WO2024130049A1

WO2024130049A1 - Methods for determining a position of a droppable object in a wellbore

Info

Publication number: WO2024130049A1
Application number: PCT/US2023/084156
Authority: WO
Inventors: Artem Valeryevich Kabannik; Arkady Yurievich Segal
Original assignee: Schlumberger Technology Corporation; Schlumberger Canada Limited; Services Petroliers Schlumberger; Schlumberger Technology B.V.
Priority date: 2022-12-15
Filing date: 2023-12-15
Publication date: 2024-06-20
Also published as: AR131384A1

Abstract

The position of a droppable object (e.g., a cementing plug or dart) inside a tubular body (e.g., a casing string) may be determined by analyzing harmonic acoustic signals generated by the droppable object as it passes through the tubular body. The harmonic acoustic signals are in the form of pressure waves. The pressure data are recorded by a pressure data acquisition system and processed mathematically to estimate a two-way travel time of the harmonic acoustic signals, and determine the acoustic signal propagation velocity, thereby allowing determination of the droppable object position.

Description

Methods for Determining a Position of a Droppable Object in a Wellbore Cross Reference Paragraph [0001] This application claims the benefit of Russian Non-Provisional Application No. 2022133126, entitled "Methods for Determining a Position of a Droppable Object in a Wellbore," filed December 15, 2022, the disclosure of which is hereby incorporated herein by reference. Technical Field [0002] The present disclosure relates generally to cementing operations. In particular, the disclosure relates to using pressure pulses to determine the positions of wiper plugs and drillpipe darts inside a casing string. Background [0003] During the construction of underground wells, it is common, during and after drilling, to place a tubular body such as a liner or casing, secured by cement pumped into the annulus around the outside of the tubular body. The cement serves to support the tubular body and to provide isolation of the various fluid-producing zones through which the well passes. This latter function prevents cross-contamination of fluids from different layers. For example, the cement prevents formation fluids from entering the water table and polluting drinking water, or prevents water from passing into the well instead of oil or gas. Furthermore, the cement sheath helps prevent corrosion of the tubular body. [0004] The cement placement process is known in the industry as primary cementing. Most primary cementing operations employ the two-plug cement-placement method. Figure 1 shows a typical wellsite configuration 100 for a primary cementing operation. A cementing head 101 is situated on the surface, and a casing string 103 is lowered into a borehole 102. As the casing string 103 is lowered into the borehole 102, the casing string interior fills with drilling fluid 108. The casing string is centered in the borehole by centralizers 104 attached to the outside of the casing string. Centralizers are placed in critical casing sections to prevent sticking while the casing is lowered into the well. In addition, they keep the casing string in the center of the borehole to help ensure placement of a uniform cement sheath in the annulus between the casing and the borehole. The bottom end of the casing string is protected by a guide shoe 105 and a float collar 109. Guide shoes are tapered, commonly bullet-nosed devices that guide the casing toward the center of the hole to minimize hitting rough edges or washouts during installation. The guide shoe differs from the float collar in that it lacks a check valve. The check valve in a float collar can prevent reverse flow, or U-tubing, of fluids from the annulus into the casing. Inside the cementing head 101 are a bottom cementing plug 106 and a top cementing plug 107. The cementing plugs, also known as cement wiper plugs or wiper plugs, are elastomeric devices that provide a physical barrier between different fluids as they are pumped through the casing string interior. Most cementing plugs are made of a cast aluminum body with molded rubber fins. [0005] The goals of the primary cementing operation are to remove drilling fluid from the casing interior and borehole, place a cement slurry in the annulus, and leave the casing interior filled with a displacement fluid such as brine or water. The bottom cementing plug 106 separates the cement slurry from the drilling fluid, and the top cementing plug 107 separates the cement slurry from the displacement fluid. [0006] Cement slurries and drilling fluids are usually chemically incompatible. Commingling may result in a thickened or gelled mass at the interface that would be difficult to remove from the wellbore, possibly preventing the placement of a uniform cement sheath throughout the annulus. Therefore, in addition to using wiper plugs, engineers employ both chemical means to maintain fluid