Heart Shape to Fracture Distance: Characterizing Hydraulic Fracture Propagation before Hits

Unconventional reservoirs, known for their low permeability and porosity, have been continuing to make a significant contribution to global hydrocarbon exploitation over the last decade. Horizontal drilling and multistage hydraulic fracturing have emerged as the predominant techniques for the development of these reservoirs. The multistage completions create multiple hydraulic fractures in a horizontal well, enabling maximization of hydrocarbon exploitation from the low-permeability reservoirs (Daneshy 2011). In addition, it is a common practice to drill multiple nearby horizontal wells to enhance overall production. However, this approach can lead to undesired fracture connections between adjacent wells, often referred to as parent-child well interference, which can potentially result in diminished performance of the unconventional reservoirs. Therefore, optimizing well spacing and implementing a suitable completion design are crucial to minimizing the negative impact of fracture interference. To mitigate fracture interference between adjacent wells, a quantitative characterization of fracture extents generated is essential. Stress fields induced by hydraulic fracturing have been studied extensively in various contexts, demonstrating the critical role of monitoring the fracture propagation (Mills et al. 2004). However, an in-situ measurement of stress fields is necessary to quantify the fracture mechanics and geometry precisely.

Distributed fiber-optic sensing technology has been found useful and applied for the completion and production monitoring in horizontal unconventional wells (Molenaar et al. 2012). Distributed fiber-optic sensing technology utilizes laser energy scattered by the impurities within fiber to measure temperature and strain. Recent studies illustrate that the low-frequency (LF; <0.1 Hz) component of the distributed acoustic sensing (DAS) (LF-DAS) signal recorded at an offset monitor well provides critical information about fracture geometry and propagation (e.g., Jin and Roy 2017; Liu et al. 2020, 2022). Ugueto et al. (2019) presented field examples of LF-DAS in wells stimulated for various completion systems and demonstrated how to use LF-DAS response to evaluate stimulation efficiency. Wu et al. (2020) demonstrated through field LF-DAS data that crosswell far-field strain measurements can be used to determine fracture azimuth, the width of the fractured region, and fracture height with a high degree of accuracy and resolution. In addition to its application to unconventional reservoirs, distributed fiber-optic sensing monitoring is also applicable to carbon capture and storage projects. Unlike injecting hydraulic fracturing fluid in an unconventional case, in this case, carbon capture and storage projects involve injection of CO₂ under its supercritical state into subsurface (Ning and Tura 2023). DAS-derived strain measurements, applied to carbon capture and storage installations, can be used to monitor unintentionally created hydraulic fractures and deformation along pre-existing faults, which can be used to prevent induced seismicity and model deformation in reservoir formation (i.e., fractures created by injection and CO₂ plume migration). However, crosswell strain measurement has its own limitations. For example, it only records the formation deformation at the monitor well location. If a fracture does not intersect the monitor well, the fracture geometry and propagation information can be mainly inferred from the stress field it creates.

An LF-DAS recorded hydraulic fracture propagation signal is considered as a vital tool for fracture monitoring and diagnostics in the field (Jin and Roy 2017). One field example of LF-DAS data from HFTS2 is shown in the top panel of Fig. 1 . The bottom panel of Fig. 1  shows the pumping curves with respect to stimulation time. These measurements are from Stage 8 of Treatment Well B2H, monitored by B3H. Fracture start time and hit time are interpreted to be 660 seconds and 2,650 seconds, respectively. The top panel shows a typical extending signal, known as the heart-shaped signal, before the fracture hit. Red colors indicate extensional strain rate, and blue colors indicate compressional. The heart-shaped signal extension zone in the strain rate waterfall plot corresponds to the progressively narrowing extension zone detected by the fiber before fracture hit.

