Also available in PDF format:
Measurement of Velocities

Introduction:

It is difficult to sample on the scale of a reservoir and it is often both cost prohibitive and impossible to recover core for the purposes of reservoir characterization. The petroleum engineer must therefore seek indirect measurements of petrophysical properties. This is nothing new. We measure resistivity to evaluate porosity, permeability and water saturation for example. These measurements are non-unique and thus we seek multiple independent measurements, using different techniques to reduce the non-uniqueness. The measurement of the speed of sound through a rock yields an index called velocity. This velocity depends upon both the elastic modulus and density of the rock.   If the rock is isotropic, homogeneous and linearly elastic then there are only two possible types of waves which can travel through the rock: 1) a compressional, acoustic or sonic wave and 2) a shear wave. Modern logging tools measure both compressional, Vp, and shear wave, Vs, velocities in the borehole which are used to calculate Young’s modulus and Poisson’s ratio required in hydraulic fracture design, to detect hydrocarbons and to estimate formation porosity. Seismic exploration uses both compressional and shear waves to define and map reservoir boundaries, detect hydrocarbons in place and monitor changes during production (4D-seismic reservoir management).  These technologies have the attractive feature of mapping petrophysical and fluid variations on the scale of the reservoir. Elastic properties, moduli and velocities are also used to model reservoir subsidence and assess borehole stability and sanding potential.

Velocity Measurement:

The actual measurement of velocities is quite simple although details of the techniques are slightly different for hard and soft rock. One imparts an impulsive stress at one end of the sample and measures the time it takes to travel the length of the sample. With very high frequency signals and the absence of noise the process is actual quite simple. To calculate velocities one then divides the sample length (mm) by the travel time (msec) to arrive at a velocity in mm/msec or km/sec. Complications arise when high frequencies are attenuated and the signal arrives in the presence of noise or other earlier signals.  The type of imparted stress dictates the type of wave which propagates. There are two waves we are interested in measuring, a compressional and shear wave. The speed of the compressional or P-wave is about twice that of shear. The ramifications of this are that the shear wave arrives amidst reverberations of the earlier P-wave making its detection more difficult.

A practical implementation of an ultrasonic technique used to measure velocities is presented below.

Figure 1: A schematic of the Pulse Transmission configuration. One transducer acts as a source and at the opposite end, a second acts as a receiver. In this arrangement, there are spacers between the transducers and the samples. These spaces are referred to as “end caps” and for hard rock measurements are made from titanium.

Pulse transmission is a simple technique for measuring the velocity of waves through rocks, fluids and other materials. The transducers on the left in Figure 1 act as transmitters. When a voltage is applied to the piezoelectric crystals, they expand creating a mechanical or acoustic wave. How one excites the transducers often impacts signal quality. Accepted methods of applying the stimulus is to impart a pseudo-delta function, a spike form a capacitor discharge for example, a gated square wave or a gated single cycle sine wave. The gated sine wave produces the cleanest and best signals for analysis. We actually sweep frequencies above and below resonance to increase the bandwidth somewhat.

The piezoelectric crystals used in our system have a natural resonance frequency of 1MHz for P and 500 KHz for S. However, once these crystals are bonded to the end caps their resonance is reduced considerably; typically to 500 KHz for P and about 250-300 KHz for S.  The shear wave transducers are also polarized meaning that the sense of shear particle motion is controlled and oriented. Two orthogonally polarized shear transducers are bonded in both transmitter and receiver assemblies to titanium end caps. Using a multiplexer, we can activate each of the P, S1 and S2 crystals independently. S1 and S2 refer to the orthogonally polarized shear. A sequence of measurements at each pressure records the following transmitter-receiver pairs: P-P, S1-S1, S2-S2, S1-S2 and S2-S1. If a material is isotropic then no signals should be detected on either the S1-S2 or S2-S1 configurations. Presence of signal is a direct indicator of elastic anisotropy. Waves propagate the distance of the sample and impinge on the piezoelectric crystals at the opposite end of the sample. The distortion causes the active crystal to output a voltage which is amplified and digitized or recorded on the oscilloscope and stored in the computer.  If we start the recording precisely when the voltage is applied to the transmitted crystal and then measure the time, Dt, required to traverse a sample of length, L, we can calculate the associated velocity:

This velocity corresponds to either a compressional wave if we excite the P-wave transducer or a shear wave if we excite the S-wave transducer. We note from Figure 1 that part of the path from transmitter to receiver is through the end caps. Since we are only interested in the speed through the sample, the transit times through the end caps must be subtracted from the total travel time to arrive at the travel time through the sample, Dt:

The end cap delays are called “transducer delays”. These can be obtained by simply putting the transmitted and receiver end caps in contact and measuring the delay times. End cap delays are determined from the intercept of a best fit line to travel times versus sample length. Precisely machined aluminum calibration samples are used in the calibration. Each end cap has an integral integrated circuit which posses a unique serial number. A file containing the serial numbers and delays is read when the acquisition system reads the serial number before beginning velocity measurements. Thus there is no operator intervention in assigning proper end cap delays.

Making measurements at pressure:

Velocity measurements can be made as a function of temperature, confining pressure and pore pressure. They are also made at various saturation states. To facilitate this, a sample is typically mounted in an impermeable jacket (see Figure 2) and inserted in a pressure vessel and connected to an external pore pressure reservoir. Computers are used to control the pressures independently.

