CA1316589C - Method of identification of well damage and downhole irregularities - Google Patents

Abstract

METHOD OF IDENTIFICATION OF
WELL DAMAGE AND DOWNHOLE IRREGULARITIES

Gary R. Holzhausen Gregory S. Baker Howard N. Egan ABSTRACT OF THE DISCLOSURE
A method of testing a water or petroleum well to obtain information about well features, especially that part of the well beneath the ground surface, is described. The method is especially useful in detecting problems or irregularities in the well such as stuck tools, casing damage, damaged zones in an uncased well, or debris in a well.
The method involves creating oscillations in the fluid pressure in the well. Transducers measure the pressure oscillations. The measured pressure oscillations are used to determine resonant frequencies.
The measured resonant frequencies are used to determine the characteristic impedance and the depth of each reflector in the well, after removing resonances caused by known reflectors. It is then possible to determine the nature of each unknown reflector based on suspected problems or the well's operational history.

Description

- J
SFP~M-749 1311 ~5~9 1 ~ETBOD OP ID~NTIFICATION OF

2 WELL D~MAGE AND DO~N~OLE IRREGUL~RITI~S

s 6 BACKGROUND OF T~E INVENTION
_ . . .
7 This invention i~ in the field of petroleum and 9round 8 water engineering. More specifically, it i3 in the fi~ld of 9 operation and maintenance of wells, including oil wells, ga~
wells and water wells of all types.
11 Some of t~e problems or irregularities encountered with 12 wells include, but are not limited to, the following:
13 ~ Stuck tools or equipment ~nfi6h~) in the wellO
14 Closed vs. open perforations.
. Casing damage in the form of local Gollapse or 16 shearing that cause a change in the ~ros~-17 sectional area of the well, 18 o Casing damage in the Çorm of corro~ion or breakage 1~ that causes a widening of the diameter of ~he interior of the well.
21 ~ A break in the casing that cau~es the fluid in the 22 well to be hydraulically eoupled with the fluid 23 surrounding and outside of the well.
24 . The contact between two dissimilar fluids in the ~5 well, e.g. oil above water.
26 . ZQne~s) at whieh there is no CemeDt bonding the 27 casing to the surrounding form~tion rock, causing 28 the casi~g to be more compliant than in other 29 locations where it is supported by cement.
Zones at which fractures or highly permeable 31 material intersect the well.

3~ Zones which, in an uncased well, have been washed 33 out, caved in or otherwise enlarged with respect 34 to the normal well diameter.
o The bottom of the open portion of a well that has 36 been partially filled with debris (e.g., ~and, 37 mineral seale, pieces of metal). Older methods 38 require the lowering of a wire line into the well SFP/M-749 1316 ~ 8 9 to find the botto~. ~he pre~ent ~ethod find~ the bottom quickly and effectively, withou~ lowering anything into the vell.

4 The prior art method~ of dealing with the above li~ted problems typically include ~ell-known methods such as 6 wireline logging and lowering into the ~ellbore of ~pecial tools to obtain information or samples of materials. ~11 8 these methods have in common the need to send spe~ial 9 equipment down the wellbore, during which time the well ~us~
be out of operation. Moreover, with regard to ~ome of these 11 well problems, ~here i~ no prior art method to obtain 12 definitive information as to the ex~ct nature or location o 13 the problem. This lack of a definitive method results in an 14 expensive and time-consuming trial-and-error approach to solving some of the typical well proble~s.
16 Copendin~ Applications NoO 06/841,645 and 17 No. 06/841,644 describe methods somewhat related to that of 18 the present application. However both these Application~
19 disclose methods or analyzing Peatures external to the well 2~ such as hydraulic fractures. In contrast, the present 21 application discloses a method of dealing with features that 22 are in or immediately adjacent to the wellbore.

2g SUMMARY OF THE INVENTION
Th~ invention consists of a new process for testing a 26 well to obtain information about the physical condition of 27 the interior o the well and areas immediately adjoining the 28 well, particularly the invisible portion bene~th the ground 29 surface. The process of the invention obtains this information quickly and reliably. It yield~ economic 31 benefits to the owner of a well by providing information 32 that can be used to increase well productivity and to avoid 33 or correct well damage. Furthermore, the speed with which 34 the method can be applied minimizes revenue loss from "downtime." That is, the present invention shortens the 36 time a well must be out of operation in comparison with 37 conventional well testing and evaluation method~. The 38 method of the present invention provides information 1 heretofore unavailable So the well owner, increasing h~a/her 2 alternatives for completing and maintaining the well in a safe and environmentally sound, yet profitabl~, mannerO
4 The method of ~he present invention uses the properties S of pressure waves traveling in a fluid to evaluate dow~hole 6 conditions. ~he low cost and relative ease with ~hich ~he required pressure waves can be generated, recorded and 8 interpret~d are valuable features of the invention.
9 A great many features of ;mportance cau~e a downgoing ~ wave to be partially reflected. The method of the prRsent 11 invention locates these fea~ures, using the resonant 12 ~requencies present in the well. The invention further 13 establishes the relative value of the characteri~tic 14 impedance a~ the feature (greater or le58 than the characteristic impedance of the well). It further provides 16 a process for evaluating the magnitude of the characteristic 17 impedance of the feature using free-oscillation decay 18 rates. It also provides a process for evaluating the 19 hydraulic cross-sectional area or wavespeed of the well at the feature. By the process of elimination, the inven~ion 21 indicates the possible physical e~planation for tbe downhole ~2 impedance change, e.g., a stuck tool, a sheared casing, a 23 hole wash out, a bad cement job, a contact between 24 dissimilar fluids in the well, etc.
This process of the inYention includes the following ~6 steps (some of which may be omitted depending on the 27 application):
28 1. Positioning one or more transducers in or on the 29 well in order to measure free or forced oscillations of pressure. Normally, these will be pressure transducers.
31 However, the use of accelerometers, strain gauges or 32 velocity transducers (geophones) may in some caseq 33 effectively measure the frequency of passing pressure waves 34 and therefore serve as a suitable substitute for, or supplement to pressure transducers.
36 2. Filling the well with fluid until a positive 37 pressure is attained at all points in the well (positive 38 pressure is a pressure greater than atmospheric).