Several studies have utilized characteristics of the heart-shaped signal for fracture propagation characterization. Zhu et al. (2022) developed a machine-learning-based method to automatically detect heart-shaped signals in LF-DAS data. Haffener et al. (2022) investigated whether the hydraulic fracturing in the Meramec and Wolfcamp formations created new fractures or reactivated existing ones. This was achieved by using the existence of heart-shaped signal diagnostics to differentiate new fracture creation vs. old fracture reactivation. The study is significant for understanding the effectiveness of hydraulic fracturing operations; however, it did not characterize the heart-shaped signal quantitatively. Leggett (2023) introduced a method for detecting fracture tip propagation using the heart-shaped signal. This study involved calculating fracture propagation velocities near the monitor well and using them to estimate the final fracture extent. Leggett’s approach utilized a zero strain-rate location method, initially based on radial fractures (Leggett et al. 2022) but later updated for rectangular fractures. It is advantageous due to its simplicity and efficiency in analyzing numerous fracture stages. However, it assumes simplified fracture geometries and relies on empirical and preassumed shape factors. In addition, the zero strain-rate location can be difficult to pick in the field data due to low signal-to-noise ratio.

In this study, we use the heart-shaped signal before the fracture intersects the monitor well to estimate fracture tip-to-well distance. First, we conduct a comprehensive numerical study of the influence of fracture geometry and monitor well offset on the heart-shaped signals using 3D DDM. A detailed numerical analysis leads to the derivation of a simple linear relationship and enables the straightforward estimation of the fracture tip distance toward the monitor well. This linear relationship is then verified by classical analytical solution of the fracture tip stress field, followed by a discussion of its limitations. We applied this method to the HFTS2 field data set to test its robustness. The linear relationship we found in this study enhances our capability and accuracy to estimate fracture propagation, offering a novel approach for optimizing hydraulic fracturing operations and improving the accuracy of subsurface monitoring techniques.

DDM is an efficient method for computing fracture-induced elastic rock deformation at any arbitrary location in an infinite homogeneous elastic domain (Crouch et al. 1983). For crack-like geometries, the boundary $S$ can be considered as two surfaces $S±$⁠. The displacement components at an arbitrary point $ξ$⁠, induced by displacement discontinuities located at point $η$ over the fracture surface $S$⁠, can be expressed as (Shou 1993; Wu 2014):

where $ui-$ and $ui+$ are the displacement components of surfaces $S-$ and $S+$⁠, respectively. The quantities $Tji(ξ,η)$ represent the influences of a displacement discontinuity in the $i$th direction at $η$ on the displacement in the $j$th direction at $ξ$⁠. To numerically solve Eq. 1, the boundary surface is discretized into $N$ elements. Assuming constant displacement discontinuities over each element $Sm$⁠, Eq. 1 can be written as

The displacement discontinuities over each element can be computed analytically for any shape of the fracture geometry. This study investigates two types of fracture geometry in 3D—ellipsoid and cuboid, with constant fracture propagation velocity. For cuboid fracture geometry, the fracture width and height remain constant, whereas the length is a function of time and fracture propagation velocity. For ellipsoid fracture geometry, we only discretize the fracture surface into small elements along the fracture length and assume constant height and width for each fracture element. In the Cartesian coordinate system, which is represented as

for height and

for width, where $H$ and $W$ are ratios of fracture height and width to fracture length $L$ and $x$ refers to the current position with respect to $L$⁠. Note that the fracture length and propagation direction are along the x-direction, and the monitor well is in the y-direction. A schematic cartoon illustrates the DDM approach for obtaining displacement in ellipsoid (Fig. 2a)  and cuboid (Fig. 2b)  fracture geometries. Once the fracture geometry is defined, the displacements and stresses at any arbitrary point over the computational domain can be calculated analytically.

