In recent years, horizontal hydraulic fracturing has been developed rapidly. A series of fracturing technologies have emerged ^[1,2], such as multi-cluster fracturing with tight spacing, stacked multi-zone fracturing, etc. Although the technologies have been developed quickly with increasing production, fracturing design is still challenging due to the poor understanding of fracture propagation geometry and lack of solid evidence for the design of the operation parameters, such as the cluster spacing and fluid volume ^[3,4]. The real-time monitoring of hydraulic fractures is a crucial method to understand fracture geometry and optimize fracturing design ^[5].

The real-time monitoring of hydraulic fractures includes near-well monitoring and monitoring far from the well. The near-well monitoring method mainly includes distributed temperature sensing (DTS) and distributed acoustic sensing (DAS). They can be used to evaluate the difference in fluid distribution near the perforated hole or near-wellbore formation, but it is difficult to obtain the fracture geometry ^[6,7]. Monitoring far from the well includes micro-seismic, tiltmeter, electro-magnetic, and fiber optic strain monitoring. With the first three techniques, we can get the fracture strike or stimulated res-ervoir volume. Still, it is challenging to properly diagnose fractures with several meters spacing in a cluster ^[8,9]. Distributed optical fiber strain monitoring method has the advantages of accurate locating, and can be used for distributed real-time monitoring, and can obtain rich information. It is an effective technique to diagnose hydraulic fractures precisely for multi-cluster staged fracturing with tight spacing ^[10].

Distributed optical fiber strain monitoring obtains the optical fiber strain during the fracturing process through the low-frequency (less than 0.05 Hz) filtering of the acoustic signal and then realizes the real-time tracking of fracture propagation ^[11]_. The fiber optic cable is usually pre- consolidated in the cement ring of the monitoring well. A laser pulse is sent down in the monitoring well during the fracturing process. The backscattered laser pulse returns to the interrogator units and is demodulated into strain data. Then the fiber strain and strain rate during the fracture propagation can be obtained to diagnose fracture propagation ^[10,11]. Shell, Schlumberger, and other petroleum companies have conducted distributed optical fiber strain monitoring tests in shale reservoirs during horizontal fracturing in North America. The field studies confirmed that the distributed fiber strain technology could effectively identify multi-fracture propagation and inter-well pressure channeling ^[12,13,14]. Distributed optical fiber strain monitoring in a horizontal well in Montenegro shale in Canada demonstrated that hydraulic fractures are usually planar fractures, which propagate in the direction of maximum principal stress ^[12]. Distributed optical fiber strain shows excellent application prospects, but its diagnosis mechanism of "strain and its strain rate evolution characteristics during fracturing" is unclear ^[10,11,12]. The strain interpretation during the fracturing is mainly based on 2D or pseudo-3D (P3D) models, although it is a 3D problem in practice ^[15,16,17]. Therefore, it is urgent to build a more accurate model to study optical fiber strain and strain rate evolution during fracture propagation ^[15,16,17].

In this study, we proposed a forward model of fiber strain and strain rate induced by fracture propagation based on a planar 3D (PL3D) multi-Fracture Growth Model coupled with wellbore and perforation. A numerical scheme for calculating fiber strain and strain rate caused by fracture propagation was presented. Evolution characteristics of optical fiber strain signals during single-cluster, multi-cluster, and multi-stage fracturing operations can be numerically analyzed to real the evolution mechanism of fiber strain and strain rate induced by multi-fracture propagation. The present study provided a theoretical basis for diagnosing fracture propagation by distributed optical fiber strain signals.

During fracturing, fracture deformation might induce additional stresses in the formation, called stress interference or stress shadow. The optical fiber cemented in the cement ring can sense the change of in-situ stress. According to the change of in-situ stress, the ground strain and fracture geometry can be inferred. Without considering the strain transfer loss from the rock to the fiber, the fiber strain is equal to rock strain; thus, we focus on the mechanism of rock strain evolution during hydraulic fracture propagation. A forward model for simulating fracture propagation is essential for interpreting optical fiber strain analysis. Note that the rock strain produces the fiber strain, so the strain transfer loss does not influence our analysis of fiber strain evolution ^[11]. Fig. 1 shows the geometric model of the fiber strain evolution induced by fracture growth. The forward model contains two parts: (1) PL3D multi-Fracture Growth Model coupled with wellbore and perforation; (2) fiber strain model. The primitive signals measured by optical fiber are fiber displacement. The fiber strain can be calculated according to the fiber displacements of distributed monitoring points. Thus, we need to calculate fiber displacement by the fiber strain model firstly.