~FP/M-7~3 ~. 316 ~ 8 9 1 3. Creating free or forced oscillations of pre~su~e 2 in ~he fluid in the well. Free o~cilla~ions are genera~ed 3 by per~urbing the fluid by rapi~ly op2ning and clo8ing a 4 valve to release a 3ma11 amount of fluid, r~pidly S pressurizing the well using compressed air, or employing 6 other techniques known in the art. ~orced o~cillation~ ~re 7 generated by the cyclic action of a pump or ot~er dev;ce that can oscillate the fluid at a point in ~he well 9 (typically the wellhead) at a controlled frequencyO
4. Measuring and recording the resulting pre~sure 11 oscillations~ or the frequency of oscillations, at one or 12 more point5 in the w~ll.
~3 5. Determining the velocity of presure waves in the 14 fluid in the well. This is done by using the appropriate equation for wavespeed, as is well know~ in the art, or by 16 measuring the sonic travel time to and from a refl~ctor at a 17 known distance and dividing ~his distance by one half the 18 travel time, or by determining the resonant frequency 19 corresponding to a known reflector at a known distance.
6. Determining the resonant frequencies present in 21 the pressure oscillations in the well.
22 7. Computing the reso~ant fre~uencies produced by 23 known features in or near the well at different depths.
24 Such features might include packers, casinq diameter changes and the bottom of the well. Separating these frequencies ~6 from other "unexpected~ resonant fre~uencies in the well.
27 8. Determining which of the "unexpected~ resonances 28 are related harmonically to one another, i.e., which one~
29 originate from the same reflectors.
9. Determining whether the ~unexpected~ resonances 31 from each discovered reflector di~play even or odd 32 harmonics-33 10. Using the equations in this disclosure infra 34 determining the distance from the wellhead to each 3S "unexpected" reflector.
36 11. For each NunexpectedN reflector, determining 37 whether its characteristic impedance i8 greater or les~ than 38 that of the wellbore itself (this determination is mad0 by _ g _ ~3~6~8~
SFP/M~749 1 observing whether the harmonics are odd or eve~.
2 12. Based on the known history of the well, evaluating 3 the characteri~tics of the ~unexpected" reflçc~or in a 4 manner that will provide diagno~tic information about the condition of the well. Fo~ exampl~:
6 7 If a tool has recently become ~uck in the well 7 and a high-impedance reflector has been found, one 8 can reasonably conclude that the depth to thiR
9 reflector is the depth tD the tool.
a If wells in a eertain vicinity are known to fail 11 by casing narrowing or ~hearing, and i~ ~ high-~2 impedance reflector has been found, one can 13 reasonably conclude that the depth to this 14 reflector i5 the depth to the point of c~ing damage.
16 e If the boundary between low-den~ity oil floating 17 on higher-density salt water in a well is sought, 18 and if a low-impedance reflector has been found, 19 one can conclude that this is the depth to the boundaryO
21 ~ If a well has been logged and been found to be in 22 good condition, but it is ~uspected that a section 23 of casing has no cement behind it, the method of 24 the invention can be u~ed to find thi~ section by finding a low-impedance reflector.
26 13. Determining the magnitude of the characteristic 27 impedance of ~he downhole feature from the decay rate of 28 free oscillations from the feature.
29 14. Estimating the hydraulic cross section or wavespeed at the feature from the magnitude of the feature's 31 characteristic impedance.
32 The invention therefore deals with the use of the 33 resonant properties of a well (oil, gas, brine or other 34 chemical solutions, or water) to locate and evaluate downhole features critical to the safe and profitable 36 operation of a well. It is the novel use of the principles 37 of resonance in 3 practical testing process that 38 distinguishes this invention ~rom th~ prior ast. The ~ 3 1 ~ 70128-155 significant and novel aspects of this invention include at least the following:
1. Systematically using the resonant properties of a well to characterize well features. This is a great advance over conventional methods, which rely on time-consuming and expensive wireline logging methods and diffusive pressure analysis le.g.
pressure transient analysis).
2. Differentiating features of interest from known features that are not of interest. This is done by separating and identifying the various resonant frequencies present in any well. The known resonances are then discarded. The remaining resonances are analyzed to obtain the needed information.
3. Distinguishing high-impedance from low-impedance features. This allows differentiating a well enlargement from a well narrowing, a stuck tool from a poorly cemented section of casing, an oil-water contact from a section of well narrowed by a coating of scale, etc.
4. Evaluating the cross-sectional area or the wave-speed characteristics of downhole features.

5. A full methodology which includes how to acquire the data and how to interpret the data.

6. A methodology usable whether the bottom of the well is open, closed, or partially open, or whether the wellbore is cased, uncased, or partially cased, or whether the wellhead is open, closed, or partially open.

7. A methodology usable with both liquid and gas-filled wells.

8. A methodology usable with wells filled with a plural-ity of ~luids such as oil and water.
According to a broad aspect of the invention there is provided a method of using the resonant properties of a well to characterize well features comprising the steps of: creating pressure oscillations in a fluid in the well; determining the resonant frequencies present in the pressure oscillations; com-puting the resonant frequencies produced by any known well feature; separating resonant frequencies produced by the known feature from the remaining resonant frequencies; determining which of the remaining resonant frequencies originate from a particular reflector; and determining characteristics of the particular reflector from the remaining resonant frequencies ori-ginating from the particular reflector.
According to another broad aspect of the invention there is provided a method of using the resonant properties of a well to characteri2e well features comprising the steps of:
creating pressure oscillations in a fluid in the well; measuring the pressure oscillations; determining the resonant frequencies present in the pressure oscillations; for a particular well fea-ture, determining from the resonant frequencies whether a characteristic impedance of the well feature is greater or less than a characteristic impedance of the well.
According to another broad aspect of the invention there is provided a method of using the resonant properties of a - 6a -~3~6~9 70128-155 well to characterize well features comprising the steps of:
creating pressure osci]lations in a fluid in the well; measuring the pressure oscillations; determining the resonant frequencies present in the pressure oscillations; computing the resonant frequencies produced by any known well feature; determining which of the resonant frequencies originate from a particular well feature; and for a particular well feature, determining whether a characteristic impedance of the well feature is greater or less than a characteristic impedance of the well.
According to another broad aspect of the invention there is provided a method for characterizing well features com-prising the steps of: creating pressure oscillations in a fluid in the well; measuring the oscillations, determining resonant frequencies present in the oscillations; and determining the characteristics of at least two well features located at different levels in the well from the resonant frequencies.
According to another broad aspect of the invention there is provided a method for characterizing well features comprising the steps of: creating pressure oscillations in a fluid in the well; measuring the oscillations; and determining the characteristics of at least two well features of different types from the resonant frequencies.
According to another broad aspect of the invention there is provided a method of using the resonant properties of a well to characterize well features comprising the steps of:
positioning at least one transducer in the well, filling the wel~

- 6b -7012~-155 ~6~

with fluid so as to obtain a positive pressure at all points in the well; creating pressure oscillations in the fluid; measurlng the amplitude of the pressure oscillations with the transducer;
determining the wavespeed of the pressure oscillations; determin-ing the resonant frequencies present in the pressure oscillations;
computing the resonant frequencies produced by any known well feature; separating the resonant frequencies produced by the known features from the remaining resonant frequencies; determining which of the remaining resonant frequencies originate from a particular reflector; determining whether the resonant frequencies from the particular reflector are even or odd harmonics; and for the particular reflector, determining whether the character-istic impedance of the reflector is greater or less than the characteristic impedance of the well.
According to another broad aspect of the invention there is provided a method for characterizing features in a well comprising the steps of: creating pressure oscillations in a fluid in the well; determining a spectrum of resonant frequencies present in the pressure oscillations; and determining character-istics of at least two well features from the spectrum of resonant frequencies.
According to another broad aspect of the invention there is provided a method for characterizing well features comprising the steps of: filling the well with a fluid until a positive pressure is attained at all points in the well; deter-mining a spectrum of resonant frequencies present in pressure ~ 31~8~ 70128-155 oscillations in the fluid; and determining characteristics of at least two well features from the resonant frequencies spectrum.
According to another broad aspect of the invention there is provided a method for characterizing well features, comprising the steps of. determining the velocity of pressure waves in a fluid in the well; and determining characteristics of at least two well features from the velocity of the pressure waves.
According to another broad aspect of the invention there is provided a method for characterizing well features, com-prising the step of: separating resonant frequencies of any known features in the well from other resonant frequencies in the well; and determining characteristics of at least two unknown well features from the other resonant frequencies.
According to another broad aspect of the invention there is provided a method for characterizing well features, comprising the steps of: determining whether the resonant fre-quencies of the features are even or odd harmonics; and deter-mining the characteristics of at least two features from whether their xesonant frequencies are even or odd harmonics.
According to another broad aspect of the invention there is provided a method for characterizing at least two well features, comprising the steps of: determining the number of the harmonic of a resonant frequency associated with the features;
and determining characteristics of the features from the number of the harmonic.

- 6d -~316~ 7012~-155 According to another broad aspect of the invention there is provided a method for determining the distances from a wellhead to at least two well features, comprising the steps of:
determining the resonant frequencies present in pressure oscil-lations in a fluid in the well; determining wavespeeds in the fluid in the well; and determining the distances from the resonant frequencies and wavespeeds.
According to another broad aspect of the invention there is provided a method for characterizing at least two well features, comprising the steps of: determining the magnitude of the characteristic impedances of the features; and determining characteristics of the well features from the magnitude of the characteristic impedances.
According to another broad aspect o~ the invention there is provided a method for characterizing at least two well features, comprising the steps of: estimating the hydraluic cross section at the depths of the features or the cross sectional area of the features themselves from the magnitude of the character-istic impedances of the features; and determining characteristics of the well features from the hydraulic cross section or the cross sectional area.
According to another broad aspect of the invention there is provided a method of characterizing well features, com-prising the steps of: estimating a wavespeed in a fluid in the well at at least two of the features, and determining the char-acteristics of the two well features from the wavespeed.

