar in shape and behavior to
fracture pressure-difference derivatives; consequently, the deformation-difference derivative can
potentially validate fracture closure and the end of storage distortion identified using G-function
derivative analysis and the log-log storage diagnostic.
Laboratory Experiments of Fracture Closure
Numerous researchers have reported experiments of fracture propagation and closure under controlled
laboratory conditions. Cheung and Haimson (1989) looked at existing fracture closure interpretation
techniques, while also developing a new model of their own, for identifying fracture closure pressure
from pressure falloff data. Their experimental method required multiple injection-falloff sequences with
the wellbore pressure falling to below 1 MPa before each subsequent injection, and their experiments


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the wellbore pressure falling to below 1 MPa before each subsequent injection, and their experiments
focused as much on identifying the fracture re-opening pressure as the fracture closure pressure. Fig. 16
contains a graph of pressure and the Gdp w/dG derivative for injection ‘a’ of Test #10 for a Niagara dolomite
(E > 6.3 × 106 psi [43.7 GPa]) fracture closure experiment. Injection ‘a’ was the first fracture-injection/
falloff in the sample, and it was the only injection/falloff sequence with sufficient analyzable before-
closure data. For example, only about 10 seconds of before-closure pressure data are available for analysis
in Fig. 16, which is barely sufficient to identify trends. The raw experimental data were not available;
however, Cheung and Haimson's Fig. 2 was digitized using publicly-available software (Rohatgi 2017),
which provided the pressure and time data required to construct Fig. 16. While digitizing historical publicly-


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which provided the pressure and time data required to construct Fig. 16. While digitizing historical publicly-
available experimental data is sometimes necessary (and worthwhile), there is inevitably a loss of resolution;
consequently, the data scatter shown in Fig. 16 may not have existed in the original raw pressure and time
recording.
Figure 16—Niagara Dolomite Test #10 Injection ‘a’. After Cheung and Haimson (1989).
The graph illustrates a clear "hump" in the Gdp w/dG derivative, which based on Fig. 1, is interpreted
as pressure-dependent leakoff. de Pater et al (1994) have found pressure-dependent leakoff in laboratory
fracturing experiments of moderately hard sandstone, so the result is anticipated even though the sample
contains no obvious natural fractures or fissures. The closure pressure is interpreted to be 13.2 MPa, which
is within 12% of the imposed 15 MPa minimum horizontal stress. While the interpretation reasonable, it is


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is within 12% of the imposed 15 MPa minimum horizontal stress. While the interpretation reasonable, it is
nonunique because of limited before-closure data and a noisy Gdp w/dG derivative. With very few before-
closure data points, and since virtually all the before-closure data is affected by pressure-dependent leakoff,
the log-log storage diagnostic is of limited value for a fracture closure interpretation in this case.


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20 SPE-187038-MS
When reviewing laboratory data, it is useful to have fracture dimensions measured during closure to help
correlate the pressure falloff to measured fracture dimensions. While Cheung and Haimson reported acoustic
emissions during closure, fracture dimensions during closure were not reported. de Pater, Desroches,
Groenenboom, and Weijers (1996) presented measurements of both pressure falloff and fracture dimensions
with respect to time for both sandstone (E ≅ 3 × 106 psi [20.4 GPa]) and cement paste. Fig. 17 contains a
graph of pressure, dp w/dG, and Gdp w/dG versus the G-function and a graph of fracture width at the wellbore,
fracture radius, and Gdp w/dG versus the G-function. The data were digitized from Fig. 3 and Fig. 10 of de
Pater, Desroches, Groenenboom, and Weijers (1996) for the sandstone sample.
Figure 17—G-function derivative analysis and fracture dimensions during closure for SNV03. After de Pater, et al (1996).


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Figure 17—G-function derivative analysis and fracture dimensions during closure for SNV03. After de Pater, et al (1996).
G-function derivative analysis suggests fracture closure was not observed through the end of the recorded
pressure falloff data as indicated by the Gdp w/dG derivative following a straight line extrapolated from the
origin. The imposed stress was 23 MPa, which coincides with the last recorded pressure. A definitive closure
identification using G-function derivative analysis would have required recording the pressure falloff below
23 MPa; however, the Gdp w/dG derivative curve is consistent with the observed closure. In other words,
closure was not observed during the falloff, which is confirmed by the Gdp w/dG derivative curve.
Pressure-dependent leakoff is once again indicated during the falloff of the sandstone sample by the
Gdp w/dG derivative "hump." Fig. 17 also shows that during the Gdp w/dG derivative pressure-dependent


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Gdp w/dG derivative "hump." Fig. 17 also shows that during the Gdp w/dG derivative pressure-dependent
leakoff signature, both the fracture width and fracture radius were decreasing. As previously discussed, a
decreasing radius during the falloff increases fracture stiffness, which should result in decreasing storage
and a changing compliance signature of the Gdp w/dG derivative, which is opposite of what is observed. In
both the Niagara dolomite and the Colton sandstone samples, any variable storage/changing compliance
signature that should be observed in the Gdp w/dG derivative, is masked by the pressure-dependent leakoff.
The fracture width change and radius change during closure confirm that the fracture was indeed closing
from the fracture tip back to the wellbore, which is consistent with the closure process defined by Hayashi
and Haimson (1991) and "progressive fracture closure" as defined Zanganeh, Clarkson, and Hawkes (2017).


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and Haimson (1991) and "progressive fracture closure" as defined Zanganeh, Clarkson, and Hawkes (2017).
However, the experiment also clearly shows that the minimum horizontal stress is not equivalent to the
pressure at tip-closure. At fracture-tip closure, the experimental data show a pressure of about 30 MPa, but
the imposed horizontal stress was only 23 MPa; thus, the conclusions of Zanganeh, Clarkson, and Hawkes
are not supported by experimental data in a hard rock sample. In hard rock, a fracture closure pressure


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SPE-187038-MS 21
corresponding to the minimum horizontal stress is correctly identified by the traditional Gdp w/dG derivative
interpretation and the log-log storage diagnostic when the fracture closes to essentially the residual width
at the wellbore.
Dong and de Pater (2007) presented a fracture closure experiment of a soft unconsolidated sand (E ≅
250,000 psi [1.7 GPa]), which makes a very interesting comparison with the hard-rock cases noted above.
Fig. 18 contains a graph of pw, dp w/dG, and Gdp w/dG versus the G-function along with a log-log graph of
the td[p e – pw(t)]/dt derivative versus time. A Gdp w/dG derivative interpretation suggests fracture closure
is observed at G = 2.09 and a closure pressure of 3 MPa after periods of pressure-dependent and normal
leakoff; however, the imposed stress during the experiment was 7 MPa. The td[p e – pw(t)]/dt derivative also
shows the effects of pressure-dependent leakoff and fracture storage distortion until fracture closure was


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shows the effects of pressure-dependent leakoff and fracture storage distortion until fracture closure was
observed. Fig. 19 is reproduced from Dong and de Pater (2007) and shows several CT scans of the hydraulic
fracture during the pressure falloff. The CT scans shows that the fracture remains open with perhaps a small
decrease in aperture as the pressure declines below the imposed horizontal stress; however, it also shows
that the fracture closes as the pressure decreases to about 3 MPa. After the pressure falls below the imposed
minimum horizontal stress, fracture closure begins at the tip and proceeds back to the wellbore until a
residual fracture width is observed. The CT scans graphically demonstrate that the Gdp w/dG derivative and
the log-log td[p e – pw(t)]/dt derivative accurately reflect fracture closure, that is, the methods can identify the
decrease to a residual fracture width and volume, but for soft rock, the closure pressure does not correspond


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decrease to a residual fracture width and volume, but for soft rock, the closure pressure does not correspond
to the minimum horizontal stress.
Figure 18—Soft unconsolidated sand closure experiment CTC08. After Dong and de Pater 2007.


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22 SPE-187038-MS
Figure 19—CT scans of fracture closure in a soft unconsolidated sand. After Dong and de Pater (2007).
Fig. 19 is also an excellent visualization of fracture-storage distortion. As previously noted, during
a fracture-injection/falloff test, storage has both wellbore and fracture components. In the laboratory
experiments, wellbore storage is minimized, so the primary contributor to storage distortion of the falloff
data is fracture closure when fluid is "squeezed" from the fracture into the formation as closure proceeds.
Figs. 18 and Fig. 19 viewed in conjunction demonstrate that the storage distortion observed in the td[p e
– pw(t)]/dt derivative is purely a function of the closing fracture. When the fracture finally closes, storage
distortion of the pressure falloff dissipates. The observation holds for field and laboratory data and is
independent of rock hardness.
Fig. 20 is an example provided by van Dam and de Pater (2017) of a closure experiment in a harder


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independent of rock hardness.
Fig. 20 is an example provided by van Dam and de Pater (2017) of a closure experiment in a harder
sample (E ≅ 1.3 × 106 psi [8.9 GPa]). The example was provided as a "blind test" where the imposed
stress remained unknown until after an analysis was completed, and the examples illustrates the difficulty
of interpreting fracture closure from laboratory data. The Gdp w/dG derivative shows a pressure-dependent
"hump," a variable storage/changing compliance "dip," and a period of normal leakoff. A changing-
compliance interpretation results in a closure pressure of 20.16 MPa, which is 26% too high, and as was
demonstrated in the field experiments, a strict changing-compliance closure interpretation as advocated by
McClure, Jung, Cramer, and Sharma (2015), does not provide the correct closure pressure and minimum
horizontal stress. A classical interpretation resulted in a closure pressure of 14.32 MPa, and while the


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horizontal stress. A classical interpretation resulted in a closure pressure of 14.32 MPa, and while the
classical interpretation is within 10.5% of the correct answer (16 MPa), it illustrates the difficulty in drawing
the correct Gdp w/dG derivative straight line during a G-function derivative analysis without a clear and
distinct deviation of the Gdp w/dG derivative off the straight line. It also shows that picking the end of storage
distortion using the td[p e – pw(t)]/dt derivative is helpful, but it is also not distinct since closure is observed
as storage distortion dissipates and not necessarily at the apex of the td[p e – pw(t)]/dt derivative. In other
words, despite all attempts to make fracture closure identification objective using the Gdp w/dG and td[p e –
pw(t)]/dt derivatives, the pressure falloff trends from laboratory experiments are often non-distinct and the
interpretation subjective, which can introduce some interpretation error—for example, 12.5% for the hard


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interpretation subjective, which can introduce some interpretation error—for example, 12.5% for the hard
dolomite and 10.5% for the hard sandstone.


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SPE-187038-MS 23
Figure 20—Laboratory fracture-closure experiment (GNV06b) in moderately hard rock (E = 1.3 × 106 psi [8.9 GPa] and v = 0.21).
Conclusions
Both laboratory and field experiments of hydraulic fracture closure with measurements of fracture
dimensions during the pressure falloff have been examined and the results compared to interpretations of
closure pressure from the pressure falloff and the relationship to the minimum horizontal stress. In two
hard-rock microdarcy sandstone reservoirs, field experiments of fracture closure monitored by downhole
tiltmeters suggested the following.
•Fracture re-opening experiments using bottomhole pressure and tiltmeter deformation
measurements suggest a B-Sand fracture re-opening pressure of (p c)ReOpen  ≈ 2,800 psi and a C-Sand
re-opening pressure of (p c)ReOpen  ≈ 2,900 psi.
•Fracture closure pressure interpretations using G-function derivative analysis and the log-log


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re-opening pressure of (p c)ReOpen  ≈ 2,900 psi.
•Fracture closure pressure interpretations using G-function derivative analysis and the log-log
storage diagnostic suggest a fracture closure pressure approximately equivalent to the fracture re-
opening pressure. In the B-Sand, the falloff-interpreted closure pressure was 2,855 psi ≤ ( pc)2017 ≤
2,893 psi. However, the C-Sand injection/falloff sequences did not allow bottomhole pressure to
decline below pw(t) = 2,956 psi, but we can confidently state that the falloff analysis in the C-Sand
suggests a fracture closure pressure of (p c)C-Sand < 2,956 psi.
•Fracture closure pressure when interpreted using G-function derivative analysis and the log-
log storage diagnostic was invariant over multiple injection/falloff sequences, including different
fluids, fluid volumes, and rates, and invariant over time.
•The fracture-compliance method interpretations of the Gdp w/dG derivative in the B-Sand injections


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•The fracture-compliance method interpretations of the Gdp w/dG derivative in the B-Sand injections
suggested a fracture closure stress of 3,208 psi ≤ ( pc)ΔCbc ≤ 3,330 psi, which are inconsistent with
the small-volume stress tests, holistic G-function derivative analysis, and the fracture re-opening
experiments.
•Tiltmeter measurements during a pressure falloff do not exhibit an identifiable and definitive
closure event.
•Tiltmeter deformation-difference derivative curves are generally similar in shape and behavior to
fracture pressure-difference derivatives; consequently, the deformation-difference derivative can
potentially validate fracture closure and the end of storage distortion identified using G-function
derivative analysis and the log-log storage diagnostic.
Laboratory experiments of fracture closure in soft rock (E < 7.25 × 105 psi [5.0 GPa]) through hard
rock (E ≅ 6.3 × 106 psi [43.7 GPa]) demonstrate that G-function derivative analysis and the log-log storage


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rock (E ≅ 6.3 × 106 psi [43.7 GPa]) demonstrate that G-function derivative analysis and the log-log storage
diagnostic interpretations of pressure falloff data can generally identify hydraulic fracture closure although