1.2.1. Fluid flow model in the wellbore

Fracturing fluid is dynamically partitioned into each cluster through the wellbore. The fluid flow in the wellbore should satisfy the flow conservation and pressure continuity conditions.

First, the total injection rate is equal to the sum of the flow rate of each fracture, namely:

Second, the pressure should be continuous, namely:

The perforation friction can be calculated by referring to reference [18]. The flow friction resistance from the heel-side fracture to the k^th fracture along the wellbore is:

The fluid flow regime in the wellbore includes laminar flow, transition flow, and turbulent flow. An approximate formula considering the whole regime is used to calculate friction resistance coefficient along the wellbore ^[19]:

where ${{\Theta }_{1}}$ and ${{\Theta }_{2}}$ are expressed as ^[19]:

1.2.2. Solid-fluid coupling model

The model requires to be simplified on fracture geometry calculation to simulate hydraulic fracture growth. The practical Fracture Growth Models are mainly the P3D model (e.g., FracproPT, Meyer, and Kinetix software) and the PL3D model (e.g., Gohfer and StimPlan software) ^[20,21]. A fully 3D model is very time-consuming, which is unfavorable for the engineering design. Fractured core observation ^[22], fracture monitoring ^[11], and fracture history matching ^[23] of fracturing show that the hydraulic fracture is mostly planar. Therefore, we use the PL3D model to simulate fracture growth, considering the computation accuracy and efficiency.

The relationship between fluid pressure in the fracture and fracture width can be given based on the discontinuous boundary elements of 3D displacement ^[24]:

The continuity equation of fluid flow in the fracture considering laminar flow in the fracture and fluid leak-off into the rock is:

Combining Eqs. (6) and (7), we can get the solid-fluid coupling equation for the fracture width and pressure ^[25]. The ordinary differential equation of fracture width can be obtained by using the finite volume discretization on a structured mesh:

Eq. (8) is the coupled equation for the solid-fluid coupling system during fracture growth. The implicit numerical method requires extensive iterations to solve the nonlinear equations, making the implicit way less efficient. In the study, an explicit algorithm for solving Eq. (8) is adopted by letting θ=0 ^[26]. A 2^nd-order accuracy Runge-Kutta-Legendre (RKL) method improves the computation efficiency to enlarge the stability domain. The method avoids numerous iterations resulting from solving nonlinear equations and increases the computation efficiency by 10 times that of the implicit method ^[27].

1.2.3. Coupling of fluid flow in the wellbore and multi-fracture growth

The fluid flow in the wellbore and multi-fracture growth is iteratively coupled by the bottom pressure and flow rate. The discretized equation for flow rate in the wellbore is:

Eq. (9) is a system of nonlinear equations with N+1 unknowns. The component form of Eq. (9) is:

The inlet pressure of each fracture can be calculated by solving the coupled equations of solid and flow in multiple fractures. The flow rate of each fracture can be solved using the inlet pressure by Eq. (10). The process loops until the inlet pressure and the flow rate converge.

1.2.4. Fracture boundary calculation

The fracture fronts are constructed by the element- based method. When the stress intensity factor or the critical width of the tip elements satisfies the propagation criterion, the non-activated elements adjacent to the tip element update as new tip elements. A fine mesh is needed to capture the fracture tip if the linear elastic fracture mechanical criterion is used. Therefore, the tip asymptotical solutions are adopted as the propagation criterion to reduce computation burden without loss of accuracy. The tip asymptotical solutions are equivalent to linear fracture mechanical criterion with an increased validity region of 10%-20% fracture length ^[28]. Therefore, the grid size can be enlarged to reduce computation burden and time.

The fiber displacement is equal to stratum displacement without considering the strain transfer loss. The fiber is a thin and long material that only with axial deformation. The distributed fiber strain monitoring obtains fiber displacement by monitoring the laser phase shift (Fig. 2), then the fiber strain can be calculated by the displacement distribution. The gauge length for the optical fiber measurement is usually 5-10 m, so the calculated strain and strain rate are average values in the gauge length.

