Automated Water Hammer Analysis for Fracture Parameter Inversion Using High-Frequency Shut-In Pressure Signals During Hydraulic Fracturing

Abstract

Hydraulic fracture geometry is of great importance for evaluating stimulation effectiveness and supporting the efficient development of unconventional oil and gas reservoirs, and it can be estimated from field shut-in water hammer signals. However, field signals are commonly characterized by strong noise, pronounced non-stationarity, strong dependence on manual extraction of effective response segments, and limited automation in inversion analysis. To address these issues, this study develops an integrated automated interpretation framework for shut-in water hammer analysis, which combines an adaptive shape-preserving Kalman filter for non-stationary signal denoising, an automatic response segment identification method, and a particle swarm optimization-based inversion strategy for fracture geometry estimation. The framework is validated using field high-frequency pressure data from hydraulically fractured wells. The results show that the proposed denoising method improves the signal-to-noise ratio from 11.99 dB to 25.05 dB while preserving key transient features. The response segments can be extracted efficiently, with runtimes of 0.84–1.22 s and onset errors within 0–5 s. For a representative fracturing stage, the relative errors of the inverted fracture half-length and fracture height are 6.21% and 3.04%, respectively. The proposed framework provides a low-cost and field-applicable tool for fracture evaluation and engineering decision-making.

1. Introduction

With the growing importance of unconventional oil and gas resources as a major contributor to global hydrocarbon reserve replacement and production growth, hydraulic fracturing has become a key technique for the effective stimulation of unconventional reservoirs [1,2,3]. The geometric characteristics of hydraulic fractures, such as half-length, height, and volume, directly govern post-fracturing well productivity and ultimate recovery [4]. Therefore, developing an efficient, reliable, and field-applicable method for fracture parameter monitoring and interpretation is of great significance for evaluating fracturing performance and optimizing subsequent development strategies [5,6]. At present, fracture monitoring mainly relies on techniques such as microseismic monitoring, distributed fiber-optic sensing, and downhole imaging. Although these methods have played an important role in characterizing fracture propagation and geometry [7,8,9,10], they often suffer from practical limitations, including operational complexity, high cost, and limited suitability for large-scale real-time field applications [11]. In contrast, water hammer-based diagnostics, which utilize the transient wellhead pressure response after pump shut-in, can directly take advantage of high-frequency pressure data recorded during fracturing operations without requiring additional downhole tools. This approach therefore offers several notable advantages, including ease of implementation, low interpretation costs, and strong engineering applicability [12,13]. Because the propagation of water hammer waves within the coupled wellbore–fracture system is highly sensitive to bottom boundary conditions and fracture properties, these transient signals provide a solid theoretical basis for fracture parameter inversion [14,15,16,17,18,19]. Despite the initial application of water hammer analysis in oil and gas development, several critical challenges still hinder its engineering deployment for the interpretation of high-frequency field pressure signals. First, field water hammer signals are commonly contaminated by strong noise and non-stationary disturbances, and conventional filtering methods often struggle to suppress noise while preserving transient waveform characteristics [20]. Second, the extraction of effective water hammer response segments still relies heavily on manual experience, and a unified, repeatable, and automated identification method remains lacking, making it difficult to process large volumes of data from multiple stages or wells efficiently [21,22]. Third, fracture parameter inversion is intrinsically a strongly nonlinear and highly constrained optimization problem. Existing studies have not yet established an integrated automated interpretation framework that effectively couples signal preprocessing, event identification, and physics-constrained inversion, thereby limiting both the efficiency and robustness of inversion results [23,24].

To address these challenges, this study develops an integrated and automated water hammer-based framework for fracture parameter interpretation using high-frequency shut-in pressure signals acquired during hydraulic fracturing. The framework combines a coupled wellbore–fracture forward model, an adaptive shape-preserving Kalman filter for denoising non-stationary signals, an objective response segment identification scheme constrained by pressure derivatives, multi-window statistics, and signal envelopes, and a global optimization-based inversion strategy for fracture geometry estimation. In this way, the proposed method enables a seamless transition from raw pressure data to physics-constrained fracture interpretation, reducing subjective intervention and improving the efficiency, robustness, and engineering applicability of water hammer analysis under field conditions.

2. Model Description

2.1. Water Hammer Model

The propagation of water hammer waves in the wellbore is essentially governed by the combined effects of fluid mass conservation and momentum conservation. Its transient response is jointly controlled by wellbore geometric parameters, fluid compressibility, tubing elasticity, and bottomhole boundary conditions. For transient flow propagating along the wellbore axis, the classical water hammer theory can be employed to establish the coupled governing equations for pressure and flow rate, thereby describing the propagation and attenuation of pressure disturbances immediately following pump shut-in.

The transient pressure behavior in the wellbore is governed by the one-dimensional momentum and continuity equations [25]:

$${\displaystyle \frac{\partial Q}{\partial t}}+gA{\displaystyle \frac{\partial H}{\partial x}}+{\displaystyle \frac{fQ\left|Q\right|}{2DA}}=0$$

(1)

$${\displaystyle \frac{\partial H}{\partial t}}+{\displaystyle \frac{a^2}{gA}}{\displaystyle \frac{\partial Q}{\partial x}}=0$$

(2)

where $a$ denotes the wave velocity, m/s; A is the cross-sectional area of the pipe, m²; Q is the flow rate, m³/s; and D is the pipe diameter, m.

$$H={\displaystyle \frac{P}{\rho \mathrm{g}}}+z$$

(3)

where H is the hydraulic head, m; and z is the elevation above the reference datum, m. The water hammer wave velocity in the wellbore can be expressed as follows [26]:

$$a=\sqrt{{\displaystyle \frac{K/\rho }{1+(K/E)\psi }}}$$

(4)

where K denotes the bulk modulus, GPa; ρ is the fluid density, kg/m³; E is the Young’s modulus of the pipe wall, GPa; and Ψ is a dimensionless parameter introduced to account for different structural conditions, including rigid pipes, thick-walled elastic pipes, thin-walled elastic pipes, and tunnels passing through hard rock. It is determined by the following equation:

$$\psi ={\displaystyle \frac{D}{e}}(1-v^2)$$

(5)

where e is the pipe wall thickness (mm), and ν is the Poisson’s ratio of the pipe material. Equations (1) and (2) constitute a pair of linear hyperbolic partial differential equations and therefore can only be solved numerically. To this end, the method of characteristics is employed to transform the governing partial differential equations into four ordinary differential equations, which are subsequently solved using the finite difference method. The resulting formulations can be written in terms of the characteristic equations along the C⁺ and C⁻ lines, as given in Equations (6) and (7).

$$C^{+}: H_i^{t+\Delta t}=H_{i-1}^t-B\left(Q_i^{t+\Delta t}-Q_{i-1}^t\right)-r{Q}_{i-1}^t\left|Q_{i-1}^t\right|$$

(6)

$$C^{-}: H_i^{t+\Delta t}=H_{i+1}^t+B\left(Q_i^{t+\Delta t}-Q_{i+1}^t\right)+r{Q}_{i+1}^t\left|Q_{i+1}^t\right|$$

(7)

where B denotes the characteristic impedance of the pipe, as given by $B={\displaystyle \frac{a}{gA}}$; r denotes the pipe resistance coefficient, as defined by $r={\displaystyle \frac{f∆x}{2gD{A}^2}}$; and f is the friction factor appearing in Equation (8):

$${\displaystyle \frac{1}{\sqrt{f}}}=-2\log \left[{\displaystyle \frac{1}{3.7065}}({\displaystyle \frac{\epsilon }{D}})-{\displaystyle \frac{5.0452}{\mathrm{Re}}}\left(\log \left[{\displaystyle \frac{1}{2.8257}}{\left({\displaystyle \frac{\epsilon }{D}}\right)}^{1.1098}+{\displaystyle \frac{5.8506}{{\mathrm{Re}}^{0.8981}}}\right]\right)\right]$$

(8)

where D is the pipe diameter, m; ε is the pipe roughness; and Re is the Reynolds number. To fully characterize the coupled wellbore–fracture transient response after pump shut-in during hydraulic fracturing, both the wellhead flow boundary condition and the bottomhole fracture boundary condition must be specified.