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24 SPE-187038-MS
the trends in the falloff data are sometimes non-distinct and the interpretations subjective. Examination of
laboratory experiments of fracture closure also suggest the following.
•Measurements of fracture dimensions and pressure during closure show that fracture closure begins
at the fracture tip and proceeds to the wellbore, which confirms the closure process of Hayashi and
Haimson (1991); however, the pressure at tip closure did not correspond to the minimum horizontal
stress.
•Although changing compliance was observed in samples based on measurements of fracture
dimensions and pressure, the fracture closure pressure corresponding to changing compliance did
not correspond to the minimum horizontal stress.
•Fracture closure is observed as storage distortion of the pressure falloff dissipates, which
corresponds to closure at the wellbore and a near-residual fracture volume.
•In soft rock, like unconsolidated sand, G-function derivative analysis and the log-log storage


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•In soft rock, like unconsolidated sand, G-function derivative analysis and the log-log storage
diagnostic correctly interpret fracture closure, but the closure pressure does not correspond to the
minimum horizontal stress.
•In hard rock, G-function derivative analysis and the log-log storage diagnostic correctly interpret
fracture closure, and the fracture closure pressure is approximately equal to the minimum horizontal
stress.
Acknowledgment
We appreciate Hans de Pater providing the raw pressure and time data for CTC08 and the blind test.
Nomenclature
Af= fracture area, L2
B= formation volume factor, L3/L3
cf= compressibility of fluid in fracture, Lt2/m
cw= compressibility of fluid in wellbore, Lt2/m
C= wellbore and fracture storage, L4t2/m
Cf= time-dependent fracture compliance, L4t2/m
Cs= slug-test storage coefficient, L4t2/m
Ct= system compliance, L4t2/m
E= Young's modulus, m/Lt
E'= Plane-strain modulus, m/Lt2
E(m) = complete elliptical integral, dimensionless


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Ct= system compliance, L4t2/m
E= Young's modulus, m/Lt
E'= Plane-strain modulus, m/Lt2
E(m) = complete elliptical integral, dimensionless
F= hypergeometric function, dimensionless
g= loss-volume function, dimensionless
G= G-function, dimensionless
h= height, L, ft
hf= fracture height, L, ft
k= permeability, L2, md
Lf= fracture half-length, L, ft
p= pressure, m/Lt2
pws= wellbore pressure with variable flow rate, m/Lt2
pwc= wellbore pressure with constant flow rate, m/Lt2
q= flow rate, L3/t
Sf= fracture stiffness, m/L4t2
Seff= system stiffness, m/L4t2


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SPE-187038-MS 25
sfs= fracture-face skin, dimensionless
t= time, t
tLfD= Dimensionless time, dimensionless
Vw= wellbore volume, L3
Vf= fracture volume one wing, L3
Greek
αN= fracture growth exponent, dimensionless
Δ= difference, dimensionless
μ= viscosity, m/Lt, cp
τ= variable of substitution, dimensionless
ϕ= porosity, dimensionless
Γ= Euler gamma function, dimensionless
Subscripts
c= constant-rate
ΔCbc = changing-compliance closure
D= dimensionless
e= end of injection
final = last recorded
i= property evaluated at initial conditions
0= end of injection
s= slug test
w= wellbore
r= reservoir
ReOpen = observed re-opening
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SPE 16417 SPE/DOE 
Society of 
Petroleum Engineers U.S. Department 
of Energy 
Modified Fracture Pressure Decline Analysis Including 
Pressure-Dependent Leakoff 
by J. L. Castillo* Dowell Schlumberger 
*Member SPE 
Copyright 1987, Society of Petroleum Engineers 
This paper was prepared for presentation at the SPE/DOE Low Permeability Reservoirs Symposium held in Denver, Colorado, May 18-19, 1987. 
This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the 
author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the 
author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers 
presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is


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presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is 
restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of 
where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836. Telex, 730989 SPEDAL. 
ABSTRACT 
Nolte developed an analysis of the pressure 
decline following a fracturing treatment to provide an 
estimate of fluid efficiency, fluid loss coefficient, 
and fracture geometry. This analysis has become an 
industry standard for evaluating and improving the 
design of fracturing treatments. The derivations by 
Nolte assume the leakoff coefficient is a pressure­
independent constant. This assumption is valid when 
leakoff is controlled by a compressible filter cake. 
In cases where leakoff is primarily controlled by


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leakoff is controlled by a compressible filter cake. 
In cases where leakoff is primarily controlled by 
filtrate viscosity, by an incompressible filter cake, 
or by reservoir permeability and compressibility, the 
leakoff coefficient is pressure dependent, and, there­
fore, declines during the closure period. Under these 
circumstances, direct application of the Nolte tech­
nique may not be appropriate, and could cause an 
optimistic estimate of fluid efficiency and leakoff 
coefficient. 
This paper introduces a new plot for fracture 
pressure decline analysis that determines the pressure 
parameters (ISIP, Pc, P*) required in the Nolte leak­
off calculations. The derivative of this plot may be 
used to identify the extension period, the closure 
period, and to assess the influence of pressure depen­
dent 1 eakoff on fracture pressure decline. In order 
to account for pressure dependent 1 eakoff, equations 
are presented where the leakoff coefficient is treated


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to account for pressure dependent 1 eakoff, equations 
are presented where the leakoff coefficient is treated 
as a mathematical variable.· The equations describe 
simple linear relations applicable to the fracture 
closure period. When these relations are plotted on 
References and illustrations at end of paper. Cartesian coordinates, the closure period is recog­
nized by a straight line with slope proportional to 
the 1 eakoff coefficient. As shown in the examples, 
this modified approach is easily applied and provides 
new insight to the fracture closure period. 
INTRODUCTION 
The methodology currently used in fracture pres­
sure decline analysis employs type-curve matching to 
determine a match pressure (P*) that is proportional 
to the leakoff coefficient.1'2 Data used in the curve 
match must correspond to the closure period. This 
period is usually interpreted as the 11linear11 region 
of a square-root of time plot or as the half-slope


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period is usually interpreted as the 11linear11 region 
of a square-root of time plot or as the half-slope 
region of a log-log plot.3,4 Analysis of the square­
root plot or the log-log plot, however, is often 
inconclusive. To aid identification of the closure 
period, the closure pressure is usually determined 
prior to a calibration treatment from a stress test 
(e.g. pump-in flow-back test). 
A new plot is introduced that has the unique 
advantage of providing all of the pressure variables 
(ISIP, Pc, P*) required in decline analysis. This 
plot is more easily applied than the Nolte type 
curves, and is superior to the square-root of time 
plot because it uses a more precise relationship 
between decline pressure and time. When the deriva­
tive of this plot is graphed against time, fracture 
extension after shut-in, fracture closure, and pres­
sure dependent 1 eakoff can be observed. This plot 
enhances interpretation and identification of the


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sure dependent 1 eakoff can be observed. This plot 
enhances interpretation and identification of the 
closure period. However, by no means should use of 
this plot replace stress tests. 
When analysis of the derivative plot indicates a 
dependence of leakoff rate on pressure, a technique is 
presented to account for a declining leakoff coeffi­
cient during closure. 
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2 MODIFIED FRACTURE PRESSURE DECLINE ANALYSIS 
INCLUDING PRESSURE-DEPENDENT LEAKOFF SPE 16417 
DEVELOPMENT OF THE 11G-FUNCTION11 PLOT 
Nolte1'2 has shown that, during closure, the 
pressure difference between two shut-in times can be 
expressed as 
1r•C•f •It 
__ ...__P _ _._p • G(to,t*o) 
2 •cf 
(l~t> ~t*) ( 1) 
where, ~t* is a reference time since shut-in, C is the 
leakoff coefficient, fp is the ratio of permeable area 
to gross area, tp is pump time, and t0 is dimension­
less time (M/t ). Equations for cf (fracture's 
compliance for wfdth) and G (dimensionless difference 
function) are given in Appendix A. 
A Nolte type-curve1 is constructed by selecting a 
dimensionless reference time (t11*) prior to closure 
and plotting G against t0 on log-log coordinates. The 
same reference time is used to plot ~p against ~t · on 
log-log coordinates. A curve match is obtained by 
overlaying the curves, aligning the x axes so that ~t 
corresponds to t0, and shifting along the y axis.


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overlaying the curves, aligning the x axes so that ~t 
corresponds to t0, and shifting along the y axis. 
Once a match is obtained, a match point is selected, 
and the Nolte match pressure (P*) is computed from 
~P/G. The match pressure is equivalent to 
TI•C•f ·It P* = p . p (2) 
2•cf 
When compliance can be inferred, the leakoff 
coefficient can be estimated from Equation 2. 
The new plot proposed here makes note of the fact 
that Equation 1 is a linear equation. Substituting 
Equation 2 into Equation 1, and selecting a reference 
time of ~t* = 0.0 results in 
P(~t) = P*•G(tD,O) + P(~t* = 0) (3) 
This equation is in the familiar form of y = mx + 
b. Thus, a Cartesian plot of decline pressure against 
G (referred to below as the 11G-Function11 plot) should 
result in a straight line during the closure period 
with slope equal toP* andy-intercept equal to a 
theoretical pressure at shut-in. The closure period 
is recognized by linearity; deviation from linearity


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theoretical pressure at shut-in. The closure period 
is recognized by linearity; deviation from linearity 
identifies the closure pressure. From this single 
plot, then, all of the pressure parameters ( Pc, P*, 
!SIP) required in the analysis calculations are 
inferred. 
Interpretation of the G-Function plot is enhanced 
by differentiating Equation 3 to give 
dP 
P* (4) 
dG 
A graph of dP/dG against ~t (referred to as the 
G-Function derivative plot) should provide a horizon­
tal line during the closure period with a constant y­
axis value of P*. The derivative, dP/dG, may be 
computed by using linear regression on a small number 
of data points (e.g. 5); the ~t value that corresponds 
to dP/dG is a time average of the points used. Example 1 
Figures 1 and 2 are the G-Function and G-Function 
derivative plots, respectively, for a case where a 
crosslinked HPG fluid was used in a gas well. Figure 
2 shows the G-Funct ion plot is 1 i near ( i .e. constant


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crosslinked HPG fluid was used in a gas well. Figure 
2 shows the G-Funct ion plot is 1 i near ( i .e. constant 
slope) from 10 to 32 minutes. The non-linearity from 
shut-in to 10 minutes is interpreted as fracture 
extension after shut-in, while the non-linearity after 
32 minutes implies that the fracture is closed. 
An estimate of a theoretical !SIP is obtained by 
drawing a line through the linear portion of the G­
Function plot and extrapolating to the y-axis (as 
indicated in Figure 1). The difference between the 
theoretical !SIP and the observed !SIP corresponds to 
a decrease in fracture width resulting from continued 
fracture extension after shut-in. Use of the theoret­
i c a 1 I S I P i n t he Nolte e f f i c i en c y c a 1 c u 1 at i o n 2 s h o u 1 d 
provide a lower bound estimate of fluid efficiency. 
Figure 3 is the square-root of time plot using 
the same data as above; Figure 4 is the derivative 
plot of the square-root of time. As shown by Figure


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the same data as above; Figure 4 is the derivative 
plot of the square-root of time. As shown by Figure 
4, the square-root plot is never linear. Instead, the 
square-root plot exhibits an inflection when fracture 
closure occurs. The inflection is observed as a 
change in slope (from + to -) in the square-root 
derivative plot. The implication of this example is 
that the square-root of delta time is not directly 
proportional to the pressure decline during fracture 
closure. 
The G-Functi on plot, therefore, represents the 
closure period more accurately than the square-root of 
time plot since G is linearly related to the pressure 
decline. Both the G-Function plot and the square-root 
of time plot are enhanced by plotting their deriva­
tives against time. 
PRESSURE DEPENDENT LEAKOFF 
Nolte1'2 developed Equation 1 from an assumption 
that the leakoff coefficient is independent of varia­
tions in pressure. This assumption was shown to be


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that the leakoff coefficient is independent of varia­
tions in pressure. This assumption was shown to be 
valid when leakoff is controlled by a compressible 
filter cake.2 When leakoff is primarily controlled by 
an incompressible filter cake, by filtrate viscosity, 
or by the reservoir, the leakoff coefficient is depen­
dent on pressure.2,s During fracture closure, there­
fore, the leakoff coefficient declines as the pressure 
in the fracture declines. As indicated by Equations 2 
and 4, a declining 1 eakoff coefficient should result 
in a declining G-Function derivative plot. 
Example 2 
Figures 5 and 6 are a G-Functi on plot and a G­
Function derivative plot, respectively, for a case 
where an oil reservoir was fractured with a waterfrac 
fluid (10 lb polymer per 1,000 gallons). The leakoff 
control in this case is primarily due to the reser­
voir's compressibility and permeability; thus, the 
1 eakoff coefficient is pressure dependent. Ex ami na­


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voir's compressibility and permeability; thus, the 
1 eakoff coefficient is pressure dependent. Ex ami na­
tion of the G-Function derivative plot in Figure 6 
indicates a slope change at five minutes that corres­
ponds to fracture closure. The square-root plot in 
Figure 7 also exhibits an inflection at five minutes 
(as indicated in Figure 8). Observe that the G-Func­
tion derivative plot in Figure 6 is not horizontal 
during the closure period. This implies that, when 
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SPE 16417 J. L. CASTILLO 3 
1 eakoff is pressure dependent, the 1 i near relation 
between decline pressure and the G-Function does not 
hold. In these cases, Equation 3 falls short of 
accurately predicting fracture pressure decline. 
Equations that account for pressure dependence 
are presented in the next section. These equations 
were developed from an assumption of a direct rela­
tionship between leakoff rate and a pressure-dependent 
leakoff coefficient. A more comprehensive relation­
ship, however, between leakoff rate and pressure 
should be derived using superposition of the total 
pressure hi story in the fracture. 6 Thus, the tech­
nique proposed below provides a first-order correction 
for pressure-dependent leakoff. 
SUMMARY OF EQUATIONS FOR PRESSURE DEPENDENT LEAKOFF 
Determination of Pressure Dependent Leakoff 
Coefficient 
The general form of the 1 eakoff coefficient as a 
function of fracture pressure above reservoir pressure 
can be expressed as,s