Set the fracture element number to be M at time t, the displacement of each element can be analytically deter-mined based on the displacement discontinuity fundamental solution, then the displacement distribution along the optical fiber can be calculated by superposing the solutions of the M elements ^[24]:

where I₁ and I₂ are kernel functions, expressed as:

where r and operator || is:

where$a=0.5\Delta x,\ b=0.5\Delta y$

According to the relationship between displacement and strain, the fiber (axial) strain can be calculated with the central differential method as:

The strain rate is given according to strains at the adjacent time as:

Eqs. (14) and (15) are all change rates of discrete data, which exhibit a certain roughness. In the study, the Gaussian filtering method is used to smooth the strain and strain rate data.

When the distributed fiber strain displacement is continuous, the fiber displacement calculated by Equation (14) is the stratum displacement. When the distributed fiber strain displacement is discontinuous (such as when the fractures extend to the fiber), the displacement near the fracture plane is discontinuous. The fiber strain calculated from the fiber displacement at the discontinuous part is not equal to the stratum strain. In contrast, it is still the stratum strain at the continuous part.

The numerical model contains three modules, namely the wellbore model, fracture growth model, and fiber strain model. The wellbore and fracture growth models are coupled using the flow rate and inlet pressure in each fracture. After the fracture width and geometry are determined in each time step, the fiber strain and strain rate are calculated by the fiber strain model (Fig. 3).

The fracture growth model has been verified from experimental results, analytical solutions, and previously published numerical results in reference [26]. The fiber strain is computed using the stratum displacement; thus, the displacement is further validated in the study. The analytical solutions of the displacements induced by a Penny-fracture under constant internal pressure are ^[29] :

The fracture is discretized by a square grid. The fracture radius R is 100 m, rock elasticity modulus E is 35 GPa, Poisson's ratio υ is 0.2, and the fluid pressure in the fracture p is 1 MPa. Different mesh sizes of Δx of 10, 5, and 2 m are implemented to test the mesh sensitivity for displacement calculation. The Eq. (16) is solved by using Simpson numerical integration. Fig. 4 compares analytical solutions with the numerical solutions of displacement at r_p=50 m. The results show that the relative error of the numerical solutions for displacements are within 5%, indicating the numerical model in this study for displacement calculation is reliable and accurate.

To study the strain and strain rate during fracture growth, the fracture growth of a single cluster case and five-cluster case are conducted. The basic input parameters are listed as follows. The reservoir thickness is 50 m, inter-layer stress difference is 5 MPa, elasticity modulus is 32.0 GPa, Poisson's ratio is 0.2, fracture toughness is 0.5 MPa•m^0.5, and injection rate is 2.4 m³/min and 12 m³/min for the single cluster case and five-cluster case, respectively. The fracturing fluid viscosity is 10 mPa•s, perforation diameter is 12 mm, perforation abrasion coefficient is 0.8, perforation number in each cluster is 12, cluster spacing is 10 m, operation time is 120 min in which the injection time is 100 min, and shut-in time is 20 min. The in-situ stress distribution and wellbore placement are shown in Fig. 5. In Fig. 5, the perforation is at a depth of y=0 m. The lateral sections of the two horizontal wells are parallel with the well spacing of 200 m. The vertical depth difference is 0, and the wellbore inner diameter is 10.48 cm, and the roughness of the inner wall of the wellbore is 1 μm. The gauge length for the distributed fiber strain sensor is 5 m. It should be noted that the leak-off coefficient is a critical parameter influencing fracture length but is hard to be accurately determined due to the uncertain reservoir heterogeneity and natural fractures ^[23]. In the numerical tests, the leak-off coefficient is 2×10^-4 m/min^0.5.

The range of fiber strain variation during the fracture propagation exhibits a magnitude difference ^[30]. To show the strain and strain rate evolution clearly, the nano- strain ε_n(ε_n=10⁹ε) and nano-strain rate $\ {{\dot{\varepsilon }}_{\text{n}}}$ ($\ {{\dot{\varepsilon }}_{\text{n}}}={{10}^{9}}\dot{\varepsilon }$) in log scale are used to present the performance plots of strain and strain rate evolution with fracturing time. The nano-strain and nano-strain rate in log scale is formulated, respectively:

The strain and strain rate in the log scale is called log strain and log strain rate, respectively, for convenience.

Firstly, we analyze the fiber strain and strain rate evolution during single fracture propagation. The fracture height is confined in the pay zone due to the high inter-layer stress difference. The fracture half-length can be calculated based on fracture area (Fig. 6).