- 6e -;.
, -~316~3 7012~-15~

Accordin~ to another broad aspect of the invention there is provided a method for characterizing well features, comprising the steps of: measuring the amplitude of pressure oscillations in a fluid in the well; and determining character-istics of at least two well features from the amplitude.
According to another broad aspect of the invention there is provided a method for characterizing well features, comprising the steps of: creating pressure oscillations in a fluid filling the well; and characterizing at least two well features from the pressure oscillations.
BRIEF DESCRIPTION OF THE DRAWINGS
Figure 1 shows a schematic view of a well.
Figure 2 shows a wellhead pressure plot, for the case - 6f -~3~ 8~

where the we~ ead and bottala of the well ~re clo~ed and 2 there is a ref lection point in the well having a 3 characteristic impedance g~ea~er than that of the well.
4 ~IG. 3 shows a frequency domain plo~ of ~he data oE
S FIG. 2, versus decibels.
6 FIG. 4 shows a frequency domain plot of wellhead 7 impedance for forced oscill~tion.
~ FIG. 5 shows a wellhead pressure plot, or the case 9 where there is a reflection point in the well having a charact@ristics impedance lo~er than that of the well.
11 FIG. 6 shows a frequen~y domain plot of the data of 12 FIG. ~, versus decibels.
13 FIG. 7 ~hows a frequency domain plot of wellhead 14 impedance for forced oscillations, for the ca~e ~hown in FIG. 5.
16 FIG. 8 shows a wellhead pressure plot for the case 17 where the bottom of the well is open and there is a 18 reflection point in the well having a characteri tic 19 impedance greater than that of the well.
FIG. 9 shows a frequency domain plot versus decibels oE
21 the data in FIG. 8.
22 FIG. 10 shows a frequency domain plot of wellhead 23 impedance for forced oscillations, for the case shown in 24 PIG. 8.
FIG. 11 shows a wellhead pressure plot for the case 2S where the bottom of the well is open and for a reflector of 27 im~edance less than that of the wellbore.
28 FIG. 12 shows a freguency domain plot versus decibels 29 of the data in FIG. 11.
FIG. 13 shows a frequency domain plot of wellhead 31 impedance for forced oscillations, for the case shown in 32 FIG. 11.
33 FIG. 14 shows a wellhead pressure plot for a well with 34 high viscosity fluid.
FIG. 15 shows a f requency domain plot versus decibels 36 of the data in FIG. 14.

SFP/M--749 13~6~9 DETAILED DESCRIPTION OF_~E l~NVENTION
2 BASIC coNcEplrs 3 A liquid-filled or gas-filled well i8 a fluid ~y~te~O
4 ~n impul~e generated in the 1uld in a well creates a 5 preqsure wave (also referred to as a sound wave or a ~onic 6 wave) that ~ravels through the fluid in the well UJltil it 7 encounter~ an impedance change in the well, where it i B wholly or partially refleeted. The mo t c~mmon ~our~e~ Of 9 impedance c~ange are changec in the cross-seotional area of the well or changes of the wave3peed in the fluid in the 11 well. In the cases of a constant pressure boundary or a 12 zero flow boundary in the well ~typically at the bottom~ the 13 reflection is total except for possible fric~ional and 14 elastic losses into the casing and Rurrounding rock. In all other cases, there is a partial reElection, wi~h part of the lS wave continuing in the original propagation direction and 17 part being reflected back toward the point of origin.
1~ A re~lected wave returns in the direction of its 19 origin. If the origin was at the wellhead, the reflected wave will soon arrive back at the wellhead, where it again 21 encounters an impedance change and is again reflected~ ~pon 22 reaching the original downhole reflection point, the wave 23 once again i3 reflected toward the wellhead. This 24 propagation-reflection process continues in this manner until the wave is fully damped by energy losses at partial-26 reflection points and by frictional and elastic losses into 27 the surrounding solid media. Because the velocity of the 28 wave is the same each time it traverses the well between the 29 two re1ection points, the travel time between the point~ is proportional to the distance between them. If the pressure 31 is recorded as a function of time at any point in the well, 32 the pressure will be seen to fluctuate periodically as the 33 wave passes by the recording point. The frequency of tbese 34 fluctuations is therefore inversely proportional to the distance ~etween the two reflection points.
36 The frequencies at which pressure oscillations occur in 37 a well that is perturbed with a ~udden impulse are termed 3~ the resonant frequencies of the well. The resonant behavior S~P/~ 7~9 ~ 3 ~ S ~ ~ 9 1 of a pre~surized 1uid-illed wellbore i~ determined by ~he 2 location of reflec~ors in the well, and al80 by ~he ~la~tic 3 properties of the well casing and surrounding sock, and by 4 the rheology of the fluid in the well. Resonant behavior changes whenever the characteri~tics of ~he wel~or~ change, ~ either intentionally or through unforeseen problem~.
7 Pro~lems can range Erom bad cement jobs to ~tuck downhole ~ tool~. One can, however, predict the r~sonant behavior of a 9 wellbore with known geometry. Deviations from this predictable behavior can ~herefore be assigned to deviati4ns 11 in ~he ~nown well characteri~tics, especially ~he pre~enc2 12 of new or unexpected downhole reflec~ors. The~e deviations 13 are used in the method of the present invention to deter~ine ~4 the depths in the well of the downhole features that correspond to these reflectors. They are further u~ed to 16 differentiate one type of reflector from another.
17 To understand the ~ethod of the present invention, it 18 is useful to review the hydraulic principles of r2~0nance 19 that describe the resonant behavior of a well or pipeline.
2~ In the following paragraphs are presented these principles 21 for the cases of free oscillation and forced oscillation~
22 both of which are used in the present invention.

The concept of hydraulic impedance is useful for 26 describing w~ve propagatiQn and reflection in a well.
27 Hydraulic impedance Z is a comples-valued variable which is 28 defined for any point, ~, in a pipeO It expresses the 29 relationship between the oscillatory hydraulic head ~ and flow Q at that point:

32 Z(x) = Q ~ iw~ (1) 34 where i = ~ is the circular frequency in radians per second and ~ is the phase difference in seconds between head 36 and flow (see "Impedance of ~ydraulic Fractures: . , . n by 37 G.R. Holzhausen and R.P. Gooch, SPE~DOE 13892, Soc. Petr.
38 Eng. ~ DOE Joint Sympoaium on Low Permeability Re~ervoirs, S~P/M-749 131 6 ~ g 9 Denver, ~ay 1985; and ~The ~ffect of ~ydraulic Pressure Growth on Free O~cillation of ~ellbore Pre~sure" by 3 G.R. ~olzhausen and ~.P. Gooeh, Proceedings of 26th U.S.
4 Symp. on Rock Mechanic~, Rapid City, 5.D., pages 621-631, S June, 1985 and ~luid Transi~nts, E.B. ~iley and 6 V.~. Streeter, F~B Pres~ bte that head ~ i~ related to 7 pressure P by the for~ula P = pgR wher2 p i~ fluid ~a~q 8 density an~ 9 is gravitational acceleration. Freguensy ~ i5 9 rela~ed to frequency f tcycles per second, or hertz) and to wave period W a~ follows: r = ~/2s and ~ = 2~J~ - ~/r.
11 Anoth~r uReful definition is the property known ~5 12 characteristic impedance. CbaracteristiC impedance Zc is a 13 property of any uniform hydraulic conduit, In the mo~t 14 general sen~e, it can be written as (~iley and Streeter, 1~82):

17 ~ = 1 (2) 18 c s 19 which is a function of the propagation constant ~2, 21 ~2 = Cs(Ls ~ R) (3) 23 the resis~ivity R, or frictional resistance per unit lengtb 24 of condui t, 26 R = 32`~2 ( 4 ) 27 gAD
28 the fluid capacitance C, 30 ~ = 8~ (5) 31 a 32 and fluid inertance L, 8A (6) 36 The complex frequency, ~, is defined as ~ + iw. The real 37 part of s, ~, indicate~ the rate of decay of the amplitude 38 of oscillations. The imaginary part w determines the SPP~ 7~9 ~3~

1 frequency of oscillatlons. Other variable~ in the equ~tion~
2 ~bove are fluid wavespeed a~ kinem~tic viscosity ~, th~
3 fluid-filled cross-sectional area of the pipe A (rePerr~d to 4 as Hhydraulic cross section~ el~ewhere in this di~clo~ure) and hydraulic diameter D.
6 In a typical wellbore R <c L, ~o that the role of friction in determining the nature of re~onance i~
8 negligible. For example, in a well filled with water at a 9 temperature of 70 F, ~ = 0.00015 ~ec/ft3 while L = 0.075 sec2/f~3. Thus, resis~ivity ~ay be set to zero abovet 11 allowing simplified expressioDs for ~ and Z~:

