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\begin{matrix} f_{s} = 1 / T_{s}, f_{K} = \frac{k}{N} F_{s} & (4) \end{matrix}

In at least one embodiment, a fifth step of the algorithm includes generating or computing partial energies of the return signals at the selected time segments ST1 and ST2 from the spectrogram data of waveforms x_CB_l_,R_i, for each known cement-bonding condition l=1, 2, . . . L. In at least one embodiment, this fifth step may be represented as in equations (5) and (6).

$\begin{matrix} E_{{CB}_{l}, R_{i}, ST 1} = \sum_{k = 0}^{N_{P} - 1} S_{{CB}_{l} - R_{i}} (m_{ST 1}, k) & (5) \end{matrix}$

$\begin{matrix} E_{{CB}_{l}, R_{i}, ST 2} = \sum_{k = 0}^{N_{P} - 1}, S_{{CB}_{l} R_{i}} (m_{ST 2}, k) & (6) \end{matrix}$

In at least one embodiment, a sixth step of the algorithm includes establishing or identifying time-dependent frequency response patterns and location-dependent waveform propagation patterns for each known cement-bonding condition l=1, 2, . . . L. In at least one embodiment, this is represented as in Equation (7).

E_P,CB_l=[E_CB_l_,R₁_,ST1,E_CB_l_,R₂_,ST1, . . . ,E_CB_l_,R₁_,ST1,E_CB_l_,R₁_,ST2,E_CB_l_,R₂_,ST2, . . . ,E_CB_l_,R₁_,ST2] (7)

In at least one embodiment, the first set of steps may be completed in advance of a well inspection to be performed using a well inspection tool.

In at least one embodiment, a second set of steps of the algorithm (such as the seventh step onwards), as described below, is to be performed using a well inspection tool that is sent into a downhole environment. This allows for processing, in part, unknown return signals to be used with pattern matching to the patterns of known return signals corresponding to the known cement bonding conditions from the first set of steps of the algorithm. As such, the second set of steps of the algorithm are associated with some steps illustrated in the method 400 of FIG. 4. In at least one embodiment, a well inspection tool sent into a downhole environment can transmit an acoustic signal into a well structure as in steps 402 of method 400.

In at least one embodiment, the second set of steps in the algorithm starts with a seventh step of the algorithm that allows loading measurement data x_depth_j_,R_i, where j=1, 2, . . . J, of an array of receivers Ri, where i=1, 2, . . . I, and where such measurement data is acquired from a downhole environment of a well or well structure. In at least one embodiment, such a step of the algorithm includes receiving return signals from the well structure at an array of receivers on the well inspection tool from which measurement data is retrieved and loaded to at least one memory to enable one or more processors to perform the algorithm herein.

In at least one embodiment, an eight step of the algorithm includes selecting a depth point [js, je] for diagnosing cement-bonding condition of the downhole environment of the well. In at least one embodiment, therefore, measurement data may be collected at multiple depth points, and then certain depth points of interest may be analyzed by such an algorithm. In at least one embodiment, however, it is possible to only obtain measurement data at the certain depth points of interest. In at least one embodiment, a ninth step of the algorithm includes setting a depth counter as represented in Equation (4).

j_C=j_s (8)

In at least one embodiment, the seventh to ninth steps of the algorithm may be cumulatively represented as step 404 of method 400 in FIG. 4.

In at least one embodiment, a tenth step of the algorithm includes performing STFT on the waveforms [Image Omitted] acquired by the array of receivers Ri, where i=1, 2, . . . I, at the depth depth_jc. In at least one embodiment, this is done as per Equations (8) and (10) for each unknown cement-bonding condition. In at least one embodiment, this tenth step is to generate or compute respective spectrograms [Image Omitted] (m, k) as per equation (9), which collectively provides the spectrogram data. In at least one embodiment, this tenth step of the algorithm may be represented as step 406 of method 400 in FIG. 4. In at least one embodiment, therefore, this tenth step performs the Short-Term Fourier Transform (STFT) on individual ones of the return signals to generate individual ones of spectrograms that are cumulated to generate the spectrogram data.

