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# Porosity evaluation with acoustic logging

Acoustic logging tools can assist in evaluating porosity because the compressional velocity of sound in fluid is less than the velocity in rock. If there is pore space in the rock, and it is fluid-filled, the acoustic energy will take longer to get from the transmitter to the receiver (i.e., low velocity indicates high porosity). The recorded velocity or travel time represents the sum of the velocity of:

• Solid part or framework of the rock (i.e., the rock matrix)
• Rock lining the pores
• Fluid filling the pore space

In turn, travel time in the rock matrix, Δtma , is influenced by variations in[1]:

• Lithology (i.e., the chemical composition)
• Confining pore pressure (i.e., compaction)

## Calculating porosity

These factors are related through an empirical relationship known as the Wyllie time-average equation.[2] When the velocity (transit time, Δt, or travel time, t) of the rock matrix and borehole fluids are known, porosity can be computed the following ways (Eq. 1 to Eq. 4).

In terms of velocity, v:

....................(1)

where

• ϕ = fractional porosity of the rock
• v = velocity of the formation (ft/sec)
• vf = velocity of interstital fluids (ft/sec)
• vma = velocity of the rock matrix (ft/sec)

In terms of transit time (Δt):

....................(2)

or

....................(3)

where

• Δt = acoustic transit time (μsec/ft)
• Δtf = acoustic transit time of interstitial fluids (μsec/ft)
• Δtma = acoustic transit time of the rock matrix (μsec/ft)

(See Table 1[3]for typical values of Δtma and Δtf.)

In terms of travel time:

....................(4)

The velocity of most borehole and reservoir fluids (except gas) does not vary greatly:

• A fluid velocity (Δtf) of 189 μsec/ft (5,300 ft/sec) is generally assumed for fresh drilling fluids
• A slightly lower value, 185 μsec/ft, is used for salt muds.

Fluid type becomes more of a concern when oil-based mud (OBM) is used if the formation of interest is not invaded or if invasion is very shallow. The lithology must be known or estimated in order to select the appropriate matrix velocity.

The Wyllie equation represents consolidated and compacted formations. In poorly consolidated or unconsolidated rocks, a correction factor is necessary (Eq. 5). Also, the presence of shale or clay within the sand matrix will increase Δ t by an amount proportional to the bulk-volume fraction of the clay. An empirical equation is used for calculating porosity in sandstones in which adjacent shale values (Δtsh) exceed 100 μsec/ft (Eq. 6):

....................(5)

where the compaction correction factor Cp is

....................(6)

where

• Δtsh = specific acoustic transit time in adjacent shales (μsec/ft),
• 100 = acoustic transit time in compacted shales (μsec/ft)

The shale compaction coefficient (C) generally ranges from 1.0 to 1.3, depending on the regional geology.

The highest velocities observed in sandstones approach 20,000 ft/sec (50 μsec/ft), but most sandstones have a lower matrix velocity. Velocities in adjacent shales are used to adjust the matrix velocity for sands with velocities lower than 18,000 ft/sec. Table 2 provides guidelines for selecting the appropriate value of Δtma. If the lithology of carbonate rocks can be reasonably estimated and if the porosity distribution is fairly uniform, the Wyllie time-average formula can provide reliable determination of porosity for this group.

A second empirical velocity/porosity relationship, the Raymer-Hunt-Gardner equation,[4] was introduced to correct for observed anomalies and shortcomings of the Wyllie time-average formula (Eq. 7). It provides improved porosity correlation over the entire porosity range and is applicable to both consolidated and some unconsolidated formations, thus eliminating the need for a compaction correction. However, in high-porosity, unconsolidated, and uncemented (slow) rocks, neither of the empirical velocity/porosity transforms may be adequate[5]

....................(7)

where α = (Δtma/2Δtf) − 1.

Graphical solutions for both algorithms for a sandstone matrix are compared in Fig. 1. One caveat regarding the use of empirically derived porosity transforms: they do not account for all the factors influencing acoustic velocity. Consequently, these relationships may not be valid for all reservoirs.

In fast formations, the shear velocity can also be used for porosity evaluation in a manner similar to that described above for compressional velocity.[6] Further, the combination of compressional and shear slowness can provide an enhanced porosity determination.[7] Using shear velocity for porosity determination offers several distinct advantages because shear velocity is:

• Generally more sensitive to porosity than compressional velocity
• Insensitive to gas effects
• Less affected by borehole washout
• Able to be used to replace nuclear porosity in some situations[8]
• Manageable because porosity evaluation can be conducted in cased hole using dipole tools (or in some cases, monopole-array tools using special processing)[9][10]

These velocity/porosity methods are for clean (shale free), water-filled formations. The calculated apparent porosity must still be corrected for the volume of pore-filling material (shale). If the formation contains shale or dispersed-clay particles, or is hydrocarbon bearing and invaded to only a very shallow depth, corrections to the basic log data are necessary before reasonable porosity values can be calculated.

Because shale transit times range from 62 to 167 μsec/ft, failure to correct for the presence of shale may result in overly optimistic porosity calculations. The acoustic measurement is also influenced by the way the shale is distributed within the sandstone reservoirs. The fraction of shale, or shale volume, can be estimated using a combination of log measurements that are influenced by shale, such as:

• Neutron porosity
• Density
• Gamma ray
• Spontaneous potential (SP)

Chartbook nomograms developed for porosity determination include graphical solutions for both undercompaction and shale volume.

