ELASTIC CONSTANTS BASICS
This page discusses how well logs are used to determine the mechanical properties of rocks. These properties are often called the elastic properties or elastic constants of rocks. The best known elastic constants are the bulk modulus of compressibility, Young's Modulus (elastic modulus), and Poisson's Ratio. The dynamic elastic constants can be derived with appropriate equations, using sonic log compressional and shear travel time along with  density log data.

Dynamic elastic constants can also be determined in the laboratory using high frequency acoustic pulses on core samples. Static elastic constants are derived in the laboratory from tri-axial stress strain measurements (non-destructive) or the chevron notch test (destructive).

Elastic constants are needed by five distinct disciplines in the petroleum industry:
1. geophysicists interested in using logs to improve synthetic seismograms, seismic models, and interpretation of seismic attributes, seismic inversion, and processed seismic sections.
2. production or completion engineers who want to determine if sanding or fines migration might be possible, requiring special completion operations, such as gravel packs
3. hydraulic fracture design engineers, who need to know rock strength and pressure environments to optimize fracture treatments
4. geologists and engineers interested in in-situ stress regimes in naturally fractured reservoirs
5. drilling engineers who wish to prevent accidentally fracturing a reservoir with too high a mud weight, or who wish to predict overpressured formations to reduce the risk of a blowout.

The elastic constants of rocks are defined by the Wood-Biot-Gassmann Equations. The equations can be transformed to derive rock properties from log data. If crossed dipole sonic data is available, anisotropic stress can be noticed by differences in the X and Y axis displays of both the compressional and shear travel times. When this occurs, all the elastic constants can be computed for both the minimum and maximum stress directions. This requires the original log to be correctly oriented with directional information, and may require extra processing in the service company computer center.

Elasticity is a property of matter, which causes it to resist deformation in volume or shape. Hooke's Law, describing the behavior of elastic materials, states that within elastic limits, the resulting strain is proportional to the applied stress. Stress is the external force (pressure) applied per unit area, and strain is the fractional distortion which results because of the acting force. The modulus of elasticity is the ratio of stress to strain.

Three types of deformation can result, depending upon the mode of the acting force. The three elastic moduli are:

Young's Modulus,
1: Y = (F/A) / (dL/L)

Bulk Modulus,
2: Kc = (F/A) / (dV/V)

Shear Modulus,
3: N = (F/A) / tanX

Where F/A is the force per unit area and dL/L, dV/V, and tanX are the fractional strains of length, volume, and shape, respectively.

Another important elastic constant, called Poisson's Ratio, is defined as the ratio of strain in a perpendicular direction to the strain in the direction of extensional force,
4: PR = (dX/X) / (dY/Y)

Where X and Y are the original dimensions, and dX and dY are the changes in x and y directions respectively, as the deforming stress acts in y direction.

Elastic properties are measured in the laboratory using triaxial stress tests (static measurements) and by measuring bulk density and acoustic travel time with a high frequency impulse (dynamic testing). Both arre done under representative overburden pressure.

The general procedures for triaxial compressive test are:
1. A right cylindrical plug is cut from the sample core and their ends ground parallel according to International Society for Rock Mechanics (ISRM) and American Society for Testing and Materials (ASTM) standards.  A length to diameter ratio of 2:1 is recommended to obtain representative mechanical properties of the sample, which is also recommended by ASTM and ISRM.  Physical dimensions and weight of the specimen are recorded and the specimen is saturated with simulated formation brine.
2. The specimen is then placed between two plate and a heat-shrink jacket is placed over the specimen.
3. Axial strain and radial strain devices are mounted in the endcaps and on the lateral surface of the specimen, respectively.
4. The specimen assembly is placed into the pressure vessel and the pressure vessel is filled with hydraulic oil.
5. Confining pressure is increased to the desired hydrostatic testing pressure.
6. Measure ultrasonic velocities at the hydrostatic confining pressure.
7. Specimen assembly is brought into the contact with a loading piston that allows application of axial load.
8. Increase axial load at a constant rate until the specimen fails or axial strain reaches a desired amount of strain while confining pressure is held constant.
9. Reduce axial stress to the initial hydrostatic condition after sample fails or reaches a desired axial strain.
10. Reduce confining pressure to zero and disassemble sample.

