Once the primary cementing of a well has been completed, it is necessary to verify the quality of the cement bond between the casing and the formation. Determination of the cement thickness is performed to ensure the isolation of different zones.
This article discusses the theories used in cement thickness determination, the development of a new thickness algorithm, the implementation of the algorithm into a computer program and a discussion of the results for two wells.
CEMENT THICKNESS AND WAVE THEORY
In the acoustics field, the cement bond log (CBL) has been the principle diagnostic tool of cementing conditions. This article examines a new technique to overcome the limitations, expense, and lack of precise results involved with the known methods describing cement thickness. The technique is based on analysis of openhole acoustic data and casedhole CBL results. The advantage of the proposed method is that it utilizes existing services, which are normally run as part of a well program.
SEISMIC WAVE PROPAGATION
The wave equation is the basis for propagation of a disturbance in an elastic medium. There are four physical equations that are the basic relationships to derive the wave equation. They consist of continuity, state, motion and force. Their combination yields the wave equation for a perfect nonviscous fluid. Wave theory and ray theory are both used to solve the wave equation.
Interpretation of seismic reflection and refraction utilizes ray theory and the travel times along ray paths. The transmission of seismic energy through solids includes the shear wave, as well as the compression wave propagation. Boundary waves are also included, although they are rapidly attenuated. The wave velocities in any given medium decrease in the following order: compression wave, shear wave and boundary waves.
Compression waves are a particular type of longitudinal wave whose direction of propagation is parallel to the direction of particle displacement. The direction of propagation is away from the source. Gases, liquids and solids have a tendency to oppose compression; therefore, compression waves can be propagated through them. This article deals primarily with the compression wave.
The travel time for reflected waves is determined using the image source solution. A critical angle of refraction solution is used to determine the refracted wave travel time.
CEMENT THICKNESS DETERMINATION
The previously discussed theory for seismic reflection and refraction are applied to the ray path analysis for a multilayer system in a cased oil well. Thickness determinations have been applied by seismologists and geophysicists to determine the thicknesses of horizontal layers by refraction and reflection of lowfrequency waves. In this article, the same seismic principles are applied on a smaller scale, in a vertical multilayer system, to determine an unknown factor—namely, the cement thickness, in the annulus, between the casing and the formation wall.
Cement bond log. The CBL is based on the amplitude attenuation of the first arrival of an acoustic signal caused by the energy transfer from the pipe and cement to the formation. The wave train (seismogram) is the summation of all the waves arriving at the receiver from different paths. The attenuation of the casing signal is considered to be the indication of a bonding condition between the casing and the formation. The casing signal is caused by the casing ray of interest comprising the mud path, casing segment traversed by the extensional wave in steel, and the return mud path. Laboratory tests have determined that the attenuation of acoustic waves in bonded pipe is directly proportional to the percentage of the circumference around the pipe having a good bond.
Openhole sonic log. This service is run as a formation evaluation survey, which directly yields the interval transit time in microseconds per foot. This is done in the open hole with a fluid and formation interface. The openhole measurement provides the transit time for the ray path analysis used to obtain the cement thickness.
Openhole caliper. An openhole caliper is run to provide a measurement of the borehole diameter. The “effective caliper” is used to describe the caliper reading after the casing has been landed. It provides the space between the casing and the borehole wall. An assumption is made that the casing has been landed with centralizers and is roughly centered in the borehole.
Interpretation. The combination of the openhole data and the casedhole data permit the cement thickness measurement to be made in a bonded condition. The interpretation consists of the analysis of the complete wave train; the amplitude attenuation of the first arrival to determine bonding; the openhole log to determine formation transit time; and picking the correct bond path arrival on the wave train.
Sample charts simulating data for different arrival times may be constructed using the computer program developed to implement this methodology. These charts aid in the interpretation of the data signature. Knowledge of the transit times involved will give the interpreter a good idea of the arrival times for each wave component.
For example, with a bond tool having a 5ft spacer, the transit times are:
Casing = 57 µsec
Mud = 166 µsec
Cement = 83 µsec
The openhole sonic shows a formation having an interval transit time of 50 µsec. The following arrival times may be approximated:
Casing = 5 × 57 = 285 µsec
Mud = 5 × 166 = 830 µsec
Formation = 5 × 50 = 250 µsec
This would mean that the arrival of the formation signal could be ahead of casing time. When performing the interpretation, the algorithm developed herein selects a reflection or a refraction, based on the openhole comparison.
The cement transit time is considered to be 83.3µsec for various types of cement—after allowing a minimum curing time of 48 hr to achieve sufficient compressive strength. If the transit time of the formation is greater than that of the cement, the arrival seen on the wave train is from a reflection at the cement and formation boundary. Refraction is not allowable, whenever the velocity of sound in layer n is greater than the velocity in the adjacent n+1 layer.