The LF-DAS measurements are equivalent to linearly scaled directional strain rates along the fiber over a predefined gauge length (Jin and Roy 2017; Ugueto et al. 2019). To be consistent with field measurements, the strain $εyy$ and strain rate $ε˙yy$ along the monitor well are calculated as

where superscripts $i$ and $n$ represent the location index (i.e., discrete sensor locations) and time index, respectively. $uy$ is the displacement component along the $y$-direction, and G$L$ is the gauge length, which is 5 m in this numerical study. The choice of gauge length creates a subtle impact on spatial resolution. However, it will not impact the locations of local maxima discussed in the next paragraph, as it can be considered as a moving average filter with averaging length of the choice of gauge lengths. As Eq. 4 shows, strain is computed by taking the spatial derivative of recorded displacement over a predefined gauge length along the fiber. In addition, strain rate is computed by taking the temporal derivative with respect to strain (Eq. 6). In this study, strain rate is used for both simulation and field data application, and we assume the fiber is fully coupled with formation.

Fig. 3  shows an example of simulated LF-DAS response, where blue color indicates compression and red color indicates extension strain rate of the fiber. In this case, the treatment and monitor wells are positioned at the same depth, with 200-m horizontal offset, and parallel to each other. The length of the monitor well extends to 400 m, and the fracture hits the monitor well at 200 m after 50 minutes of injection, with a fracture propagation speed of 4 m/min. The fracture has an elliptical shape with a maximum height of 60 m and a maximum width of 0.01 m. Black crosses show the locations of local maxima at each timestep, which we refer to as peaks of the heart-shaped signal. The distance between two peaks at the same timestep is referred to as peak distance (⁠$dpeak$⁠). The LF-DAS response shown in Fig. 3  is used as the base case in this study to investigate the relationship between fracture tip distance to the monitor well, $dfrac$⁠, and peak distance, $dpeak$⁠. The simulated fracture intersects the monitor well when d_peak = d_frac = 0 m, which also indicates convergence of two peaks along a single spatial trace at a timestep.

The relationship between $dpeak$ and $dfrac$ from base case is shown in Fig. 4 . This figure illustrates a strong and linear correlation between $dpeak$ and $dfrac$ from 0 m to 150 m. The trend appears to be nonlinear after d_frac ≥ 150 m, where the fracture tip distance is much larger than the fracture dimension.

To further validate these observations and understand the impact of fracture geometry (including height, width, and shape) and the vertical offset between the monitor and the treatment well on the characteristics of heart-shaped signal, a comprehensive set of 126 numerical simulations was conducted. The parameters varied across these simulations, including fracture height (30 m, 60 m, and 90 m), fracture width (0.001 m, 0.01 m, and 0.1 m), vertical offset (0–60 m in 15-m increments and 60–120 m in 30-m increments), and both rectangular and elliptical shapes for the fracture geometry. Fig. 5  shows a schematic view of fracture dimensions and treatment-monitor spatial relationship in a cross section. The blue circle illustrates the monitor well’s location.

Fig. 6  presents a comparison of the impact of fracture height on the relationship between $dpeak$ and $dfrac$⁠, with the vertical offset and fracture width held the same as the base case (i.e., maximum width = 0.01 m, vertical offset = 0 m). In the case of elliptical fractures (Fig. 6a ), an identical relationship between $dpeak$ and $dfrac$ can be identified regardless of the fracture heights. For rectangular fractures (Fig. 6b ), a similar correlation is observed as fracture height varies. However, in this case, $dpeak$ has a minor sensitivity to the fracture height, showing a small spread of values for $dfrac$ larger than 25 m. The correlation between $dpeak$ and $dfrac$ for both elliptical and rectangular cases with different fracture heights shows a similar trend as the base case (Fig. 4) , while elliptical cases show no sensitivity to fracture height.

To understand fracture width impact imposed on the relationship between $dpeak$ and $dfrac$ for both elliptical and rectangular fractures, the fracture height and well vertical offset remain the same as the base case (i.e., maximum height = 60 m, vertical offset = 0 m). Fig. 7  shows the results for three different maximum fracture widths: 0.001 m, 0.01 m, and 0.1 m. For both elliptical and rectangular fracture shapes with varying fracture width, $dfrac$ values overlap each other regardless of the width of the fracture and follow the same trend. This is expected as the DDM assumes a linear relationship between rock and fracture deformations, and fracture width only changes the magnitude of the strain rate signal, not the spatial distribution.