Figure 2: Transducer-sample assembly for insertion into a pressure vessel. Impermeable jacket prohibits the confining fluid from entering the sample. Pore pressure ports through the end caps allow external control of pore fluid pressure.

Our typical measurement suite occupies 9 pressure points while increasing confining pressure to a maximum of 10,000 psi. If a saturated measurement is desired, the sample is evacuated and saturated with an appropriate brine for 24 hours at 1000 psi prior to insertion. Once inside the jacketed assembly, pore pressure is increased after the initial application of confining pressure and the sample is soaked at several hundred psi to guarantee saturation. Pore pressure, Pp, is then reduced to zero and the velocities are measured at the prescribed confining pressures, Pc. Given normal pressure gradients, samples are generally run to effective pressures appropriate for their depth of burial. Effective pressure, Pe, is: Pe = Pc – nPp. Without additional knowledge n is assumed to be 1.

Error Analysis:

Velocities are described by the following formula

The error in a velocity then has components in the sample geometry and our ability to measure it accurately as well as the travel time through the sample. Remember that the measurement of Dt for S is hampered by 1) lower frequency signals, 2) P-wave energy, and 3) transducer ringing which makes the arrival totally uncertain.

The variance, s², in V becomes:

The standard error in the presence of random errors and independence of Dt and L is:

The error depends directly upon the magnitude of the velocity and inversely with sample length and travel time. This suggests that there is a minimal sample length required and this length should be increased with faster samples. This makes it difficult to work on sidewall cores, especially in carbonates. Furthermore, there is no rule of thumb of 1% for Vp and 2% for Vs as often published in the literature. Typical errors in length are ±0.001” (0.0254 mm) for properly prepared samples and the error in Dt for high quality signals is around  ± 0.080 msec. Even if these are constants, the error in velocity will depend on sample length, velocity magnitude and travel time and thus will not and can not be the same for all samples!

Using these values, a standard sample length of 1” (25.4 mm) and assuming measured velocities to be 10,000 ft/sec (3.048 mm/msec) and 20,000 ft/sec (6.096 mm/msec) the minimum velocity errors are: 1% and 2% , respectively. Halving the sample length doubles these errors!  On the other hand making the samples too long introduces new modes and leads to more multiple inference with the shear arrivals.

Soft Rock Considerations:

For hard rock, the end cap material is titanium or sometimes aluminum. The acoustic impedance of these materials is not terribly different than that of harder rocks so the signal insertion losses are not a problem. The signal insertion is the amount of energy coupled from the piezoelectric crystal into the sample. However, when dealing with soft, unconsolidated rock, the acoustic impedance contrast is quite large making it difficult to couple energy into the sample. To minimize this energy loss we use completely different endcap materials which match more closely the acoustic impedance of the material but reduce the maximum testing pressures.

A second consideration is the larger change in sample dimensions when the material is soft. We instrument the assembly with an LVDT to measure the length changes directly for soft sediments.

Coupling energy into the system is an art which is best achieved when the two surfaces are polished flat and smooth so that a lapped bond can be made. The coupling is a direct reflection of the sample preparation. This is in part limited by grain size and grain pull out. P-wave coupling is generally much more efficient than shear wave coupling and can be substantially improved with a thin film of soap, a couplant, between the transducer assembly and the sample. Shear requires a more viscous couplant and is more sensitive to surface irregularities than the P-wave. Often thermally softened polymers such as polystyrene serve as an efficient shear-wave couplant. Again the art is in making these improvements second order effects in the actual velocity measurement. Note that this coupling issue and its repeatability cause first order problems in attempts to make attenuation measurements.

Linear Elasticity:

If a rock is isotropic and linearly elastic then one can relate the velocities and moduli through the following relationships:

Knowing any two moduli or in this case knowing Vp and Vs and density allows us to calculate any of the other moduli. In these equations, K is the bulk modulus, G is the shear modulus and r is the bulk density.  Since Vp depends upon the sum of K and 4/3 G and Vs only depends upon G, the magnitude and hence speed of Vp must be greater than Vs.   We can rewrite these equations to show that:

The bulk modulus is used to evaluate reservoir drive and compaction problems. The shear modulus is very useful in evaluating sanding problems and borehole collapse potential. In designing hydraulic fractures, knowledge of E, Young’s modulus and n, Poisson’s ratio are required. By running a “sonic” logging tool we can measure Vp and Vs. We could combine these with density to calculate E and n as follows:

Note that the formula for E is a bit complicated and that one generally calculates K and G, the bulk and shear moduli first.

We already know that bulk density, r, has an explicit porosity dependence:

This would suggest that velocities, Vp and Vs, have both a porosity and fluid sensitivity.  In addition, velocities also depend on moduli and moduli depend upon both magnitude and shape of the pores. This combined dependence has been exploited to great advantage in using velocities to predict porosity, fluid saturation and mineralogy. Velocities, principally through moduli are very sensitive to matrix properties, If clays are grain supporting, the resultant moduli are decreased proportionally.

Send mail to csondergeld@ou.edu with questions or comments about this web site.
Last modified: 08/14/04