At the wellhead, the flow rate is assumed to decrease linearly to zero during pump shutdown. Accordingly, the wellhead boundary condition can be defined in terms of the variation in flow rate, as expressed in Equation (9).

$$Q_1^t=Q_0\tau (t)$$

(9)

where Q(t) denotes the wellhead boundary flow rate, which varies with time during pump shut-in. By incorporating the wellhead flow boundary condition, the corresponding characteristic equations can be solved simultaneously to obtain the transient pressure response at the wellhead boundary. Based on the prescribed wellhead boundary flow rate, the hydraulic head at the wellhead boundary can then be calculated by combining Equations (7) and (9).

To simplify the bottomhole boundary, multiple fractures are represented by an equivalent single-fracture system. Following the bottomhole fracture boundary formulation proposed by Carey et al. [14], the fracture boundary condition can be modeled by an equivalent electrical circuit consisting of constants R, C and I.

Accordingly, the boundary condition is expressed as follows:

$$\rho g\Delta H=RQ+{\displaystyle \frac{1}{C}}{\displaystyle \int Qdt+I\frac{dQ}{dt}}$$

(10)

The relationships among R, C and I are given by Equations (11)–(14). In the RCI-equivalent circuit model, R, C, and I correspond to the resistance, capacitance, and inductance elements in an electrical circuit, respectively. In the hydraulic fracturing model, R characterizes the resistance effect generated during fluid flow in the near-wellbore region, mainly reflecting near-wellbore frictional losses (Pa·s/m³); C represents the storage and elastic response capacity of the fracture system, corresponding to the capacitance associated with fracture compliance (m³/Pa); and I characterizes the inertial effect of fluid flow in the wellbore–fracture coupled system, i.e., the inertance of the system (Pa/(m³/s²)).

$$R={\displaystyle \frac{\Delta P_{nwf}}{Q_0}}$$

(11)

$$C={\displaystyle \frac{\Delta V}{\Delta P}}$$

(12)

$$Q=C{\displaystyle \frac{dP}{dt}}$$

(13)

$$\Delta P=RQ=I{\displaystyle \frac{dQ}{dt}}$$

(14)

where Q₀ is the injection rate before pump shut-in, m/s, ΔP_nwf is the near-wellbore frictional pressure drop, MPa, ΔV is the change in fracture volume, m³, and ΔP is the change in fracture pressure, MPa. The average net pressure ΔP₀ is defined as the difference between the average fracture pressure ($P_{BH}-\Delta P_{nwf}$) and the minimum horizontal principal stress ($S_{H\min }$), MPa.

$$\Delta P_0=(P_{BH}-\Delta P_{nwf})-S_{H\min }$$

(15)

The average net pressure is related to the fracture dimensions through the following equation, and the fractures are classified into long fractures (2L_f/h_f ≥ 1) and short fractures (2L_f/h_f < 1) [27].

$$\Delta P_0={\displaystyle \frac{4}{{\pi }^2}}{\displaystyle \frac{E}{1-{\nu }^2}}E(m)w\max ({\displaystyle \frac{1}{L_f}},{\displaystyle \frac{2}{L_f}})$$

(16)

E(m) is the complete elliptic integral of the second kind, where the definition of m is given in Equation (18).

$$E(m)={\displaystyle {\int }_0^1\frac{\sqrt{1-m^2{t}^2}}{1-t^2}dt}$$

(17)

$$m=1-{\left({\displaystyle \frac{\min (\frac{1}{L_f},\frac{2}{L_f})}{\max (\frac{1}{L_f},\frac{2}{L_f})}}\right)}^2$$

(18)

The formulas for calculating fracture length, width, and height are given below.

$$L_f=\sqrt{{\displaystyle \frac{CI\Delta P_0}{\rho }}}$$

(19)

$$h_f=\left\{\begin{array}{ll}{\displaystyle \frac{4E`E(m)C}{{\pi }^2{L_f}^2}}\hfill & (\mathrm{short} \mathrm{fracture})\hfill \\ \sqrt{{\displaystyle \frac{4E`E(m)C}{{\pi }^2{L}_f}}}\hfill & (\mathrm{long} \mathrm{fracture})\hfill \end{array}\right.$$

(20)

$$w={\displaystyle \frac{\rho L_f}{I{h}_f}}$$

(21)

$$E`={\displaystyle \frac{E}{1-{\nu }^2}}$$

(22)

where L_f is the fracture half-length, m, h_f is the fracture height, m, and w is the fracture width, m.

Previous studies have demonstrated that water hammer models based on RCI boundary conditions can effectively characterize the effects of the bottomhole fracture system on pressure wave propagation and attenuation, and have been successfully applied to the analysis of fracture length, attenuation behavior, and multi-cluster fracturing responses [28]. Related studies further indicate that water hammer signals are highly sensitive to fracture geometry and near-wellbore boundary conditions, and generally show good agreement with monitoring results obtained from microseismic and other diagnostic techniques [29]. Therefore, the integration of water hammer dynamic models with automated signal identification and global optimization-based inversion methods provides both a sound theoretical foundation and practical engineering feasibility for the quantitative interpretation of hydraulic fracture parameters in field applications.

2.2. Adaptive Kalman Filter-Based Shape-Preserving Denoising

Field water hammer signals recorded after pump shut-in during hydraulic fracturing typically exhibit pronounced non-stationary characteristics. At the initial stage of pump shut-in, the pressure usually undergoes a rapid drop, followed by a stage of continuously decaying oscillations, and finally transitions gradually into a relatively stable recovery process. Meanwhile, measured data are often contaminated by high-frequency random noise, local spike outliers, and abnormal disturbances, which can easily obscure the effective transient features of the original signal. If feature identification and parameter fitting are performed directly on the raw signal, significant errors may be introduced. The classical Kalman filtering method has a solid theoretical foundation in time-series signal processing and achieves optimal state estimation of the system through recursive updating [30]. However, this method usually assumes that the process noise covariance and observation noise covariance are constant, that is, the system approximately satisfies stationary conditions. In the case of water hammer signals, which represent a typical non-stationary scenario, fixed noise parameters are difficult to adapt to the signal characteristics at different stages. During periods of abrupt transient variation, the filter response may lag behind the actual signal, leading to amplitude attenuation; during relatively stable stages, insufficient noise suppression may occur. Therefore, it is necessary to introduce an adaptive improvement to the conventional Kalman filter.