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Coefficient 
The general form of the 1 eakoff coefficient as a 
function of fracture pressure above reservoir pressure 
can be expressed as,s 
C(z,t) = M • (~Pf,r(z,t) )u (5) 
The ~pf r(z,t) term is the difference between 
fracture pres~ure (Pf) and reservoir pressure (Pr) at 
a given position in the fracture (z) at a given time 
(t), i.e.: 
~pf r(z,t) = Pf(z,t) -Pr = , 
Pn(z,t) + Pc -Pr 
where, 
Pn = net pressure above closure 
Pc = closure pressure (6) 
The coefficients, M and u, are functions of the 
leakoff mechanism, and are given below for three cases 
of 1 eakoff control where pressure-dependence is si g­
nificant:2,5 
I: Reservoir Permeability and Compressibility 
u = 1 
M = .001183 • lkr • $r • cr I ~r 
where, 
kr = permeability to reservoir fluid, md 
$r = porosity containing reservoir fluid 
cr = compressibility of reservoir fluid, 
~r = reservoir fluid viscosity, cp 
II: Viscosity of Fluid Filtrate 
u = 112 
M = .001483 • v'kf • $f I ~f psi-1 where,


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~r = reservoir fluid viscosity, cp 
II: Viscosity of Fluid Filtrate 
u = 112 
M = .001483 • v'kf • $f I ~f psi-1 where, 
kf =permeability to fracture fluid filtrate, md 
$f =porosity containing fracture fluid filtrate 
~f = fracture fluid filtrate viscosity, cp 
III: Incompressible Filter Cake 
u = 112 
M = function of filter cake properties 
An estimate of the total effective leakoff coef­
ficient as a function of wellbore net pressure (Pnw) 
can be approximated by 
C( t) = M • ( Pnw( t) • 8 + pc -pr )u (7) 
where the 8 term is defined by Nolte as the ratio of 
average net pressure in the fracture to well bore net 
pressure.1'2 Values for 8 are provided in Appendix A 
(8 = 8p during pumping, 8 = 8s during shut-in). 
Equation 7 can be used to estimate C when M and u have 
been determined. The exponent u is selected based on 
the supposed primary 1 eakoff mechanism. The coeffi­
cient M may be graphically determined as discussed in 
the next sections.


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the supposed primary 1 eakoff mechanism. The coeffi­
cient M may be graphically determined as discussed in 
the next sections. 
Determination of M for Reservoir Leakoff Control 
The linear equation below was derived (see Appen­
dix B) for the case where the primary mechanism of 
1 eakoff control is from the reservoi r• s permeabi 1 ity 
and compressibility (i.e. u = 1): 
-1r•M•f •8s·lf 
ln[Pnw( ~t) •8s + Pc -Pr] = p p • G(t0,0) 
2 •Cf 
+ ln[Pnw(O) •Ss + Pc -Pr] (8) 
A plot of ln[Pnw(~t)•8s + Pc -Pr] against G 
should yield a straight line during closure with slope 
proportional to M and a y-intercept from which a 
theoretical ISIP can be inferred. After determining 
the slope and y-i ntercept of this plot, M can be 
calculated from 
M - -SLOPE •2 •cf 
lT•f 0 8S •ft p p 
and a theoretical ISIP can be calculated from 
exp(yint) -Pc + Pr 
ISIP = -------- + Pc 
8S (9) 
(10) 
Notice that calculation of the term ln[Pnw(~t) •8s 
+ Pc -Pr] requires an estimate of closure pressure


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ISIP = -------- + Pc 
8S (9) 
(10) 
Notice that calculation of the term ln[Pnw(~t) •8s 
+ Pc -Pr] requires an estimate of closure pressure 
and reservoir pressure. Ideally, closure pressure 
will have been estimated from a stress test and reser­
voir pressure wi 11 have been estimated from a tran­
sient pressure test. 
Determination of M for Filtrate Viscosity or Incom­
pressible Filter-Cake Leakoff Control 
The equation below was derived (see Appendix B) 
for the case where the primary mechanism of 1 eakoff 
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4 MODIFIED FRACTURE PRESSURE DECLINE ANALYSIS 
INCLUDING PRESSURE-DEPENDENT LEAKOFF SPE 16417 
control is from the viscosity of fracture fluid fil­
trate or from an incompressible filter cake (i.e. u ~ 
1/2): ~ 
[Pnw(~t)•as + Pc-Pr]1/2 = 
+ [Pnw(O)•as + Pc ~ Pr]1/2 ( 11) 
A plot of [Pnw(~t) •as + Pc -Pr]1/2 against G 
should yield a straight line during closure with slope 
proportional to M and a y-intercept from which a 
theoretical ISIP can be inferred. After determining 
the slope and y-i ntercept of this plot, M can be 
calculated from 
SLOPE•4•cf 
M = -~~~--..:.-
'JT•f •as·lt p p (12) 
and a theoretical ISJP can be calculated from 
2 (yint) -Pc + Pr 
ISIP = -----....,..-.- + Pc (13) 
as 
Notifh that cal cul ati on of the term [Pnw( 4t) • as + 
Pc -PrJ requires an estimate of closure pressure 
and reservoir pressure. 
Fluid Efficiency 
The following equation was derived to include 
pressure dependent l eakoff effects in Nol te• s fluid


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and reservoir pressure. 
Fluid Efficiency 
The following equation was derived to include 
pressure dependent l eakoff effects in Nol te• s fluid 
efficiency calculation 2 based on dimensionless closure 
time (t0c): 
.,.b -~2 -4ac 
eff = (14) 
2a 
where, 
a gdl gdu 
b gdu 2. gdl + 1 -Ks/ j c = gdl 1 (15) 
and, 
gdl gl(tDc) 
--"--~ and gdu (16) 
gl ( O) 
Ks = Nolte•s correction for spurt (usually 1.0) 
Also, 
j (17) Pnws = wellbore net pressure at shut-in 
m = slope of Nolte-Smith plot 
Pc = closure pressure 
Pr = reservoir pressure 
The terms g1 and gu are the lower and upper 
bound, respectively, for Nolte•s dimensionless loss­
volume function.1'2 Equations for these functions can 
be found in Appendix A. Note that when pressure­
dependence becomes negligible, the exponent u ap­
proaches 0 and j in Equation 17 approaches unity. 
APPLICATION 
Recall that Example 2 was characterized by a 
declining leakoff coefficient during closure (Figure


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APPLICATION 
Recall that Example 2 was characterized by a 
declining leakoff coefficient during closure (Figure 
6). Since the reservoir in this example was liquid 
filled, and since the wall-building properties of the 
fluid were negligible, it was inferred that leakoff 
control was primarily due to the reservoir. There­
fore, Equation 8 was used to construct the specialized 
plot in Figure 9. The derivative plot (Figure 10) 
shows that Figure 9 is linear during the closure 
period. This implies that Equation 8 pro vi des a 
linear relation during closure between functions of 
pressure and time when leakoff is reservoir con­
trolled. Table 1 compares analysis results between 
Equations 1 and 2 (pressure independence) and Equa­
tions 7, 8, and 9 (pressure dependence). Notice that 
if data near closure had been used in the Nolte type­
curve match, a P* value of 1,270 psi would have been 
estimated resulting in a C value of .0038. This value


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curve match, a P* value of 1,270 psi would have been 
estimated resulting in a C value of .0038. This value 
is 25% less than the value calculated from Equation 7 
when pressure-dependence is considered. 
CONCl-USIONS 
A new plot (G-Function plot) has been introduced 
that is superior to the square-root of time plot. The 
square-root plot uses an approximate relationship 
between pressure and time, whereas the G-Function plot 
uses a more precise relationship (gquation 5), When 
the derivative of the G-Function plot is graphed 
agC!i nst time, phenomena such as fracture extension 
after shut-in, fracture closure, and pressure-depen ... 
dent 1 eakoff can be observed, To account for pres­
sure-dependence, first-order approximation functions 
are provided (Equations 8, 11) to correct for a de­
clining leakoff coefficient during closure. These 
functions tend to linearize the relation between 
pressure and time during the closure period. As shown


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functions tend to linearize the relation between 
pressure and time during the closure period. As shown 
in Example 2, ignoring pressure dependence can result 
in an optimistic estimate of the leakoff coefficient. 
NOMENCLATURE 
a = viscosity degradation coefficient 
Af = fracture area, ft2 
cf = fracture compliance for width, ft/psi 
C = leakoff coefficient, ft/lrnln 
E1 = plane-strain modulus, psi 
eff = 
f = p 
f(tp) 
g (to) = fluid efficiency 
ratio of permeable area to gross area 
dimensionless loss-rate function 
dimensionless loss-volume function 
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SPE 16417 J. L. CASTILLO 5 
G(t0,t*0) dimensionless difference function 
Hg =gross fracture height, ft 
!SIP = instantaneous shut-in pressure, psi 
k = permeability, md 
L = fracture half-length, ft 
M = leakoff proportionality constant 
n• behavior index of fracturing fluid 
P pressure, psi 
Pc closure pressure, psi 
Pr = reservoir pressure, psi 
Pn = net pressure above closure, psi 
Pnw = wellbore net pressure, psi 
P* = Nolte Match Pressure, psi 
~t = delta time since shut-in, minutes 
~t* = reference time since shut-in, minutes 
t = p pump time, minutes 
to = dimensionless delta time ( t.t/tp) 
u = exponent for leakoff 
Vi = injected volume , ft 3 
w = average fracture width, ft 
z ::: position in fracture 
S = ratio of average net pressure to wellbore 
net pressure 
~P(~t,~t*) = pressure difference between ~t and ~t* 
~Pf,r =difference between fracture and reservoir 
pressure 
~s = Nolte•s correction for spurt 
~ = porosity 
~ = viscosity, cp 
REFERENCES


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~Pf,r =difference between fracture and reservoir 
pressure 
~s = Nolte•s correction for spurt 
~ = porosity 
~ = viscosity, cp 
REFERENCES 
1. Nolte, K. G.: 110etermination of Fracture Parame­
ters from Fracturing Pressure Decline Analysis, .. 
paper SPE 8341 presented at the 1979 SPE Annual 
Technical Conference and Exhibition, Las Vegas, 
Sept. 23-26. 
2. Nolte, K. G.: 11A General Analysis of Fracturing 
Pressure Decline Analysis With Application to 
Three Models,11 SPEFE (Dec. 1986) 571-583. 
3. 
4. 
5. Elbel, J. L. et al.: 11Stimulation Study of Cot­
tage Grove Formation, .. J. Pet. Tech. (July 1984) 
1199-1205. 
Nolte, K. G.: 11Fracture Design Considerations 
Based on Pressure Analysis ,11 paper SPE 10911 
presented at the SPE Cotton Valley Symposium, 
Tyler, May 1982. 
Howard, G. C. and Fast, C. R.: 
ing, Monograph Series, SPE, 
( 1970) , Vo 1 • 2. Hydraulic Fractur­
Richardson, Texas 
6. Nolte, K. G.: Personal communication, Feb. 1987. APPENDIX A


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( 1970) , Vo 1 • 2. Hydraulic Fractur­
Richardson, Texas 
6. Nolte, K. G.: Personal communication, Feb. 1987. APPENDIX A 
Calculation of Fracture Compliance (cf): 
The fracture•s compliance for width is given by 
Nolte as: 
'IT. ss 
cf = -~ • 
2 •E I r- -
Hg PK 
2 •L 
32 
--z•L 
31T --KGD 
Radial (A1) 
where E1 is the plane-strain modulus, Hg is gross 
fracture height, Lis hydraulic half-length, and Sis 
the ratio of average net pressure in the fracture to 
the well bore net pressure at shut-in. Nolte reports 
values of s during closure (ss) and during pumping 
( Sp) as f o ll ows : 
[2n+2)./(2n.+3+a~· PK Ss 0. 9 KGD 
0.925 Radial 
and, 
n• 
a ~2n+2)/(3n+3+aJ.· PK·. Sp = ~ 0.85 KGD 
? Radial 
Behavior index of fracture fluid 
Nolte viscosity degradation correction 
0 for no degradation 
1 for moderate degration 
2 for severe degradation (A2) 
(A3) 
The Sp value for the radial model is expected to 
be much less than unity.2 
Once efficiency has been determined, the following