 Depth (m) Confining Pressure (psi) Compressive Strength (psi) Static Young's Modulus (x106 psi) Static Poisson's Ratio XX51.50 3850 63359 8.70 0.40 XX61.15 3850 56831 5.75 0.36 XX71.15 3850 56026 5.79 0.34 XX05.20 3850 50910 5.08 0.39

Static elastic properties measured with triaxial stress test

 Depth Bulk Ultrasonic Wave Velocity Dynamic Elastic Parameter m Density g/cc Compressional ft/sec  usec/ft Shear ft/sec usec/ft Young's Modulus (x106 psi) Poisson's Ratio Bulk Modulus (x106 psi) Shear Modulus (x106 psi) XX51.50 2.81 20161 49.60 10760 92.94 11.39 0.30 9.53 4.38 XX61.15 2.57 15829 63.18 9555 104.66 7.68 0.21 4.46 3.16 XX71.15 2.66 17226 58.05 10299 97.10 9.30 0.22 5.57 3.81 XX05.20 2.64 16451 60.79 9763 102.43 8.31 0.23 5.10 3.38

Dynamic elastic properties measured with ultrasonic impulse in the lab. Note differences between static and dynamic values. Elastic properties from log analysis models match lab dynamic data better than static data.

Dynamic elastic properties calculated from density and sonic log data, showing close match to dynamic data from lab measurements (coloured dots). Lab data is from table shown above. Note synthetic sonic and density plotted next to measured log curves (Tracks 2 and 3), showing reasonably small differences due to minor borehole effects. Synthetic curves can repair worse logs or even replace missing curves.

ELASTIC CONSTANTS THEORY
The velocity of sound in a rock is related to the elastic properties of the rock/fluid mixture and its density, according to the Wood, Biot, and Gassmann equations.

The composite compressional bulk modulus of fluid in the pores (inverse of fluid compressibility) is:   ____1:   Kf = 1/Cf = Sw / Cwtr + (1 - Sw) / Coil
_OR 1a: Kf = 1/Cf = Sw / Cwtr + (1 - Sw) / Cgas

The pore space bulk modulus (Kp) is derived from the porosity, fluid, and matrix rock properties:
2: ALPHA = 1 - Kb / Km
3: Kp = ALPHA^2 / ((ALPHA - PHIt) / PHIt / Kf )

The composite rock/fluid compressional bulk modulus is:
4: Kc = Kp + Kb + 4/3 * N

Compressional and shear velocity (or travel time) depend on density and on the elastic properties, so we need a density value that reflects the actual composition of the rock fluid mixture:
5:  DENS = (1 - Vsh) * (PHIe * Sw * DENSW + PHIe * (1 - Sw) * DENSHY + (1 - PHIe) * DENSMA)
+ Vsh * DENSSH

Compressional velocity (Vp) and shear velocity (Vs) are defined as:
6: Vp = KS4 * (Kc / DENS) ^ 0.5
7: Vs = KS4 * (N / DENS) ^ 0.5

Although it is not a precise solution, we often invert equations 5 and 6 to solve for Kb and N from sonic log compressional and shear travel time values.

WHERE:
ALPHA = Biot's elastic parameter (fractional)
Cgas = gas compressibility
Coil = oil compressibility
Cwtr = water compressibility
DENS = rock density (Kg/m3 or g/cc)
DENSW = density of fluid in the pores (Kg/m3 or g/cc)
Kb = compressional bulk modulus of empty rock frame
Kc = compressional bulk modulus of porous rock
Kf = compressional bulk modulus of fluid in the pores
Km = compressional bulk modulus of rock grains
Kp = compressional bulk modulus of pore space
N = shear modulus of empty rock frame
PHIt = total porosity of the rock (fractional)
Sw = water saturtation (fractional)
Vp = compressional wave velocity (m/sec or ft/sec)
Vs = shear wave velocity (m/sec or ft/sec)
Vp = Stoneley wave velocity (m/sec or ft/sec)
KS4 = 68.4 for English units
KS4 = 1.00 for Metric units

The Biot-Gassmann approach looks deceptively simple. However, the major drawback to this approach is the difficulty in determining the bulk moduli, particularly those of the empty rock frame (Kb and N), which cannot be derived from log data. Murphy (1991) provided equations for sandstone rocks (PHIe < 0.35) that predict Kb and N from porosity:
8: Kb = 38.18 * (1 - 3.39 * PHIe + 1.95 * PHIe^2)
9: N   = 42.65 * (1 - 3.48 * PHIe + 2.19 * PHIe^2)