$\begin{matrix} X_{{depth}_{j_{c}}, R_{i}} (m, k) = \sum_{n = 0}^{N - 1} x_{{depth}_{j_{c}}, R_{i}} (n) g (n - m) W_{N}^{kn} & (9) \end{matrix}$

$\begin{matrix} S_{{depth}_{j_{c}}, R_{i}} (m, k) = {❘ X_{{depth}_{j_{c}}, R_{i}} (m, k) ❘}^{2} & (10) \end{matrix}$

$\begin{matrix} x_{{depth}_{j_{c}}, R_{i}} (n) = x_{{depth}_{j_{c}}, R_{i}} ({nT}_{s}), W_{N}^{kn} = e^{- j \frac{2 π}{N} kn} & (11) \end{matrix}$

$\begin{matrix} f_{s} = 1 / T_{s}, f_{K} = \frac{k}{N} f_{s} & (12) \end{matrix}$

In at least one embodiment, an eleventh step of the algorithm includes computing the partial energies of the signals at the selected time segment ST1 and ST2 from the spectrogram data of the waveforms x_depth_j_,R_ifor each known cement-bonding condition. In at least one embodiment, this eleventh step may be represented as in equations (13) and (14) and as a determination of the short-term power spectra in step 408 of method 400 in FIG. 4.

$\begin{matrix} E_{{depth}_{j_{c}, R_{i}, ST 1}} = \sum_{k = 0}^{N_{p} - 1} S_{{depth}_{j_{c}, R_{i}}} (m_{ST 1}, k) & (13) \end{matrix}$

$\begin{matrix} E_{{depth}_{j_{c}, R_{i}, ST 2}} = \sum_{k = 0}^{N_{p} - 1} S_{{depth}_{j_{c}, R_{i}}} (m_{ST 2}, k) & (14) \end{matrix}$

In at least one embodiment, a twelfth step of the algorithm includes establishing or identifying time-dependent frequency response patterns and location-dependent waveform propagation patterns for each known cement-bonding condition at each depth_jc. In at least one embodiment, this is represented as in Equation (15) and in step 410 of the method 400 in FIG. 4.

$\begin{matrix} E_{P, {depth}_{j_{c}}} = E_{{depth}_{j_{c}, R_{1}, ST 1}}, E_{{depth}_{j_{c}, R_{2}, ST 1}}, \dots & (15) \end{matrix}$

$[\begin{matrix} \dots, & E_{{depth}_{j_{c}, R_{I}, ST 1}}, & E_{{depth}_{j_{c}, R_{1}, ST 2}}, & E_{{depth}_{j_{c}, R_{2}, ST 2}}, & \dots, & E_{{depth}_{j_{c}, R_{I}, ST 2}}] \end{matrix}$

In at least one embodiment, a thirteenth step of the algorithm includes generating or computing cross correlation coefficients ρ_j_c_,l, between the extracted pattern response of the array of receivers Ri, where i=1, 2, . . . I, at depth_jcand the pattern responses of the array of receivers Ri, where i=1, 2, . . . , I, for the known cement bonding conditions. In at least one embodiment, the cross correlation may be represented as in Equation (16).

$\begin{matrix} ρ_{j_{C}, l} = \frac{E_{P, {depth}_{j_{c}}} E_{P, {CB}_{l}}^{T}}{{(E_{P, {depth}_{j_{c}}} E_{P, {depth}_{j_{c}}}^{T})}^{1 / 2} {(E_{P, {CB}_{l}} E_{P, {CB}_{l}}^{T})}^{1 / 2}}, where l = 1, 2, \dots L & (16) \end{matrix}$

In at least one embodiment, a fourteenth step of the algorithm includes searching, identifying, or determining the maximum ρ_j_c_,maxof the cross correlation coefficient sequence

$ρ_{j_{c}, l'} l = 1, 2, ..., L} .$

In at least one embodiment, this information can then be used to determine the cement-bonding condition [Image Omitted] at a intended depth in the downhole environment corresponding to the maximum of the cross correlation coefficient sequence.