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(A3) 
The Sp value for the radial model is expected to 
be much less than unity.2 
Once efficiency has been determined, the following 
equation from Nolte2 can be used to estimate fracture 
half-length (L): 
r-2 ~ 
Hg •L PK 
Vi •eff•E1 2 
ss . 2 •Hg •L KGD (A4) 
16 3 -·L Radial 
31T 
where, '---
injected vo 1 ume 
net pressure immediately after shut-in 
Once compliance has been determined, average 
hydraulic width in the fracture at shut-in can be 
estimated from: 
(A5) 
Calculation of the G-Function: 
The G-Function is calculated from Nolte•sl,2 
dimensionless loss-volume function (g) as follows: 
4 
• [g(t0) .. g(O)] (A6) 
'IT 
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6 MODIFIED FRACTURE PRESSURE DECLINE ANALYSIS 
INCLUDING PRESSURE-DEPENDENT LEAKOFF SPE 16417 
where g is defined as 
(A7) 
The upper bound (gu) and lower bound (g1) for g are 
gu(X) 4/3{(l+X)1•5 -x1•5} (eff+l) 
(AB) 
g1(x) (l+X)sin-l(l+X)-.5 + x·5 (eff+O) 
APPENDIX B 
Determination of the M Coefficient: 
Nolte2 developed the following equation for calcu­
lating leakoff rate during closure: 
2•C•f •Af 
q( t) p 
~ 
eff+l 
(Bl) 
eff+O 
By substituting Equation 7 for C and replacing the 
bracketed term in Equation Bl by f(tD) results in 
q(t) = 
2•M•{Pnw(t)•Ss + Pc-Pr}u •f •A -------------l.:...p____;_f • f(tD) 
~ (B2) If fracture area (Af) is assumed to remain essen­
tially constant during closure such that leakoff area 
does not change, and such that the fracture• s com­
pliance for width remains constant, then the rate of 
change of fracture volume can be related to the rate 
of change of average fracture width (w): 
dw 
-Af ·-= q(t) 
dt (B3)


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change of fracture volume can be related to the rate 
of change of average fracture width (w): 
dw 
-Af ·-= q(t) 
dt (B3) 
Based on Nolte• s definition of compliance, the 
rate of change in average fracture width can be re­
lated to the rate of change in net pressure: 
dw dPnw 
(B4) 
dt dt 
Substituting B3 and B4 into B2 cancelling Af, and 
rearranging terms results in an equation that relates 
rate of change in pressure to dimensionless time: 
-dPnw 
dt 2•M•fp•{Pnw•Ss + Pc-Pr}u 
Cf•~ 
Separating variables results in: 
-dPnw 2•M•f ·It 
_ ____.,P _ _._P • f ( t D) d t D 
cf {Pnw•Ss + Pc -Pr}u (B5) 
(B6) 
Integrating B6 between shut-in (tD=O) and some 
later time using u=l and u=l/2 provides the relations 
presented in this paper. 
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SPE 1 6 4 1 ~ I 
TABLE 1 
PKN MODEL 
Pressure Pressure 
Parameter Independence Dependence 
efficiency 0.137 0.113 
derivative 1,510 psi 0.73 
@ shut-in (see Figure 6) (see Figure 10) 
derivative 1,270 psi 0.73 
@ closure (see Figure 6) (see Figure 10) 
.0038 ( ft/ /min) 
(near closure) 
Leakoff Coeff. .0051 ( ft/ /min) 
(C) 
.0045 (ft/lmin) 
(near shut-in) 
Half-Length 154 ft 127 ft 
(see Equation A4) (see Equation A4) 
Mean Width .112 in. .112 in. 
@ shut-in (see Equation A5) (see Equation A5) 
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6,000 
5,aoo 
,_ 
·;;; 
~ 5,600 
LaJ 
l:t: 
::::> 
Vl 5,400 
Vl 
LaJ 
l:t: a.. 
5,200 
5,000 
0 0.5 1.0 1.5 2.0 
G FUNCTION 
Figure 1. Pressure vs. the G-Function (Example 1). 
~ 600 
a. End of -Extension Fracture Closed 
LaJ / > 
~ 500 
> 
~ 
LaJ c 
1-400 
z 
0 a.. 
I 
LO 300 
(!) 
"C 
"200~----~----~------~----~----~----~ 
~ 0 10 20 30 40 50 
·DELTA TIME SINCE SHUT-IN (min) 
Figure 2. G-Function Derivative Plot (Example 1). 
5,aoo 
,_ 
-;;; 
~ 5,600 
LaJ 
l:t: 
:::::J 
~ 5,400 
LaJ 
l:t: a.. 
5,200 60 
5,ooo o'-----'-----2~---'-3 ---.l...4----.J...5 ___ .J...6 ___ _._7 _ ____. a 
SQUARE ROOT OF DELTA TIME 
Figure 3. Square-Root Plot (Example 1). SP£ 1 6 4l7i I 
1-200 a 
....J 
D.. Fracture Closed 
1-a:= a I 
Vl 
1.&.. 150 
a 
LaJ > i= 
~ > 100 ~ 
LaJ c 
1-z 
0 a.. 
I 
LO 10 20 30 40 50 60 
DELTA TIME SINCE SHUT-IN 
Figure 4. Square-Root Derivative Plot (Example 1). 
6,200 
6,000 
,_ 
"iii 
~ 5,aoo 
LaJ 
l:t: 
::::> 
Vl 5,600 Vl 
LaJ 
l:t: a.. 
5,400


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Figure 4. Square-Root Derivative Plot (Example 1). 
6,200 
6,000 
,_ 
"iii 
~ 5,aoo 
LaJ 
l:t: 
::::> 
Vl 5,600 Vl 
LaJ 
l:t: a.. 
5,400 
5,200 L..._ __ ___l ____ __.J.. ___ .....J.... __ .....J._ ___ ~---...l------' 
0 0.1 0.2 0.3 0.4 .0.5 0.6 0. 7 
G FUNCTION 
Figure 5. Pressure vs. the G-Function (Example 2). 
~ 2,500 .--..----------------------------------, 
a. -
LaJ > i= 2,000 
~ 
~ 
LaJ c 
1-1,500 
z 
0 a.. 
I 
LO 1,000 
(!) 
"C ' 500 a.. 
"C 0 
Figure 6. Fracture Closed 
2 3 4 5 6 7 a 9 10 11 12 
DELTA TIME SINCE SHUT -IN (min) 
G-Function Derivative Plot (Example 2). 
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6,000 -"iii 
~ 5,800 
LW 
~ 
:::J 
Vl 5,600 Vl 
LW 
[t: 
D... 
5,400 
..... 350 0 
.....1 
D... 
..... [t: 
0 300 Vl 
La.. 
0 
w 250 > 
~ ct > 
~ w 200 c 
..... :z 
0 
D... 
I Ill 
Figure 0.5 1.0 1.5 2.0 2.5 3.0 3.5 
SQUARE ROOT OF DELTA TIME 
Figure 7. Square-Root Plot (Example 2). 
2 3 4 5 6 7 a 9 10 11 12 
DELTA TIME SINCE SHUT -IN 
8. Square-Root Derivative Plot (Example 2). 
281 SP'E 1 6 4 1 ?1 ~ 
7.5 :z 
0 
~ 7.4 u :z 
:::J 
La.. 
..... 7.3 
:z-wu 
~:::J 7.2 
W~:t: n...o 
~La.. 7.1 
w 
[t: 
:::J 7.0 Vl 
Vl w 
[t: 6.9 a.. 0 0.1 0.2 0.3 0.4 0.5 0.7 0.6 
G FUNCTION 
Figure 9. Pressure Function (u=l) vs. the G-Function 
(Example 2) • 
z 1.4 
0 
~ u z 1.2 :::J 
La.. 
La.. 
0 1.0 
Wll 
>:::J 
~[t: 
~0 o.a 
~I.&.. 
w c 
..... 0.6 z 
0 
D... 
I 0.4 Ill 0 
Figure 10. 2 3 4 s s 7 a 9 10 11 12 
DELTA TIME SINCE SHUT -IN (min) 
Pressure Function (u=l) Derivative Plot


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..... 0.6 z 
0 
D... 
I 0.4 Ill 0 
Figure 10. 2 3 4 s s 7 a 9 10 11 12 
DELTA TIME SINCE SHUT -IN (min) 
Pressure Function (u=l) Derivative Plot 
( Ex amp 1 e 2 ) • Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/87LPR/All-87LPR/SPE-16417-MS/3233786/spe-16417-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS
Best Practices in DFIT Interpretation - Comparative Analysis of 80 DFITs in
the Canadian Montney Shale Play
Claudio Virues , Alexandra Robertson, and Bruce Sendecki , Alberta Energy Regulator
Copyright 2023, Society of Petroleum Engineers DOI 10.2118/215017-MS
This paper was prepared for presentation at the 2023 SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 16 - 18 October 2023.
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Abstract
Current practice for conducting minifrac or diagnostic fracture injection tests has been changing with time.
The Alberta Energy Regulator (AER) has a pressure and deliverability testing directive that has guidance
on those tests. This guidance was updated in late 2021. Many operators from the Canadian Montney shale
play submit minifracs to AER, 80 of these minifracs were summarized statistically in this paper. Closure,
reservoir pressure, and permeability were the main analyzed parameters in the paper scope.
Requirements for pressure and deliverability tests are set out by this newly updated directive. Industry
uses minifrac in horizontal wells with multistage fracture treatments when developing unconventional


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uses minifrac in horizontal wells with multistage fracture treatments when developing unconventional
reservoirs. Closure, initial reservoir pressure is commonly being determined with the common well test.
Holistic or tangent vs. compliance method are compared for closure pressure in this paper. Readers are
eager to replicate results, so for After Closure Analysis (ACA) methods equations that govern the analysis
are provided in detail.
Published or commercially available holistic and compliance models are compared for closure pressure.
Published or commercially available ACA Soliman and Nolte methods are compared for permeability
and reservoir pressure. Horizontal wells in 80 unconventional shale field case studies are presented. The
fact that the signature of closure pressure is not apparent is a significant finding. Similarity of closure
pressure outcomes or estimates for tangent vs compliance is remarkable. Differentiation of reservoir


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pressure outcomes or estimates for tangent vs compliance is remarkable. Differentiation of reservoir
pressure determination for either linear or radial flow are observed. Interpretation on net pressure impact is
evaluated and ultimately differentiation on ACA permeability determination is assessed. Analogous shale
plays operators in USA and current Canadian Montney Shale play operators will find practical and direct
implications. To calculate effective fracture half length and optimize fracture design and well spacing
accurate permeability values are needed. To design hydraulic fractures with geomechanics applications
accurate closure pressure values are essential. To conduct Rate Transient Analysis (RTA) for resources and
reserves assessment in similar unconventional shale plays accurate initial reservoir pressure as an input is
fundamental.
Comparing analytically closure pressure methods in unconventional shale plays is novel. In addition,


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fundamental.
Comparing analytically closure pressure methods in unconventional shale plays is novel. In addition,
practicing engineers or well testing interpreters will find the comparison of different after closure analysis
relevant.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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2 SPE-215017-MS
Introduction
The AER requires pressure data to better manage the Montney resource play by understanding regional
pressure trends. Pressure analysis is also important to confirm the resource estimate used for annual reserves
reporting in ST98 Alberta Energy Outlook. As part of contributions towards modernization, the AER has
revised the pressure and deliverability testing of oil and gas wells directive to include a minifrac testing and
submission requirement. To ensure licensees and operators could submit minifrac results and data, AER
revised the PAS file in early 2020.
Horizontal multistage wells with important volumes of treatment fluids are being used by industry in
the development of unconventional or tight reservoirs. In these reservoirs, build up tests are not providing
a representative initial reservoir pressure post-completion mainly due to extended shut-in time needed.


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a representative initial reservoir pressure post-completion mainly due to extended shut-in time needed.
Instead, minifrac, fracture injection or fall-off test has become the go to well test as far as determination of
initial reservoir pressure in unconventionals.
Proppant-less minifracs are normally executed before the principal hydraulic fracture treatment.
Minifrac's intent is generate a short fracture while injecting in order for the formation to be broken down.
Therefore, while the pressure is falling down, the closure of the short fracture could be observed. Small
injected volumes of typically water, normally less than 16 m3, are used in these low rate tests. Less than
15 minutes are normally used for total injection to be concluded, with the pressure falling (fall-off) over
typically a period of two weeks.
During injection and fall-off periods the data on pressure is collected. Pressure is uploaded, smoothed


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typically a period of two weeks.
During injection and fall-off periods the data on pressure is collected. Pressure is uploaded, smoothed
and filtered for analysis in order to extract stress estimates, pore pressure and permeability. While assisting
production and pressure analysis, being inputs for reservoir models, aiding to reserves quantification or
stimulation assessment, determination of pore pressure and permeability is extremely important. Once
pressure data is analyzed and interpreted, leak-off coefficients, gradient of fracture pressure and closure of
fracture pressure or minimum principal stress (or S min) could aid fracture design.
Geology of Montney
In late 2013, NEB, AER and BC MoNGC assessed collaboratively the Montney geological formation's
saleable, and not conventional oil resources.
In Alberta and British Columbia in the 1950's, industry was focusing only on dolostone/sandstone