When the injection time is 43 min, the fracture half-length is 200 m. The fracture tip reaches the fiber in monitoring well. When the injection time is 100 min, the pump is stopped. The fracture continues to propagate for about 6 min and with a half-length of 320 m eventually. Fig. 7 shows the fracture geometries when the injection time is 100 min and 120 min. After the pump ends, the maximum fracture width decreases by about 1.5 mm.

Fig. 8 shows the distributed optical fiber strain evolution and the bottom hole pressure during single fracture propagation. The plot of log strain and fracturing time reflects the variation of fiber strain distribution along the fiber axial direction (z coordinate) with time (Fig. 8a). The blue color in Fig. 8 represents compressive strain, and the red color represents tensile strain. It can be seen that the fiber strain during fracture propagation can be divided into three stages: (1) The strain increasing stage (stage A in Fig. 8), the fiber exhibits tensile strain, and the strain is relatively weak; (2) the strain shrinkage convergence stage (stage B in Fig. 8), the tensile strain region is reduced; (3) The strain straight line convergence stage (stage C in Fig. 8), the optical fiber tensile strain signal shrinks into a straight line, the width of the convergence band is a gauge length (5 m). The compressive region besides the band is the stress shadow area. After injection for 100 min and the pump stopped, no obvious change in strain signal was shown on the fiber strain distribution plot. The bottom hole pressure increased slightly during the fracture propagation, consistent with the bottom hole pressure variation for the PKN (Perkins-Kern-Nordgren) Fracture Growth Model. After the pumping stopped, the bottom hole pressure dropped rapidly.

To further analyze the change of fiber strain during single fracture propagation, three measuring points in the fiber are selected to study the strain evolution clearly (Fig. 8b). It can be observed that the optical fiber shows tensile strain before the fracture hits the optical fiber monitoring well (0-43 min). When the fracture hits the optical fiber, the tensile strain suddenly increases and rapidly transforms into the compressive value. The compressive strain increases continuously before the pumping ends, while the increase rate gradually decreases. After pumping ends, the compressive strain rises for about 6 min and then decreases until 120 min.

Fig. 9 shows the evolution of distributed optical fiber strain rate and bottom hole pressure during single fracture propagation. The relationship of log strain rate and fracturing time reflects the strain rate change of each measuring point (z axial coordinate) of the distributed optical fiber with injection time (Fig. 9a). The blue color is the negative strain rate, and the red is the positive strain rate. It can be seen from Eq. (15) that a negative strain rate denotes an increase in compressive strain or a decrease in tensile strain, and a positive strain rate denotes an increase in tensile strain or a reduction of compressive strain. The fiber strain rate variation during a fracture propagation includes four stages: (1) The strain rate increasing stage (stage A in Fig. 9a), the fiber strain rate at this stage is positive; (2) The strain rate shrinkage convergence stage (stage B in Fig. 9a). At this stage, the strain rate of the optical fiber has a heart-shaped convergence feature, which is consistent with the stress characteristics around the fracture tip. Therefore, this feature can clearly show the time of inter-well pressure channeling; (3) The strain rate straight line convergence stage (stage C in Fig. 9a). Due to the discontinuity of the displacement on both sides of the optical fiber, the strain rate on both sides of the convergence band is negative, which is the stress shadow area; (4) The strain rate reversal stage after the pumping stop (stage D in Fig. 9a). The positive strain rate of the fiber changed to negative strain, and the negative strain rate turned into a positive strain rate at the last stage. This four-stage pattern of strain rate is consistent with the field study of Ugueto et al. ^[12], which is based on field monitoring results, indicating that the fiber strain forward model in this study is reasonable.

Similarly, three optical fiber measurement points were selected to analyze the strain rate evolution with operation time (Fig. 9b). When the fracture is far away from the optical fiber monitoring well, the strain rate is weak, and it is a positive strain rate. After the fracture extends to the optical fiber, the positive strain rate reaches a peak, and then the strain rate drops to a negative value, and it starts to rise after reaching the peak value. After the ending of the pump, the strain rate signal increases rapidly and turns into a positive strain rate signal. Because only one fracture extends to the optical fiber monitoring well, the strain rate has only one obvious peak. At the same time, the peak of the optical fiber measuring point closer to the fracture is more obvious.