13 s 14 ~ ~ a (7) 15[No Eguation 8~ (8) 17 2c 8A (9) 19 Thus, for practical considerations, characteristic impedance in a ~ection of a well is determined by the fluid wavespeed 21 and diameter in that section.
22 .
23 CONTROL OF ~AVE REFLECTIO~ BY CEARACTERISTIC I~PEDANCE
24 It is known in hydraulics (Wiley and Streeter, 1982) and also in electrical tran~mission line theory 26 ~Electromagnetic Wave Propa~ation, D. W. Dearholt and W. R~
McSpadden, McGraw-~ill, 1973) that changes of char~cteristic 28 impedance within a pipe or in a tran~mission line cause 29 reflections of oscillatory energy. The amount of the enery in a pressure wave that is reflected, and the amount that is 31 transmitted past the reflection point, are determined by the 32 reflection and propagation coefficients. These coefficient~
are defined by the characteristic impedances of the section 34 pipe on opposite sides of the reflection point. For a wave travelling down pipe section 1 toward pipe section 2, the 36 reflection coefficient r is:
38 r = ~o2 ~cl (10) SFP/M~743 ~ 3 ~ fi ~ ~ ~

2 and the propagation coeffi~ien~ p i8 ~2 ' ~c 6 The reflec~ion coeffi~ient i~ the ratio of the amplitude o~
7 the reflected wave to ~hat of the incident wave. The 8 propagation coefficien~ is the ratio of the amplitude o~ the 9 wave propaga~ed beyond the reflection point to the amplitude of the incident wave. Excluding losses from fluid friction 11 and elastic wave propagation through the casing, the sum of 12 the reflected and the ~rans~itted waves is equal to the 13 amplitude of the initially incident wave.
14 It is seen from these last two equation~ and rom the simplified eguation for characteristic impedance that the 16 amount of energy reflected and propagated as highly 17 dependent on fluid wavespeed and cross-sectional areas in 18 adjacent sections of pipe. These parameters determine not 19 only the amount of energy reflected, but also whether it will be reflected with a positive or negative siyn. If the 21 second section of pipe has a larger diameter or a ~lower 22 wavespeed than the first, a wave fro~t encounterin~ the 23 boundary will be reflected back up pipe 1 with a negative 24 rather than a positive amplitude. A cross-sectional area reduction in pipe section 2, with no wavespeed change, 26 produces a positive reflection. Thus, a tool stuck in the 27 well will produce a positive reflection becaus* it reduces 28 the cross-sectional area of the fluid in the wel~
29 Similarly, kinkingt shearing or other de~ormation of the well casing that serves to reduce the area will produce 31 positive reflection of a wave traveling in the well.
32 Buildup of minerals (scale) on the inside of a well casing 33 will also narrow the diameter and produce a positive 34 reflection. On the other hand, a widening caused by severe corrosion of a section of the casing will produce a negative 36 reflection. Similarly, if a well is uncased and a section 37 of ofter material ha~ wa3hed away, giving that ~ection a 38 larger diameter, a negative reflection will al90 be - 12 - ~

, SFP/M~7~9 ~ ~316~89 1 produced.

EFFECTS 0~ DOWN~OLE IRREGUhARITIES ON ~AVESP~ES IN ~LLS
4 Fluid wavespeed in a confined conduit, such as a well, is determined primarily by the bulk ~odulus and density of the fluid and by the compressibility of the conduit.
Conduit compressibility is primarily a function of the 8 elastic modulus of the conduit material, the ~all thickness 9 of the pipe and whether or not it is semented to and supported by the urrounding rock material. The degree to 11 which the pipe can expand longitudinally is al80 a faetor in 12 determining wave~peed.
13 Equation~ for computing wavespeed are well known in the 14 art and can be found in ater_ammer AnalY i8, John Parmakian, Dover, 1953, Chap. III and Wiley and Streeter, 16 1982. For purposes of demonstrating the method of the 17 present invention below are presented wavespeed e~uations 18 for boundary conditions typically found in oil, gas and 19 water wells. The invention i5 not li~ited, however, to the specific boundary conditions represented by the following 21 wavespeed equations.
22 In general, the equation for wavespeed a in a pipe or 23 well casing is:

26 a = ~ ~12) 27 where d is pipe diameter, e pipe wall thickness, E Young'c 28 modulus of the pipe wall material (typically on the order of 29 4.32xlO9 pounds per square foot tpsf) for steel), K bulk modulus for the fluid in the well (about 43.2x106 psf for 31 water), ~ the specific weight of the fluid (about 62.4 32 pounds per cubic ft. for water) and ~ Poisson's ratio of the 33 pipe wall material labout 0.~ for steel). The term c i~ a 34 coefficient that depends on the elastic boundary conditions of the well.
36 When the pipe or casing is fixed at the upper end but 37 not the lower end, and is not cemented to the ~urrounding 3~ rock (thi~ is commonly the case for a tubing string in a SFP/M 749 ~ 3 :16 ~ 8 9 1 wel~):
2 c = 4 - ~ (13~
3 For a well casing or tubing ~ring ~hat is ~upported ~t both 4 ends 80 that it cannot ~ove longitudinally, but that iB not cement~d ~o the ~ormation between these support point87 6 c = 1 _~2 (1~) 7 ~or a well casing that is ce~ented unifor~ly to the rock around it (the rock having the shear modulus Gj 9 c = Ee/~Gd ~ ~e) (15) Finally, wavespeed a in an uDcased well is given by ~he 11 equation 13 a ~ 16) 14 ~ g(~ ~ G) Reflections caused by ~avespeed contrast~ rev~al many 16 important features within a well. For e~ample, the contact 17 between two liguid~ of different den ities in a well, quch 18 as ~ column of oil floating on a column of water, produce~ a 19 reflection because of a wavespeed contrast. The quality of the cement bond between casing and the surrounding rock 21 formation is an important question in the petroleum and 22 ground water engineering. The unexpected absence of a 23 cement bond because of a poor cementing treatment is a 24 common cause of expensive problems. These problems range from failed hydraulic-fracturing treatments (fr3cturing 26 fluid flows up the cas~ng-formation annulus rather than into 27 a fracture) to contamination of ground water supplies by the 28 migration of ~alt water brines along the open annular ~pace 29 between the casing and the rock. Various well logging techniques have been developed to evaluate cement bond 31 quality, but they are expensive and time-consuming to 32 perform. Comparing equation 13 to equation 15 above reveals 33 that the wa~espeed in an uncemented section of a cased well 34 is slower than in a section that is cemented to the surrounding rock. Thus, a wave traveling down a well in 36 which the casing is firmly bonded to the surrounding rock 37 will undergo a negative partial reflection when it enters a 38 portion of the well without cement behind the casing. Tbis ~ 3 ~ 9 SFP/~-749 1 effect is extrem21y useful ~or identifying interv~ls l~c~ing ~ in cement beore proble~ develop at a later d~te.

4 USE OF FREE OSCILLATIONS OF PhESSURE ~o L~AT~ AND EVALUATE
~WN~OLE FEA~RES
6 When the pressurized fluid in a well is abruptly 7 perturbed, the re~ulting pressure os~illations oc~ur at the 8 re~onant frequencie~ of the well. The downhole featur~es 9 that one wishes to locate are points of characteri~ti~
impedance contrast. The locations of these featureQ
11 determine some of the resonant frequencie~ that are pre3ent 12 in the well. To develop the procedure for locating these 13 features using free-oscillation behavior, one begins with 14 the well-known hydraulic transfer equation~ (~ylie and Streeter, 1982~;
16 ~D = ~ cosh(~ QUZCS1nh~ 17) 18 QD = QU COSh(~ inh( ~) (18) 19 c where ~ is the length of a uniform section of well or pipe~
21 and the U and D subscripts refer to upstream twellhead) and 22 downstream reflection points, i.e., poi~ts where there i~ a 23 change of characteristic impedance. If t~e well con~i~t~ of 24 a single uniform section of pipe, then the U and D
subscripts refer to the wellhead and the bottom of the well 26 re6pectively. If the well consists of two or more difer~nt 27 sections, then the V and D subscripts refer to the upstream 28 and downstream end of a particular ~ection. The~e 29 expressions use the characteristic impedance o the pipe to express the relationship between the head and di~charge of 31 one end of a pipe section to the head and discharge at the 32 other end.
33 For demonstration purposes, let the boundary condition 34 at the upstream end of a section of well be zero flow (zero discharge). This condition describes a closed wellhead, or 3~ example. The same approach as followed below can be applied 37 when the wellhead is open or partially open, allowing some 38 flow in or out of the well. The methodology of the SFP~-749 1 316 ~ 8 9 1 nece~sary mathematical derivations for each of th2~e ca8e8 is well known in the ar~. Por a z~ro-flow up~tream boundary ~ = 0 (19) 4 The downstream boundary condi~ion is pecified for the most general ca~e ~s the hydraulic impedance at the down~tream 6 ~erminus g D QD ~20) 10 The combination of equations 17 through 20 yields the 11 relation 12 ZDsinh(~Q~ + Zccosh(~) = 0 ~21j or i~ put in exponential for~
14 e2~(Zc ~ ZD) + ~Zc ZD) (22 15 The real part of this equation is 18 (e2~Q/a) cos ~2 ~) ZD - Zc ~23) 19 The imdginary part of this equation i~