separation. Chemical washes and spacer fluids may be pumped between the cement slurry and drilling fluid. These fluids have the added benefit of cleaning the casing and formation surfaces, which is helpful for achieving good bonding with the cement. [0007] Figure 2 shows a chemical wash 201 and a spacer fluid 202 being pumped between the drilling fluid 103 and the bottom cementing plug 106. Cement slurry 203 follows the bottom cementing plug. The bottom cementing plug has a membrane that ruptures when it lands at the bottom of the casing string, allowing cement slurry to pass through the bottom cementing plug and enter the annulus (Fig.3). [0008] Once a sufficient volume of cement slurry has been pumped to fill the annular region between the casing string and the borehole wall, the top cementing plug 107 is released, followed by the displacement fluid 301. The top cementing plug 107 does not have a membrane; therefore, when it lands, hydraulic communication is severed between the casing interior and the annulus (Fig. 4). After the cementing operation, engineers wait for the cement to set and develop strength—known as "waiting-on-cement” (WOC). After the WOC time, further operations such as drilling deeper or perforating the casing string may commence. [0009] Conventional cementing plugs are pumped directly from the surface because they pass through one pipe with a continuous inside diameter (ID). Liners, on the other hand, do not begin at the surface; instead, they are run downhole on the drillstring to the setting depth. Liners typically have a much larger ID than the drillstring; as a result, a single cementing plug cannot be pumped from the surface. Therefore, the displacement is performed by two plugs. One plug, known as the drillpipe dart, is located in the surface cementing equipment. The second plug is either attached to the bottom of the liner setting tool assembly, or the top of the liner setting tool assembly. The second plug is called a liner wiper plug. [0010] After the cement has been pumped in the liner and the drillstring, the drillpipe dart is released from the surface cementing equipment. When the drillpipe dart reaches the top of the liner, it latches into the liner wiper plug. Both the drillpipe dart and the liner wiper plug then become a single divider between the cement slurry and the displacement fluid. This arrangement may be seen in extended-reach wells and multistage cementing applications. [0011] Additional information concerning cementing plugs, drillpipe darts and primary cementing operations may be found in the following publications. Leugemors E et al.: “Cementing Equipment and Casing Hardware,” in Nelson EB and Guillot D (eds.): Well Cementing–2^nd Edition, Houston, Schlumberger (2006) 343–458. Piot B and Cuvillier G: “Primary Cementing Techniques,” in Nelson EB and Guillot D (eds.): Well Cementing–2^nd Edition, Houston, Schlumberger (2006) 459–501. Trogus M: “Studies of Cement Wiper Plugs Suggest New Deepwater Standards,” paper SPE/IADC-173066-MS, presented at the SPE/IADC Drilling Conference and Exhibition, London, UK, 17–19 March 2015. [0012] Deviations from the idealized cementing operation depicted above may occur. Possible reasons include borehole rugosity leading to inaccurate displacement volume calculations, pump rate fluctuations, differences between nominal and actual casing geometry, lost circulation, casing deformation and fluid loss. Note that the borehole rugosity, lost circulation and fluid-loss affect the top-of-cement depth in the annulus, not the depth of the droppable object inside the casing. With these uncertainties, operators and engineers are motivated to achieve real-time monitoring of cementing plug positions. [0013] Furthermore, when the cement slurry is pumped into the annulus around the outside of the casing string, it is undesirable to overdisplace and allow displacement fluid to enter the annulus. The over displacement is commonly called a “wet shoe” and may result in contamination or absence of cement in the casing section between the float collar and the casing shoe after the primary cementing. Repairing the poor cement isolation caused by over displacement may require a costly remedial squeeze cementing operation. [0014] Traditionally, to determine when to stop the cement displacement operation, the top cement plug position is tracked volumetrically by dividing the displaced volume by the casing internal cross-sectional area. However, the volumetric method is prone to uncertainties related to displacement fluid compressibility, pressure pump inefficiency, flowmeter inaccuracy, and variance in casing joint diameters. With this imprecise volumetric plug tracking method, operators may prefer to stop at the calculated displacement and add half of shoe track volume at the maximum. This reduces the wet shoe risk but may result in an excessive volume of cement inside the casing that takes more time to drill out or for production casing where