 RECOMMENDED PARAMETERS: Water Salinity Cf psi-1 Kf psi Cf GPa-1 Kf GPa 5000 0.0000040 250000 0.580 1.723 35000 0.0000039 270270 0.537 1.862 200000 0.0000027 344828 0.420 2.376 Oil Depth 2000 ft 610 m 0.0000085 117647 1.233 0.811 4000 ft 1220 m 0.0000095 105263 1.378 0.725 8000 ft 2440 m 0.0000116 86207 1.683 0.594 12000 ft 3660 m 0.0000135 74074 1.959 0.510 Gas Depth 2000 ft 610 m 0.001250 800 181.422 0.006 4000 ft 1220 m 0.000510 1961 74.020 0.014 8000 ft 2440 m 0.000180 5556 26.124 0.038 12000 ft 3660 m 0.000100 10000 14.513 0.069

INITIAL CONSIDERATIONS
In the following equations, DENS, DTC, and DTS are measured log values. DENSMA, DTCMA, and DTSMA are rock matrix values which can be assumed from the lithology description or can be derived from normal log analysis methods using porosity derived from density neutron log data.

Biot's original paper in 1956 pointed out that sonic velocity varied with frequency and described a low frequency case (typically 5 to 35 KHz under normal reservoir conditions) and high frequency case (typically 100 KHz to 1 MHx). Logging tools usually operate in the low frequency range and conform to Biot's low frequency case except in high porosity (> 35%).

Sonic velocity measurements made under laboratory conditions are usually made at 1 MHz because the core plugs are small and the high frequency has a short enough wavelength to fully penetrate the sample. R. A. Anderson's paper in 1984 gave graphs of both high and low frequency data versus Wyllie porosity. By comparing the response of the two frequencies, we can create equations to convert high frequency data to equivalent low frequency (logging tool) values. Travel times taken at high frequency are too fast (DTShi is too low).
1: DTScor = (DTShi - KS1) * 1.25 + KS1
2: DTCcor = (DTChi - KC1) * 1.02 + KC1

WHERE:
DTCcor = compressional sonic corrected for high frequency effect (usec/ft or usec/m)
DTChi = lab measured compressional sonic reading (usec/ft or usec/m)
DTScor = shear sonic corrected for high frequency effect (usec/ft or usec/m)
DTShi = lab measured shear sonic reading (usec/ft or usec/m)

 Sandstone Limestone Dolomite English Metric English Metric English Metric KS1 88.8 291 90.2 296 79.2 260 KC1 55.5 182 47.5 155 44.0 144

Use ONLY to convert lab measured high frequency (1 MHz) sonic data to equivalent low frequency sonic log data.

These new values of DTS and DTC should be substituted for the original measured lab data in the following sub-sections. The correction for DTC is very small and often ignored.

Frequency and fluid effects on Sonic travel time (Anderson, 1984)

In gas zones only, the density log and the compressional sonic log data must be corrected to a liquid filled state. The sonic reads too high and density too low due to the gas effect. If a full blown log analysis is available, density and sonic can be back-calculated from the porosity and lithology, provided that reasonable gas corrections were made in that analysis. Another approach is to use log data from a nearby wet or oil bearing zone in an offset well.

The following equations will also provide better data than the raw log data in gas zones:
3: DENScor = DENS + 0.5 * PHIe * Sgxo * (DENSMA - DENSW)
4: DTCcor = DTC + 0.5 * PHIe * Sgxo * (DTMA_C - DELTW)
5: DTScor = DTS

WHERE:
DENScor = density corrected for gas effect (gm/cc or Kg/m3)
DENS = density log reading (gm/cc or Kg/m3)
PHIe = effective porosity (fractional)
Sgxo = gas saturation near the well bore (fractional)
default = 0.80 for sonic, 0.70 for density log
DENSMA = matrix density (gm/cc or Kg/m3)
DENSW = water density (gm/cc or Kg/m3)
DTCcor = compressional sonic corrected for gas effect (usec/ft or usec/m)
DTC = compressional sonic log reading (usec/ft or usec/m)
DTCMA = compressional sonic travel time in matrix rock (usec/ft or usec/m)
DTScor = shear sonic corrected for gas effect (usec/ft or usec/m)
DTS = shear sonic log reading (usec/ft or usec/m)
DELTW = sonic travel time in water (usec/ft or usec/m)

These new values of DENS and DTC should be substituted for the original log data in the following sections. Gas correction on DTS is very small so no correction is usually applied.