21 3in(2aQ) = 0 (24) 23 The values of ~ which satisfy equation 24 define all the 24 possible free-oscillation frequencies that may occur in the wellboreO Two sets of solutions for equation 22 exiæt, they 26 are:

27 for ZD < Zc 29 ~ = 2~1n(z ~ z ) ~25) 31 [No E~uation 26] (26) 33 n a n = 1,3~. (27) and for 2D ~ Zc 337 ~ ~ 2~ ln(z , z ) ~2~) . - 16 -:~3~6~
1 ~o Equation 291 (29~

3 ~ n a n =2,4........................... (30) 4 In practice, ~ can be determined easily Ero~ a record of free oscillations of pressure at the wellhead. ~o do 6 this, the pre~sure data i~ transformed into the frequency 7 domain using a Fast-FourierJTransform (FFT~ alogrithm or 8 another time-domain-to frequency-domain conver3ion, a~ i3 9 common in the art. The output of the FFT gives the distri~ution of power in the frequency domain. ~hen 11 plotted, this output has distinct peaks in various parts of 12 the spectrum which correspond to the resonant ~requencies of 13 the well.

14 ~ith the method of the present invention an observation of the frequency spectrum allows an immediate interpretation 16 of the location and nature of a downhole reflector. IE odd 17 harmonics are observed (n=1,3..) this indicate~ that the 1~ hydraulic impedance at the reflector i~ leSB than the 19 characteristic impedance of the wellbore. Even harmonics (n=2,4..) indicate that the impedance of the downhole 21 reflector is greater than the wellbore's characteri tic 22 impedance. ~f the change is solely the resul~ of a change 23 in casing diameter, and the free o~cillation frequencies are 24 odd harmonics, equation 27 indica~es that the casin~
diameter increases at the reflector~ In contrast, even 26 harmonics indicate a decreasing casing diameter.
27 The distance between the wellhead and the downhole 28 reflector is determined directly from the measured 29 frequencies of free oscillation using equation 27 or 30, a~
appropriate. First, the har~onic number n i8 selec~ed rom 31 an inspection of the data, then the frequency ~ and 32 wavespeed a are substituted into the appropriate equation~
33 The depth to the reflection point ~ is then computed.
34 In using the method of the present invention it is necessary to differentiate resonances produced by the 36 downhole features under investigation from re~onances that 3~ would otherwise occur in the well. The ~trongest resonances 3B in most wells are caused by reflections off the bottom of SFP/M-749 1 316 ~ 8 9 1 the well. Becau~e the dept~ to the botto~ i8 known in ~o~t 2 cases, the resonances caused by the bottom ~re ea~ily 3 computed u~ing equation~ 27 or 30. If the bottom i~ a dead 4 end, i.e., a no-flow boundary, equation 30 i~ ~ed to compute it~ re~onan~ frequeDcie~ ~O If it i~ a con~tant ~ pressure boundary (at least during the period over which the 7 measurementS are made, norEally several ~econds) t ~guation 27 is u~e~ to compu~e the frequencie~ w, This procedurg of 9 computing depth to the botto~ is explained more fully in copendins U.5. Patent Applications No. 06/841,645 and 11 06/841,64~. Other known features in a well, such as a 12 small-diameter ~nipple" or a small-diame~er casing liner, 13 can be identified in the same manner and thereby 14 differentiated from unknown features under inve~tigation.
Figs. 1-13 illustrate ~he use of the method of th~
16 present invention for location of a downhole impedance 17 contr~st and the differentiation of this contras~ fro~ the 18 harmonics caused by wave reflection at the bottom of She 19 well,. FigO 1 ~hows a well (total depth = 2201 Et.) in which there is a characteristic impedance contrast at a ~1 depth oP 1000 to 1001 feet. Otherwise, the well is of 22 uniform characteristic impedance over its entire depth.
23 Figs. 2, 3 and 4 show pressure oscillations and resonant 24 behavior for the case in which the wellhead and the bottom of the well have much higher impedances than the 26 characteristic impedance of the well itself. ~oth are 27 effectively no-flow boundaries and the characteristic 28 impedance Zc2 of the short central section i5 al~o greater 29 than the characteristic impedance (Zcl = Zc3~ of ~he rest of the well. Fig. 2 shows the pressure oscillations that occur 31 at the wellhead after the ~ell is perturbed with an initlal 32 impulse, Fig. 3 is a frequency domain plot of the pressure 33 data in Fig. 2, showing the resonant peaks from the bottom 34 of the well and from the re~lector at 1000 fto In ~ig. 3 the resonances from the bottom of the ~ell are clearly 36 visible. The lowest-frequency peak (n = 2) is the 37 fundamental frequency of the entire length of the well. It5 38 higher-order harmonics (n - 4,6,8) are also clearly vi3ible SFP/~-749 ~316~8~

1 in Pig. 3. If it i~ de~ired ~o find the hydr~ulic bottom of 2 a well ~hat had been parti~lly filed with debris, one 3 eubstitu~es the ~requencies of the~e pe~k~ along with their 4 harmonic numbers n and the fluid wave~peed into equation 30, S ~olving for the well depth 1. However, if one is looking 6 for the depth of the impedance contra~t in the well, fir~t ? one removes the resonances caused by the bottom of the well from further consideration. Then one processe~ the 9 remaining resonant frequencies.
Taking the wavespeed for the well as 5000 ft/~ec, it is 11 found that the lowest frequency from the reflector i8 2rS
12 hz, corresponding (from equation 30, setting n = 2) to a 13 reflector depth of 1000 ft. Note that there is a lower 14 frequency at 2.08 hz, which re~ults from re~ona~ce between the reflector and the bottols o the well. Solving for the 16 length of well Q in which the 2.08 hz resonance occur~
17 (using equation 30 and settiDg n - 2 and u = 5000 ft/~ec), 18 it is found that Q = 1200 feet, which is e~actly the 19 distance between the reflector and ~he bottom o the well.
From the frequency information alone one can say with 21 certainty that the reflector is ei~her 1000 ft. deep or 1200 22 ft. deeP.
23 It is possible now to conclusively establi~h reflector 24 depth by examining the time series plot of pressure oscillations (Fig. 2) o The f irst perturbation of the ~6 wellhead pressure after the initial impulse comes from the 27 wave reflec~ed up from the point of impedance contra~t in 28 the well. By measuring the time between the ~tart of the 29 initial impulse and the arrival of the reflection, one can 30 f ind the depth to the reflection point. Prom Fig. 2 it i~
31 seen that this time is 0.4 seconds. The two way travel 3~ distance (down and back) is 0.4 seconds x 5000 ft/sec = 2000 33 f t . The distance to lthe reflection point is therefore one-34 half this amount, or 1000 feet.
Now consider a well in which the reflection point in 36 the middle ha~ a lower characteri~tic impedance than the 37 rest of the well. Again, the boundary condition at the 38 wellhead and at the bottom is highly re~tricted flow or no ~FP/M-7~g ~3~ ~8~