the plan is to move to perforation immediately after displacement; additional drilling runs may be necessary to clean out the set cement. Brief Description of the Drawings [0015] Figure 1 shows a typical wellsite configuration during a cementing operation. [0016] Figure 2 shows a cementing operation in progress. The bottom cementing plug has been released, separating the cement slurry from chemical washes, spacer fluids and drilling fluid. [0017] Figure 3 shows a cementing operation in progress. The bottom cementing plug has landed on the float collar. A membrane in the bottom cementing plug ruptures, allowing cement slurry to enter the annulus between the casing string and the borehole wall. [0018] Figure 4 shows a completed cementing operation. Cement slurry fills the annulus, both cementing plugs have landed on the float collar, and the interior of the casing string is filled with displacement fluid. [0019] Figure 5 is an illustration of pressure waves in a wellbore. [0020] Figure 6 is an illustration of resonant and anti-resonant zones in pressure waves as a droppable object travels down casing. [0021] Figure 7 is an example of a plug depth as a function of the internal casing volume. [0022] Figure 8 illustrates the change of surface pressure amplitude and periodicity resulting from a top plug entering a region of larger inside casing diameter during a cementing operation. [0023] Figure 9 shows measured data versus model predictions as a plug travels down casing and passes through a region of larger inside casing diameter during a cementing operation. [0024] Figure 10 shows predicted reflection time trajectories and an estimated reflection time distribution during a cementing operation as illustrated in Fig.8. [0025] Figure 11 illustrates a family of trajectories τ(t;dV,C) and probabilities of travel time density distributions P_τ(t) during a cementing operation as illustrated in Fig.8. [0026] Figure 12 illustrates predicted travel time distributions when the casing internal diameter changes at 3000 m. [0027] Figure 13 illustrates a volumetric depth function for an example well with 5-in. casing and a depth of 5791 m. [0028] Figure 14 illustrates cement displacement data from real job, including surface pressure and flowrate obtained by flowmeter and pump strokes; as well as spectrogram and cepstrogram of the surface pressure. [0029] Figure 15 illustrates harmonic amplitudes recorded during the cementing operation as illustrated in Fig.8. [0030] Figs. 16–18 show correction volumes and correction factor probability distributions as well as predicted reflection time trajectories and reflection time and displaced volumes probability distributions for several time intervals. [0031] Figure 19 shows a combined reflection time probability disribution. In the beginning and in the end it coincides with the ones revealed by the cepstrum. In the middle of displacement it shows much more pronounced plug trajectory than the one obtained with the cepstrogram. Summary [0032] In an aspect, embodiments relate to methods for determining a position of a droppable object inside a casing string. The droppable object (e.g., a cementing plug or dart) is placed inside casing string filled with a first fluid. A displacement fluid is pumped behind the droppable object, causing the droppable object to travel through the interior of the casing string. The hydraulic pumps generate a harmonic acoustic signal in the form of pressure waves that travel down the casing and reflect from the droppable object. The pressure data are recorded and transmitted to a pressure data acquisition system. The pressure data are then processed mathematically to estimate a two-way travel time of the harmonic acoustic signal reflected by the droppable object. The propagation velocity of the acoustic signal is determined, followed by determining the position of the droppable object in the casing string. Detailed Description [0033] At the outset, it should be noted that in the development of any such actual embodiment, numerous implementations—specific decisions must be made to achieve the developer's specific goals, such as compliance with system related and business related constraints, which will vary from one implementation to another. Moreover, it will be appreciated that such a development effort might be complex and time consuming but would nevertheless be a routine undertaking for those of ordinary skill in the art having the benefit of this disclosure. In addition, the composition used/disclosed herein can also comprise some components other than those cited. In the summary of the disclosure and this detailed description, each numerical value should be read once as modified by the term "about" (unless already expressly so modified), and then read again as not so modified unless otherwise indicated in context. Also, in the summary of the disclosure and this detailed description, it should be understood that a concentration range