To address the above issues, this study proposes an adaptive Kalman filtering method driven by local variation intensity. Based on the local rate of change in the pressure signal, a dimensionless indicator reflecting the dynamic characteristics of the signal is constructed, and the noise parameters in the filtering process are dynamically adjusted accordingly, thereby achieving stage-wise adaptive optimization of the filtering performance.

(1)

Data preprocessing

Let the original pressure time series be defined as follows:

$$P(t_i), i=1,2,\dots ,N$$

(23)

Here, t_i denotes the sampling time, p(t_i) is the corresponding wellhead pressure value, and N is the total number of sampling points. To correct isolated spike outliers that may occur in the acquired data, a Hampel filtering method based on the local median and median absolute deviation (MAD) is first employed for robust preprocessing. For the i-th sampling point, within a local window Ω_i centered at that point with a half-window length of k, the local median is defined as follows:

$${\tilde{p}}_i=\mathrm{median}\{p(t_j)\mid j\in {\mathrm{\Omega }}_i\}$$

(24)

where ${\tilde{p}}_i$ denotes the local median within the window corresponding to the i-th point, $\mathrm{median}\{·\}$ represents the median operator, $p(t_j)$ denotes the j-th pressure value within the window, and Ω_i is the moving window set centered at the i-th point.

The local scale estimate is given by the following:

$${\sigma }_i=1.4826\cdot \mathrm{median}(|p(t_j)-{\tilde{p}}_i|,j\in {\mathrm{\Omega }}_i)$$

(25)

where ${\sigma }_i$ denotes the locally robust estimate of the standard deviation, MPa, 1.4826 is the scaling factor used to convert the MAD into an estimate of the standard deviation, and $p(t_j)-{\tilde{p}}_i$ represents the residual relative to the median, MPa.

If a sampling point satisfies the outlier criterion, it is identified as an isolated outlier and replaced with the local median; otherwise, the original value is preserved.

$$|p(t_i)-{\tilde{p}}_i|>n_{\sigma }{\sigma }_i$$

(26)

$n_{\sigma }{\sigma }_i$ is the threshold coefficient used for outlier identification and is typically chosen in the range of 3–5.

After this treatment, a pre-cleaned pressure series, p_pre(t_i) is obtained, namely, the pressure series after outlier correction, MPa. The purpose of this step is to remove local spike outliers without distorting the main pressure fluctuation pattern, thereby preventing anomalous points from being amplified during subsequent differentiation and threshold-based identification.

(2)

Construction of the variation intensity indicator

To characterize the variation intensity of the water hammer signal at different stages, the rate of pressure change between two adjacent sampling instants is adopted in this study as the basic measure of local variation intensity. This indicator can reflect the dynamic differences in the pressure signal during transient abrupt changes, oscillatory decay, and stable recovery stages. A larger rate of change indicates that the signal is more likely to be in a genuine transient response stage near that instant, whereas a smaller rate of change suggests that the signal tends to become stable and that the proportion of noise components may be higher.

Therefore, after obtaining the preprocessed series, a local variation intensity indicator is further constructed to enable the subsequent adaptive adjustment of filtering parameters. The adjacent difference and the time step are defined as follows:

$$\Delta p_i=p_{\mathrm{pre}}(t_i)-p_{\mathrm{pre}}(t_{i-1})$$

(27)

$$\Delta t_i=t_i-t_{i-1}$$

(28)

where Δp_i and Δt_i denote the pressure difference (MPa) and the time interval (s) between two adjacent sampling points, respectively. Accordingly, the absolute value of the local pressure-change slope can be expressed as follows:

$$s_i=\left|{\displaystyle \frac{\Delta p_i}{\Delta t_i}}\right|$$

(29)

where s_i denotes the absolute value of the pressure change rate, MPa/s.

Considering that field signals often contain outliers and non-Gaussian disturbances, the median s_med and the median absolute deviation s_mad of this sequence are further calculated to improve robustness against abnormal local fluctuations. Based on these quantities, a normalized motion score is constructed as follows:

$$m_i={\displaystyle \frac{s_i-s_{\mathrm{med}}}{3s_{\mathrm{mad}}}}$$

(30)

where s_med denotes the median of the slope sequence, and s_mad denotes the median absolute deviation of the slope sequence.

It is then constrained to the interval [0, 1], as follows:

$$m_i=\min (1,\max (0,m_i))$$

(31)

where m_i is the normalized variation intensity indicator, ranging from 0 to 1, and min and max denote the truncation functions. A larger value of m_i indicates that the signal changes more abruptly in the vicinity of that time instant, and is therefore more likely to correspond to a genuine transient response rather than a stable background segment. In the present algorithm, this indicator is used to dynamically adjust the process noise and observation noise in the Kalman filtering procedure.

(3)

Adaptive updating of Q and R

On this basis, an adaptive adjustment model for the process noise and observation noise is established as follows:

$$Q_k=Q_0·(1+\alpha m_k)$$

(32)

$$R_k=R_0·\max (\beta ,1-\gamma m_k)$$

(33)

where Q₀ and R₀ are the baseline process noise and observation noise parameters, respectively; α and γ are adjustment coefficients; and β is the lower-bound constraint parameter for the observation noise. The core idea of this adaptive strategy is as follows: when the signal varies abruptly, Q_k is increased to enhance the filter’s responsiveness to state variations, while R_k is reduced to increase confidence in the observations, thereby avoiding excessive smoothing of the true oscillatory signal. In contrast, when the signal tends to become stable, Q_k is reduced and R_k is increased so as to strengthen the denoising capability.

(4)

Standard Kalman filtering recursion

After the dynamic updating of the noise parameters is completed, the filtering process still follows the standard recursive Kalman filtering framework, including two stages: state prediction and measurement update.

$${\widehat{x}}_{k|k-1}={\widehat{x}}_{k-1|k-1}$$

(34)

$$P_{k|k-1}=P_{k-1|k-1}+Q_k$$

(35)

$$K_k={\displaystyle \frac{P_{k|k-1}}{P_{k|k-1}+R_k}}$$

(36)

Through the above adaptive mechanism, the filter can dynamically match signal characteristics across different time scales, thereby enabling shape-preserving denoising of transient water hammer signals. Compared with the conventional Kalman filter with fixed parameters, the proposed method significantly reduces high-frequency noise interference while preserving the amplitude and phase characteristics of pressure oscillations, thus improving the signal-to-noise ratio and interpretability of the signal.

Overall, the proposed adaptive Kalman filtering method based on local variation intensity can adjust the filtering parameters in real time according to the dynamic variation characteristics of the pressure signal, achieving a better balance between suppressing high-frequency noise and preserving effective transient features. Compared with fixed-parameter filtering, the present method is able to dynamically match signal variation characteristics over time scales and effectively suppress high-frequency noise and local abnormal disturbances while retaining the principal oscillation period, amplitude, and phase information of the water hammer signal. This method provides higher-quality input data for the subsequent automatic identification of effective water hammer segments and inversion of fracture parameters, and serves as a key preprocessing step in the overall automated interpretation workflow.

