Validation of Estimating Stress from Fracture Injection Tests Using Continuous Wavelet Transform with Experimental Data ^†

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 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.

1. Introduction

Using the continuous wavelet transform (CWT) as a fracture closure detection technique 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 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 detection 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 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 introduced 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 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

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 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.

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 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 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.

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]).

$$\overline{W}=\frac{P-S_{hmin}}{S_f}$$

(1)

where $S_f$ is the fracture stiffness, $\overline{W}$ is the average fracture width, and $S_{hmin}$ 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 ($S_{hmin}$) is reopened, the normal stress acting on the fracture will exceed $S_{hmin}$, 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]).

The asperities contact at a critical fracture width is $\overline{W_c}$. If the fracture were perfectly smooth, then $\overline{W_c}$ would 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 $\overline{W_c}$ indicates 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, 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 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 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, $S_{Hmax}$, is estimated to be 34 MPa (4931 psi), while the minimum horizontal stress, $S_{hmin}$, 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 $S_{hmin}$. 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.

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. Injection duration, total injection volume, average injection rate, and shut-in duration for E1-I 164 tests (Guglielmi et al. (2022) [8]).

Name	Injection Duration (min:sec)	Injection Volume, L	Avg. Injection Rate, L/min	Shut-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

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.

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.

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 (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.

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) 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. Injection duration, total injection volume, average injection rate, and shut-in duration for TV4100 tests (Guglielmi et al. (2022) [8]).

Name	Injection 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

shows the Injection duration, total injection volume, average injection rate, and shut-in duration for TV4100 tests.

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 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.

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.

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 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.

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. Comparison of the closure events period of the four EGS Collab tests.

Fracture Closure Pressure Detection Technique	E1-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 Technique	Start 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 Method	Rapid 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)
Fracture closure pressure using Tangent Method	Less 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 method	No closure signature	No closure signature	No closure signature	18.6 MPa (2700 psi)

. 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].

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 increasing and does not reach a peak by the end of the test. The dP/dG curve showed a monotonic 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 continued 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 of the SIMFIP. These results indicate that conventional techniques may fail in analyzing complex or non-ideal tests.

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.

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).

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 conventional 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.

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 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.

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% 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 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.

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 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 26 illustrates the application of the continuous wavelet transform (CWT) technique 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.

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 fracture closure detection technique as per

Table 4. Comparison of the closure events period of the four EGS Collab tests.

Test	Pressure at Start of Closure Event, MPa (psi)	Pressure at Start of Closure Event, MPa (psi)	Period of Closure Event, min	Number 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
TV4100 Test 7	19.03 (2760)	18.61 (2700)	42	10

. 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.

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:

• The signal average energy using the CWT closure detection technique is a reliable mathematical independent magnification for the pressure decay. This has been confirmed using the SIMFIP tool.
• 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.
• 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”.