In some producing regions, producibility indexes based on the volume of shale in producing sandstone reservoirs have been developed. The fraction of total porosity occupied by dispersed clay (q factor) is empirically related to effective and total porosity and production characteristics. Local experience is used to create permeability cutoffs using the q factor (Fig. 2).[11]

Acoustic travel time in gas and oil is higher than in water. The presence of unflushed hydrocarbons in an interval can result in high values of apparent formation porosity. Commonly used correction factors are 0.9 in oil zones and 0.7 in gas zones.[1] More recently, a gas-zone porosity-correction technique using shear slowness has been developed.[12]

An additional empirical velocity-porosity predictive model has recently been proposed and is still in the experimental phase.[13][14]

Carbonate and complex lithology reservoirs are generally comprised of varying proportions of:

• Limestone
• Dolomite
• Chert
• Quartzite
• Occasionally, evaporites

The primary influences on porosity in these rocks are lithology and pore type. Generally, any shale present is in dispersed form and in small amounts that do not significantly impact porosity calculations. Acoustic porosity is a measure of the primary or intergranular (matrix) porosity. In dual-porosity reservoirs, the secondary porosity (e.g., isolated pores, vugs, and fractures) may significantly influence the rock-pore distribution, but may be overlooked by acoustic-log measurements. This topic is the subject of ongoing research.[15] In contrast, nuclear-porosity devices, such as density and neutrons, measure total porosity. The difference between the nuclear-porosity and acoustic-porosity measurements is an approximation of the secondary porosity.

## References

1. Tixier, M.P., Alger, R.P., and Doh, C.A. 1959. Sonic Logging. J Pet Technol Trans., AIME, 216: 106.
2. Wyllie, M.R.J., Gregory, A.R., and Gardner, L.W. 1956. Elastic Wave Velocities in Heterogeneous and Porous Media. Geophysics 21 (1): 41–70. http://dx.doi.org/10.1190/1.1438217
3. Carmichael, R.S. ed. 1982. Handbook of Physical Properties of Rocks, Vol. 2, 1-228. Boca Raton, Florida: CRC Press Inc.
4. Raymer, L.L., Hunt, E.R., and Gardner, J.S. 1980. An Improved Sonic Transit Time-to-Porosity Transform, paper P. Trans., 1980 Annual Logging Symposium, SPWLA, 1–12.
5. Dvorkin, J. and Nur, A. 1998. Time-Average Equation Revisited. Geophysics 63 (2): 460–464. http://dx.doi.org/10.1190/1.1444347
6. Medlin, W.L. and Alhllall, K.A. 1992. Shear-Wave Porosity Logging in Sands. SPE Form Eval 7 (1): 106-112. SPE-20558-PA. http://dx.doi.org/10.2118/20558-PA
7. Krief, M. et al. 1990. A New Petrophysical Interpretation Using the Velocities of P and S Waves (Full Waveform Sonic). The Log Analyst 31 (6): 355–369.
8. Spears, R., and Nicosia, W.: "Porosity Estimation from Shear Wave Interval Transit Time in the Norphlet Aeolian Jurassic Sandstone of Southern and Offshore Alabama," paper N, Trans., 2003 Annual Logging Symposium, SPWLA, 1–12.
9. Valero, H.-P., Skelton, O., and Almeida, C.M. 2003. Processing of Monopole Sonic Waveforms Through Cased Hole, paper BH 1.1. Expanded Abstracts, 2003 Annual Meeting Technical Program, SEG, 285–288.
10. Tello, L.N. et al. 2004. Through-Tubing Hostile-Environment Acoustic Logging Tool, paper Y. Trans., 2004 Annual Logging Symposium, SPWLA, 1–11.
11. Fertl, W.H. 1981. Openhole Crossplot Concepts A Powerful Technique in Well Log Analysis. J Pet Technol 33 (3): 535-549. SPE-8115-PA. http://dx.doi.org/10.2118/8115-PA.
12. Smith, J., Fisburn, T., and Bigelow, E. 1996. Acoustic Porosity Corrections for Gas and Light Hydrocarbon-Bearing Sandstones, paper AAA. Trans., 1996 Annual Logging Symposium, SPWLA, 1–9.
13. Knackstedt, M.A., Arns, C.H., and Pinczewski, W.V. 2003. Velocity-Porosity Relationships, Part 1: Accurate Velocity Model for Clean Consolidated Sandstones. Geophysics 68 (6): 1822–1834. http://dx.doi.org/10.1190/1.1635035.
14. Knackstedt, M.A., Arns, C.H., and Pincsewski, W.V. 2005. Velocity-Porosity Relationship: Predictive Velocity Model for Cemented Sands Composed of Multiplemineral Phases. Geophysical Prospecting 53 (3): 349–372.
15. Kazatchenko, E., Markov, M., and Mousatov, A. 2003. Determination of Primary and Secondary Porosity in Carbonate Formations Using Acoustic Data. Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, 5-8 October 2003. SPE-84209-MS. http://dx.doi.org/10.2118/84209-MS

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