listed or described as being useful, suitable, or the like, is intended that any and every concentration within the range, including the end points, is to be considered as having been stated. For example, “a range of from 1 to 10” is to be read as indicating each and every possible number along the continuum between about 1 and about 10. Thus, even if specific data points within the range, or even no data points within the range, are explicitly identified or refer to only a few specific points, it is to be understood that inventors appreciate and understand that any and all data points within the range are to be considered to have been specified, and that inventors possessed knowledge of the entire range and all points within the range. [0034] This disclosure pertains to detecting the position of droppable objects in a casing string or liner during a well cementing operation. The droppable objects may comprise top or bottom cementing plugs and drill pipe darts. The presently disclosed method is based on analyzing noise from reciprocal pumps that are used to place wellbore fluids in subterranean wells. [0035] A method and system for locating downhole objects that reflect a hydraulic signal are disclosed in the patent application WO 2018/004369. The monitoring of the well is based on cepstral analysis of pressure data recorded at the wellhead. It is designed to locate downhole objects that reflect a hydraulic signal. A hydraulic signal is detected by a pressure sensor, then the pressure data are processed to obtain their properties such as tube wave reflection times. One (but not the only) method of obtaining such information is a cepstrum analysis. The cepstrum analysis is widely used in various applications, for example for hydraulic fracturing operations monitoring. The cepstrogram allows detection of objects that reflect the hydraulic signal. This method for hydraulic fracturing operations uses hydraulic signal sources including the water hammer effect, noise from surface or submersible pumps and perforating events. [0036] US Patent 6401814 B1 discloses a method for locating a cementing plug in a subterranean well during cementing operations using pressure pulse reflections. Once generated, pressure pulses are transmitted through displacement fluid, reflected off the cementing plug and, finally, received by a pressure sensor. A location of the plug is calculated from reflection time and pressure pulse velocity in the given media. The method of generating and transmitting of pressure pulse through the fluid in a casing string comprises momentarily opening a valve installed in the flowline of the well. Other methods of pressure pulse generation include an air gun, varying the pump’s engine speed or disengaging the pump. [0037] US Patent 5754495 discloses a method for acoustic determination of the length of a fluid conduit. It comprises constructing a pressure containment system, connecting pressure sensors, filling the system with a fluid, generating a pressure pulse, measuring a pressure pulse traveling to the distal end of the fluid conduit, and calculating the length of the fluid conduct. In the embodiment, a tube wave is generated by a sudden release of pressure in the well through a valve. [0038] US Patent 4819726 discloses a method for indicating the position of a cement wiper plug prior to its bottomhole arrival. It comprises an apparatus that includes a section of pipe string with an interior shearable, temporary means of restricting the motion of the cement wiper plug through the section of pipe string. The arrival of the cementing plug at the shearable, temporary restriction means in a pipe string is sensed by an increase in pipe string pressure at the surface and monitored by a pressure sensor. [0039] US Patent 9546548 discloses a deviсe and method of use for cement sheath analysis based on acoustic wave propagation. It consists of an acoustic wave detection apparatus, comprising a fiber optic cable drawn down in a well, an optical source and a data acquisition system. The acoustic source produces a compressional wave in a casing string. The pressure in the annulus is determined as the cement slurry sets, and this pressure is compared to the maximum formation pressure as an indication of whether the cement had set to a strength, enough to maintain an effective formation-to-casing seal across the annulus. [0040] There are several methods to track droppable objects based on the analysis of pressure pulses as the droppable objects encounter variations of internal casing diameter as they travel down the well. [0041] There are several methods to track the top plug based on the analysis of the pressure pulses generated by the top cementing plug passing the casing collars with a negative or positive change of inner cross-sectional dimension. US 2021/0062640 (“Methods for Determining a Position of a Droppable Object in a Wellbore’), based on cepstrum analysis of high resolution cement head