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saleable, and not conventional oil resources.
In Alberta and British Columbia in the 1950's, industry was focusing only on dolostone/sandstone
Montney reservoir for oil and gas discovery. Siltstone, which has far greater volume of rock with oil and
gas in it but less permeable, surrounded the standard dolostone/sandstone reservoir at the time. Montney
siltstones had to wait until 2005 to be developed, when drilling horizontals and multistage hydraulic
fracturing came roaring.
The Montney geological formation (Lower Triassic) is vast, approximately ~131,000 km2 (Figure 1).
Its thickness is roughly on average 200 m, while thinning out at its eastern and northeastern flanks while
spreading to over 300 m on the west before commencing with outcrops in the Rocky peaks.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 3
Figure 1—Position of the Montney Formation in the subsurface of Alberta and British
Columbia. (Source: The Ultimate Potential for Unconventional Petroleum from the
Montney Formation in Alberta and British Columbia energy briefing note, November 2013)
Methods
The minifrac analysis is performed before and after closure:
•Pre or before Closure Analysis (PCA)
•(ACA) or After Closure Analysis
Pre-Closure Analysis (PCA)
The subsequent parameters are extracted from the Before Closure Analysis:
•Closure Pressure of the fracture (p c)
•Instantaneous Shut-In Pressure (ISIP)
ISIP = Final Bottomhole Injection Pressure minus Friction Component
•Gradient of ISIP
Gradient of ISIP = ISIP divided by Depth of the Formation
•Net of the Fracture Pressure (Δp net)
Δp net = ISIP minus Closure PressureDownloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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4 SPE-215017-MS
•Fluid Efficiency: accumulated volume within the fracture divided by the total fluid injected.
This paper will review two methods to pick closure:
1.McClure et al (2019)
2.Barree et al (2009)
After Closure Analysis (ACA)
After Closure Analysis is completed on fall-off data compiled after fracture closure. The workflow is akin
to conventional pressure transient analysis.
There are two ACA techniques discussed in this paper: Nolte and Soliman/Craig
Pre-Closure – "Compliance" procedure from McClure et al. (2019)
Figure 2  shows field and simulation data with indication of closure from the tangent method (blue) and
indication of closure from the fracture-compliance method (orange).
Figure 2—Identification of closure from the tangent and compliance
method in field and simulation data (Source: SPE Journal 21)
Diagnostic plots.   Figure 3  (left) reveals a DFIT conducted in the Montney formation. The plot shows


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Diagnostic plots.   Figure 3  (left) reveals a DFIT conducted in the Montney formation. The plot shows
measured calculated bottomhole pressure (BHP). When calculating fracture closure pressure, a hydrostatic
adjustment would be added to convert from wellhead pressure (WHP) to BHP.
Figure 3—BHP Pressure (blue) and injection rate (orange) versus elapsed
time in hours (left) and BHP (blue) and dP/dG (red) versus G-time (right)Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 5
Figure 3  (left) reveals pressure (blue) and injection rate (orange) versus time during injection and initial
shut-in.
Figure 3  (right) reveals pressure BHP (blue) and dP/dG (orange) versus G-time.
Data requirements for applying the procedure from McClure et al. (2019)
Identifying the contact pressure.   Figure 4 exhibits 2 cases of G-function plots where this key signature
is clear or "adequate" or "strong."
Figure 4—G-function plots where McClure closure signature is clear.
("Adequate" (left) vs. "Strong" (right) indication of McClure closure.
Figure 5  indicates G-function plots from two Montney DFITs. The DFIT on the left can be classified as
displaying "weak" indication of closure. The separation between the minimum and maximum dP/dG is not
good. Appraising stress with these types of tests is considered best practice.
Figure 5—"Weak" (left) vs. "No" (right) indication of McClure closure


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good. Appraising stress with these types of tests is considered best practice.
Figure 5—"Weak" (left) vs. "No" (right) indication of McClure closure
The plot on the right can be characterized as displaying "no" indication of closure. The dP/dG plot is
diminishing monotonically downward without modulating upward. Best practice is not to carry on with
estimating stress from these types of tests.
Estimating effective Instantaneous Shut-in Pressure (ISIP).   Figure 6  displays how to detect contact
pressure and determine effective ISIP.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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6 SPE-215017-MS
Figure 6—Identification of contact pressure and effective ISIP (Source: URTeC-2019-123)
Minifrac Observations from real data – McClure et al (2019)
Estimating effective ISIP and S min.   In Figure 7 , contact happens around G-time of ~7.8, contact pressure
or closure pressure estimation is 53,908 kPa and for the S min estimation a subtraction of 515 kPa (75 psi)
from fracture closure will be required. The effective ISIP, ~56,000 kPa, is assessed by identifying the point
of minimum dP/dG and extrapolating the pressure versus G-time curve back to zero from this point. The
effective ISIP is an assessment for the pressure in the far-field fracture at shut-in. This is the pressure actually
present in the fracture that is opening up and spreading the crack.
Figure 7—Well 6 – Identification of contact pressure, minimum dP/dG and effective ISIP
Pre-Closure – "Holistic" procedure from Barree et al. (2009)


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Figure 7—Well 6 – Identification of contact pressure, minimum dP/dG and effective ISIP
Pre-Closure – "Holistic" procedure from Barree et al. (2009)
Estimating closure pressure and Smin.   Barree et al. (2009) proposed penciling in a straight line from
the origin to the tangent to the curve, near the peak, and recognizing this as the time of fracture "closure",
believed to be equal to S min (see Figure 8 ). This is the so-called "tangent" method. Closure event is also
marked by a straight vertical dashed line in Figure 8 .Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 7
Figure 8—G-function plot (Source: SPE 107877)
Supplementary pre-closure interpretations from the "tangent" method.
Leak-off types.
1.Normal Leak-off (NL)  (Figure 9 , left)
2.Transverse Fracture Storage/Fracture Height Recession (TFS/FHR) ( Figure 9, right)
3.Pressure Dependent Leak-off (PDL): (Figure 10, left)
4.Fracture Tip Extension (FTE)  (Figure 10 , right).
Table 1  shows a summary of the leak-off types for all 80 field case studies.
Figure 9—Normal leak-off (left) and transverse fracture storage/fracture height recession leak-off (right) – Source: S&P GlobalDownloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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8 SPE-215017-MS
Figure 10—Pressure dependent leak-off (left) and fracture tip extension leak-off – Source: S&P Global
Table 1—Summary of the leak-off types and whether McClure Closure was observed.
Leak-Off Type McClure Closure Pressure
Well 1 TFS / FHR n/a
Well 2 NORMAL (NL) YES
Well 3 TFS / FHR YES
Well 4 TFS / FHR n/a
Well 5 PDL n/a
Well 6 TFS / FHR YES
Well 7 TFS / FHR YES
Well 8 TFS / FHR YES
Well 10 PDL n/a
Well 11 TFS / FHR YES
Well 12 TFS/FHR n/a
Well 13 PDL n/a
Well 14 PDL n/a
Well 15 PDL n/a
Well 16 PDL n/a
Well 17 PDL n/a
Well 18 TFS / FHR n/a
Well 19 TFS / FHR n/a
Well 20 TFS / FHR n/a
Well 21 PDL n/a
Well 22 TFS / FHR n/a
Well 25 TFS / FHR YES
Well 26 TFS / FHR n/a
Well 27 TFS / FHR YES
Well 28 PDL n/a
Well 29 TFS / FHR n/a
Well 30 TFS / FHR n/aDownloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 9
Leak-Off Type McClure Closure Pressure
Well 31 TFS / FHR YES
Well 32 TFS / FHR YES
Well 33 NORMAL (NL) n/a
Well 35 TFS / FHR n/a
Well 36 TFS / FHR n/a
Well 37 TFS / FHR YES
Well 38 TFS / FHR n/a
Well 39 TFS / FHR n/a
Well 40 TFS / FHR n/a
Well 41 NORMAL (NL) n/a
Well 42 TFS / FHR n/a
Well 44 NORMAL (NL) n/a
Well 45 TFS / FHR n/a
Well 46 NORMAL (NL) n/a
Well 47 TFS / FHR n/a
Well 48 TFS / FHR YES
Well 49 TFS / FHR YES
Well 50 NORMAL (NL) YES
Well 52 TFS / FHR YES
Well 53 TFS / FHR YES
Well 54 FTE YES
Well 55 TFS / FHR YES
Well 56 TFS / FHR YES
Well 57 TFS / FHR YES
Well 58 TFS / FHR YES
Well 59 NORMAL (NL) n/a
Well 60 TFS / FHR YES
Well 61 TFS / FHR YES
Well 62 TFS / FHR YES
Well 63 TFS / FHR YES
Well 64 TFS / FHR YES
Well 65 TFS / FHR YES
Well 66 FTE YES
Well 67 TFS / FHR YES
Well 68 TFS / FHR YES
Well 69 TFS / FHR YES
Well 70 TFS / FHR YES
Well 71 TFS / FHR YES
Well 72 TFS / FHR YES


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Well 66 FTE YES
Well 67 TFS / FHR YES
Well 68 TFS / FHR YES
Well 69 TFS / FHR YES
Well 70 TFS / FHR YES
Well 71 TFS / FHR YES
Well 72 TFS / FHR YES
Well 73 TFS / FHR YESDownloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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10 SPE-215017-MS
Leak-Off Type McClure Closure Pressure
Well 74 TFS / FHR YES
Well 75 TFS / FHR YES
Well 76 TFS / FHR YES
Well 77 TFS / FHR YES
Well 78 TFS / FHR YES
Well 79 TFS / FHR YES
Well 80 TFS / FHR YES
The pivot chart in Figure 11  shows how many leak-off types show indication of McClure closure pressure
vs. when it is not available (n/a). McClure closure is not found in pressure dependent leak-off.
Figure 11—Pivot chart for leak-off types
Minifrac Observations from real data - Barree et al. (2009)
Well 44 - Normal Leak-off G-function Analysis.   The field case in Figure 12 reveals some minor deviation
from the perfect constant leak-off but is a notable example of the anticipated curve shape with a clear-cut
indication of closure at Gc = 7.157.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 11
Figure 12—Well 44 - G-function
Well 44- Normal Leak-off Sqrt(t) Analysis.   The plot in Figure 13 , reveals that the slope of the pressure
curve begins low, then rises and gets a maximum rate of decline at the inflection point, then diminishes
again after closure.
Figure 13—Well 44 - Sqrt(t) plot
Results
Established industry methodologies were compared. No one method is recommended over another. The
analysis performed demonstrates the ranges of outcomes between the alternative approaches.
The results are summarized in Table 2 . Thereafter, we discuss the clarity of the compliance closure
response, a comparison between closure and net pressure from holistic vs. compliance method, estimation
of permeability and pressure, and rate transient analysis implications.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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12 SPE-215017-MS
Table 2—Summary of the statistical comparisons, conditions for inclusion of a test in the comparison, and key findings
Comparisons Condition for Inclusion Key Findings
Comparison of DFITs showing strong,
adequate, weak, or no "compliance" method
indication of closureNo condition; all DFITs were
included20% showed strong indication of closure,
~24% showed adequate indication of closure,
~9% showed weak indication of closure and
~48% showed no indication of compliance
closure.
Comparison of closure pressure
estimated from "compliance" versus
"tangent" methodsMust have strong, adequate "compliance"
indication of closureThe tangent method and compliance
methods provide similar closure pressure within
8% difference.
Comparison of net pressure Must have strong or adequate
‘compliance’ indication
of closureThe net pressure estimated using tangent
closure and ISIP is 10x higher than that estimated
based on compliance closure and effective
ISIP, or an average of ~17,000 kPa.


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closure and ISIP is 10x higher than that estimated
based on compliance closure and effective
ISIP, or an average of ~17,000 kPa.
Comparison of permeability Must have either achieved radial
and/or linear flowsPermeabilities estimates were of the same
order of magnitude for Nolte and
Soliman/Craig.
Comparison of pore pressure Must have either achieved radial
and/or linear flowsPore pressure estimates were of the same
order of magnitude for Nolte and
Soliman/Craig.
Comparison of Xf √k  and
Af√kMust have at least achieved
linear flowEstimates are heavily dependent on assumptions
of net pay. RTA estimates for Af√k
were significantly higher, ~300x.
Discussion
Clarity of the "compliance" closure response
Figure 14  shows the fraction of tests with strong, adequate, weak, and no "compliance" method indication
of closure on the dP/dG plot.
Figure 14—Fraction of tests with strong (yellow), adequate (grey),
weak (orange) and no (red) compliance indication of closure.