In very slow formations, where shear travel time was impossible to measure on older sonic logs, this formula is used to calculate shear travel time (DTS) from Stoneley travel time:
6: DTS = (DENS / DENSW * (DELTst ^ 2 - DELTW ^ 2)) ^ 0.5

The dipole shear sonic log has reduced the need for this calculation, as it sees shear waves better than older array sonic logs. This new value of DTS should be substituted for the original log data in the following sub-sections.

When lithology is known from sample descriptions or from detailed log analysis, the shear travel time or velocity can be predicted from the porosity, lithology, and elastic constants from tables or from the following approximation:
8: DTS = Sum (Vi * DTSMAi) + KS7 * PHIe

WHERE:
KS7 = 1150  for Metric units (usec/m)
KS7 = 350  for English units (usec/ft)            1.6 – 1.8 for
Vi = volume of each mineral in the matrix rock (fractional)
DTS_MAi = shear travel time in each mineral (usec/ft or usec/m)

This is an empirical approximation and KS7 may be varied by calibrating to available DTS log data.

In rough hole conditions where sonic and density may have problems, it may be necessary to create synthetic sonic and density curves based on a competent log analysis that did not use the bad data as inputs to the log analysis model. To calibrate the synthetic curves, we usually calculate them over the entire interval of interest. In good hole conditions, the synthetic curves should match the measured curves. If they do not, either the original log analysis is a poor model or the parameters selected for the synthetic calculation are not appropriate. Ibxe rge oarameters and model are tuned, the synthetic curves can be generated even in wells where there are no measured sonic or density data. An example is shown in the previous image in Tracks 2 and 3. The equations needed are:
9:  DENSsyn = Vsh * DENSSH + DENS1 * Vmin1 + DENS2 * Vmin2 + DENS3 * Vmin3 + PHIe * SW * DENSW

+ PHIe * (1 - SW) * DENSHY
10:  DTCsyn = Vsh * DTCSH + DTC1 * Vmin1 + DTC2 * Vmin2 + DTC3 * Vmin3 + PHIe * SW * DTCW

+ PHIe * (1 - SW) * DTCHY
11:  DTSsyn = Vsh * DTSSH + DTS1 * Vmin1 + DTS2 * Vmin2 + DTS3 * Vmin3 + PHIe * SW * DTSW
+ PHIe * (1 - SW) * DTSHY

Where:
DFNSsyn, DTCsyn, and DTSsyn are synthetic density, compressional sonic, and sher sonic
DENSx, DTCx, and DTSx are density and sonic parameters for the specific mineral and fluid terms

Shale values are chosen by observation og the appropriate log interval. Standard parameters for other minerals  (which may need tuning) are:
Quartz    Calcite  Dolomite    Anhydrite   Water       Oil           Gas
DENS              2650       2710        2870           2980         1000        800        350 - 650
DTC                  182         155          144            160            650       700      2000 - 3000
DTS                  289         290          236            280           1150     1250      3000 - 4500

CALCULATING THE ELASTIC PROPERTIES

Shear Modulus is defined as the applied stress divided by the shear strain.

For rock with porosity:
1: N = KS5 * DENS / (DTS ^ 2)

For rock with no porosity:
2: DENSMA = (DENS - PHIt * DENSW) / (1 - PHIt)
3: DTSMA = DTS / (1 - PHIt)
4: No = KS5 * DENSMA / (DTSMA ^ 2)

WHERE:
KS5 = 13400 for English units
KS5 = 1000 for Metric units

If the rock is anisotropic, both N and No can be calculated for the minimum and maximum stress directions by using DTSmin and DTSmax from a crossed dipole shear sonic log.

Density is in gm/cc, travel time is in usec/ft, and N is in psi * 10^6 for English units. Density is in Kg/m3, travel time is in usec/m, and N is in Giga-Pascals (10^9 Pa or GPa) for Metric units.

For quicklook analysis, charts may be faster than a calculator:

Chart to calculate N from DENS and DTS

Poisson's Ratio is the lateral strain divided by longitudinal strain.