Transit times are used in the analysis, since velocities are not common in logging operations. Based on the data simulation used to create the cement thickness charts, cement reflection waves are expected to interfere with the attenuated pipe waves after 420 microseconds (5 × 83.3) for 5ft spacing. Should the pipe waves be completely attenuated, then the first arrival seen is that of the cement wave.
Several factors must be accounted for in the determination of the cement thickness, for either refracted or reflected waves:
 Mud delay. The acoustic signal is delayed by the travel time equation, dependent on twice the mud space, the critical angle and the mud transit time.
 Casing delay. The acoustic signal is delayed by the travel time equation, dependent on twice the casing thickness, the critical angle and the casing transit time.
 Time differential. The amount of time in microseconds, due to the delay caused by the mud, casing, and the formation, is subtracted from the seismogram time. This operation leaves the amount of time that the signal is delayed by the bonded cement.
 Cement thickness. The minimum amount of cement in the annulus between the casing and the formation wall is dependent on the time differential, the critical angle and the transit time.
CEMENTTHICKNESS EQUATIONS
Refracted waves. During an acoustic event, waves will propagate in all directions, in a spherical manner. The ray that causes a wave to propagate down the borehole wall at a 90° angle is called the ray of interest; the angle causing the vertical displacement from the previous media is called the critical angle. In a cased well, the critical angle is a function of the cement transit time and the formation transit time. The cement transit time is a known factor; the formation transit time is previously computed by the openhole sonic log.
Several additional factors must be accounted for in the determination of the cement thickness with refracted waves:
 Effective formation spacing. The length of the formation sampled by the ray of interest is dependent on the detector spacing, tool diameter, bit size and critical angle.
 Formation delay. The acoustic signal is delayed by the formation and is dependent on the effective formation spacing and the openhole formation transit time.
 CBL seismogram time. The total time that the signal is delayed in a good bonding condition is identified on the seismogram wave train as the arrival of the first compressional wave from the acoustic path of interest. (Seismogram data are also known as the Variable Density Log (VDL)).
Cement thickness can be determined by refraction and formation signal, when the formation transit time is less than the cement transit time. This would correspond to hard rock country. When the transit times are equal, it is not possible to make a calculation. The formation characteristics, in this condition, match those of cement.
Reflected waves. When the formation transit time is greater than the cement transit time, it is not possible to obtain a refracted wave at a 90° angle along the cased borehole wall. In this case, the first arrival of interest will be that of a reflected wave at the cement/formation interface. (This may be the reason that formation time is not normally seen in a cased hole logged in South Louisiana.)
Several additional factors must be accounted for in the determination of the cement thickness with reflected waves:
 Angle of reflection. The angle of reflection can be approximated as a function of the seismogram time (VDL). The seismogram time is the first of the compressive reflected waves in the cement. A formation signal will not be seen under these circumstances and the first arrival has a strong attenuation.
 CBL seismogram time (VDL). The total time that the signal is delayed in a good bonding condition is identified on the wave train as the first compressional wave arrival of the cement reflected wave.
The reflection analysis must be used in formations where the formation transit time is greater than the cement transit time. This would, generally, correspond to soft and unconsolidated formations. When the formation transit time is equal to the cement transit time, it is not possible to make the cement thickness calculation, since a reflection angle in a cased hole is not allowed where the cement and formation interface, theoretically, does not exist. In practice, however, there is a reflected signal caused by the roughness of the borehole wall.
A true mud angle algorithm was implemented to compute a cement thickness measurement, based on a mud/casing interface angle of incidence required to cause either a refraction or a reflection angle in the cement.
LOGGING EQUIPMENT AND ANALYSIS
The cement bond log system is used to present an amplitude curve to determine if there is a bond to the casing; a transit time curve, which serves as a travel time indicator to determine the efficiency of the bond; and the VDL for detailed interpretation of the cement condition. A computer program has been developed to implement the methodology described above. In the program, borehole segments are defined as lengths of pipe that have the same geometric parameters, well parameters and formation parameters.
The geometry of each borehole segment is defined by the following parameters:
 Segment ID of previous segment
 Length of segment
 Azimuth
 Inclination.
Note that each borehole segment also has the following well/formation characteristics:
 Mud transit time (µsec)
 Casing transit time (µsec)
 Cement transit time (µsec)
 Casing OD (in.)
 Casing ID (in.)
 Bit size (in.).
The well is modeled as an inverted tree, with the wellhead at the origin of a righthanded Cartesian coordinate system (the positive xaxis is due east and the positive zaxis is up). Each segment is modeled as a vector with a tail point, length, inclination, azimuth and head point. The well data, plus borehole segment geometry data, are entered in the application for each well’s borehole segment. The log data, consisting of path parameters and the actual sonic and VDL log data, are contained in a separately loaded text file.
RESULTS
Measurements were taken on three cement bond sections in two wells. All the GR CBL VDL logs were run after the 48hr curing period required for the cement to achieve a good compressive strength. The logs used were the caliper, ISF/Sonic, and CBL VDL. Well characteristics are presented in Table 1.
Table 1. Well characteristics from the two test wells. 