In real cases, monitor wells may not be located at the same depth as the treatment well. Therefore, understanding the impact of vertical offset between the two wells on the relationship between $dpeak$ and $dfrac$ extracted from the heart-shaped signal is crucial. In addition, given the symmetricity of the fracture geometry, only investigating the upper half space is sufficient. In these synthetic examples, the fracture geometry maintains the same as the base case (maximum height = 60 m, maximum width = 0.01 m, and horizontal offset = 200 m), while the monitor well’s vertical offset relative to the treatment well changes from 0 m to 60 m with a 15-m increment and from 60 m to 120 m with a 30-m increment. Fracture tip distance $dfrac$ is estimated based on the closest distance between the monitor well and the fracture edges, as illustrated in Fig. 5 . Fig. 8  shows the result for both elliptical (Fig. 8a)  and rectangular (Fig. 8b)  fracture shapes. For both shapes, $dfrac$ follows a similar trend as observed in the base case. However, as shown in Fig. 8a , $dfrac$ values less than 150 m are less sensitive to the vertical offset and maintain the same linear trend in elliptical cases. In rectangular cases, though $dfrac$ values follow a similar trend, the variations are relatively larger. It is worth noting that the cases with the vertical offset larger than the fracture height do not have values where $dfrac$ reaches zero, yet the relationship between $dfrac$ and $dpeak$ is still consistent with the base case.

To generalize the relationship between $dpeak$ and $dfrac$ with different fracture geometry and monitor well locations, we conducted a comprehensive test of all parameters. By combining the variations of tested parameters including height, width, and vertical offset, a total of 63 different model setups were tested each on elliptical and rectangular fractures, with a summary of all results shown in Fig. 9 . In both cases, a strong linear relationship between $dfrac$ and $dpeak$ can be observed in the region where $dfrac$ is less than 150 m, regardless of fracture geometry and monitor well vertical offsets. We also observe that the spread of data points in rectangular fracture cases is larger, indicating a less robust linear trend between the two variables in this case.

We further extend our testing parameters to monitor wells with different horizontal offsets, and the results from 630 model setups are concluded in Fig. 10 . In the elliptical case, a strong and consistent linear relationship between $dfrac$ and $dpeak$ can be observed until the fracture tip distance is close to the distance between the monitor and treatment wells. The linear relationship is less robust in the rectangular fracture cases, similar to the example shown in the previous cases.

Considering the actual hydraulic fractures in the subsurface have a geometry closer to elliptical cases and the observed consistency in the linear relationship between $dpeak$ and $dfrac$ for elliptical fractures in numerical modeling, despite variations in fracture geometry and monitor well locations, we derive an important linear relationship from synthetic data for elliptical cases:

This linear relationship is plotted as the red dashed line in Fig. 10 . It can be observed that the linear trend in the rectangular fracture cases has a slightly different slope (⁠$dfrac=0.92dpeak$⁠), shown as the black dashed line in Fig. 10b . In addition, we tested velocity variation impact on the obtained linear relationship as shown in Eq. 7. By only perturbing the fracture propagation velocity and maintaining the same results of parameters as in the base case, the linear relationship still holds as illustrated in Figs. S-1 through S3 of the Supplementary Material, which show the velocity of the function under a constant velocity of 3 m/min, 5 m/min, and a sine function, respectively.