1 flow. The irst reflection, arriving at the wellhead aft2r 2 the initial negative impulse, is positive ~Pig. 5) r~ther 3 than negative as was the case in ~ig. 2. Thi~ behavior is 4 consistent with a re~lectio~ coefficient with a value between 0 and -1, as predicted by equation 10. The 6 corresponding frequency domain plot (Fi~o 6) reveal~ 8V~
7 harmonies from ~he bottom of the well as in th~ previou~
example. The undamental harmonic (n = 2) fro~ the 2200 ft.
9 depth is masked by the funda3ental harmollics as~ociated with resonances above and below the reflector, but the n = ~,6, 11 and 8 harmonics are clearly visible. Likewi~e the odd n = 1 12 and n = 3 harmonics from the resonances above and below the 13 reflector are distinct and allow evaluation of the depth of 14 this reflector. Using equation 27~ the two p~ssible depths to the reflector ~1000 t. and 1200 ft.) are quickly 16 evaluated. The fact that the reflection point produces odd 17 harmonics also reveals that it is a point of lower 18 charaeteristic impedance than the rest o the well. This 19 information indicates that the reflection point i5 a probable area of reduced wavespeed or greater diameter in 21 the well. As in the previous example one measures ~he ti~e 22 between the start of the initial lmpulse and the arrival of 23 the reflection to unambiguously define the depth to the 24 reflection point. One finds from Fig. 5 that thi~ time is 0,4 seconds. The two way travel distance (down and back) i~
26 0.4 seconds x soao ft/sec = 2000 ft. The distance to the 27 reflection poin is therefore one~half this amount~ or 1000 28 feet.
~9 Now consider a well as ih Pig. 1 in which the bottom i~
a constant-pressure boundary rather than a no-flow boundary, 31 i.e., ZD at the bottom of the well is close to zero. Figs.
32 8 and 9 show the pressure oscillations and resonant 33 frequencies that result from impulsive excitation of such a 34 well. As predicted by eguation 27, the reflertion off the bottom of th0 well now produces odd harmonics with a 36 fundamental frequency of 0.57 hz. This ~requency and the n 37 = 3,5, and 7 harmonic resonances clearly ~how up in the 38 frequency domain plot (Fig. 9). The other resonances are in - 2~ -SFP/M-749 1 31 6 ~ 8 9 the section of ~he well above the reflector and the ~ection below the reflector. The 1200-ft. section produce~ odd 3 harmonics and the 1000-ft~ gection pr~duces even har~onic~, 4 both of which are ~hown clearly in ~i9o 9~ ~0 de~ermine which of the~e depths i~ the depth from ~he wellhead ~o the 6 reflector, one refers to the pre~sure o~illation plot (Fig.
7 8). As in the previous e~amples, one finds that the fir~t 8 reflection arrives at the wellhead 0.4 ~econd~ af~er the 9 start of the initial impulse~ Multiplying by the wave~peed 1~ and then dividing ~y 2 yields the depth of 10~0 ft. One 11 knows that ~he characteristic impedance of the reflector i~
12 greater ~han that of the well because the harmonics between 13 ;t and the wellhead a~e even (Fig. 9) and becau~e the fir~t 14 reflected wave to the return to the well head ~Fig. 8) ha~
the same polarity a~ the initial impulse.
1~ Finally, consider a well as in Fi~. 1 in which the 17 bottom is a constant-pressure boundary and in which the 18 reflector at 1000 ft. has a characteristic impedance lower 19 than that o~ the wellbore. ~igs. 11 and 12 ~how the pressure oscillations and resonant frequencies that reqult 21 from impulsive excitation of such a well. As in the 22 previous example ~Fig. 9) the reflection off the bottom of ~3 the well produces odd harmonics with a fundamental frequency 24 (n = 1) of 0.57 hz. This freguency and the n = 3,5, and 7 harmonic resonances clearly show up in the frequency domai~
Z6 plot (Fig. 12). The other resonances are in the section of 27 the well above the reflector and the section below the 28 reflector. The 1200-ft. section produces even har~onic~ and 29 the 1000-ft. section produces odd harmonics, both of which are shown clearly in Fig. 12. To determine which of these 31 depths is the depth from the wellhead to the reflector, one 32 refers to the pressure oscillation plot (Fig. 11). As in 33 the previous examples, one finds that the first reflection 34 arrives at the wellhead 0.4 seconds after the start of the initial impulse. Multiplying by the wavespeed and then 36 dividing by 2 yields the depth of 1000 ft.
37 The free-oscillation examples above ~re for fluid 38 viscosities of 1 centipoise. Fig. 14 and 15 show free-SFP/H-749 1 3 ~ 6 ~ 8 9 oscillation behavior ~nd the requency spectrum ~or ~ well with 10 centipoise-vi~co~ity fluid, The boundary condition~
are the ~ame a in ~he la~ example: the bo~to~ i~ a 4 con~tan~pre sure boundary and the reflec~or at 1000 ft. ha~
a characteri~tic impedance lower than that of the 6 wellbore. One find~ by comparison of Fig~. lS and 12 that 7 the resonance frequencies in and for ~he two difer~nt fluid 8 viscosities is the same, proving the generality of the 9 inventive method.
The foregoing discu~sion of free oscillation~ ha~ ~hown 11 that it is possible, using the me~hod of the invention, to 12 locate a downhole reflector and to determine whether it~
13 characteristic impedance is great~r or les~ than that of the 14 well, regardless of the boundary condition at the bottom of the well. Furthermore, a plurality of re1ectors can be 16 loc~ted in a ~ingle well using the m~thod of ~he inYention, 17 and their impedances relative to the well itself can be 18 determined. The same procedure as outlined abov2 i~
19 followed, care belng taken to differentiate the harmonic frequencies of each reflector during the analysi~.
21 The quantitative magnitude of the impedance of the 22 reflector can also be determined using the method of the 23 present invention. Equations 25 and 28 relate the decay 24 rate of free oscillations to the impedance of the downstrea~ -~5 end of a section of pipe, ~D. One can generalize theQe 26 equations by writing 28 a Zc2 ~ Zc1l (31) ~ which states that ~he decay rate of free oscilla~ions is 31 proportional to the natural log of the absolute value of the 3~ reflection coefficient at the downstream end of the pipe.
33 One wishes to determine the magnitude of Zc2~ which i5 the 34 characteristic impedance of the feature of interest. One therefore must isolate the resonant frequencie~ in the 20ne, 36 say, between the wellhead and the reflection point. This 37 can be done by filtering out the other resonant Erequencies 38 in the well, using techniques well known in the art, and ~ 22 -SFP/~-749 ~316~89 1 replotting a ~ime ~0rae3 wi~h only the frequencie~ of 2 interest. ~he decay rate ~ per second (or per ~o~e 3 convenient time interYal) is then deter~ined fr~m the decay 4 of the subject o~cillations. Decay rate will be a numb~r between O (no decay) and 1 (instantaneous decay) per 6 second. A~ter ~ is found, it is ~ubstituted into equ~tion 7 31 along with the fluid wave~peed a, characteristic 8 impedance Zcl and length ~ of the section of pipe between the wellhead and the reflector. One then solve~ for ~2.
1~ Ater the value of this downhole impedance has been 11 determined in this manner, the magnitude of the ~avespeed 12 change or well cross section ohange can be estimated u~ing 13 equation 9. For example, if the reflection is cau~ed by 14 crushed or sheared casing, the wavespeed at the depth o damage will not have changed, but the cross-sectional area 16 0~ the fluid will be maller than normal. Equation 9 17 reveals the ac~ual oross-sectional area at this point.
18 Knowledge of ~he cross-sectional area can then be u~ed in 19 planning remedial action. For xample, it will deter~ine the size of wireline tools that can be lowered past that 21 point of the well. Another example is determini~g the 22 diameter of a washed-out zone in a well prior to ~ementing 23 casing into the well. ~he diameter of enlargement~ of tbis 24 sort are important in calculating the required volume of cement.