In at least one embodiment, a fifteenth step and a sixteenth step of the algorithm are to set the searched cement-bonding condition for depth_jcusing Equation (17) and to update a depth counter using Equation (18), respectively.

$\begin{matrix} {BC}_{j_{c}} = BC ❘_{ρ_{j_{c}, \max}} & (17) \end{matrix}$

$\begin{matrix} j_{c} = j_{c} + 1 & (18) \end{matrix}$

The diagnosis of the downhole environment can be stopped once the condition j_C>j_e, following updates in Equation (18), is met. In at least one embodiment, the tenth step of the algorithm is performed so that all intended depths are analyzed. In at least one embodiment, one or more steps from the thirteenth step to the sixteenth step of the algorithm may be represented in step 412 of the method 400 in FIG. 4.

In at least one embodiment, computer and network aspects 500 for a downhole system as illustrated in FIG. 5, may be used as described herein. In at least one embodiment, these computer and network aspects 500 may include a distributed system. In at least one embodiment, a distributed system 500 may include one or more computing devices 512, 514. In at least one embodiment, one or more computing devices 512, 514 may be adapted to execute and function with a client application, such as with browsers or a stand-alone application, and are adapted to execute and function over one or more network(s) 506.

In at least one embodiment, a server 504, having components 504A-N may be communicatively coupled with computing devices 512, 514 via network 506 and via a receiver device 508, if provided. In at least one embodiment, components 512, 514 include processors, memory, and random-access memory (RAM). In at least one embodiment, server 504 may be adapted to operate services or applications to manage functions and sessions associated with database access 502 and associated with computing devices 512, 514. In at least one embodiment, a server 504 may be associated with an array of receivers 508 of a downhole tool 520.

In at least one embodiment, server 504 may be at a wellsite location, but may also be at a distinct location from a well site location. In at least one embodiment, such a server 504 may support a downhole tool or wireline system 520 for analysis of a well structure in a downhole environment and to perform well structure inspection of a well 522. Such a well inspection tool 520 may be partly downhole and partly at a surface level. Such a downhole tool 520 may include a well inspection tool 518 to perform in at least part of the functions described throughout herein.

There may be a modelling system 516 within a downhole tool 520 or separate from the downhole tool 520, at a server 504. In at least one embodiment, such a modelling system may be provided to perform part of the first set of steps of the algorithm to recalibrate the downhole tool 520 using known cement bonding conditions. In at least one embodiment, the modelling system 516 may be pre-calibrated on a surface level using known cement bonding conditions or simulations (or other representations, including images) thereof. The subsystems 508, 516, 518 of the downhole tool may be encased in one or more computing devices having at least one processor and memory so that the at least one processor can perform functions based in part on instructions from the memory executing in the at least one processor. In at least one embodiment, even though illustrated together, the system boundaries of each module 508, 516, 518 may be around a distributed system having the subsystems in different geographic locations, including downhole and surface areas.

In at least one embodiment, an array of receivers 508 of a downhole tool 520 is provided to inspect a downhole environment of a well 522. In at least one embodiment, a system for determining cement bonding conditions includes a wireline system for the analysis, where such a system may be adapted to transmit, either through wires or wireless, information received therein, from an array of sensors back to the surface. In at least one embodiment, modeling performed using a modeling system 516 may be associated with calibration or pre-calibration of known cement bonding conditions and representations thereof) that may be recorded within a modeling system 516.