In reviewing the results, note that the sonic data are a function of the formation; they are not affected by the cement thickness. Conversely, the VDL response is a function of the cement thickness.
For Well A, plots of cement thickness and sonic data versus depth are presented in Figs. 1 and 2. The cause of the large VDL spikes in Fig. 2 is unknown; however, they are most likely the result of noise in the signal.

Fig. 1. Plots of cement thickness and sonic data versus depth from Well A. 


Fig. 2. Plots of cement thickness and VDL versus depth from Well A. 

For Well B, only the ISF/Sonic and the CBL were available. Since there were no caliper data available, the results were compared to the bit size used to drill the hole. Plots of cement thickness and sonic data versus depth are presented in Fig. 3 and Fig. 4 for the 10,300ft zone and Fig. 5 and Fig. 6 for the 10,400ft zone.

Fig. 3. Plots of cement thickness and sonic data versus depth from Well B’s 10,300ft zone. 


Fig. 4. Plots of cement thickness and VDL versus depth from Well B’s 10,300ft zone. 


Fig. 5. Plots of cement thickness and sonic data versus depth from Well B’s 10,400ft zone. 


Fig. 6. Plots of cement thickness and VDL versus depth from Well B’s 10,400ft zone. 

Measurement uncertainty. One of the most important quantities involved is cured cement transit time, given to be 83.3 µs/ft or 6.9 µs/in. A variation of 1 µs/in. in the cement transit time would be considered reasonable. This yields a 0.140in. uncertainty per inch of cement. The value of cement transit time does change slightly with types of cement and curing time, resulting in a 0.3in. uncertainty per inch of cement. The fact that the cement transit time is not exactly known could partially account for the difference in measurement of cement thickness by the cement bond log, as compared to the openhole caliper.
CONCLUSION
An algorithm using ray and wave theory has been developed to perform an acoustical determination of the cement sheath thickness in cased oil wells. The algorithm implements a different approach that also takes into account reflection analysis, due to the particular geological configuration affecting cased well measurements.
Cement thickness determination is a decisionmaking tool, when it comes to deciding on performing remedial jobs in new and/or mature wells, saving the operator both time and money, in addition to minimizing risks and production problems.