2.3. Automatic Identification Algorithm

Conventional water hammer signal analysis usually relies on manually extracting effective response segments from the pressure–time curve, followed by model fitting and interpretation. This process is not only time-consuming, but also strongly dependent on the interpreter’s experience, resulting in considerable subjectivity. Since the triggering of a water hammer event is typically accompanied by a pronounced pressure drop, the first-order derivative of pressure with respect to time can serve as an important basis for identifying candidate onset points. However, relying solely on negative derivatives is insufficient to distinguish genuine water hammer events from ordinary local fluctuations. To address this issue, this study introduces multi-window statistical criteria on the basis of derivative detection, so as to jointly constrain the pressure characteristics before and after candidate time instants. Considering that the water hammer phenomenon essentially originates from the short-term propagation of pressure waves induced by abrupt changes in fluid velocity, an automated identification algorithm is developed based on pressure-derivative features, local statistical characteristics, and oscillatory attenuation behavior to detect and extract physically meaningful effective water hammer response segments. The algorithm can automatically determine the triggering time, principal oscillation interval, and termination position of a water hammer event, and can further screen and score candidate segments, thereby enabling the automatic identification of multiple water hammer signals.

(1)

Adaptive filtering

The field-measured pressure signal, p(t) is first filtered to improve its signal-to-noise ratio, thereby reducing the influence of sensor noise and high-frequency interference on the pressure data.

$$\tilde{p}(t)=S\left\{p(t_{pre})\right\}$$

(37)

In this equation, P(t) denotes the original pressure signal, MPa, while $\tilde{p}(t)$ represents the smoothed pressure signal, MPa. S{·} is the smoothing operator, which is implemented in this study using an adaptive Kalman filter.

(2)

Detection of candidate water hammer event segments

After shape-preserving denoising is completed, the time derivative of the pressure curve used for identification is calculated as follows:

$${\left({\displaystyle \frac{dp}{dt}}\right)}_i\approx \nabla p_{\mathrm{smooth}}(t_i)$$

(38)

where dp/dt is the time derivative of pressure. The derivative reflects the transient rate of pressure change and serves as an important basis for identifying the triggering time of a water hammer event. However, negative slope alone is insufficient to effectively distinguish genuine water hammer events from ordinary fluctuations. Therefore, multi-window statistical criteria are further introduced in this study. For each candidate time instant t_i, a preceding window, a subsequent short window, and a persistence window are defined, denoted by W_pre, W_post and W_sus, respectively, and the following quantities are defined:

$${\overline{p}}_{\mathrm{pre}}(i)={\displaystyle \frac{1}{\left|W_{\mathrm{pre}}\right|}}{\displaystyle \sum _{j\in W_{\mathrm{pre}}}{p}_{\mathrm{smooth}}}(t_j)$$

(39)

$${\overline{p}}_{\mathrm{post}}(i)={\displaystyle \frac{1}{\left|W_{\mathrm{post}}\right|}}{\displaystyle \sum _{j\in W_{\mathrm{post}}}{p}_{\mathrm{smooth}}}(t_j)$$

(40)

$${\overline{p}}_{\mathrm{sus}}(i)={\displaystyle \frac{1}{\left|W_{\mathrm{sus}}\right|}}{\displaystyle \sum _{j\in W_{\mathrm{sus}}}{p}_{\mathrm{smooth}}}(t_j)$$

(41)

where ${\overline{p}}_{\mathrm{pre}}$ is the mean pressure before triggering, MPa ${\overline{p}}_{\mathrm{post}}$ is the mean pressure within the short window after triggering, MPa. ${\overline{p}}_{\mathrm{sus}}$ is the mean pressure within the persistence window, MPa. W_pre, W_post and W_sus are the corresponding time windows, and ∣W∣ denotes the window length.

In addition, the standard deviation of the preceding window is defined as follows:

$${\sigma }_{\mathrm{pre}}(i)=\mathrm{std}\{p_{\mathrm{smooth}}(t_j),j\in W_{\mathrm{pre}}\}$$

(42)

where ${\sigma }_{\mathrm{pre}}$ denotes the standard deviation of the pre-trigger window.

Meanwhile, three types of pressure drop indicators are defined as follows:

$$\Delta p_{\mathrm{short}}(i)=p_{\mathrm{smooth}}(t_i)-p_{\mathrm{smooth}}(t_{i+\Delta n})$$

(43)

$$\Delta p_{\mathrm{mean}}(i)={\overline{p}}_{\mathrm{pre}}(i)-{\overline{p}}_{\mathrm{post}}(i)$$

(44)

$$\Delta p_{\mathrm{sus}}(i)={\overline{p}}_{\mathrm{pre}}(i)-{\overline{p}}_{\mathrm{sus}}(i)$$

(45)

Here, Δp_short denotes the instantaneous pressure drop, MPa, which characterizes the intensity of the local abrupt drop; Δp_mean represents the mean pressure difference over a short time scale before and after triggering and is referred to as the mean pressure drop MPa, and Δp_sus characterizes whether the signal remains in a sustained oscillatory attenuation state after the event, and is referred to as the oscillatory-stage pressure drop, MPa. Δn is the short-window length, expressed as the number of sampling points.

If a candidate point i satisfies the following conditions, then it is identified as a valid candidate triggering point.

$${\left({\displaystyle \frac{dp}{dt}}\right)}_i\le {\theta }_{\mathrm{slope}}$$

(46)

$${\overline{p}}_{\mathrm{pre}}(i)\ge {\theta }_{\mathrm{pre}}$$

(47)

$${\sigma }_{\mathrm{pre}}(i)\le {\theta }_{\mathrm{std}}$$

(48)

$$\Delta p_{\mathrm{short}}(i)\ge {\theta }_{\mathrm{short}}$$

(49)

$$\Delta p_{\mathrm{mean}}(i)\ge {\theta }_{\mathrm{mean}}$$

(50)

$$\Delta p_{\mathrm{sus}}(i)\ge {\theta }_{\mathrm{sus}},$$

(51)

The manually extracted results were jointly determined by field engineers with practical interpretation experience based on the onset of the pressure drop and the first complete oscillation cycle, and were therefore used as a practical reference baseline for evaluating the algorithmic identification results, rather than as absolute ground truth. The rationale for imposing the above threshold conditions is that a genuine water hammer event usually occurs after a relatively high and stable pressure plateau, and is accompanied by a pronounced short-term pressure drop followed by sustained oscillatory attenuation until the signal gradually approaches a low-pressure stable state. Therefore, compared with a single pressure-threshold method, the multi-window joint constraint is better able to capture the full structural characteristics of the water hammer formation process.

To avoid repeated identification of multiple neighboring points around the same water hammer event, candidate points with time intervals smaller than a prescribed merging threshold are grouped into the same candidate cluster. The composite score of a candidate point is defined as follows:

$$S_i=\Delta P_{\mathrm{s}\mathrm{u}\mathrm{s}}(i)+0.5\Delta P_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}(i)+0.3|{({\displaystyle \frac{dP}{dt}})}_i|$$

(52)

S_i denotes the score assigned to each candidate onset point. Within each candidate cluster, only the point with the highest score is retained as the final trigger point. This scoring function does not merely emphasize abrupt changes in the pressure derivative; instead, it comprehensively accounts for the persistence of the post-drop response, the contrast between the pre- and post-event mean values, and the instantaneous rate of change. As a result, the selected trigger point is more likely to correspond to the dominant true onset of the water hammer event. This is consistent with the algorithmic procedure in which adjacent candidate points are grouped and the highest-scoring point within each cluster is chosen to define the start of the water hammer segment.