In summary, when a fracture hits the fiber, the performance plots of fiber strain or strain rate shrink to a straight line convergence band, with the width of theoretically the gauge length (5 m in the study). Two sides of the convergence band are the compressive strain or negative strain rate regions, indicating both fiber strain and the strain rate distribution show the stress shadowing region. The sign of the fiber strain does not change after the ending of the pump, but the sign of the strain rate reverses. According to the strain or strain rate pattern characteristics, the inter-well pressure channeling can be identified, and the fracture propagation velocity and the injected fluid volume can be estimated.

Fig. 10 shows the half-length of 5 clusters of fractures in a stage change with injection time. It can be seen that the heel-end fracture (HF1) has the largest length, while the middle fracture (HF3) has the smallest, showing a phenomenon of "heel-end dominant" growth for multi- cluster fracturing. Field near-well DAS (distributed acoustic sensing) and perforation abrasion imaging have confirmed this phenomenon ^[9]. From the fracture geometry at the end of injection and operation (Fig. 11), it can be seen that the width of the fracture is reduced after the pump is stopped due to fluid leak-off, but the width of the exterior fracture is still greater than the width of the middle fracture.

Fig. 12 shows the evolution of fiber strain and the bottom hole pressure change during the growth of five clusters of fractures in a stage. The performance plot of fiber strain during the propagation of the five clusters of fractures can also be divided into three stages (Fig. 12a): (1) The strain increasing stage (stage A), the fiber shows tensile strain which becomes stronger during operation. (2) Strain shrinkage convergence stage (stage B), due to the different propagation velocities of multiple fractures, the heel-end fracture (HF1) first reaches the monitoring well. Therefore, the convergence point first appears at the HF1 position. (3) Strain straight line convergence stage (stage C), each band appears corresponding to a fracture. Each convergence strip corresponds to a fracture, and it converges in turn according to the speed of fracture expansion. After the ending of the pump, the fiber strain sign of the multi-cluster fractures does not reverse. The time for each fracture reaching the fiber monitoring well can be obtained from Fig. 12a, and then the non-uniform propagation of multiple fractures can be evaluated. In the numerical analysis, HF1 reaches the optical fiber earliest, and HF3 is the latest, and the time difference between HF1 and HF3 reaching the monitoring well is 35 min. The growing velocity difference between HF1 and HF3 can be estimated. The strain curve of three optical fiber measurement points (Fig. 12b) shows that the distributed optical fiber strain presents multiple peak points, consistent with the uneven propagation of multiple c fractures.

The strain rate during the propagation of the five clusters of fractures in a stage also includes 4 stages (Fig. 13a): (1) The strain rate increasing stage (stage A); (2) The strain rate shrinkage convergence stage (stage B); (3) The straight-line convergence stage of strain rate (stage C). At this stage, the heel-end fracture (HF1) first extends to the fiber in the monitoring well, followed by HF5, HF2, HF4, and HF3. When the fracture reaches the fiber, a "heart-shaped" pattern in strain rate appears. At this stage, the tensile strain rate convergence band appears "break line" (sign change in strain rates). For example, when HF2 reaches the optical fiber, a negative strain rate straight line convergence band appears in the tensile strain rate convergence band of HF1, and the duration of negative strain rate lasts for about 15 min and then turns to the positive strain rate. Fig. 13a shows no fractures without propagation during the injection process. Therefore, the "break line" in the band indicates the strain change resulting from uneven fracture propagation and stress interference. The field monitoring presented by Ugueto et al. ^[15] also displays a relatively common phenomenon of "break line." The simulation analysis in this study explains the mechanism of "break line"; (4) The strain rate reversal stage after the stopping of the pump (Stage D). At this stage, the positive strain rate bands corresponding to fractures change into negative strain rate bands, and the other positions reverse from negative strain rate to positive strain rate. From the strain rate change of the three optical fiber measurement points (Fig. 13b), the strain rate characteristics of stages A, B, and D are the same as that of a single fracture case. However, in stage C, the non-uniform propagation of multiple fractures causes multiple peaks of the strain rate. An obvious strain rate peak will be generated when the single fracture extends to an optical fiber monitoring well since there is only one fracture. For the multi-fracture propagation, when the subsequent fractures reach the optical fiber monitoring well, the fiber strain at the measuring point includes the tensile strain near the tip of the fracture and the compressive strain generated by the fracture that has reached the fiber monitoring well. Consequently, although all five fractures hit the optical fiber monitoring well, the number of strain rate peaks may be less than five times.