When shear velocity or shear travel time is available:
For rock with porosity:
1: R = Vp / Vs
OR 2: R = DTS / DTC
3: PR = (0.5 * R^2 - 1) / (R^2 - 1)

For rock with no porosity:
4: Ro = DTSMA / DTCMA
5: PRo = (0.5 * Ro^2 - 1) / (Ro^2 - 1)

If the rock is anisotropic, P can be calculated for the minimum and maximum stress directions by using DTSmin and DTSmax from a crossed dipole shear sonic log. PRmax comes from DTSmin and vice versa.

When shear travel time is not known, which is the case in the vast majority of older wells, a value for Poisson's ratio can be estimated. The usual estimate for Poisson's ratio in shaly sands is:
6: PR = 0.125 * Vsh + 0.27

This was developed in the US Gulf Coast and the parameters might need some adjustment in other areas.

A table of values for other rock types is shown later in this section.

If good conventional and shear seismic data are available, then Poisson's ratio can be derived continuously from seismic data. This is sometimes referred to as “seismic petrophysics”.

For quicklook analysis, use this chart for Poisson’s Ratio:

Chart to calculate PR from DTC and DTS

A plot of Poisson's ratio versus compressional velocity, below, shows the effect of lithology and gas. Values for Poisson's ratio are also listed in Table 1 near the end of this Chapter.

Poisson’s ratio versus lithology

In the absence of good shear sonic data, Poison's Ratio can be estimated from the graph below, based on known or assumed lithology (courtesy Barree and Associates).

Correlations of Poisson's Ratio versus DTC

The equations on this graph are:
7: PRshl = min(0.5,max(0,0.000 000 086 754 * DTC^3 - 0.000 044 154 * DTC^2 + 0.008 587 * DTC - 0.155))
8: PRqrtz = min(0.5,max(0,0.000 000 126 482 * DTC^3 - 0.000 058 769 * DTC^2 + 0.010 703 * DTC - 0 .296))
9: PRlime = min(0.5,max(0,-0.000 000 341 745 * DTC^3 + 0.000 117 836 * DTC^2 - 0.011 609 * DTC + 0.646))
10: PRdolo = min(0.5,max(0,-0.000 002 394 128 * DTC^2 + 0.000 708 300 * DTC + 0.2281355))
11: PRcoal = min(0.5,max(0.32,0.000 000 334 448 * DTC^3 - 0.000 083 251 * DTC^2 + 0.004 122 * DTC + 0.478))
12: PR = SUM(Vi * PRi)

Where:
DTC = sonic travel time (usec/ft)
Vi = volume of each component of the ock (fractional)
PRi = Poisson's Ratio of each component (fractional)

NOTE: Be sure to convert metric DTC values to Engliah units.

A high Poisson’s ratio indicates high stress level, which in turn indicates possible boundaries to a hydraulic fracture. Low Poisson’s ratio indicates weak zones which may not constrain the frac job, resulting in communication to undesired formations. Most shales constrain fractures but some may not do so. Two to three meters of rock with a Poisson's Ratio greater than 0.26 is the minimum needed to constrain a typical hydraulic fracture.

Gas zones, where the sonic compressional data has not been corrected for gas, will show abnormally low Poisson's ratio.

Poisson’s ratio is used to predict fracture pressure gradient in consolidated formations (Section 20.10).

 Typical Poisson's Ratio values Sandstone 0.18 - 0.22 deeper, highly compacted, cemented Sandstone 0.22 - 0.40 shallow, uncompacted, poorly cemented Siltstone 0.20 - 0.28 Shale 0.26 - 0.40 Limestone 0.310 Dolomite 0.283 Anhydrite 0.319

A more detailed list is provided in Table 20.01.

Bulk Modulus is the hydrostatic pressure divided by volumetric strain.

For rock with porosity:
1: Kb = KS5 * DENS *(1 / (DTC^2) - 4/3 * (1 / (DTS^2)))

For rock with no porosity:
2: DENSMA = (DENS - PHIt * DENSW) / (1 - PHIt)
3: DTCMA = (DTC - PHIt * DTW) / (1 - PHIt)
4: Km = KS5 * DENSMA / (1 / (DTCMA ^ 2) - 4/3 * (1 / (DTSMA^2)))

WHERE:
KS5 = 13400 for English units
KS5 = 1000 for Metric units

If the rock is anisotropic, both Kb and Km can be calculated for the minimum and maximum stress directions by using DTSmin and DTSmax from a crossed dipole shear sonic log.