As time progresses, the oscillation energy of the water hammer gradually decays. When the pressure amplitude falls below a prescribed threshold and subsequently remains at a relatively low level, the event is considered to have ended. The amplitude threshold ${\theta }_A$ is defined as follows:

$$|\tilde{p}(t)-\overline{p}|<{\theta }_A,\forall t\in [t_p,t_e]$$

(53)

where t_e denotes the end time of the water hammer event, s, and $\overline{P}$ represents the average pressure during the stabilized stage, MPa. ${\theta }_A$ is the amplitude threshold, which may be defined as a certain percentage of the peak amplitude.

(3)

Envelope-assisted identification of peaks and troughs

Because field pressure signals are often accompanied by local noise disturbances, directly identifying peaks and troughs from the raw signal may easily lead to misjudgment. Therefore, an envelope-assisted identification strategy is introduced in this study to improve the robustness of peak and trough detection. Specifically, the upper and lower envelopes of the signal can be constructed using either the Hilbert transform or a local-extrema connection approach. The decay trend of the envelopes is then used to identify the true dominant oscillation peaks while suppressing false peaks caused by noise or local perturbations. This treatment helps improve the stability of water hammer segment boundary determination and key feature extraction.

$$p_{env}^{\pm }(t)=\overline{p}\pm |H[\tilde{p}(t)]|$$

(54)

Here, $H\left[·\right]$ denotes the Hilbert transform operator, and $H\left[\tilde{P}\right(t\left)\right]$ represents the instantaneous amplitude of the signal. The decay trend of the envelope helps determine te and facilitates the elimination of false peaks caused by noise or local dissipation.

2.4. Automated Fitting and Inversion Algorithm

To quantify the agreement between the simulated results and the field measurements, this study constructs an objective function based on the sum of squared errors between the simulated pressure series and the measured pressure series. The identified effective water hammer response segment is taken as the calculation interval, and the optimal model parameters are obtained by minimizing the deviation between the two series over the entire time sequence. When necessary, weighting coefficients can also be introduced according to sampling reliability or the importance of specific time periods, so as to enhance the sensitivity of the objective function to key characteristic intervals.

To enable automatic fitting of field-measured pressure data, a global optimization-based parameter inversion strategy is employed, in which model parameters are identified by minimizing the discrepancy between simulated and measured pressure responses. The optimization framework is defined as follows. Let P(t_i), i = 1, 2, …, N, denote the measured pressure series, and P_sim(t_i; θ) the simulated pressure obtained from the numerical model, where θ = [R,C,I] is the parameter vector to be inverted. The fitness function is defined as a weighted sum of squared errors between the simulated and measured pressures, providing a quantitative measure of the agreement between model predictions and field observations. After the optimal RCI-related parameters are obtained, they are mapped to the corresponding fracture geometric parameters through the model relationships, and the fracture half-length, fracture height, and fracture width are then calculated according to Equations (19)–(21).

$$J(\theta )={\displaystyle \frac{1}{N}}{{\displaystyle \sum _{i=1}^N[P_{sim}(t_i,\theta )-P(t_i)]}}^2$$

(55)

where J(θ) denotes the objective function value; P_sim(t_i, θ) represents the simulated pressure obtained from the model under the parameter set θ, MPa; P(t_i) is the measured pressure at time, MPa; and w_i is the weighting coefficient, which can be determined based on the confidence level of the measurement points or the sampling frequency.

$${\theta }^{*}=\arg \underset{\theta \in \mathrm{\Omega }}{\underbrace{\min }}J(\theta )$$

(56)

The optimization objective is expressed as min{θ ∈ Ω} J(θ), where Ω denotes the feasible domain of parameters, representing the upper and lower bounds of the search space.

Global optimization algorithms, including the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Simulated Annealing (SA), can all be abstracted into the following unified iterative form:

$${\theta }^{k+1}=\mathrm{\Phi }({\theta }^k,J({\theta }^k),{\xi }^k)$$

(57)

where θ^k denotes the parameter vector at the k-th generation or iteration; represents the corresponding objective function value; ξ^k denotes a stochastic factor or perturbation introduced by the algorithm; and $\mathrm{\Phi }(\cdot )$ is the update operator specific to each optimization algorithm.

Given the strong nonlinearity and non-uniqueness of the water hammer inversion problem, the solution may deviate from actual engineering conditions if no reasonable constraints are imposed on the parameter search ranges. Therefore, physically acceptable upper and lower bounds are specified for each parameter to be inverted. These parameter ranges are mainly determined based on field operation data, wellbore structural parameters, fundamental reservoir characteristics, and relevant literature, so that the search space retains sufficient flexibility while avoiding obviously unrealistic solutions. In addition, the inversion results must satisfy the basic physical constraints imposed by fracture geometric relationships and model boundary conditions, so as to prevent parameter combinations that achieve numerical fitting but lack engineering interpretability. By introducing parameter bounds and physical constraints, the stability and reliability of the inversion results can be improved.

To address the strong nonlinearity, multiple local optima, and pronounced parameter coupling in water hammer parameter inversion, this study employs the Particle Swarm Optimization (PSO) algorithm for global search [31]. PSO is a typical swarm intelligence optimization method with relatively simple parameter settings, easy coupling with numerical forward models, and strong global search capability in complex multimodal search spaces. It is therefore well suited for constrained optimization problems such as shut-in water hammer curve fitting [32]. During the iterative process, each particle updates its velocity and position according to its own historical best position and the global best position of the swarm, and progressively approaches the optimal solution.

The particle swarm optimization algorithm iterates as follows:

$$\left\{\begin{array}{l}{v}_i^{k+1}=w{v}_i^k+c_1{r}_1(p_i-x_i^k)+c_2{r}_2(g-x_i^k)\\ x_i^{k+1}=x_i^k+v_i^{k+1}\end{array}\right.$$

(58)

Here, w is the inertia weight, c₁ is the cognitive acceleration coefficient, and c₂ is the social acceleration coefficient. r₁ and r₂ are random numbers uniformly distributed within the interval [0, 1]. v_i^k+1 represents the velocity of the i-th particle in the k + 1-th iteration, while x_i^k+1 denotes the spatial position of the i-th particle in the k + 1-th iteration.

In summary, this study employs the Particle Swarm Optimization (PSO) algorithm to perform parameter searching for the water hammer model and to achieve automatic fitting to the field-measured pressure data. During the iterative process, each particle updates its velocity and position according to its own historical best position and the global best position of the swarm, and gradually converges toward the minimum value of the objective function. By coupling PSO with the above wellbore–fracture coupled forward model, automatic matching between the simulated and measured pressure curves can be achieved over the effectively identified pressure segment, thereby yielding the optimal estimates of the fracture geometric parameters.

The workflow diagram of the automatic water hammer pressure analysis algorithm, including the automatic recognition algorithm and automatic fitting algorithm, is shown in Figure 1.