Building on the empirical findings in Eq. 7, our next step involves contextualizing the result within the framework of classical fracture mechanics. Here, we refer to the classical Kolosov-Muskhelishvili formulas for solving general plane elasticity problems. In the application to crack problems, in particular, an open-crack problem, the Westergaard function is more convenient for finding near-fracture-tip stress field for an infinite plane under uniform biaxial stress (Westergaard 1939). In our case, the measured strain component is parallel to the width direction (i.e., $y$-direction), and the value of $εyy$ can be derived from the original Westergaard function as

where $K1$ is the fracture intensity factor for open cracks, $r$ and $θ$ are the polar coordinates converted from Cartesian coordinates $x$ and $y$⁠, fracture length is along the x-direction, width (opening) is along the y-direction, $υ$ is the Poisson’s ratio, and $εyy$ is the corresponding strain field along the $y$-direction. To convert the spatial distribution of the strain field into distriubuted strain sensing strain rate measurements, we assume the fracture propagates at a constant velocity $v$ and apply the differential chain rule:

From Eq. 9, we can see that the strain rate should have the same spatial distribution as the strain spatial gradient along the $x$-direction, which is plotted in Fig. 11a . The extracted relationship between $dpeak$ and $dfrac$ for the analytical solution is shown in Fig. 11b . The orange line in Fig. 11b  indicates a line with a slope of 0.8. The analytical solution illustrates a similar linear relationship to the one we obtained from DDM. However, different elastic parameters impact the slope. Table 1  shows the impact of Poisson’s ratio. The average Poisson’s ratio for shale formation is 0.26, with a corresponding slope around 0.8. If the target formation has a different Poisson’s ratio, the slope of the obtained linear relationship can be found in the table with the corresponding Poisson’s ratio. According to the third equation in Eq. 8, Young’s modulus is a scalar factor, and it will change the magnitude of the heart shape rather than impose an impact on the slope of the obtained relationship.

Eq. 8 presents the near-tip solution for a 2D elliptical fracture, assuming infinite fracture length and height. This assumption aligns with the results from the DDM models, where the linear relationship is only valid when the fracture tip is in close proximity to the monitoring well, and the fracture dimensions are sufficiently large. Another insight from this analytical solution is that the different slopes obtained from the elliptical and rectangular cases are more likely due to the variations of fracture width rather than height along the fracture length. The analytical solution with infinite fracture height is more consistent with the elliptical cases than the rectangular cases despite the fact that the fracture height of rectangular fractures is usually much larger than that of elliptical fractures at the fracture tip.

The straightforward linear relationship between $dpeak$ and $dfrac$ allows the estimation of fracture tip locations before the fracture intersects the monitor well. Because elliptical fractures are usually considered closer to actual hydraulic fracture geometry in the subsurface, we use the elliptical fractures to obtain the linear relationship. In addition, for rectangular fracture shape as an oversimplified end-member case, we use Eq. 7 to estimate fracture tip distances in both synthetic and field data examples, and we use the estimation errors in the rectangular fracture cases to serve as the upper bound estimation of the errors. Because in real situations, the shape of hydraulic fracture should be between elliptical and rectangular.

We first test the estimation results for the cases with a 200-m horizontal well offset. The estimation result $dfracestimated$⁠, compared with the true values $dfractrue$⁠, is shown in Fig. 12 . The estimation is accurate (<5 m error) for $dfrac$ smaller than 150 m. Beyond that, the prediction tends to underestimate fracture tip distances. Based on the well geometry, the fracture dimension (length in this case) can be calculated from the distance between the wells and the tip distance to the monitor well. For $dfrac>150$m, the fracture length is less than 50 m. This implies that the tip distance estimation is less accurate where the tip distance is much larger than the fracture dimension, as suggested by the analytical solution. For the rectangular fracture cases, the estimation errors are in general much larger (Fig. 12b)  due to a slightly different slope between $dfrac$ and $dpeak$⁠, and more variations in the relationship. However, for near-tip region (⁠$dfrac<150$ m), the estimation error is less than 20 m in most cases.