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Figure 14—Fraction of tests with strong (yellow), adequate (grey),
weak (orange) and no (red) compliance indication of closure.
Closure pressure from the compliance method and tangent method
Figure 15  shows a comparison of the dP/dG and G*dP/dG plots from a Montney DFIT.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 13
Figure 15—dP/dG as a function of G-time (left) and G*dP/dG as a function of G-time
(right) plots from Montney DFITs (Source: Discussion of the Paper SPE-187038-MS)
The methods yield similar results as far as closure pressure is concerned. compliance closure pressure is
~8% higher for this dataset. Table 3 displays a comparison between compliance closure pressure (McClure
et al 2019 ) vs. the holistic closure pressure.
Table 3—Summary of closure pressure values
Compliance Closure
Pressure (kPa)Holistic Closure
Pressure (kPa)Delta Closure
Pressure (Xx)Delta Closure
Pressure (kPa)
Well 1 n/a 49898
Well 2 40,742 38721 1.1x 2021
Well 3 43,600 39441 1.1x 4159
Well 4 n/a 48658
Well 5 n/a 37136
Well 6 53,908 50979 1.1x 2929
Well 7 45000 40154 1.1x 4846
Well 8 40000 38306 1.0x 1694
Well 9 n/a 46274
Well 10 n/a 40993
Well 11 25500 22849 1.1x 2651
Well 12 n/a 48941
Well 13 n/a 42201
Well 14 n/a 50861
Well 15 n/a 46387
Well 16 n/a 32003
Well 17 n/a 62545
Well 18 n/a 43727


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Well 12 n/a 48941
Well 13 n/a 42201
Well 14 n/a 50861
Well 15 n/a 46387
Well 16 n/a 32003
Well 17 n/a 62545
Well 18 n/a 43727
Well 19 n/a 36657
Well 20 n/a 48642
Well 21 n/a 44425
Well 22 n/a 39274
Well 23 n/a 49202
Well 24 n/a 32177Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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14 SPE-215017-MS
Compliance Closure
Pressure (kPa)Holistic Closure
Pressure (kPa)Delta Closure
Pressure (Xx)Delta Closure
Pressure (kPa)
Well 25 42000 39822 1.1x 2178
Well 26 n/a 40033
Well 27 48000 42779 1.1x 5221
Well 28 n/a 41951
Well 29 n/a 41981
Well 30 n 39572
Well 31 41000 37289 1.1x 3711
Well 32 43000 40722 1.1x 2278
Well 33 n/a 50727
Well 34 n/a 31641
Well 35 n/a 45609
Well 36 n/a 48477
Well 37 52000 49933 1.0x 2067
Well 38 n/a 50366
Well 39 n/a 37965
Well 40 n/a 35101
Well 41 n/a 58321
Well 42 n/a 45132
Well 43 n/a 40327
Well 44 n/a 46743
Well 45 n/a 40408
Well 46 n/a 43833
Well 47 n/a 41213
Well 48 52811 48611 1.1x 4200
Well 49 41000 37157 1.1x 3843
Well 50 50,000 49563 1.0x 437
Well 51 n/a 53873
Well 52 47000 43747 1.1x 3253
Well 53 46250 44040 1.1x 2210
Well 54 45500 43341 1.0x 2159
Well 55 44000 39140 1.1x 4860
Well 56 43000 40475 1.1x 2525
Well 57 59500 53083 1.1x 6417
Well 58 44500 38359 1.2x 6141
Well 59 n/a 55895
Well 60 42000 38344 1.1x 3656


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Well 56 43000 40475 1.1x 2525
Well 57 59500 53083 1.1x 6417
Well 58 44500 38359 1.2x 6141
Well 59 n/a 55895
Well 60 42000 38344 1.1x 3656
Well 61 47000 43332 1.1x 3668
Well 62 42000 37941 1.1x 4059
Well 63 37500 36070 1.0x 1430
Well 64 41000 37720 1.1x 3280Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 15
Compliance Closure
Pressure (kPa)Holistic Closure
Pressure (kPa)Delta Closure
Pressure (Xx)Delta Closure
Pressure (kPa)
Well 65 50020 45652 1.1x 4368
Well 66 39000 37827 1.0x 1173
Well 67 46000 42301 1.1x 3699
Well 68 40000 36382 1.1x 3618
Well 69 44000 40475 1.1x 3525
Well 70 44000 40934 1.1x 3066
Well 71 45000 41406 1.1x 3594
Well 72 41500 39774 1.0x 1726
Well 73 44000 39551 1.1x 4449
Well 74 44500 39371 1.1x 5129
Well 75 47000 42233 1.1x 4767
Well 76 46000 41633 1.1x 4367
Well 77 47000 43466 1.1x 3534
Well 78 43500 39560 1.1x 3940
Well 79 46000 42845 1.1x 3155
Well 80 42500 38906 1.1x 3594
Average 44,472 42,667 1.08x 3,419
Estimating Permeability and Pore Pressure – Nolte and Soliman/Craig
Nolte ACA Derivative.   Figure 16  represents the log-log diagnostic plot for well 2 used to confirm fracture
closure and identify the flow regimes developed after closure.
Figure 16—Well 2 - Log-log derivative (diagnostic) plot – Nolte radial flow


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closure and identify the flow regimes developed after closure.
Figure 16—Well 2 - Log-log derivative (diagnostic) plot – Nolte radial flow
Nolte ACA Linear Flow.   If impulse linear flow is identified in the derivative plot, the fluid loss coefficients,
CR and CT, can also be estimated.
The equations 1 , 2, 3 and 4 governing impulse linear flow are defined as:Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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16 SPE-215017-MS
(1)
(2)
(3)
Figure 17  represents the Well 2 fall-off data using Nolte's linear time function.
Figure 17—Well 2 - Fall-off data using minifrac linear time function for Nolte
Nolte ACA Radial Flow.   The equations 4 , 5, 6 and 7 governing impulse radial flow are defined as:
(4)
(5)
(6)
(7)
Figure 18  represents the Well 3 fall-off data using Nolte's radial time function.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 17
Figure 18—Well 3 - Fall-off data using minifrac radial time function for Nolte
Soliman/Craig ACA Derivative.   Figure 19  represents the log-log diagnostic plot for well 3 used to confirm
fracture closure and identify the flow regimes developed after closure.
Figure 19—Well 3 - Log-log Derivative (diagnostic) plot – Soliman/Craig Analysis radial flow
Soliman/Craig ACA Radial Flow.   The equations 8 , 9, 10 and 11 governing radial flow are defined as:
(8)
(9)
(10)
(11)Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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18 SPE-215017-MS
Figure 20  represents the Well 3 fall-off data using Soliman/Craig's radial time function. When radial flow
is achieved after closure, this plot yields a constant slope. By fitting a straight line through the radial flow
trend, the analysis yields a kh of 0.0353 md.m and k = 1.26E-03md using net pay h = 28.1m.
Figure 20—Well 3 – Soliman/Craig analysis minifrac radial flow
Table 4  summarizes the results of the 80 field cases.
Table 4—Summary of the permeability values for Nolte and Soliman/Craig After Closure Analysis for radial flow and key findings
Nolte Permeability
Radial Flow
(mD)Soliman/Craig
Permeability Radial
Flow (mD)Key Findings
Well 1 3.96E-04 3.69E-04
Well 2 7.67E-05 1.89E-04
Well 3 7.30E-04 7.40E-04
Well 4 4.65E-03 3.72E-03
Well 5 4.30E-02 Radial flow not
achieved (Soliman/Craig)
Well 6 1.80E-04 1.90E-04
Well 7 1.00E-02 9.10E-05
Well 11 3.80E+00 3.24E+00
Well 12 5.97E-03 6.00E-03
Well 14 1.41E-02 1.29E-02
Well 15 1.66E-03 1.60E-03


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Well 7 1.00E-02 9.10E-05
Well 11 3.80E+00 3.24E+00
Well 12 5.97E-03 6.00E-03
Well 14 1.41E-02 1.29E-02
Well 15 1.66E-03 1.60E-03
Well 16 8.07E-02 2.84E-02
Well 20 3.10E-03 3.16-03
Well 23 6.20E-03 2.00E-03
Well 24 4.39E-02 4.36E-02
Well 25 2.46E-04 2.46E-04
Well 26 1.40E-04 1.60E-04
Well 29 3.15E-04 3.04E-04
Well 30 4.23E-03 Radial flow not
achievedDownloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 19
Nolte Permeability
Radial Flow
(mD)Soliman/Craig
Permeability Radial
Flow (mD)Key Findings
(Soliman/Craig)
Well 33 1.00E-07 9.70E-08
Well 34 1.58E-02 1.60E-02
Well 35 1.50E-01 4.00E-02
Well 36 3.98E-03 2.30E-03
Well 37 8.45E+00 5.69E+00
Well 38 1.03E-02 3.00E-03
Well 39 3.20E-04 3.30E-04
Well 41 2.50E-02 Radial flow not
achieved
(Soliman/Craig)
Well 42 1.60E-03 1.60E-03
Well 43 3.10E-01 3.08E-01
Well 44 3.70E-03 3.80E-03
Well 46 3.10E-04 3.10E-04
Well 48 2.30E-02 2.24E-02
Well 50 1.02E-03 Radial flow not
achieved
(Nolte)
Well 62 2.40E-03 2.30E-03
Well 65 1.20E-03 9.90E-04
Well 80 3.96E-04 3.69E-04
The stacked chart in Figure 21  was used to display the relationship of part to whole over Nolte vs Soliman/
Craig categories.


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The stacked chart in Figure 21  was used to display the relationship of part to whole over Nolte vs Soliman/
Craig categories.
Figure 21—Stacked area chart for Nolte vs. Soliman/Craig permeability estimates using radial flow.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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20 SPE-215017-MS
The scatter chart in Figure 22  was used to compare Nolte and Soliman/Craig sets of permeability values
for radial flow. It shows the one-to-one relationship between both sets of values.
Figure 22—Scatter Chart for Nolte vs. Soliman/Craig permeability estimates using radial flow.
The histogram in Figure 23  shows the frequency of occurrence of values within both data sets, Nolte,
and Soliman/Craig. Ranges of values are grouped into bins.
Figure 23—Histogram for Nolte vs. Soliman/Craig permeability estimates using radial flow.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 21
A comparison of pore pressure estimated using Nolte and Soliman/Craig is shown in Table 5.
Table 5—Summary of the pore pressure for Nolte and Soliman/Craig ACA for radial and linear flow.
Nolte ACA Pore
Pressure Radial
Flow (kPa)Nolte ACA Pore
Pressure Linear
Flow (kPa)Soliman/Craig ACA
Pore Pressure Radial
Flow (kPa)Soliman/Craig ACA
Pore Pressure Linear
Flow (kPa)
Well 1 28,013 26,000 28,058 26,560
Well 2 31,187 30,562 31,700 30,467
Well 3 29,735 29,188 29,781 29,287
Well 4 40,377 39,912
Well 5 29,120 29,210
Well 6 43,432 41,413 42,350 41,182
Well 7 31,105 31,101
Well 8 24,639 23,193
Well 11 17,014 15,059 16,895 13,749
Well 12 37,390 36,302 37,397 35,362
Well 14 41,783 40,870
Well 15 39,140 39,133 35,699
Well 16 31,830 31,799 31,674
Well 17 82,400 79,110
Well 18 30,397 30,400
Well 19 27,448 27,370
Well 20 36,971 35,796 37,022 35,228
Well 21 31,085 31,102
Well 22 29,108 28,689
Well 23 41,763 41,758 41,257
Well 24 25,981 25,985
Well 25 26,870 26,871


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Well 21 31,085 31,102
Well 22 29,108 28,689
Well 23 41,763 41,758 41,257
Well 24 25,981 25,985
Well 25 26,870 26,871
Well 26 28,784 28,385 28,844 28,361
Well 27 31,126 31,117
Well 28 26,359
Well 29 31,789 31,780
Well 30 29,287 25,429
Well 31 26,249 26,242
Well 32 30,940 30,937
Well 33 43,429 43,436
Well 34 25,428 25,438
Well 35 34,831 34,287 34,416 34,241
Well 36 41,675 38,289 40,729 35,101
Well 37 42,424 41,407 42,117 41,159
Well 38 46,953 44,777 44,217 38,430
Well 39 28,342 28,077 28,361 28,036Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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22 SPE-215017-MS
Nolte ACA Pore
Pressure Radial
Flow (kPa)Nolte ACA Pore
Pressure Linear
Flow (kPa)Soliman/Craig ACA
Pore Pressure Radial
Flow (kPa)Soliman/Craig ACA
Pore Pressure Linear
Flow (kPa)
Well 40 24,911 22,801 24,866 22,676
Well 41 43,230
Well 42 37,073 37,116 30,547
Well 43 27,083 25,622 27,084 25,924
Well 44 37,914 37,058 37,920 27,154
Well 46 31,267 31,262
Well 47 29,967
Well 48 38,105 36,533 38,072 37,830
Well 49 27,647
Well 50 32,097 32,142 33,210
Well 51 44,104 43,188
Well 52 27,197
Well 53 27,741 27,711
Well 54 30,607
Well 55 30,273
Well 56 29,024
Well 57 39,255 37,361
Well 58 25,492
Well 59 24,103
Well 60 29,127 29,106
Well 61 32,655 32,593
Well 62 28,748 28,740
Well 63 26,154 23,335 26,162 23,276
Well 64 28,904 28,896
Well 65 38,074 35,506 37,819 34,912
Well 66 25,348 25,385
Well 67 29,848
Well 68 29,482 29,454
Well 69 29,024
Well 70 25,001 24,986
Well 71 27,632 27,564
Well 72 28,010 27,998
Well 73 28,908
Well 74 26,322 26,286
Well 76 27,551 27,559
Well 77 28,927


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Well 70 25,001 24,986
Well 71 27,632 27,564
Well 72 28,010 27,998
Well 73 28,908
Well 74 26,322 26,286
Well 76 27,551 27,559
Well 77 28,927
Well 78 30,112 30,087
Well 79 28,700 28,693
Well 80 24,518 24,504Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 23
The clustered column chart in Figure 24 shows slightly higher estimates for radial flow vs. linear flow
for Nolte and Soliman/Craig methods.
Figure 24—Clustered column chart for Nolte vs. Soliman/Craig pressure estimates for linear and radial flow.
Net pressure from the compliance method and tangent method
The ‘compliance net pressure’ is calculated as the ‘effective ISIP’ minus the estimate for compliance
closure pressure for the compliance closure the ‘holistic net pressure" is calculated as the ‘ISIP’ minus the
estimate for holistic closure pressure. It is an estimate for the pressure driving crack propagation through
the formation – the amount that pressure exceeds closure pressure.
The holistic net pressures are on average about ~10x higher (~17,000 kPa higher). This comparison uses
the ~44% of the tests for which the compliance method closure pick was assessed as being "strong" or