3. Base Case

3.1. Synthetic Ideal Water Hammer Signal

To verify the denoising and identification capability of the proposed method for water hammer signals under complex noise conditions, an idealized water hammer pressure response signal was first constructed to simulate the typical transient variation process of wellhead pressure after pump shut-in during hydraulic fracturing. The ideal signal consists of three stages: an initially relatively stable pressure plateau, a rapid pressure drop triggered by pump shut-in, and the subsequent decaying oscillation process. To better approximate actual hydraulic fracturing shut-in conditions, local fracture response disturbances were introduced into the ideal response at 13.4 s and 13.5 s to reflect the influence of bottomhole boundary condition changes on the local characteristics of the pressure curve. The resulting ideal water hammer signal can not only represent the basic dynamic characteristics of shut-in water hammer, but also retain a certain degree of complexity, making it suitable for subsequent algorithm performance evaluation (show in Figure 2).

On this basis, to simulate the random noise interference commonly present in field monitoring data, random noise was superimposed onto the ideal water hammer signal to generate a noisy water hammer signal, as shown in Figure 3. It can be observed that, after noise contamination, both the overall amplitude and local details of the signal are significantly disturbed, and the original water hammer oscillation characteristics are partially obscured. In particular, during the decay stage, high-frequency random fluctuations cause the signal to exhibit pronounced nonphysical oscillations.

3.2. Field Case Description

Well A is a newly drilled horizontal development well located in the Qiulin nose structure in the transitional zone between central and western Sichuan Basin. The target interval is the second member of the Shaximiao Formation, and the treated interval extends from 2502.0 to 3735.0 m, with a total length of 1233.0 m. Log interpretation indicates that the well has favorable reservoir quality and good potential for hydraulic fracturing stimulation.

Well A was completed using a plug-and-perf multistage, multi-cluster hydraulic fracturing design, with a total of 12 fracturing stages. Microseismic monitoring was conducted during the treatment to evaluate fracture geometry and stimulation effectiveness. In addition, the block where Well A is located is characterized by relatively gentle structural conditions, limited fault development, and comparatively good reservoir homogeneity, providing favorable geological conditions for fracture propagation analysis and interpretation of the monitoring results.

3.3. Fixed Input Parameters and Inversion Bounds

The field data used in this study mainly consist of three categories. The first category is the high-frequency wellhead pressure data recorded during the shut-in stage after hydraulic fracturing, which are used for water hammer signal denoising, automatic identification of effective response segments, and fracture parameter inversion, and thus constitute the core dataset of this study. The second category includes wellbore configuration and fracturing treatment parameters, such as well depth, horizontal section length, tubing diameter, wall thickness, pumping rate, shut-in parameters, and stage-and-cluster design. These data are used to establish the coupled wellbore–fracture water hammer forward model and serve as known inputs or constraints for parameter inversion. The third category is the interpreted microseismic monitoring results, which are used to validate the inverted fracture half-length and fracture height.

The basic input parameters used in the model calculations are listed in

Table 1. Input parameters for base cases.

Parameters	Value Range
Vertical length of pipe (m)	2319
Horizontal length of pipe (m)	1416~3735
Pipe diameter (mm)	139.1
Pipe thickness(mm)	12.7
Young’s modulus of pipe (GPa)	206
Poisson’s ratio of casing pipe	0.3
Wall roughness of casing pipe (mm)	0.015
Resistance (Pa·s/m3)	0.5~3 × 107
Capacitance (m3/Pa)	1~5 × 10−5
Poisson’s ratio of rock	0.17~0.22
Minimum Principal Stress (Mpa)	49
Fluid density (kg/m3)	1000
Rock fracture pressure (MPa)	72.9~79.1
Friction coefficient	0.05
Pump time (s)	5672~9428
injection rate (m3/min)	18
ISIP (MPa)	40
Wave speed (m/s)	1416
Inductance (Pa/(m3/s2))	1~5 × 10−5
Young’s modulus of rock (GPa)	38~46

, including true vertical depth, horizontal well length, pipe diameter, pipe roughness, fluid density, elastic parameters of the tubing string, shut-in time, injection rate, and initial pressure conditions. These parameters jointly determine the pressure wave propagation characteristics in the wellbore and the boundary response of the bottomhole fracture system, and therefore provide the basis for forward simulation and parameter inversion. Fracture geometry and related boundary parameters are determined within physically reasonable ranges using a global optimization method. Based on the field data from Well A, the fixed input parameters include wellbore parameters and rock mechanical parameters, with their value ranges given in Table 1. The RCI boundary conditions are calculated using empirical fracture parameters derived from hydraulic fracturing data in the Qiulin area.

4. Result

4.1. Evaluation of Denoising Performance Using Synthetic Signals

The noisy signal was processed using the proposed adaptive filtering method, and a comparative analysis of the signals before and after filtering was performed. The results are shown in Figure 4a. As can be clearly observed, the pre-filtered signal is strongly affected by noise, making the local peaks and fluctuation structures difficult to identify accurately. After adaptive filtering, however, the signal becomes considerably smoother overall, with high-frequency noise effectively suppressed, while the principal oscillatory characteristics and decay trend of the water hammer response are well preserved.

Further examination of the locally enlarged view shows that the adaptive filtering method exhibits good stability in preserving the phase and amplitude characteristics of the signal, without introducing any obvious phase lag or amplitude attenuation. This is of great significance for the subsequent identification of water hammer characteristic parameters, such as oscillation period and attenuation coefficient. In addition, the comparison of the signal-to-noise ratio before and after filtering indicates that the proposed method can effectively suppress random noise while preserving the physical information contained in the original signal to the greatest extent possible, thereby achieving a good balance between noise reduction and waveform fidelity.

As shown in Figure 4b, the noisy signal exhibits strong high-frequency fluctuations, whereas the denoised curve closely follows the local variation trend of the ideal signal. Near local inflection points, the filtered result does not show obvious excessive lag or severe distortion, indicating that the adaptive Kalman filtering method has strong detail-preserving capability. It can not only recover the overall trend of the signal, but also retain local dynamic variations that are meaningful for subsequent feature identification.

As shown in Figure 5a, the original error exhibits a relatively large magnitude and strong dispersion. After denoising, the error converges significantly, with a markedly reduced amplitude and a distribution more concentrated around zero. This indicates that the deviation between the filtered signal and the ideal signal is substantially reduced, demonstrating good stability and accuracy of the denoising result.

As can be seen from Figure 5b, the signal-to-noise ratio (SNR) before denoising is 11.99 dB, which increases to 25.05 dB after denoising. This indicates that the filtering process improves the SNR by 13.06 dB, corresponding to an increase of 108.9%.

4.2. Evaluation of Denoising Performance Using Field High-Frequency Signals

The proposed adaptive filtering method was further applied to the processing of field-measured high-frequency water hammer signals, and the results are shown in Figure 6. It can be clearly observed that the original signal contains strong high-frequency random fluctuations and local spike interference. In particular, during the pressure drop stage and the early oscillation stage, the superposition of noise partially obscures the true water hammer response characteristics, resulting in poor signal readability.

These results indicate that the adaptive Kalman filtering method can effectively suppress random noise and local spike interference while preserving the main transient features of field water hammer signals. This improved signal quality provides a clearer basis for subsequent event identification and parameter inversion.