27 USE OF FO~CED 05CILLATIONS OF PRESS~RE TO LOCATE AND
2 8 EVALUATE DO~NHOLE F} :ATURl~S
29 A steady oscillatory flow may be forcibly created at the wellhead (or anywhere in a well) by the action of a 31 pump. Such oscillations are termed forced oscillation~
32 because the pump determines their frequency. Forced 33 oscillation conditions are also advantageously used in the 34 present invention to locate and characterize downhole feature~.
36 Under these conditions the method of the present 37 invention takes the following approach. The impedanc~
38 tran~fer equation ~Wylie and Streeter, (1982) 5 ~ ~
701~-155 ZD + Zc tanh ~
1 + ZD tanh (~Q) (32) c glves the hydraul.tc lmpe~ance at the upstream end of a pipe in terms of the hydraullc impedance at the downstream end. One can use thls e~pression to ~e~ermlne the resonan~ ~requencles of a slngle section of a well, or of any entire well composed of a series of discrete sectlons.
When the pump is turned on, slnusoldal oscillatlons develop at each point ln the system at the frequency of the forcing function and will not decay in time, l.e., a = 0. The propagatlon constant becomes ,~,= i~
~ (33) The magnitude of the hydraulic lmpedance ZU at the wellhead ls found from the amplitudes o~ prassure and discharge oscillations at the wellhead according to the formula: H
IzU l= QU (34) Combining the expressions for ZU and ~ with the definitlon of the magnltude of a complex number results in the following expresslon for the impedance at the upstream boundary (typically the wellhead) where the forcing function, l.e., pumplng action, is belng applied.

~(ZD (1 + tan(~~ )) 2 +(Zctan~ ~ - z tan(~a~)) 1 +(Z tan (~a3Q))2 If ~ and ZU are known, the downstream impedance is the only varlable in equation 35 and lt can be found with the use of an iteratlve solution.
If ~ ls not known beforehand, which would be the case when trying to locate an unanticipated reflector, another approach ls needed in order to solve for both ~ and ZD To SFP/~749 1 316 ~ 8 9 1 determine t~e~e value~ one ~uqt ~ind the resonant 2 frequencies of the ~ys~em. æt the resonant ~requencies ~he 3 impedance at the wellhead is at a local ~axi~m in the 4 frequency domain. Becau~e the flow per pump ~troke iB
essentially constant under forced-oscillation condition~, 6 equation 34 teaches that the amplitude oE the pre~sure 7 oscillations is therefore al50 at a local maximum at each o 8 the resonant frequencies of the well. These local ma~ima of 9 pressure correspond to local maxima of impedance u~ing the technique of the inven~ion. The resonant frequencie~ are 11 therefore found by varying the pump frequency over a 12 suitable r~nge (i.e., by varying the forcing function) and 13 measuring the wellhead pressure oscillation~ a~ each 14 frequency. Pressure 06cillation amplitude i5 then plotted as a function of frequency. This can be done ~anually fro~
16 the pres~ure-time data or by processing the pres~ure-time 17 history into frequency domain information using t~chnigue~
18 well known in the art of signal processing.
19 The impedance transfer equation 35 was u~ed to plot the ~0 absolute value of the wellhead impedance versus the 21 frequency of the forcing func~ion for the wellbore geometry 22 shown in ~ig. 1 and ~he four sets of ~oundary conditions 23 described in the foregoing discussion of free 24 oscillations. The well i5 composed of three hydraulic elements, each with a different length: a 1000-ft. section 26 is uppermost, a l-ft. section i~ below it, and a 1200-ft.
27 section is at the bottom. Equatio~ 35, which i~ for a 28 single section of pipe, can be used for more complicated 29 geometries (such as this example well) by treating the well as a series of pipes. The downstream impedance of one pipe 31 is simply the upstream impedance of the next lower pipe.
32 This construction results in a set of equations that i~
33 solved simultaneously to define the frequency respon~e 34 characteristic of the well.
3S Figs. 4, 7, 10 and 13 show the wellhead impedance 36 magnitudes Eor the four sets of boundary condition~.
37 Comparison with Figs. 3, 6, 9 and 12, which are the 38 corresponding frequency domain plots for free oscillation~, SFP/M-7~9 ~ 3 ~ 9 1 reveal~ the 6a~e re~onant frequencies that w~ found for the 2 free-oscilla~ion ca~es. Thus, the re~onant fr~quencie~ o a 3 pipe or well ~ubjected to forced o~cillation are ident~cal 4 to the resonant frequencies found from free o~cillation S behavior~ There are 8ma11 differenc~ in ~he relatl~e 6 amplitudes of ~he frequency peaks for forced 48cillati9n~
? compared to free oscillations. The~e amplitude difference~
~ ari~e from the frequencies contained in the impul~e that ~as 9 used to numerically generate the free oscillations. ~he ~ energy in this impulse was not distributed evenly among the 11 several resonant freguencies. In the 6imulation of forced-12 oscillation behavior (Figs. 4, 7, 10 and 13) the energy 13 distribution was more uniform acro~s the frequency spectru~.
14 The forced-osoillation analysis means that, if the lS hydraulic impedances at both ends of a hydraulic Plement are 16 greater or less than the characteri~tic impedance of the 17 element, the resonant frequencies of the ele~ent are:

19 w = n2~a n = 2,4.. (36) 21 ~owever, iE the hydraulic impedance at one end is gre~ter ~2 than-the characteristic impedance of the element, and at the 23 other end it is less, the resonant frequencies of the 24 element are:
26 ~ = 2I n = 1,3.................................. (37) 2B These equations are the same as equations 30 and 27 which 29 were developed for free-oscillation condition~. Thu~, it i8 also possible to use forced-oscillation measurements to 31 determine the distances ~ to unanticipated downhole 32 impedance contrasts by: 1. measuring their resonant 33 frequencies, 2. determining ~hether the harmonics are odd or 34 event, and 3. then using the last two eguations to evaluate ~.
36 After resonant frequencies w and length~ ~ have been 37 determined, the characteristic impedance tZD in equation 35) 38 o~ the reflector of int~rest can also be found. Thi~ i8 5FP/M-749 1 31 6 ~ 8 9 1 done by ~ub3tituting these ~alues along with the up~re2~
2 impedance, ZU~ at re30nance in~o equa~ion 35 and ~olving for 4 ZD. After ZD has been found in thi~ manner, egu~tion 9 1 used to estimate the hydraulic crons ~ection or wave~p2ed 6 change at the reflector, and thi~ information i~ applied ~o determine the cause of the downhole feature (stuck tool, pinched casing, wa~h ou~, ~ad cement job, etc.).
~ he foregoing description is not intended to be construed in a limiting ~ense. Various modification~ o the disclosed embodiment, as well as other embodi~ents of the invention, will be apparent to persons skilled in the art.
13 The invention is therefore to be limited only by the 14 claims that follow.

~3 2g

Claims (71)

1. A method of using the resonant properties of a well to characterize well features comprising the steps of:
creating pressure oscillations in a fluid in the well;
determining the resonant frequencies present in the pressure oscillations;
computing the resonant frequencies produced by any known well feature;
separating resonant frequencies produced by the known feature from the remaining resonant frequencies;
determining which of the remaining resonant frequen-cies originate from a particular reflector; and determining characteristics of the particular reflector from the remaining resonant frequencies originating from the particular reflector.

2. A method of using the resonant properties of a well to characterize well features comprising the steps of:
creating pressure oscillations in a fluid in the well;
measuring the pressure oscillations;
determining the resonant frequencies present in the pressure oscillations;
for a particular well feature, determining from the resonant frequencies whether a characteristic impedance of the well feature is greater or less than a characteristic impedance of the well.

3. A method of using the resonant properties of a well to characterize well features comprising the steps of:
creating pressure oscillations in a fluid in the well measuring the pressure oscillations;
determining the resonant frequencies present in the pressure oscillations;
computing the resonant frequencies produced by any known well feature;
determining which of the resonant frequencies ori-ginate from a particular well feature; and for a particular well feature, determining whether a characteristic impedance of the well feature is greater or less than a characteristic impedance of the well.

4. A method for characterizing well features comprising the steps of:
creating pressure oscillations in a fluid in the well;
measuring the oscillations;
determining resonant frequencies present in the oscillations; and determining the characteristics of at least two well features located at different levels in the well from the resonant frequencies.

5. A method for characterizing well features comprising the steps of:
creating pressure oscillations in a fluid in the well;
measuring the oscillations; and determining the characteristics of at least two well features of different types from the resonant frequencies.

6. The method of Claim 1, further comprising the step of filling the well with fluid until a positive pressure is attained at all points in the well, prior to the first step of determining.

7. The method of Claim 1, further comprising the step of determining the velocity of pressure waves in the fluid in the well, after the first step of determining.