Density is in gm/cc, travel time is in usec/ft, and Kb is in psi * 10^6 for English units. Density is in Kg/m3, travel time is in usec/m, and Kb is in Giga-Pascals (10^9 Pa or GPa) for Metric units.

If you like quicklook charts, here is one for Kb:

Chart for calculating Kb from P and N

Bulk Compressibility is the inverse of Bulk Modulus.

For rock with porosity:
1: Cb = 1 / Kb

For rock with no porosity:
2: Cm = 1 / Km

This term is called rock compressibility and abbreviated Cr in some literature.

If the rock is anisotropic, both Cb and Cm can be calculated for the minimum and maximum stress directions by using DTSmin and DTSmax from a crossed dipole shear sonic log.

N and Cb predict sanding (sand production) in unconsolidated formations. When log analysis shows sanding may be a problem, sand control methods (injection of plastic or resin or gravel packing) can be initiated. Sanding is not a problem when N > 0.6*10^6 psi. in oil or gas zones. High water cuts increase the likelihood of sanding. This threshold corresponds to Cb of 0.75*10^-6 psi^-1. N/Cb > 0.8*10^12 psi^2 is a more sensitive cutoff than either N or Cb cutoffs. High N/Cb ratios indicate low chance for sanding. A good cement job is also needed to reduce sanding.

Biot's Constant is the ratio of the volume change of the fluid filled porosity to the volume change of the rock when the fluid is free to move out of the rock (ie. the hydraulic pressure remains unchanged)..

For rock with porosity:
1: ALPHA = 1 - Kb / Km
OR 2: ALPHA = 1 - Cm / Cb

For rock with no porosity, Kb = Km so ALPHA = 0.

If shear travel time is unavailable, this empirical relation may be useful:
3: ALPHA = 1 - (1 - PHID) ^ KS8

where KS8 has the range 2 to 3, with KS8 = 3 most often used.

Biot's Constant versus porosity

In the absence of good shear sonic data, Biot's Constant can be estimated from the graph above, based on known or assumed lithology (courtesy Barree and Associates). This graph suggests KS8 in the previous equation is greater than 2.0. The empirical straight line fit to the data is:
4: ALPHA = 0.62 + 0.935 * PHIe

Young's Modulus is applied uni-axial stress divided by normal strain.

For rock with porosity:
1: Y = 2 * N * (1 + PR)

For rock with no porosity:
2: Yo = 2 * No * (1 + PRo)

If the rock is anisotropic, Y can be calculated for the minimum and maximum stress directions by using DTSmin and DTSmax from a crossed dipole shear sonic log when calculating N and P.

Young's modulus calculated from log data is often called the dynamic Young's modulus, Ydyn.

Young’s modulus is used in the fracture width (aperture) calculation in fracture design software.

Here is the quicklook chart for Young’s modulus:

Chart to calculate Y from P and N

In the absence of good shear sonic data, Young's Modulus can be estimated from the graph below, based on known or assumed lithology (courtesy Barree and Associates). The ordinate on this graph is Young's Modulus divided by density (gm/cc), so multiply the Y axis value by density to obtain Y.

Young's Modulus versus DTC for various lithologies

The equations on the above graph are:
3: Yshl = 6.894 * DENS * (0.000 000 100 214 * DTC^4 - 0.000 050 013 * DTC^3 + 0.009 417 * DTC^2
- 0.806 315 * DTC + 27.30)
4: Yqrtz = 6.894 * DENS * (0.000 000 099 297 * DTC^4 - 0.000 049 604 * DTC^3 + 0.009 3678 * DTC^2
- 0.807 280 * DTC + 27.68)
5: Ylime = 6.894 * DENS * (0.000 000 037 682 * DTC^4 - 0.000 019 762 * DTC^3 + 0.003 996 * DTC^2
- 0.380 084 * DTC + 14.97)
6: Ydolo = 6.894 * DENS * (0.000 000 084 048 * DTC^4 - 0.000 041 6941 * DTC^3 + 0.007 775 * DTC^2
- 0.659 893 * DTC + 22.59)
7: Ycoal = 6.894 * DENS * (0.000 001 498 * DTC^3 - 0.000 588 141 * DTC^2 + 0.069 142 * DTC - 1.84)
8: Ydyn = SUM(Vi * Yi)

Where:
DTC = sonic travel time (usec/ft)
DENS = density (g/cc)
Vi = volume of each component of the ock (fractional)
Yi = Young's Modulus of each component (GPa)
Ydyn = dynamic Young's Modulus of rock (GPa)

NOTE: Be sure to convert metric DTC and DENS values to Engliah units. Results are in GPa. Divide by 6.894 to get psi * 10^6.