4.3. Performance Evaluation of the Identification Algorithm

After denoising of the pressure signal, the effective water hammer segments in the field shut-in pressure curves were further identified automatically. The identification results for a single water hammer signal are shown in Figure 7, while the results for multiple water hammer signals are presented in Figure 8. The results demonstrate that the proposed identification algorithm can accurately locate the key onset point at which the signal enters the principal oscillatory response after the pressure drop, and can reasonably determine the subsequent decay interval and the terminal stage approaching stability. For signals with relatively clear onset characteristics, the algorithm can rapidly identify the starting position of the water hammer event. For signals with long attenuation durations and gradually decreasing amplitudes at late times, the algorithm can appropriately extend the identified segment by incorporating local stability criteria, so as to preserve the principal dynamic information as much as possible.

From the perspective of time-domain correspondence, the automatically identified water hammer segments show good consistency with the typical transient response locations in the original pressure curve. The identified onset point is generally located near the initial stage of rapid pressure decline followed by oscillatory response, which helps preserve the first complete fluctuation feature. The identified end point is mostly located near the stage where the oscillation amplitude has significantly decayed and the curve gradually approaches a stable plateau, thereby reducing the influence of steady tail-end noise on subsequent inversion. For signals with relatively strong noise or obvious local disturbances, the algorithm can still achieve effective screening by relying on derivative features, multi-window statistical criteria, and envelope constraints, demonstrating good anti-interference capability.

Further analysis of Figure 8 shows that when multiple shut-in-induced transient responses exist within the same field pressure record, the proposed method is able not only to identify a single water hammer event, but also to automatically separate multiple independent effective water hammer response segments from a long time series, while determining the triggering point, peak position, and termination position of each segment individually. This indicates that the algorithm has multi-event identification capability under complex field conditions, thereby avoiding the conventional practice of manually browsing the data and extracting segments one by one based on experience. For multi-event water hammer signals, differences in background pressure level, superposed local disturbances, and attenuation behavior commonly exist among different events, so a single-threshold method is prone to missed detections, false detections, or mixing of adjacent events. In contrast, the present method combines derivative-based abrupt-change detection with multi-window statistical constraints and further screens candidate segments using envelope attenuation characteristics, enabling better physical consistency and time-domain separability of the identified water hammer segments.

It should be emphasized that the capacity for automatic identification of multiple water hammer events constitutes one of the main innovations of this study. Previous water hammer analyses have mostly focused on the processing of single and isolated response events, with the identification process relying heavily on manual experience, which makes them difficult to apply to long-duration, high-frequency, and noisy field pressure records containing multiple transient events. In contrast, the proposed method enables the automatic extraction of multiple effective water hammer signals from continuous pressure records, thus providing a basis for batch parameter inversion and dynamic analysis of fracture responses at different time instants. In particular, under field hydraulic fracturing conditions, different shut-in stages or operational transitions may correspond to multiple usable water hammer events. Automatic identification and segment-by-segment analysis of these events can improve data utilization, enhance the continuity of fracture parameter interpretation, and provide richer time-series information for characterizing the dynamic fracture propagation process.

In addition, the multi-event identification results in Figure 8 indicate that each identified segment preserves the principal oscillation period and its attenuation process well, suggesting that the algorithm is capable not only of detecting water hammer events, but also of accurately extracting the effective analysis interval. This is particularly important for subsequent model fitting, because if the starting point is identified too late, the first principal oscillation information may be missed; if the end point is selected too far, a large amount of steady tail-end noise unrelated to water hammer dynamics may be introduced, thereby reducing the stability of parameter inversion. Therefore, the proposed multi-event automatic identification method improves not only the efficiency of signal recognition, but also the consistency and reliability of the input data for subsequent fracture parameter inversion.

To quantitatively evaluate the computational efficiency and identification consistency of the proposed algorithm, batch automatic identification was further performed on the shut-in signals from all fracturing stages of Well A, and the computation time and onset-position differences for each stage were statistically analyzed. The results are shown in Figure 9. As shown in Figure 7a, the computation time is relatively concentrated across different stages, mainly ranging from 0.84 to 1.22 s, with an interquartile range of approximately 0.89–1.08 s and a median of about 1.00 s, indicating good computational stability of the algorithm under different stage conditions. Most stages can be identified within approximately 1 s, demonstrating that the proposed method is capable of meeting the demand for rapid processing and batch analysis of multi-stage high-frequency pressure data in field applications.

As shown in Figure 9b, the onset difference is generally within the range of 0–5 s, with an interquartile range of approximately 1.0–3.5 s and a median of about 2.5 s. The mean value is close to the median, suggesting that the distribution is relatively stable and does not exhibit a pronounced skewed pattern. For most stages, the onset difference is within 1–4 s, with only a few stages approaching the upper bound. Considering that field shut-in water hammer signals are commonly affected by noise interference, local abnormal disturbances, and inherently indistinct onset boundaries, while manually interpreted results themselves also involve a certain degree of subjectivity, this level of discrepancy is considered acceptable for engineering applications. Overall, these results suggest that the proposed automatic identification algorithm shows reasonable consistency with the manually interpreted reference in determining onset positions and can effectively extract meaningful water hammer response segments.

Overall, the identification results show that the proposed method can extract effective water hammer segments with short computation time and acceptable onset error. This supports its use for batch processing of multi-stage field pressure data.

4.4. Validation in the Inversion Algorithm

After the effective water hammer response segment was obtained through automatic identification, fracture parameter inversion was further carried out based on the coupled wellbore–fracture water hammer model and a global optimization algorithm. Figure 10 illustrates the inversion process for one fracturing stage. Specifically, Figure 10a shows the field shut-in pressure curve, Figure 10b shows the effective water hammer segment obtained by automatic identification, and Figure 10c presents the fitting result between the simulated shut-in water hammer curve and the measured pressure curve. It can be seen that the simulated curve reproduces well the principal oscillation period and the overall attenuation trend of the measured curve after the pressure drop, indicating that the proposed forward–inverse framework can effectively capture the main dynamic response characteristics of the coupled wellbore–fracture system.

Based on the inversion results for this fracturing stage, the fracture half-length and fracture height were determined to be 289.8 m and 44.6 m, respectively. Compared with the microseismic interpretation results, which gave a fracture half-length of 309 m and a fracture height of 46 m and are used here as an external reference rather than absolute ground truth, the two sets of results show good overall agreement, with relative differences of 6.21% and 3.04%, respectively. The relatively small errors indicate that the proposed forward–inverse workflow can recover fracture geometry with reasonable accuracy for this stage. The result also shows that useful fracture information can be extracted from shut-in water hammer signals without additional downhole monitoring.

4.5. Field Validation

From the inversion results of all 12 fracturing stages in Well A (show in Figure 11), it can be seen that the overall errors of the proposed method for both fracture half-length and fracture height remain at relatively low levels, indicating that the established automated water hammer inversion workflow has good applicability and stability under batch processing of multiple stages. Overall, the inversion results for fracture height are more concentrated and exhibit less dispersion, whereas the error distribution for fracture half-length is relatively wider, suggesting that the half-length parameter is more sensitive to inter-stage differences and field condition variations.