pressure data and their combination with a Kalman filter: WO 2022/025790 (“Methods for Determining a Position of a Droppable Object in a Wellbore). Also, there is a method of plug tracking where the collar pulses are processed with the particle filter (unpublished). The limitation of pulse based methods is that they require change of inner cross-sectional diameter of the casing collars, to generate the pulses, that is not always available. The cepstrum based method require either pulses or broad band acoustic signal in the wellbore that is also not always available. [0042] As discussed above, the present invention relates to methods for detecting the position of downhole (or droppable) objects in a wellbore during liner or casing cementing operations. The method is based on recording high frequency pressure data in the wellbore filled with fluid and analyzing pressure oscillations generated by hydraulic pumps during cementing operations. The hydraulic pumps generate harmonic signals comprising wellbore pressure oscillations at a number of frequencies. The moving cement plug serves as a reflection boundary for the pressure signal that continuously changes the resonant frequencies of the pressure oscillations in the wellbore as the plug travels down the wellbore. The numerical model predicts the resonance response of the wellbore basing on casing geometry and pump rates with the two-way travel time of the signal as an unknown parameter. The downhole plug position is determined from the two-way travel time and the signal propagation velocity. The signal propagation (tube wave) velocity can be estimated theoretically based on the media properties or determined by calibration when the plug passes the completion components with known position, such as casing diameter changes or a landing point for a droppable object. [0043] In an aspect, embodiments relate to methods for determining a position of a droppable object inside a casing string. The droppable object (e.g., a cementing plug or dart) is placed inside casing string filled with a first fluid. The first fluid in the casing string may comprise a drilling fluid, a spacer fluid or a brine, or a cement slurry. A displacement fluid is pumped behind the droppable object, causing the droppable object to travel through the interior of the casing string. The hydraulic pumps generate a harmonic acoustic signal in the form of pressure waves that travel down the casing and reflect from the droppable object. The pressure data are recorded and transmitted to a pressure data acquisition system. The pressure data may be recorded by at least one pressure transducer. The pressure data may contain both direct and reflected harmonic acoustic signals. The pressure data are then processed mathematically to estimate a two-way travel time of the harmonic acoustic signal reflected from the droppable object. The propagation velocity of the acoustic signal is determined, followed by determining the position of the droppable object in the casing string. Pressure Waves in the Wellbore [0044] During the well cementing operation the cement slurry is forced from the casing interior into the annulus by the top cement plug, which in turn is pushed by the displacement fluid pumped with reciprocal pumps. The pumps generate harmonic pressure oscillations inside the casing ^{string. These harmonic oscillations consist of a fundamental frequency f0 and its multiples} _{2f0,3f0,4f0,... called harmonics.} [0045] The fundamental pressure oscillation frequency f0, generated by the reciprocal pump, is proportional to the pump rate: . Here, Q is the pump rate, N_pl is a number of plungers in the pump and F_K is a volume of a single pump cylinder also called a K-Factor [0046] One of these harmonic pressure oscillations with a circular frequency of ω = 2πf that will induce pressure waves (also known as “tube waves”) that propagate along the wellbore. The tube- wave distribution along the wellbore can be represented by the following sum: P(x,ω,t) = D(x,ω,t) + U(x,ω,t), where D(x,ω,t) = Aexp(iωt+ikx) is a downgoing wave, U(x,ω,t) = B exp(iωt−ikx) is an upgoing wave, A and B are complex amplitudes of the corresponding waves, x is a spatial coordinate along the well and t is a time. The k = ω/c is a wavenumber. In further considerations, the wavenumber is considered to be a real value, while the imaginary part responsible for the signal attenuation will be neglected. Finally, c is a tube wave velocity, further assumed to be a constant. [0047] An illustration of a pressure wave distribution generated by 10-Hz oscillations at some arbitrary time t = 0 is shown in Fig.5. The solid lines are real components of complex oscillatory pressure and the shaded areas illustrate the amplitudes of corresponding waves. [0048] The top cement plug forms a rigid reflection boundary for the tube waves so the spatial derivative of the pressure vanishes at the reflector: , [0049] where L is the top cement plug