To better understand the relationship between estimation error, tip distance, and fracture dimension, we performed the tip distance estimation using Eq. 7 for all horizontal offsets, with the results shown in Fig. 13 . Again, we observe good estimation results for elliptical cases when the ratio between fracture tip distance and fracture dimension is small. We observe an expected linear increase of estimation error with tip distance for rectangular fracture cases, which serves as an upper bound estimation of estimation errors.

For hydraulic fracturing operations, because the fracture dimension can be estimated using tip distance and well spacing, it is more convenient to plot the prediction error against the ratio of fracture tip distance and the treatment-monitor well distance to identify the criteria where the prediction is accurate. Fig. 14a  illustrates the relationship between the prediction error and the distance ratio for all elliptical cases. Accurate estimation with error smaller than 10 m can be achieved for all cases where the tip distance is smaller than 60% of the well distance. To make this relationship applicable for field studies where the actual tip distance is unknown, we use estimated tip distance instead of actual tip distance to calculate the ratio and find that a similar result can be observed in Fig. 13b .

The linear relationship between peak distance and fracture tip distance obtained above is further applied to the LF-DAS field measurements from the HFTS2 project. To yield a cleaner heart-shaped signal and optimize the picking accuracy, a signal-processing workflow is developed before the tip distance estimation. After data preconditioning, including amplitude normalization and de-median, a two-stage filtering strategy is adopted for noise removal. First, we applied a low-pass filter with a corner frequency of 0.05 Hz along the spatial direction to reduce the background noise. Second, we applied a 2D rank filter. Rank filter is similar to median filter, except an arbitrary percentile of input data is selected as filter output, which is 42 percentile for this particular stage. This two-stage filtering strategy yields a smooth heart-shaped signal with a stronger contrast between signal and background noise to allow automated peak-picking. Next, we picked local maxima at each timestep on the LF-DAS signals to obtain the peak distance. Finally, we utilized the relationship in Eq. 7 to estimate the fracture tip distance to monitor well.

The result of the preprocessing workflow is shown in Fig. 15 , top panel. In this case, we use the same LF-DAS data as shown in Fig. 1 , so the impact of data processing can be illustrated. After applying the proposed workflow, we can observe that the heart-shaped signal is much smoother and more prominent in contrast to the background noise. The middle panel shows the pumping data, and the bottom panel shows the fracture tip distance estimation result based on the peak distance. Note that the estimation is only applied where the ratio between the estimated tip distance and well distance is smaller than 0.6. With the Euclidean distance between B2H and B3H being approximately 205 m (Srinivasan et al. 2023), the maximum tip distance we can reliably estimate in this case is 123 m. Fig. 15c  shows the estimated fracture tip distance using Eq. 7. It is interesting to observe a near-constant fracture propagation velocity over the distance where the tip location can be reliably estimated. The linear regression result indicates the fracture propagates at around 4.5 m/min. The average fracture propagation velocity between the treatment and monitor well can also be estimated using well distance and fracture hit time (Zhu and Jin 2022), which is around 6.0 m/min for this stage. The difference between the two velocities indicates the fracture propagated much faster in the early period of the pumping stage, which could be due to a smaller fracture area. We will further investigate fracture propagation speed across different treatment-monitoring pairs to verify if there is a consistent decrease in this behavior.

The linear relationship we have identified, despite its simplicity, can be very useful in various scenarios. One application is to better characterize hydraulic fracture propagation, especially in the area between the treatment and monitor wells. We can estimate the in-situ fracture propagation velocity and compare it with different completion designs. Because of the nature of the heart-shaped signal, we can better characterize the fracture that first intersects the monitor well. Considering that the first arrival fracture is usually the fastest propagating and most dominant fracture, such characterization can help us evaluate the evenness of fluid distribution among different clusters, assuming each cluster would start its own hydraulic fracture. This approach allows us to evaluate completion efficiency more effectively using crosswell signals (Mjehovich et al. 2024). Additionally, better measurement of the in-situ propagating velocity of fractures near the monitor well can better constrain the total fracture length.