Brittleness Coefficient
A brittleness coefficient was proposed by Mullen in 2007, as shown below:
1:
Ybrit = ((Y - 1) / (8 - 1) * 100)
2: PRbrit = ((PR - 0.4) / (0.15 - 0.4)) * 100
3: BritCoeff = (Ybrit + PRbrit) / 2

Where:
Y = Young's Modulus (
psi * 10^6)
PR = Poisson's Ratio (unitless)
Ybrit = Young's Modulus for brittleness calculation
PRbrit = Poisson's Ratio for brittleness calculation
BritCoeff = Brittleness Coefficient

Modulus of compressibility

Fluid bulk modulus:
1: Kf = 1/Cf = Sw / Cwtr + (1 - Sw) / Chyd

Pore bulk modulus:
2: Kp = ALPHA^2 / ((ALPHA - PHIt) / PHIt / Kf)

For rock with porosity:
3: Kc = Kp + Kb + 4/3 * N.

For rock with no porosity, Kp = 0 and Kb = Km, and N = No, so:

4: Kc  = Km + 4/3 * No

 RECOMMENDED PARAMETERS: Water Salinity Cf psi-1 Kf psi Cf GPa-1 Kf GPa 5000 0.0000040 250000 0.580 1.723 35000 0.0000039 270270 0.537 1.862 200000 0.0000027 344828 0.420 2.376 Oil Depth 2000 ft 610 m 0.0000085 117647 1.233 0.811 4000 ft 1220 m 0.0000095 105263 1.378 0.725 8000 ft 2440 m 0.0000116 86207 1.683 0.594 12000 ft 3660 m 0.0000135 74074 1.959 0.510 Gas Depth 2000 ft 610 m 0.001250 800 181.422 0.006 4000 ft 1220 m 0.000510 1961 74.020 0.014 8000 ft 2440 m 0.000180 5556 26.124 0.038 12000 ft 3660 m 0.000100 10000 14.513 0.069

The above calculations assume that fluid compressibilities are known from lab measurements of produced fluids. In recently drilled wells, this information is not always available. It would therefore be useful to predict fluid compressibility or fluid bulk modulus and use this result to predict the fluid type in the reservoir. A method using pore bulk modulus is more convenient, and is based on some empirical evidence for sandstones.

By setting Kb = Km - 0.9 * N in equation 3, and solving for Kp:
5: Kp = Kc - Km + 0.9 * N - 4/3 * N

Interpretation is based on the following:
6: IF Kp <= 0.15 THEN Zone is gas bearing (Kp in GPa):
7: IF 0.15 < Kp < 0.35 THEN Zone is oil bearing
8: IF Kp >= 0.35 THEN Zone is water bearing

Kp is sometimes shown as Kf in the literature so be careful.

If conventional and shear seismic data are of sufficient quality to be inverted, then these same equations can be used to detect fluid type in porous sandstones

If conventional and shear seismic data are of sufficient quality to be inverted, then these same equations can be used to detect fluid type in porous sandstones.

Quicklook Bulk and Shear Modulus

Murphy (1991) provided equations for sandstone rocks (PHIe < 0.35) that predict Kb and N from porosity:
1: Kb = 38.18 * (1 - 3.39 * PHIe + 1.95 * PHIe^2)
2: N   = 42.65 * (1 - 3.48 * PHIe + 2.19 * PHIe^2)

These equations are for the water filled case and cannot be used as fluid identification, but they may have other uses.

Calibrating Dynamic to Static Constants
The mechanical properties of rocks derived from log data, or from high frequency sonic measurements in the lab are called dynamic constants. Those derived in the laboratory from stress strain tests or destructive tests are called static constants. In my opinion, lab derived dynamic shear data should be corrected for high frequency effects, as described earlier in this Chapter. This is seldom done, so it is difficult to compare dynamic log data to dynamic lab data.

Unfortunately, the difference between dynamic (well log) values and static (lab) values on cores can be quite large, leading some people to dismiss the log data as wrong or useless. What makes this worse is that fracture design software has been calibrated to static (lab derived) values, so dynamic data has to be transformed to equivalent static numbers.