In summary, the convergence bands of fiber strain and strain rate can identify the time when the fracture hits the fiber. The fracture width becomes smaller with constant strain direction after pumping end, the strain rate reverses and therefore can reflect the injection performance. The fiber strain and strain rate in the multi-fracture propagation process can be observed when the fracture hits the fiber so that the non-uniform fracture propagation can be identified, and then the fracture propagation velocity can be estimated based on the well spacing.

The five-fracture case is used to discuss the evolution of the fiber strain and strain rate when the depths of the monitoring well and the fracturing well are different. The vertical propagation of the fracture in this study can reach the position of y=25 m. If the depth difference between the monitoring well and the fracturing well is 10 m, the fracture can still hit the optical fiber in the monitoring well. Therefore, when the fractures reach the fiber, the strain and strain rate change during the entire injection process is still within the observable range of the fiber.

Fig. 14 shows the performance of strain and strain rate and bottom hole pressure during multi-cluster of fracture propagation when the depth of the monitoring well is 10 m higher than that of the fracturing well. It can be seen that strain and strain rate all show the straight-line convergence bands, and the performance of strain and strain rate are the same with the case of h_w=0 m (Figs. 12a and 13a). When the depth of the monitoring well is 30 m higher than that of the fracturing well, the fiber of the monitoring well is above the fracture during the entire injection time. Namely, the fracture cannot touch the optical fiber in the vertical direction, so the straight line convergence band of strain and strain rate can not be monitored (Fig. 15). The convergence band has a wide range, about 60-100 m. Although the signal intensity changes, it is difficult to distinguish each fracture. Hence, it is not easy to evaluate the propagation difference. Still, it can be used to judge that the fractures did not extend to the position of the fiber in the vertical direction.

We used a 2-stage 5-cluster fracturing case to examine the influence of multi-stage fracturing on the evolution of optical fiber strain and strain rate. The parameters of the two stages of fracturing were identical with those of single-stage fracturing. The distance between the toe end of the second stage and the heel end of the first stage was 20 m. The fracture geometry at the end of the operation is shown in Fig. 16. It can be seen from Fig. 16 that due to the inter-stage stress interference, the toe-end fracture (HF5) of stage 2 is suppressed by greater stress shadowing effect. The length and width of this fracture are significantly smaller than the heel-end fracture (HF1) of stage 2, which is consistent with the field monitoring result ^[7].

Fig. 17 shows the fiber strain evolution and bottom hole pressure variation during the multi-stage and multi-cluster fracturing process. During the first stage (0-120 min), the optical fiber strain signal is the same as single-stage fracturing. After the ending of the first stage fracturing, the fiber strain at the fracture extension position of the first stage is still the tensile strain band, and the outside of the band is the stress shadow area. However, there is no tensile convergence pattern during the second fracturing stage due to the influence of the stress shadow from the first stage. When the fractures in the second stage hit the optical fiber of the monitoring well, the tensile strain straight line convergence band appears. The extension pressure of the second stage fracturing is about 1.0 MPa higher than that of the first stage. Since the perforation friction and wellbore friction of two fracturing stages are close, it can be inferred that the stress interference of the first stage fracturing on the second stage is about 1.0 MPa.

Fig. 18 shows the evolution of optical fiber strain rate and bottom hole pressure during the 2-stage and 5-cluster fracturing process. The fiber strain rate evolution of the first stage fracturing is the same as that of the single-stage fracturing; during the second stage fracturing, despite the stress interference of the previous stage, the strain rate characteristics of the second stage are the same as that of the single-stage fracturing. The four strain rate stages all exist in the second stage. The strain rate of the optical fiber results from the change of the fracture geometry, so the strain rate of each stage can reflect the fracture propagation process of each stage. When the fracture size and width of the first stage do not change, the strain from the first stage to the second stage is constant, so it does not affect the rate of stain change.

In summary, for multi-stage fracturing, influenced by the stress interference of the previous fracturing stage, the fiber optic tensile strain convergence region may not appear in the subsequent fracturing stage. The fiber strain rate can effectively present the fracture propagation performance of each stage of fracturing. The evolution characteristics of the optical fiber strain rate for single- stage fracturing are applicable for multi-stage fracturing.

The fiber strain evolution induced by fracture propagation can be divided into three stages, including strain increasing, shrinkage convergence, and strain straight line convergence. The strain rate evolution can be divided into four stages, including strain rate increasing, shrinkage convergence, straight-line convergence, and reversal after the stopping of the pump. The fiber strain does not change after the stopping of the pump, but the strain rate can be reversed. The fiber strain rate reversal can reflect the injection performance.