The discovered relationship can also be used to prevent unwanted fracture hits. There are various potential scenarios for this type of use case. The first application is to minimize production degradation caused by fracture hits at parent wells. By equipping parent wells with sensing fibers, the heart-shaped signal can be utilized to forecast the timing of the fracture hits, allowing for adjustments to the pumping schedules to avoid them. However, some prior observations suggest that the heart-shaped signal may not be clearly evident for the reactivation of existing fractures (Haffener et al. 2022), which could limit the use of this method for parent well protection. The other application is to better prevent drilling hazards if hydraulic fracturing operations pose blowout risks for nearby drilling operations. As demonstrated by Stark et al. (2024), instrumented monitor wells between the treatment and drilling wells can be used to monitor fracture propagation and potentially detect fracture growth toward the drilling site. The heart-shaped signal in this case can provide early warning of the fracture propagation and better estimation of the fracture propagation speed toward the well being drilled.

One important insight from this relationship is that DSS measurements can help to determine fracture geometry and deformation without the need for the fracture to intersect the monitoring well. This has significant implications for long-term reservoir monitoring in carbon storage reservoirs. One potential scenario is to place a monitoring well in the overburden to monitor potential fracture propagation in the caprocks. Since most caprocks are low-permeability and high-ductile shale formations, they are more likely to deform aseismically, making it challenging to monitor such deformation using seismic-based methods (e.g., microseismic monitoring). On the other hand, both tensile and shear fractures create a stress field as they deform and propagate. This stress field is much larger than the fracture itself and can be captured by DSS measurements within a certain distance. The authors intend to develop a more dedicated inversion algorithm to determine fracture location, geometry, and deformation in future studies.

This study presents a novel approach that utilizes the heart-shaped signal in LF-DAS measurements to estimate the hydraulic fracture tip distance before the hydraulic fracture intersects the monitor well. This method can provide critical insight to characterize hydraulic fracture propagation. Utilizing the DDM, we have quantitatively characterized the heart-shaped signal with different fracture geometry and monitor well locations. Through extensive numerical simulations and empirical modeling, we established a robust linear relationship between the heart-shaped signal’s spatial extent and the fracture tip distance to the monitor well. We further verified the linear relationship using the classical analytical solution of a planar fracture and discussed its assumptions and limitations. The tip distance estimation based on the linear relationship was first verified with synthetic data and then applied to the field data from the HFTS2 project. The simple method we developed enables real-time monitoring and constrains hydraulic fracture propagation in the far field. The assumptions of this study are listed below:

• Linear elasticity

• Poisson’s ratio is the average ratio of shale formation, which is 0.26

• Reservoir properties need to be homogeneous and isotropic

• The monitor needs to be perpendicular to the planar fracture

The potential applications include more robust fracture characterization for unconventional and enhanced geothermal system development, and long-term reservoir monitoring for carbon sequestration projects. In summary, this work illustrated:

• A novel approach using the heart-shaped signal from LF-DAS to estimate hydraulic fracture tip distance before the fracture intersects the monitor well, enhancing real-time fracture propagation monitoring.

• Numerical simulations and empirical modeling established a robust linear relationship between the spatial extent of the heart-shaped signal and the fracture tip distance, verified by classical analytical solutions, offering a new method for hydraulic fracturing operations.

• The method’s potential applications include optimizing well spacing, enhancing unconventional and enhanced geothermal system development, and improving long-term reservoir monitoring for carbon sequestration projects.

This project was funded by the United States Department of Energy, National Energy Technology Laboratory, in part, through a site support contract. Neither the United States Government nor any agency thereof, nor any of their employees, nor the support contractor, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Original SPE manuscript received for review 3 July 2024. Revised manuscript received for review 19 September 2024. Paper (SPE 223631) peer approved 14 October 2024. Supplementary materials are available in support of this paper and have been published online under Supplementary Data at https://doi.org/10.2118/223631-PA. SPE is not responsible for the content or functionality of supplementary materials supplied by the authors.