Since the tiny core plugs used for lab work have been de-stressed and re-stressed a number of times, there is some doubt that this cycle is truly reversible, so lab measurements may not represent in-situ conditions. The difference between static and dynamic values are larger for higher porosity, which suggests that some grain bonds are easily broken by coring and subsequent testing. It might be a wise move to calibrate fracture design software to dynamic data, since this data is more readily available, and may actually have fewer inherent measurement problems.

Further, the effects of reservoir anisotropy cannot be simulated in the lab, so there is no reason to expect lab data to match in-situ log results. A possible solution to this dilemma is described in later Sections.

Errors in Poisson's ratio strongly affect calculation of in-situ closure stress.

Doug Boyd (1991) presented a summary of published Poisson's ratio data, shown at the right.

As you can see, there is no direct relation between dynamic and static Poisson' ratio. Additional reasons for the mismatch might include dry rock measurements (as opposed to restored state), moisture variations in shaly samples, closing of fractures, pore shape changes, anisotropy (especially if core plugs are used instead of whole core), inconsistent (and often unknown) experimental methods, and experimental measurement error. Results of such a comparison made today might be less erratic if anisotropic effects were reduced by use of crossed dipole shear sonic data and appropriate core handling procedures.

Calibration of local data seems possible, but there is no universal correction factor. When field measured closure stress is available from mini-fracs, a calibration of Poisson's ratio is fairly easy in a homogenous reservoir, but probably impractical in many cases.

Young’s modulus is also affected by differences between static and dynamic values. A transform published by Morales and Marcinew in 1993 is shown in the graph on the left and formulated as:

1: Yst = 10^(A + B * log(Ypsi))

WHERE:
Yst = static Young’s modulus (psi)
Ypsi = dynamic Young’s modulus (psi)

A and B are constants that depend on porosity as shown below:

 Porosity A B 0.00 - 0.10 n/a 0.10 - 0.15 2.137 0.6612 0.15 - 0.25 1.829 0.6920 > 0.25 -0.4575 0.9402

This transform was based on high frequency dynamic lab data compared to static lab data. Low frequency log data was not used so this widely used transform may have no validity for log derived rresults.

The same paper quoted other data sets and compared their data to a transform by Eissa. Note that these transforms invoke the rock density to normalize the data. The equations are:
Average of all data
2: Yst = 10^(0.05 + 0.77 * log(DENS * Ydyn))
Low porosity 10 -15%
3: Yst = 10^(0.02 + 0.77 * log(DENS * Ydyn))
Medium porosity 15 - 25%
4: Yst = 10^(-0.11 + 0.77 * log(DENS * Ydyn))
High porosity > 25%
5: Yst = 10^(-0.72 + 0.77 * log(DENS * Ydyn))

NOTE: DENS is in g/cc and Ydyn is in GPa.

Static to dynamic transforms for Young's Modulus

Lacy proposed a correlation to obtain static Young's Modulus, Yst, from Ydyn. First convert Ydyn to English units:
6: Ypsi = Ydyn / 6.894
7: Yst_shl = 6.894 * (0.0420 * Ypsi^2+0.2330 * Ypsi)
8: Yst_qrtz = 6.894 * (0.0293 * Ypsi^2 + 0.4533 * Ypsi)
9  Yst_carb = 6.894 * (0.0180 *Ypsi^2 + 0.4220 * Ypsi)
10: Yst_coa; = Ycoal
11: Yst_lacy = SUM(Vi * Ysti)

NOTE: Results are in GPa. Divide by 6.894 to get psi * 10^6.

Barba's correlation for Yst is:
12: Y99 = min(0.9,max(0.5,(Vsh + Vqrtz) * (-0.0003 * Ypsi^4 + 0.0052 * Ypsi^3 - 0.0203 * Ypsi^2 + 0.0312 * Ypsi
+ 0.4765) + ((Vlime + Vdolo) * (Ypsi * 0.85 - 0.424)) / (Vsh + Vrock)))
13: Yst_barba = Y99 * Ydyn

NOTE: Results are in GPa. Divide by 6.894 to get psi * 10^6.

Examples of Mechanical Properties Logs
The above equations can be computed continuously and presented as logs. The format and curve complement vary widely between service companies and age of log. Some logs have Metric depths but the moduli are in English units. Some are vice versa. Here are some examples.

Mechanical properties log with lithology/porosity track at the right.

Another mechanical properties log

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