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the ~44% of the tests for which the compliance method closure pick was assessed as being "strong" or
"adequate". Table 6  shows a summary of the net pressure values for Compliance and Holistic methods.
Table 6—Summary of Net Pressures using Holistic and Compliance with inputs.
Holistic
ISIP (kPa)Compliance
(Effective)
ISIP (kPa)Holistic Closure
Pressure (kPa)Compliance
Closure
Pressure (kPa)Holistic Net
Pressure (kPa)Compliance Net
Pressure (kPa)
Well 1 66,234 n/a 49,898 n/a 16,336 n/a
Well 2 67,916 45,500 38,721 40,742 29,195 4,758
Well 3 63,444 45,000 39,441 43,600 24,003 1,400
Well 4 55,564 n/a 48,658 n/a 6,906 n/a
Well 5 46,328 n/a 37,136 9,192 0
Well 6 64,666 56,000 50,979 53,908 13,687 2,092
Well 7 76,254 46,000 40,154 45,000 36,100 1,000
Well 8 47,087 42,500 38,306 40,000 8,781 2,500
Well 9 52,431 n/a 46,274 n/a 6,157 n/a
Well 10 67,785 45,000 40,993 n/a 26,792 n/a


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Well 8 47,087 42,500 38,306 40,000 8,781 2,500
Well 9 52,431 n/a 46,274 n/a 6,157 n/a
Well 10 67,785 45,000 40,993 n/a 26,792 n/a
Well 11 26,751 26,500 22,849 25,500 3,902 1,000Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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24 SPE-215017-MS
Holistic
ISIP (kPa)Compliance
(Effective)
ISIP (kPa)Holistic Closure
Pressure (kPa)Compliance
Closure
Pressure (kPa)Holistic Net
Pressure (kPa)Compliance Net
Pressure (kPa)
Well 12 56,048 54,000 48,941 n/a 7,107 n/a
Well 13 45,975 n/a 42,201 n/a 3,774 n/a
Well 14 72,438 59,000 50,861 n/a 21,577 n/a
Well 15 53,541 50,000 46,387 n/a 7,154 n/a
Well 16 32,385 32,325 32,003 n/a 382 n/a
Well 17 101,279 n/a 62,545 n/a 38,734 0
Well 18 68,256 50,000 43,727 n/a 24,529 n/a
Well 19 81,662 43,500 36,657 n/a 45,005 n/a
Well 20 55,261 52,500 48,642 n/a 6,619 n/a
Well 21 56,144 48,000 44,425 n/a 11,719 n/a
Well 22 42,903 n/a 39,274 n/a 3,629 n/a
Well 23 58,341 55,000 49,202 n/a 9,139 n/a
Well 24 37,224 n/a 32,177 n/a 5,047 n/a
Well 25 76,049 44,000 39,822 42,000 36,227 2,000
Well 26 45,008 44,000 40,033 n/a 4,975 n/a
Well 27 72,672 49,000 42,779 48,000 29,893 1,000
Well 28 73,982 n/a 41,951 n/a 32,031
Well 29 68,290 45,750 41,981 n/a 26,309 n/a
Well 30 63,821 n/a 39,572 n/a 24,249 0


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Well 28 73,982 n/a 41,951 n/a 32,031
Well 29 68,290 45,750 41,981 n/a 26,309 n/a
Well 30 63,821 n/a 39,572 n/a 24,249 0
Well 31 51,506 41,750 37,289 41,000 14,217 750
Well 32 77,376 46,000 40,722 43,000 36,654 3,000
Well 33 66,448 n/a 50,727 n/a 14,217 n/a
Well 34 42,298 n/a 31,641 n/a 10,657 n/a
Well 35 68,408 54,500 45,609 n/a 16,740 n/a
Well 36 59,604 55,750 48,477 n/a 11,127 n/a
Well 37 55,169 54,500 49,933 52,000 5,236 2,500
Well 38 54,820 54,500 50,366 n/a 4,454 n/a
Well 39 68,227 47,500 37,965 n/a 30,262 n/a
Well 40 42,880 39,500 35,101 n/a 7,779 n/a
Well 41 62,933 n/a 58,321 n/a 4,612 n/a
Well 42 52,207 51,000 45,132 n/a 7,075 n/a
Well 43 48,682 n/a 40,327 n/a 8,355 n/a
Well 44 54,807 54,000 46,743 n/a 8,064 n/a
Well 45 47,548 45,500 40,408 n/a 7,140 n/a
Well 46 67,523 n/a 43,833 n/a 23,690 n/a
Well 47 53,237 n/a 41,213 n/a 12,024 n/a
Well 48 53,638 54,000 48,611 52,811 5,027 1,189
Well 49 77,084 44,000 37,157 41,000 39,927 3,000


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Well 47 53,237 n/a 41,213 n/a 12,024 n/a
Well 48 53,638 54,000 48,611 52,811 5,027 1,189
Well 49 77,084 44,000 37,157 41,000 39,927 3,000
Well 50 52,700 52,500 49,563 48,500 3,137 4,000Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 25
Holistic
ISIP (kPa)Compliance
(Effective)
ISIP (kPa)Holistic Closure
Pressure (kPa)Compliance
Closure
Pressure (kPa)Holistic Net
Pressure (kPa)Compliance Net
Pressure (kPa)
Well 51 90,441 n/a 53,873 n/a 36,568 0
Well 52 71,593 48,250 43,747 47,000 27,846 1,250
Well 53 68,659 48,000 44,040 46,250 24,619 1,750
Well 54 77,548 49,500 43,341 45,500 34,207 4,000
Well 55 62,712 47,500 39,140 44,000 23,572 3,500
Well 56 70,992 44,250 40,475 43,000 30,517 1,250
Well 57 63,140 61,000 53,083 59,500 10,057 1,500
Well 58 57,153 46,000 38,359 44,500 18,794 1,500
Well 59 63,170 n/a 55,895 n/a 7,275 0
Well 60 67,694 43,500 38,344 42,000 29,350 1,500
Well 61 72,766 48,000 43,332 47,000 29,434 1,000
Well 62 65,552 43,000 37,941 42,000 27,611 1,000
Well 63 41,622 40,000 36,070 37,500 5,552 2,500
Well 64 79,428 44,000 37,720 41,000 41,708 3,000
Well 65 51,418 48,000 45,652 50,020 5,766 -2,020
Well 66 72,990 47,250 37,827 39,000 35,163 8,250
Well 67 60,842 47,750 42,301 46,000 18,541 1,750


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Well 65 51,418 48,000 45,652 50,020 5,766 -2,020
Well 66 72,990 47,250 37,827 39,000 35,163 8,250
Well 67 60,842 47,750 42,301 46,000 18,541 1,750
Well 68 60,527 41,000 36,382 40,000 24,145 1,000
Well 69 70,992 44,500 40,475 44,000 30,517 500
Well 70 76,096 48,000 40,934 44,000 35,162 4,000
Well 71 71,998 46,000 41,406 45,000 30,592 1,000
Well 72 69,560 44,500 39,774 41,500 29,786 3,000
Well 73 72,033 44,500 39,551 44,000 32,482 500
Well 74 78,979 45,000 39,371 44,500 39,608 500
Well 75 60,774 48,000 42,233 47,000 18,541 1,000
Well 76 70,560 47,000 41,633 46,000 28,927 1,000
Well 77 75,124 47,250 43,466 47,000 31,658 250
Well 78 71,225 47,500 39,560 43,500 31,665 4,000
Well 79 57,788 47,000 42,845 46,000 14,943 1,000
Well 80 53,923 43,000 38,906 42,500 15,017 500
Average 18,939 1,908
In hydraulic fracturing, higher net fracture pressures can have several implications:
1.Enhanced fracture growth: high net fracture pressures help create and extend fractures in the target


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1.Enhanced fracture growth: high net fracture pressures help create and extend fractures in the target
formation, allowing for greater contact with the reservoir and improved flow of oil and gas.
2.Increased production rates: by generating larger and longer fractures, high net fracture pressures can
enhance the productivity of a well, leading to increased production rates of oil and gas.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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26 SPE-215017-MS
3.Reservoir connectivity: high net fracture pressures can help establish better connectivity between the
wellbore and the reservoir, enabling the extraction of hydrocarbons from a larger area within the
formation.
4.Risk of fracture propagation: while high net fracture pressures can be advantageous, they also carry
the risk of unintended fracture propagation.
5.Hydraulic fracturing fluid requirements: achieving high net fracture pressures may require larger
volumes of hydraulic fracturing fluid, which can pose logistical challenges, increase costs and
potentially impact water availability in some regions.
Rate Transient Analysis
Rate Transient Analysis was conducted to make a comparison on values of Af√ k estimates using DFIT
ACA methodologies. The periods analyzed for RTA ranged from 1 day up to 75 days of production and
pressure data. Figure 25  shows a typical production and pressure profile for the wells analyzed.


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pressure data. Figure 25  shows a typical production and pressure profile for the wells analyzed.
Figure 25—Well 36 – Typical production and pressure profile used for rate transient analysis.
Soliman/Craig ACA Linear Flow.   The equations 12, 13, 14 and 15 governing linear flow are defined as:
(12)
(13)
(14)
(15)
Figure 26  shows minifrac linear flow plot for Soliman/Craig.Downloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 27
Figure 26—Well 3 - Soliman / Craig Analysis – Linear Flow
Table 7  shows a summary of the Soliman -Craig estimates of Xf√k, Af√k and RTA estimates of Af√k
(linear flow). Wells with no or insufficient data are shown as non/available (n/a).
Table 7—Summary of the Soliman -Craig estimates of Xf√k, Af√k and RTA estimates of Af√k (linear flow)
Soliman / Craig
Linear Flow
Xf √ k  (m*√md)Assumed h
Net Pay (m)Soliman / Craig Linear
Flow Af√k
(m2 √md)RTA Linear Flow
Af√k
(m2 √md)
Well 1 0.03 45 1.35 7,595
Well 6 0.06 15 0.9 788
Well 9 9.64 45 434 n/a
Well 12 0.39 15 5.85 13,753
Well 15 0.05 10 0.5 41,668
Well 16 2.87 15 43.05 3,970
Well 20 0.21 15 3.15 4,805
Well 21 0.34 20 6.80 n/a
Well 26 0.05 15 0.75 1,163
Well 35 5.68 20 113.6 769
Well 36 0.14 15 2.1 5,748
Well 37 70.87 20 1417 n/a
Well 38 0.17 20 3.4 31,278
Well 39 0.11 15 1.65 687
Well 40 0.2 20 4 2,842
Well 42 0.07 39 2.73 10,705
Well 43 7.55 20 151 n/a
Well 44 0.11 53 5.83 n/a
Well 46 0.07 40 2.8 4,428


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Well 40 0.2 20 4 2,842
Well 42 0.07 39 2.73 10,705
Well 43 7.55 20 151 n/a
Well 44 0.11 53 5.83 n/a
Well 46 0.07 40 2.8 4,428
Well 48 1.16 15 17.4 3,748
Well 50 0.12 20 2.4 n/aDownloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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28 SPE-215017-MS
Soliman / Craig
Linear Flow
Xf √ k  (m*√md)Assumed h
Net Pay (m)Soliman / Craig Linear
Flow Af√k
(m2 √md)RTA Linear Flow
Af√k
(m2 √md)
Well 65 0.18 17 3.06 6,224
Conclusions
This study reviews results from 80 DFITs from different operators in the Canadian Unconventional Montney
shale play. The key findings are as follows:
a.A ‘compliance method’ closure/stress estimate is not apparent in the significant majority of DFITs in
the Canadian Montney formation.
b.The compliance method and tangent method provide similar closure pressures. The compliance
closure pressure was ~8% higher than the tangent closure pressure on average. To estimate the
minimum principal stress, a subtraction of a pressure constant from the compliance closure pressure
is suggested by McClure et al. 2019 .
c.The clarity of the ‘compliance method’ closure response varies in the Montney.
d.The net pressure estimated based on tangent closure and ‘ISIP’ are ~10x higher on average than that


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d.The net pressure estimated based on tangent closure and ‘ISIP’ are ~10x higher on average than that
estimated based on compliance closure and ‘effective ISIP’.
e.7 cases were identified as normal leak-off (NL). 56 cases were identified as transverse fracture storage/
fracture height recession (TFS/FHR). 9 cases out of 80 cases were identified as pressure dependent
(PDL). 2 cases were identified as fracture tip extension (FTE). 6 cases were not clearly identified. As
expected, McClure closure was not found in PDL signatures.
f.Nolte and Soliman-Craig ACA permeability estimates were of similar order of magnitude for linear
or radial flow, respectively.
g.Nolte and Soliman-Craig ACA pore pressure estimates for linear and/or radial flow were of similar
order of magnitude for linear or radial flow, respectively. Pore pressure estimates for linear flow were
slightly lower than for radial flow.