As shown in Figure 11, for fracture half-length, the relative error among the 12 stages ranges from 0.56% to 11.2%, with a median of 2.9% and a mean of 3.82%. As shown in the boxplot, most of the data are distributed within a relatively low error range, indicating that the inverted fracture half-lengths are generally in good agreement with the microseismic interpretation results for most stages. However, the box width and overall spread of fracture half-length are both noticeably larger than those of fracture height, and one high-value outlier of approximately 11.2% is present, indicating that the half-length inversion deviation is relatively large for a few individual stages. This suggests that fracture half-length, as a geometric parameter describing fracture extension along the wellbore direction, is more susceptible to the combined influence of nonuniform fracture propagation, reservoir heterogeneity among stages, inter-cluster interference, and uncertainties in microseismic interpretation, resulting in greater variation in its error distribution.

In contrast, the inversion results for fracture height show better concentration and stability. The statistical results indicate that the relative error of fracture height ranges from 0.17% to 6.1%, with a median of 2.2% and a mean of 2.31%. The boxplot shows that most fracture height data are concentrated within a relatively narrow range, and the overall dispersion is significantly smaller than that of fracture half-length, with only a small number of high-error points. This indicates that, under the multi-stage conditions of Well A, the proposed method is more robust in the inversion of fracture height and is less affected by local abnormal stages, yielding better overall consistency than fracture half-length.

From the overall performance of the multi-stage batch inversion results, the proposed method can stably recover fracture geometric parameters under different fracturing-stage conditions, demonstrating that the integrated workflow of signal denoising–effective segment identification–model fitting–parameter inversion has good engineering applicability. Although differences exist among stages in terms of signal quality, boundary conditions, and fracture propagation behavior, the errors of fracture half-length and fracture height for most stages remain within relatively low ranges, indicating that the method has the capability for cross-stage application.

At the same time, the fact that the overall error of fracture half-length is higher than that of fracture height also reflects that different parameters exhibit different sensitivities to model assumptions and data quality. Compared with fracture height, fracture half-length is more easily affected by complex fracture propagation paths, inter-cluster competition, local heterogeneity, and interpretation errors in monitoring data, and therefore shows larger inversion dispersion. This result is consistent with the general pattern of field fracture geometry interpretation and further suggests that, although the proposed method has good interpretive capability for both types of geometric parameters, the inversion of fracture half-length still depends more strongly on high-quality water hammer signals and more stable boundary constraints.

5. Discussion

The wellhead pressure signal after fracturing shut-in exhibits pronounced non-stationary behavior, typically including rapid pressure drop, decaying oscillations, and a later stage approaching stability. Because these stages differ substantially in variation rate, amplitude, and noise level, conventional fixed-parameter filtering methods often cannot balance noise suppression and transient-feature preservation. Excessive filtering may weaken true peak–trough characteristics and introduce phase lag or distortion, whereas insufficient filtering cannot effectively remove high-frequency noise and local disturbances. To address this issue, this study employs an adaptive shape-preserving Kalman filter driven by local variation intensity. By dynamically adjusting the process and observation noise according to local signal characteristics, the method improves tracking during rapidly varying stages while enhancing noise suppression during relatively stable stages. Results from both synthetic and field high-frequency signals show that this method can suppress random noise and spike interference while preserving key features such as pressure drop onset, dominant oscillation period, peak–trough structure, and attenuation trend.

Accurate extraction of effective water hammer segments is essential for subsequent parameter inversion. Conventional processing relies heavily on manual selection of start and end points, which is inefficient and may lead to inconsistent interpretation, especially for multi-stage high-frequency field data. The automatic identification method proposed in this study combines pressure derivatives, multi-window statistical criteria, and envelope constraints to detect water hammer events based on the pattern of “stable plateau–rapid drop–sustained decaying oscillation.” This enables identification of both clear and more complex signals with gradually weakening oscillations. Batch results show second-level computation time and engineeringly acceptable onset error, indicating a reasonable balance between efficiency and accuracy.

Although the proposed forward–inverse framework can reproduce the main dynamic characteristics of measured shut-in water hammer curves and yield fracture parameters close to microseismic interpretation results, fracture inversion remains a constrained nonlinear inverse problem. The results are affected by parameter coupling, non-uniqueness, data quality, parameter bounds, search-space settings, and the selection of effective response segments. Small differences in identified start and end points, as well as noise, sampling precision, and local disturbances, may all influence the inversion results. Therefore, the estimated fracture parameters should be regarded as quantitatively constrained estimates rather than unique solutions. Microseismic results are used here only as an external reference for validation, not as absolute ground truth.

This study also has limitations. The equivalent RCI boundary model simplifies the multi-cluster fracture system as a single composite boundary, which may reduce applicability in cases with strong fracture interaction or complex fracture networks. In addition, field validation is currently based mainly on Well A, and the number of field cases remains limited.

6. Conclusions

To address the problems of strong noise, pronounced non-stationarity, reliance on manual experience for effective water hammer segment extraction, and insufficient automation in fracture parameter inversion for high-frequency pressure signals after fracturing shut-in, this study developed an automated water hammer analysis workflow integrating signal denoising, event identification, and fracture parameter inversion. The effectiveness of the proposed method was validated using both synthetic signals and field cases. The main conclusions are as follows:

(1) An adaptive Kalman filtering method driven by local variation intensity was proposed. The method can adjust the filtering parameters in real time according to the dynamic characteristics of the signal, thereby suppressing high-frequency noise and local spike disturbances while preserving key features such as the pressure drop position, dominant oscillation period, and attenuation behavior. For the synthetic signal, the signal-to-noise ratio increased from 11.99 dB to 25.05 dB after filtering, demonstrating good shape-preserving denoising capability.

(2) A multi-event automatic identification method for effective water hammer segments was developed by combining pressure derivatives, multi-window statistical criteria, and envelope constraints. Batch identification results show that the computation time was mainly distributed between 0.84 s and 1.22 s, while the onset error was generally within 0–5 s, indicating that the method can meet the engineering requirements for rapid processing of high-frequency field pressure data and batch analysis of multiple stages.

(3) An automated fracture parameter inversion workflow was established based on the coupled wellbore–fracture water hammer model and a global optimization algorithm. The field case results show that the inverted fracture half-length and fracture height were 289.8 m and 44.6 m, respectively, with relative errors of 6.21% and 3.04% compared with the microseismic interpretation results, confirming the feasibility and reliability of the proposed method for fracture geometry characterization.

(4) The batch inversion results of all 12 fracturing stages in Well A further demonstrate that the proposed method has good applicability and stability under multi-stage conditions. The relative error of fracture half-length ranges from 0.56% to 11.2%, with a median of 2.9% and a mean of 3.82%. The relative error of fracture height ranges from 0.17% to 6.1%, with a median of 2.2% and a mean of 2.31%.

(5) The inversion results for fracture height are more concentrated and stable than those for fracture half-length, whereas fracture half-length is more sensitive to inter-stage differences, signal quality, and variations in boundary conditions.

Overall, this study established an end-to-end automated workflow for shut-in water hammer analysis and provides a low-cost and practical approach for fracture evaluation in unconventional oil and gas wells. By using routinely acquired high-frequency shut-in pressure data, the proposed method has the potential to support rapid fracture assessment and engineering decision-making without requiring additional downhole monitoring tools. Future work should focus on improving boundary characterization under complex conditions and further validating the method with more field cases.