8. The method of Claim 1, further comprising the step of determining whether the resonant frequencies from the parti-cular reflector are even or odd harmonics, and what the numbers of the harmonics are.

9. The method of Claim 8, further comprising the step of determining the distance from the wellhead to the particular reflector.

10. The method of Claim 1, further comprising the step of determining the magnitude of the characteristic impedance of the particular reflector.

11. The method of Claim 10, wherein the step of deter-mining the magnitude includes observing the decay rate of free oscillations from the particular reflector.

12. The method of Claim 10, further comprising -the step of estimating the hydraulic cross section at the particular reflector from the magnitude of the characteristic impedance of the reflector.

13. The method of Claim 10, further comprising the step of estimating a wavespeed at the reflector from the magnitude of the characteristic impedance of the reflector.

14. The method of Claim 8, further comprising the step of determining that the hydraulic impedance of the reflector is greater than a characteristic impedance of the well when the reflector displays even harmonics, and that the hydraulic impedance of the reflector is less than the characteristic impedance of the well when the reflector displays odd harmonics.

15. The method of Claim 9, wherein the oscillations are free oscillations, and the step of determining the distance in-cludes calculating the distance from the frequency of the free oscillations, the harmonic number, and a wavespeed in the fluid.

16. The method of Claim 1, wherein the first step of determining includes positioning at least one transducer in the well.

17. The method of Claim 1, wherein the step of determin-ing includes positioning at least one transducer on the well.

18. The method of Claim 1, wherein the first step of determining includes positioning at least one transducer on the wellhead.

19. The method of Claim 1, wherein the first step of determining includes measuring the pressure of the oscillations.

20. The method of Claim 1, wherein the first step of determining includes measuring the frequency of the oscillations.

21. The method of Claim 1, wherein the first step of determining includes measuring at more than one point in the well.

22. The method of Claim 1, wherein the step of creating pressure oscillations includes generating free oscillations.

23. The method of Claim 22, wherein the generating of free oscillations includes rapidly opening and closing a valve to release an amount of the fluid in the well.

24. The method of Claim 22, wherein the generating of free oscillations includes pressurizing the well by use of a gas.

25. The method of Claim 1, wherein the step of creating pressure oscillations includes generating forced oscillations.

26. The method of Claim 25, wherein the generating of forced oscillations includes the cyclic action of a pump means to oscillate the fluid at a controlled frequency.

27. The method of Claim 1, wherein the wellhead is closed.

28. The method of Claim 1, wherein the wellhead is open.

29. The method of Claim 1, wherein the wellhead is par-tially open.

30. The method of Claim 1, wherein the step of computing the resonant frequencies includes performing a time-domain to frequency-domain conversion of the oscillations as measured in the first step of determining.

31. The method of Claim 1, wherein the bottom of the well is open.

32. The method of Claim 1, wherein the bottom of the well is closed.

33. The method of Claim 1, wherein the bottom of the well is partially open.

34. The method of Claim 1, wherein the well is cased.

35. The method of Claim 1, wherein the well is uncased.

36. The method of Claim 1, wherein the well is partially cased.

37. The method of Claim 1, wherein the characteristics of a plurality of reflectors are determined.

38. The method of Claim 1, wherein there are a plurality of fluids in the well.

39. A method of using the resonant properties of a well to characterize well features comprising the steps of:
positioning at least one transducer in the well;
filling the well with fluid so as to obtain a posi-tive pressure at all points in the well;
creating pressure oscillations in the fluid;
measuring the amplitude of the pressure oscillations with the transducer;
determining the wavespeed of the pressure oscillations;
determining the resonant frequencies present in the pressure oscillations;
computing the resonant frequencies produced by any known well feature;
separating the resonant frequencies produced by the known features from the remaining resonant frequencies;
determining which of the remaining resonant fre-quencies originate from a particular reflector;
determining whether the resonant frequencies from the particular reflector are even or odd harmonics; and for the particular reflector, determining whether the characteristic impedance of the reflector is greater or less than the characteristic impedance of the well.

40. The method of Claim 39, further comprising the step of determining the distance from the wellhead to the reflector.

41. The method of Claim 39, further comprising the step of determining the magnitude of the characteristic impedance of the particular reflector from the decay rate of free oscillations from the particular reflector.

42. The method of Claim 39, further comprising the step of estimating the hydraulic cross section and wavespeed at the particular reflector from the magnitude of the characteristic impedance.

43. The method of Claim 39, further comprising the steps of measuring the frequency of the pressure oscillations.

44. The method of Claim 39, wherein the wellhead is closed.

45. The method of Claim 39, wherein the wellhead is open.

46. The method of Claim 39, wherein the wellhead is partially open.

47. The method of Claim 39, wherein the step of deter-mining resonant frequencies includes performing a time domain to frequency-domain conversion of the oscillations as measured in the step of measuring.

48. The method of Claim 39, wherein the bottom of the well is closed.

49. The method of Claim 39, wherein the bottom of the well is open.

50. The method of Claim 39, wherein the bottom of the well is partially open.

51. The method of Claim 39, wherein the well is closed.

52. The method of Claim 39, wherein the well is uncased.

53. The method of Claim 39, wherein the well is par-tially cased.

54. The method of Claim 39, wherein the step of creating pressure oscillations includes generating forced oscillations.

55. The method of Claim 39, wherein the step of creating pressure oscillations includes generating free oscillations.

56. The method of Claim 39, wherein there are a plurality of fluids in the well.

57. A method for characterizing features in a well comprising the steps of:
creating pressure oscillations in a fluid in the well;
determining a spectrum of resonant frequencies present in the pressure oscillations; and determining characteristics of at least two well features from the spectrum of resonant frequencies.

58. A method for characterizing well features comprising the steps of:
filling the well with a fluid until a positive pres-sure is attained at all points in the well;

determining a spectrum of resonant frequencies pre-sent in pressure oscillations in the fluid; and determining characteristics of at least two well features from the resonant frequencies spectrum.

59. A method for characterizing well features, comprising the steps of:
determining the velocity of pressure waves in a fluid in the well; and determining characteristics of at least two well features from the velocity of the pressure waves.

60. A method for characterizing well features, compris-ing the step of:
separating resonant frequencies of any known features in the well from other resonant frequencies in the well; and determining characteristics of at least two unknown well features from the other resonant frequencies.

61. A method for characterizing well features, compris-ing the steps of:
determining whether the resonant frequencies of the features are even or odd harmonics; and determining the characteristics of at least two features from whether their resonant frequencies are even or odd harmonics.

62. A method for characterizing at least two well fea-tures, comprising the steps of:

determining the number of the harmonic of a resonant frequency associated with the features; and determining characteristics of the features from the number of the harmonic.

63. A method for determining the distances from a well-head to at least two well features, comprising the steps of:
determining the resonant frequencies present in pressure oscillations in a fluid in the well;
determining wavespeeds in the fluid in the well; and determining the distances from the resonant fre-quencies and wavespeeds.

64. A method for characterizing at least two well features, comprising the steps of:
determining the magnitude of the characteristic impedances of the features; and determining characteristics of the well features from the magnitude of the characteristic impedances.

65. A method for charactizing at least two well features, comprising the steps of:
estimating the hydraulic cross section at the depths of the features or the cross sectional area of the features them-selves from the magnitude of the characteristic impedances of the features; and determining characteristics of the well features from the hydraulic cross section or the cross sectional area.

66. A method of characterizing well features, comprising the steps of:
estimating a wavespeed in a fluid in the well at at least two of the features; and determining the characteristics of the two well features from the wave speed.

67. A method for characterizing well features, comprising the steps of:
measuring the amplitude of pressure oscillations in a fluid in the well; and determining characteristics of at least two well features from the amplitude.

68. A method for characterizing well features, compris-ing the steps of:
creating pressure oscillations in a fluid filling the well; and characterizing at least two well features from the pressure oscillations.

69. The method of Claim 1, further comprising the steps of:
determining a magnitude of a characteristic impedance of the reflector; and determining compressibility of the reflector from the magnitude of its characteristic impedance.

70. The method of Claim 1, wherein the step of creating oscillations includes the step of creating oscillations having a predetermined range of frequencies.

71. The method of Claim 39, wherein the step of creating oscillations includes the step of creating oscillations having a predetermined range of frequencies.