The modeling system 516 can communicate to a well inspection tool 518 and to an array of receivers 508 to enable determination of cement bonding conditions of a well 522 using models of known cement bonding conditions stored in the modeling system 516. For example, each model may require specific signals associated with specific ones of the cement bonding conditions to verify matching patterns using the above-described algorithm. Such signals may include acoustic signals to be applied from an acoustic transmitter of the well inspection tool 518. Detected return signals to the array of receivers 508 may be used to determine the cement bonding conditions by pattern matching using the time-dependent frequency response and using the location-dependent waveform propagation patterns from the returned signals. In at least one embodiment, therefore, the system 500 enables the at least one processor (such as from components 512, 514) to access the measurement data and to access patterns of known cement-bonding conditions to perform the pattern matching using the time-dependent frequency response and using the location-dependent waveform propagation patterns.

In at least one embodiment, such returned signals may be received in the array of receivers 508 and transmitted from there. In at least one embodiment, a server 504 may function as a setting device (with the acoustic transmitter of the well inspection tool providing the actual acoustic signal and the array of sensors receiving a returned signal). In at least one embodiment, however, the server 504 may also perform other functions described throughout herein and at least as to the algorithm described herein.

In at least one embodiment, one or more component 504A-N may be adapted to function as a signal provisioning device within a server 504 to enable the acoustic transmitter to transmit its signals at the frequency ranges described throughout herein. In at least one embodiment, one or more components 504A-N may include one or more processors and one or more memory devices adapted to function as a detector or receiver device, while other processors and memory devices in server 504 may perform other functions.

In at least one embodiment, a server 504 may also provide services or applications that are software-based in a virtual or a physical environment (such as to support the simulations referenced herein). In at least one embodiment, when server 504 is a virtual environment, then components 504A-N are software components that may be implemented on a cloud. In at least one embodiment, this feature allows remote operation of a system for determining cement bonding conditions formation using a wireline system that includes a well inspection tool, as discussed at least in reference to all the figures herein. In at least one embodiment, this feature also allows for remote access to information received and communicated between any of aforementioned devices. In at least one embodiment, one or more components 504A-N of a server 504 may be implemented in hardware or firmware, other than a software implementation described throughout herein. In at least one embodiment, combinations thereof may also be used.

In at least one embodiment, one computing device 510-514 may be a smart monitor or a display having at least a microcontroller and memory having instructions to enable display of information monitored by an array of receivers 508. In at least one embodiment, one computing device 510-512 may be a transmitter device to transmit directly to a receiver device or to transmit via a network 506 to an array of receivers 508 and to a server 504, as well as to other computing devices 512, 514.

In at least one embodiment, other computing devices 512, 514 may include portable handheld devices that are not limited to smartphones, cellular telephones, tablet computers, personal digital assistants (PDAs), and wearable devices (head mounted displays, watches, etc.). In at least one embodiment, other computing devices 512, 514 may operate one or more operating systems including Microsoft Windows Mobile®, Windows® (of any generation), and/or a variety of mobile operating systems such as iOS®, Windows Phone®, Android®, BlackBerry®, Palm OS®, and/or variations thereof.

In at least one embodiment, other computing devices 512, 514 may support applications designed as internet-related applications, electronic mail (email), short or multimedia message service (SMS or MMS) applications and may use other communication protocols. In at least one embodiment, other computing devices 512, 514 may also include general purpose personal computers and/or laptop computers running such operating systems as Microsoft Windows®, Apple Macintosh®, and/or Linux®. In at least one embodiment, other computing devices 512, 514 may be workstations running UNIX® or UNIX-like operating systems or other GNU/Linux operating systems, such as Google Chrome OS®. In at least one embodiment, thin-client devices, including gaming systems (Microsoft Xbox®) may be used as other computing device 512, 514.

In at least one embodiment, network(s) 506 may be any type of network that can support data communications using various protocols, including TCP/IP (transmission control protocol/Internet protocol), SNA (systems network architecture), IPX (Internet packet exchange), AppleTalk®, and/or variations thereof. In at least one embodiment, network(s)