On the basis of the straight line convergence band of distributed optical fiber strain and strain rate, it is possible to identify the time of fractures reaching the optical fiber and inter-well pressure channeling. At the same time, we can identify the non-uniform propagation of multiple fractures according to the time of each fracture reaching the optical fiber monitoring well and evaluate the degree of uneven fracture propagation.

When the optical fiber in the monitoring well is within the range of the fracture height, the time when the fracture extends to the optical fiber can be identified, and when the optical fiber in the monitoring well is beyond the fracture height, the straight line convergence band is not apparent.

During the multi-stage fracturing operation, influenced by the stress interference from the previous fracturing stage, the optical fiber strain of the subsequent fracturing stage may no longer show the tensile strain region as the initial stage of fracturing. However, the strain rate of the optical fiber can effectively show the fracture propagation in each stage. The evolution of fiber strain rate in single-stage fracturing is applicable for the multi-stage fracturing case.

a, b—half-length in the x and y direction, respectively, m;

A—coefficient matrix from the discretized flow equation, m/(s•Pa);

C—influence coefficient matrix, Pa/m;

C_l—leak-off coefficient, m/s^0.5;

d—distance from the fracture surface, m;

D_w—wellbore diameter, m;

E—rock elasticity modulus, Pa;

F—equations;

F_k—the k^th component of F, its unit is Pa when 1≤k≤N, and m³/s when k=N+1;

f_c—friction coefficient along the wellbore, dimensionless;

h_w—depth difference between the monitoring well and fracturing well, m;

i—serial number of mesh in x direction;

I₁, I₂—kernel function, dimensionless;

$I\left( \xi,\eta \right)$—kernel function, dimensionless;

J₀, J₁—the first kind of zero-order and first-order Bessel functions;

k—serial number of cluster;

l_w—distance from the heel-end perforation cluster, m;

L—gauge length of measuring points, m;

m—serial number of fracturing stage;

M—number of meshes;

N—number of fractures;

p—fluid pressure in the fracture, Pa;

p—matrix of fluid pressure in the fracture, Pa;

p_in,k—the inlet pressure of the k^th fracture, Pa;

p_c,k—wellbore flow friction resistance from the k^th fracture to the heel-end fracture, Pa;

p_p,k—the perforation friction resistance of the k^th fracture, Pa;

p_w—bottom hole pressure at the heel-end fracture, Pa;

Q_k—the inlet flow rate of the k^th fracture, m³/s;

Q_t—injection rate, m³/s;

r—distance from fiber measuring point to fracture element, m;

r_p—radial distance, m;

R—fracture radius, m;

Re—Reynolds number, dimensionless;

s_w—well spacing of the monitoring well and the fracturing well, m;

S—coefficient matrix of the source term, m/s;

t—time, s;

t₀—time when leak-off occurs, s;

Δt—time step, min;

u_f—axial displacement of fiber, m;

u_n, u_r—vertical and radial displacement of the circle fracture, respectively, m;

V_w—fluid flow velocity in the wellbore, m/s;

w—fracture width, m;

w—matrix of fracture width, m;

w₀—matrix of width at the previous time step, m;

x—vector of coordinate, m;

x_in,k—vector of coordinate at the injection postion, m;

x, y, z—cooridinate, m;

$\delta \left( x \right)$—Dirac function, m^-2;

Δx, Δy—grid size, m;

ε—fiber strain, dimensionless;

ε_a—wellbore inner wall roughness, m;

$\dot{\varepsilon }$—fiber strain rate, min^-1;

ε_L—strain in log scale, dimensionless;

${{\dot{\varepsilon }}_{\text{L}}}$—strain rate in log scale, dimensionless;

ε_n—nano-strain, dimensionless;

${{\dot{\varepsilon }}_{\text{n}}}$—nano-strain rate, min^-1;

θ—constant coefficient, 0≤θ≤1;

${{\Theta }_{1}}$, ${{\Theta }_{2}}$—Intermediate variables;

$\zeta $—integration variable, dimensionless;

μ—fluid viscosity, Pa·s;

v—Rock Poisson' 's ratio, dimensionless;

$\xi $,$\eta $—local coordinate of the mesh, m;

ρ—fluid density, kg/m³;

σ_h—minimum horizontal principal stress, MPa;

σ_h—matrix of minimum horizontal principal stress, MPa.