the derivative results in A exp(ikL) − B exp(−ikL) = 0, the complex upgoing wave amplitude coefficient B can be derived from A: B = A exp(2ikL). [0050] To simplify this expression, one can use the expression for the two-way travel time for the pressure signal τ = 2L/c , also known as the reflection time B = A exp(iωτ). [0051] Substitution of the expression for B into the pressure wave equation results in: P(x,ω,t) = A[exp(iωt + ikx) + exp(iωτ)exp(iωt – ikx)]. [0052] The pressure waves in the wellbore with the reflector are illustrated in the figure below. The amplitude of the oscillatory pressure signal measured at the surface (x = 0) is an absolute value :

.

[0053] Practically, the amplitude of the surface oscillatory pressure signal is obtained by computing the Discrete Fourier Transform (DFT) of the measured pressure signal, followed by taking its absolute value. The absolute value of DFT of the surface pressure signal is then evaluated at the circular frequency of interest ω. [0054] When the plug is moving, its position L and the two way-travel time τ in the above equations are actually functions of time: L(t) and τ(t), with the time variable t omitted for simplicity.

[0055] During displacement the top-plug depth monotonically increases, so the surface signal amplitude |P(0,ω,t)| changes periodically between maxima (resonant zones) and minima (antiresonant zones) as illustrated in Fig. 6. The horizontal black lines denote three subsequent positions of the downhole reflection boundary. On the left the depth of the reflector comprises the integer number of half wavelengths that results in a maximum amplitude at the surface, which manifests a resonant state of the wellbore. The next reflector position results in an intermediate amplitude of the surface pressure oscillations. On the right the number of half wavelengths between the reflector and the surface is equal to some integer number plus a quarter wavelength. In this last case, the amplitude of the surface signal is a minimum that represents the antiresonant state of the wellbore. Volumetric Plug Depth Model [0056] The surface pressure signal amplitude |P(0,ω,t)| depends on the two-way travel time τ that, in turn, depends on the plug position L. In cementing operation, the plug position L is tracked volumetrically from the measured pumping rate and internal cross-sections of the casing string. To define the volumetric plug tracking model a few definitions are introduced. The volume of the casing string above the plug as a function of its depth L is called displacement volume V_d and defined as the following integral: ^, _{where S(x) = 0.25πd} ² _{(x) is the casing string internal cross-sectional area profile as a function of} casing internal diameter d(x) measured at the distance x from the surface. The displaced volume V_d along the casing string is a piecewise linear monotonic increasing function. Therefore, there to-one correspondence between the displaced volume V_d and the plug depth L, which allows defining the plug depth to be a function of displaced volume L(V_d) as an inverse function of Vd(L). An example of the volumetric plug depth as the function L(Vd) is shown in Fig.7. Forward Model [0057] In well cementing operations the displaced volume V_d is estimated with the pumped volume V_p measured with a surface flowmeter or by a pump stroke counter. In general, the displaced volume Vd is close but not equal to the pumped volume Vp. The discrepancy between the displaced and the pumped volumes is related to displacement fluid compressibility, pressure pump inefficiency, flowmeter inaccuracy, and variance in casing joint diameters. [0058] The unknown displaced volume Vd from the measured pumped volume Vp may be expressed by the following relation: V_d = dV + C · V_p, where dV is a correction volume and C is a correction coefficient. Combining everything together results in the following expression for the surface pressure signal amplitude: .

reflection time determination purposes it is set to default constant value of c=1500 m/s. However for plug depth tracking it can be either estimated theoretically based on the media properties or determined by calibration when the plug passes the completion components with known position, such as casing diameter changes or a dart landing point. [0060] Below is a hypothetical example of surface pressure signal amplitude generated during a primary well cementing job at a constant pump rate of Q=0.005 m³/s. The correction volume in the synthetic model dV is set to 0 m³ and the correction coefficient is C = 0.95. The surface pressure amplitude responds to constant pressure oscillations at f=10 Hz generated by the hydraulic pump during top plug displacement. The casing diameter increase from 80 mm to 110 mm reduces the plug speed after passing the diameter change point, resulting in increasing periodicity of the surface pressure amplitude (Fig.8). Parameter Estimation [0061] To estimate the unknown parameters dV and C, the model predictions for the surface pressure oscillations amplitude P_pred(t,dV,C) are computed. Figure 9 is an example of comparison of the measured data with the model predictions for pairs of parameters: dV = 0 m³, C = 0.93 and dV = 0 m³, C = 0.97. Lower values of correction coefficient C correspond to smaller predicted reflection times and displaced volumes as well as longer periods between the predicted resonant states of the wellbore. Similarly, higher values of parameter C increase the predicted reflection times and displaced volumes as well as decrease the period between the resonant states of the wellbore. _{[0062] Estimation of the unknown parameters ^} ^� _{^^^ ^^^^ and ^^̂^^ is performed by minimizing the square} of residuals between measured P_meas(t) and predicted P_pred(t,dV,C) amplitudes of surface pressure oscillations: ^^� _{^^^ ^^^^ , ^^̂^^ = arg min� ^^^^} ⁽ _^^^^ ⁾ _{− ^^^^} ⁽ _{^^^^, ^^^^} ^{) 2} ^_{^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^, ^^^^�} _{^^^^ ^^^^, ^^^^} and their standard deviations ^^^^ _{^^^^ ^^^^}

probability distribution _{function ℒ� ^} ^� _{^^^ ^^^^ , ^^̂^^, ^^^^ ^^^^ ^^^^ , ^^^^ ^^^^�. Figure 10 is an example of the joint probability distribution function} _{ℒ� ^} ^� _{^^^ ^^^^ , ^^̂^^, ^^^^ ^^^^ ^^^^ , ^^^^ ^^^^�.} [0063] Each pair of parameters dV and C corresponds to the plug “trajectory” in terms of travel time τ(t;dV,C), while the corresponding value of ℒ� ^^�^^^ ^^^^ , ^^̂^^� defines a weight or likelihood of that trajectory. The probability of travel time density distribution Pτ(t) is found by integration of the family of trajectories τ(t;dV,C) weighted by L(dV,C) over parameter ranges for dV and C:

[0064] Illustrations of of travel time density distribution Pτ(t) are

[0065] To convert the reflection probability distribution into the plug position probability distribution ,one can either use the tube wave velocity estimated theoretically based on the media properties or determined by calibration when the plug passes the completion components with known position, such as casing diameter changes or a dart landing point. [0066] In the present example the diameter change at 3000 m allows the tube velocity calibration. The plug passes that point at 3150 s with the reflection time of 3.9 s. With that measurement the tube wave velocity becomes c = 2*3000 m / 3.9 s = 1538.46 s (Fig.12). Real Data Example [0067] An example is given of processing cement-displacement data during an operation in 5” casing. The casing internal diameter (ID) is 121.4 mm, the depth of the landing collar is 5791 m. The volumetric depth function for the well is shown in Fig.13. [0068] To verify if there is a reflected signal a cepstrogram is computed (Fig.14). The reflection from the plug during displacement is buried with the noise. However, there is a strong water hammer at landing showing reflection time at around 8 seconds. [0069] To determine the signal amplitude at pump noise harmonic frequencies, a spectrogram is computed. The harmonic frequencies were obtained from the recorded flow rate measured by strokes from the front and rear pumps. The harmonic amplitudes were cut from the spectrogram at corresponding frequencies (Fig.15). [0070] Correction volumes and correction factor probability distributions as well as predicted reflection time trajectories and reflection time and displaced volumes probability distributions for several time intervals are shown in Figs.16–18. [0071] The combined reflection time probability disribution is shown in Fig.19. In the beginning and in the end it conicides with the ones revealed by the cepstrogram. In the middle of displacement it shows much more pronounced plug trajectory than the one obtained with the cepstrogram. [0072] Although only a few example embodiments have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible in the example embodiments without materially departing from this invention. Accordingly, all such modifications are intended to be included within the scope of this disclosure as defined in the following claims.

Claims

Claims 1. A method for determining a position of a droppable object inside a casing string, comprising: a. placing the droppable object inside the casing string filled with a first fluid; b. pumping a displacement fluid behind the droppable object, causing the droppable object to travel through the interior of the casing string, wherein hydraulic pumps generate a harmonic acoustic signal in the form of pressure waves that travel down the casing string and reflect from the droppable object; c. recording and transmitting the pressure data to a pressure data acquisition system; d. processing the pressure data mathematically to estimate a two-way travel time of the harmonic acoustic signal reflected from the droppable object; e. determining an acoustic signal propagation velocity; f. determining a position of the droppable object from the estimated two-way travel time and the acoustic signal propagation velocity.

2. The method of claim 1, wherein the displacement fluid is pumped with at least one reciprocal pump.

3. The method of claim 1, wherein the harmonic acoustic signal is generated with at least one reciprocal pump.

4. The method of claim 1, wherein the droppable object serves as a reflected boundary for the acoustic signal, while its movement continuously changes resonant frequencies of pressure oscillations inside the casing string.

5. The method of claim 1, wherein the pressure data are recorded with at least one pressure transducer.

6. The method of claim 1, wherein the pressure data contain both direct and reflected harmonic acoustic signals.

7. The method of claim 1, wherein a mathematical method predicts a resonant response of the casing string by using the two-way travel time of the harmonic acoustic signal as an unknown parameter.

8. The method of claim 1, wherein the acoustic signal propagation velocity is determined theoretically based on media properties.

9. The method of claim 1, wherein the acoustic signal propagation velocity is determined by calibration when the droppable object passes a completion component with a known position.

10. The method of claim 9, wherein the completion component is a casing internal diameter change.

11. The method of claim 9, wherein the completion component with known depth is a droppable object landing point.

12. The method of claim 1, wherein the mathematical processing comprises computing a spectrogram.

13. The method of claim 1, wherein the harmonic acoustic signal comprises a tube wave.

14. The method of claim 1, wherein the first fluid comprises a drilling fluid, a spacer fluid or brine, or a cement slurry.