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order of magnitude for linear or radial flow, respectively. Pore pressure estimates for linear flow were
slightly lower than for radial flow.
h.As far as RTA implications, Xf√k  and Af√k  estimates from ACA methodologies are heavily dependent
on the assumption on whether the net pay has been partially or fully accessed. When Af√ k estimates
from DFIT were compared with RTA estimates, the latter values were ~7,000x higher.
i.The AER does not make recommendations or approve one single analysis method over another. The
analysis of the various methods demonstrates the range of possible outcomes and illustrates unique
properties of the Montney reservoir and can be used to optimize development.
j.The ability to independently calculate reservoir pressure across the unconventional play allows better
understanding the pressure variances of the Montney and enables more accurate resource estimation


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understanding the pressure variances of the Montney and enables more accurate resource estimation
of the play to be performed and reported. Without DFIT data, resource evaluation has been historically
more difficult.
Nomenclature
Af= fracture area, L2, m2
ct= total compressibility, Lt2/m, kPa−1
CT= total fluid loss coefficient, L/t½, m/min1/2
FL1= Nolte linear flow time function, dimensionless
FR1= Nolte radial flow time function, dimensionless
G= G-function, dimensionless
h= net pay, L, m
k= permeability, L2, mdDownloaded from http://onepetro.org/SPEATCE/proceedings-pdf/23ATCE/3-23ATCE/D031S036R004/3290732/spe-215017-ms.pdf/1 by University of Houston user on 27 October 2023


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SPE-215017-MS 29
Xf= fracture half-length, L, m
p= pressure, m/Lt2, kPa
pfo = fracture pressure measured at wellbore, m/Lt2, kPa
q= flow rate, L3/t, m3/D
Qt= total injection volume, L3, m3
t= time, hr
tp= producing time, hr
V= injection  volume, m3
π= constant, dimensionless
Δ= difference, dimensionless
µ= viscosity, m/Lt, cp
Φ= porosity, dimensionless
Subscripts
a= adjusted
c= closure
D= dimensionless
i= initial
f= fracture
p= producing
t= total
w= wellbore
References
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URTeC: 123 
 
A Collaborative Study on DFIT Interpretation: Integrating Modeling, 
Field Data, and Analytical Techniques  
Mark McClure*1, Vidya Bammidi2, Craig Cipolla3, Dave Cramer4, Lucas Martin5, Alexei A 
Savitski6, Dave Sobernheim2, and Kate Voller7; 1. ResFrac Corporation , 2. Keane 
Group , 3. Hess Corporation , 4. ConocoPhillips  Company , 5. Formerly with Apache 
Corporation, now with Marathon Oil Corporation , 6. Shell  International Exploration and 
Production Inc. , 7. Range Resources Corporation  
 
Copyright 201 9, Unconventional Resources Technology Conference (URT eC) DOI 10.15530 /urtec -2019-123 
This paper was prepared for presentation at the Unconventional  Resources Tech nology Conference held in Denver , Colorado , USA, 
22-24 July 201 9. 
The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract


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22-24 July 201 9. 
The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract 
submitted by the author(s).  The contents of this paper have not been rev iewed by URTeC and URTeC does not warrant the 
accuracy, reliability, or timeliness of any information herein.  All information is the responsibility of, and, is subject to  corrections by 
the author(s). Any person or entity that relies on any information obt ained from this paper does so at their own risk.  The information 
herein does not necessarily reflect any position of URTeC. Any reproduction, distribution, or storage of any part of this paper by 
anyone other than the author without the written consent of URTeC is prohibited.   
 
Abstract  
This paper summarizes findings from a one -year study sponsored by seven operators and service companies 
to investigate interpretation of diagnostic fracture injection tests (DFIT’s). The study combined


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to investigate interpretation of diagnostic fracture injection tests (DFIT’s). The study combined 
computational modelin g, a diverse collection of field data, and operator experience. DFIT simulations were 
performed with a three -dimensional hydraulic fracturing, wellbore, and reservoir simulator that describes 
fracture propagation, contacting of the fracture walls, and mult iphase flow. Interpretation procedures were 
applied to estimate stress, permeability, and pressure from the synthetic data. The interpretations were 
compared to the simulation input parameters to evaluate accuracy. Based on the results, new techniques 
were  developed, existing techniques were refined, and an overall interpretation protocol was developed. 
The techniques were applied to interpret over thirty field DFIT’s drawn from shale plays across the US and 
Canada, and the methods were evaluated in the con text of operator experience. The results are applicable


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Canada, and the methods were evaluated in the con text of operator experience. The results are applicable 
to fracturing tests in formations with permeability ranging from nanodarcies to 10s of microdarcies. The 
minimum principal stress is estimated by identifying the ‘contact pressure’ when the fracture w alls come 
into contact, causing fracture compliance and system storage coefficient to decrease. After the walls come 
into contact, the pressure transient is controlled by the interplay of changing fracture compliance, deviation 
from Carter leakoff, and mul tiphase flow. The contact pressure is slightly greater than the minimum 
principal stress. It can be identified from either a plot of dP/dG or a relative stiffness plot. Permeability is 
estimated  using the G -function method, a newly developed h -function met hod that accounts for deviation 
from Carter leakoff, and impulse linear flow. These  three methods, which are based on linear flow


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from Carter leakoff, and impulse linear flow. These  three methods, which are based on linear flow 
geometry,  require an estimate of fracture area . We derive equations for estimating area using mass balance 
equations, accounti ng for wellbore storage and fluid leakoff. The results from field data  show that  impulse 
linear permeability estimates are usually 2 -5 times lower than estimates derived from the G -function and 
h-function methods, apparently indicating a difference between  effective permeability during leakoff and 
permeability during flow of reservoir  fluid through the formation . Impulse radial flow regime may be used 
for estimating permeability, but  should be used with caution.  Simulation results indicate that a variety of  
processes can cause an apparent radial trend that is not actually radial flow. Simulations and field data 
indicate that ‘false radial’ is very common  in gas reservoirs and, if applied, leads to a large overestimate of


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indicate that ‘false radial’ is very common  in gas reservoirs and, if applied, leads to a large overestimate of 
permeability. Production history matc hing using overestimated permeability will underestimate fracture 
length, potentially resulting in suboptimal choices for well and cluster spacing.  
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URTeC 123   2 
1. Introduction  
1.1 Background  
Diagnostic fracture injection tests (DFIT’s) involve water injection at a rate and pressure sufficient to create 
a dominant hydraulic fracture, followed by a shut -in period lasting days, weeks, or even months. The trend 
in pressure after shut -in is analyze d to estimate stress, permeability, and/or pore pressure. Compared with 
other types of fracture injection tests, DFIT’s typically involve a longer shut -in, are performed after drilling 
is complete, are usually performed in low permeability rock, and have a  greater emphasis on estimating 
reservoir properties, such as permeability and pore pressure.  
The magnitude of the minimum principal stress, permeability, and pore pressure are key parameters in 
reservoir engineering, hydraulic fracturing, and drilling. K nowledge of the stress state is used during drilling 
(wellbore stability and mitigation of lost circulation), exploration (fault seal analysis and prediction of


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(wellbore stability and mitigation of lost circulation), exploration (fault seal analysis and prediction of 
critically stressed fractures), and hydraulic fracturing (predicting injection pressure, estima ting height 
growth, selecting proppant, and inferring net pressure) (Zoback, 2007; Economides and Nolte, 2000; Smith 
and Montgomery, 2015). Permeability estimates are used during optimization of hydraulic fracture design, 
well spacing, and cluster spacing.  Pore pressure estimates are used to delineate the extent of drainage, which 
helps with optimization of well spacing.  
Permeability is difficult to measure in unconventional formations. Permeability from core may not be 
directly applicable in reservoir eng ineering calculations because in -situ flow can be strongly influenced by 
pre-existing planes of weakness that are not sampled in the core. In addition, core measurement may be


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pre-existing planes of weakness that are not sampled in the core. In addition, core measurement may be 
impacted by changes in the rock as it is removed from the subsurface. Rate -trans ient analysis (RTA) yields 
an estimate of fracture surface area times  the square root of permeability ( 𝐴√𝑘), but cannot estimate these 
parameters separately. On the other hand, DFIT’s provide the opportunity to derive a unique, in -situ 
estimate of permeab ility. The DFIT provides a permeability estimate that is the area -weighted average of 
the permeability across the fracture surface area. It is an estimate of the effective permeability to the mobile 
reservoir phase (absolute permeability times relative per meability).  
This paper summarizes findings from a one -year study sponsored by seven operators and service 
companies. The purpose of the study was to evaluate the accuracy of existing DFIT interpretation


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companies. The purpose of the study was to evaluate the accuracy of existing DFIT interpretation 
techniques, develop new techniques if needed, and ass emble an overall interpretation procedure. The study 
integrated unique simulation capability, field data, and operator expertise. The interpretation procedure can 
be performed entirely in an Excel spreadsheet.  
The body of the paper presents a step -by-step worked example, commentary on the most important findings, 
and additional field examples. The appendix provides detailed mathematical derivations explaining the 
methods applied in the paper  
The study focused primarily on DFIT’s performed in formations with  permeability on the order of 
nanodarcies to 10s of microdarcies. With higher permeability, transients behave qualitatively differently, 
and many of the techniques in this paper cannot be applied (discussed in Section 4.1).  
1.2 Literature review


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and many of the techniques in this paper cannot be applied (discussed in Section 4.1).  
1.2 Literature review  
DFIT inte rpretation combines concepts from well test analysis and hydraulic fracture analysis. 
Conventional well testing involves production or injection of fluid at pressure below the fracture gradient. 
Pressure data is analyzed to infer permeability, skin, and ot her parameters (Theis, 1935; Horne, 1995; 
Kamal et al., 2009; Stewart, 2011; Spivey and Lee, 2013). DFIT’s involve similar physics, but with 
important differences. In conventional well tests, fractures are static. In DFIT’s, fractures are dynamic – 
changin g in size, compliance, and conductivity. As a result, c oncepts from conventional well test analysis 
are not always directly applicable to DFIT analysis .  
Hydraulic fracturing was first described in a publication by Clark (1949). In subsequent decades, it w as


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Hydraulic fracturing was first described in a publication by Clark (1949). In subsequent decades, it w as 
recognized that hydraulic fracturing could be used as a method to estimate the magnitude of the minimum Downloaded from http://onepetro.org/URTECONF/proceedings-pdf/19URTC/2-19URTC/D023S030R001/1138462/urtec-2019-123-ms.pdf/1 by University of Houston user on 27 October 2023


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URTeC 123   3 
principal stress (Hubbert and Willis, 1957; Godbey and Hodges, 1958; Kehle, 1964; Haimson and Fairhurst, 
1967; Hickman and Zoback, 1983). If fluid is  injected at pressure greater than the minimum principal stress, 
a hydraulic fracture will form and propagate through the formation. As a result, the minimum principal 
stress can be estimated from pressure measurements during and after injection (Economide s and Nolte, 
2000; Zoback, 2007; Smith and Montgomery, 2015). The uniaxial stress/strain equations can be used for 
engineering purposes to help predict stress differences between layers (Eaton, 1969; Blanton and Olson, 
1999), but fracture injection tests a re needed for calibration and to ensure accuracy.  
Nolte (1979) extended the applicability of fracturing tests by developing analytical techniques that can be 
used to estimate fluid leakoff parameters. These techniques can assess leakoff even when using com plex


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used to estimate fluid leakoff parameters. These techniques can assess leakoff even when using com plex 
fluids that develop filtercake. These tests are typically called minifrac tests (McLennan and Roegiers, 1982) 
or fracture calibration tests (Mayerhofer and Economides, 1993). The G -function developed by Nolte 
(1979) became a routine part of data plott ing and interpretation (Castillo, 1987; Barree and Mukherjee, 
1996). Mayerhofer and Economides (1993) and Mayerhofer et al. (1995) showed how to disentangle the 
effects of filtercake and formation permeability on leakoff.  
Conventional well testing is infe asible in very tight formations because of the difficulty of sustaining 
sufficiently low rates of production or injection. The “diagnostic fracture injection test” (DFIT) emerged to 
combine the benefits of a well test and a fracturing test – providing esti mates for stress, permeability, and


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combine the benefits of a well test and a fracturing test – providing esti mates for stress, permeability, and 
pore pressure (Craig and Brown, 1999). ‘Preclosure’ analysis methods are usually based on techniques from 
Nolte (1979) and Mayerhofer and Economides (1993) and Mayerhofer et al. (1995). Barree and Mukherjee 
(1996) and Ba rree et al. (2009) developed techniques for qualitative interpretation and for estimating the 
closure pressure and minimum principal stress. Impulse analysis is based on the use of impulse solutions 
from conventional well testing (Gu et al., 1993; Soliman et al., 2005; Craig and Blasingame, 2006) or the 
‘linear flow time function’ technique developed by Nolte et al. (1997). Examples of DFIT interpretation 
are provided in recent papers by Marongiu -Porcu et al. (2011; 2014), Soliman and Kabir (2012), Soliman 
and Gamadi (2012), Cramer and Nguyen (2013), Padmakar (2013), Wallace et al. (2014), Meng et al.


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and Gamadi (2012), Cramer and Nguyen (2013), Padmakar (2013), Wallace et al. (2014), Meng et al. 
(2014), Craig (2014), Zanganeh et al. (2018), and Hawkes et al. (2018).  
McClure et al. (2014; 2016) and Jung et al. (2016) proposed the compliance method of e stimating stress 
from a DFIT ( Section 3.1.2 ). The compliance method leads to an earlier, higher stress estimate than the 
commonly used tangent method (Barree et al., 2009). McClure et al. (2016) and Jung et al. (2016) performed 
detailed DFIT simulations co nsidering the effect of residual aperture fracture after the fracture walls 
contact. Simulations indicate that in low permeability formations, the contacting of the fracture walls causes 
the magnitude of the  pressure derivative to increase, resulting in a pressure signal often interpreted as height 
recession or closure of transverse fractures (Barree et al., 2009). Wang and Sharma (2017) reproduced these


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recession or closure of transverse fractures (Barree et al., 2009). Wang and Sharma (2017) reproduced these 
findings and proposed a modified approach for estimating stress called the ‘variable compliance method.’  
1.3 Defining a new term – the ‘contact pressure’  
The term ‘closure pressure’ is used widely in the literature but is rarely given a precise definition. A review 
of the literature shows that a variety of assumptions are made about ‘closure pressure ,’ and these 
assumptions  are often problematic and inconsistent. Most authors assume that the closure pressure is equal 
to the magnitude of the minimum principal stress (issues with this definition are discussed in Section A.2). 
Craig and Blasingame (2006) as sume that the compliance (or compressibility) of a ‘closed’ fracture is 
negligible (issues with this definition are discussed in Section A.2). The classical calculation of efficiency


