ROCK MECHANICS CALCULATIONS 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 triaxial stress strain measurements (nondestructive) or the chevron notch test (destructive). The elastic constants of rocks are defined by the WoodBiotGassmann 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. 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.
ELASTIC
CONSTANTS THEORY The pore
space bulk modulus (Kp) is derived from the porosity, fluid, and
matrix rock properties: The
composite rock/fluid compressional bulk modulus is:
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:
Compressional velocity (Vp) and shear velocity (Vs) are defined as: Where: The
BiotGassmann 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:
INITIAL
CONSIDERATIONS 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). Where:
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 subsections. The correction for DTC is very small and often ignored.
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 backcalculated 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: Where:
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: 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 subsections.
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: 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: Where:
CALCULATING THE ELASTIC PROPERTIES For
rock with porosity: For
rock with no porosity: Where:
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 GigaPascals (10^9 Pa or GPa) for Metric units. For quicklook analysis, charts may be faster than a calculator:
When
shear velocity or shear travel time is available: 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: 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:
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.
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).
The
equations on this graph are: 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).
Bulk Modulus is the hydrostatic pressure divided by volumetric strain. For
rock with porosity: For
rock with no porosity: Where:
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 GigaPascals (10^9 Pa or GPa) for Metric units. If you like quicklook charts, here is one for Kb:
Bulk Compressibility is the inverse of Bulk Modulus. For
rock with porosity: For
rock with no porosity: 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: For rock with no porosity, Kb = Km so ALPHA = 0. If
shear travel time is unavailable, this empirical relation may
be useful: where KS8 has the range 2 to 3, with KS8 = 3 most often used.
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:
Young's Modulus is applied uniaxial stress divided by normal strain. For
rock with porosity: 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:
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.
The equations on the above graph
are:
Fluid bulk modulus:
For rock with no
porosity, Kp = 0 and Kb = Km, and N = No, so:
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:
Interpretation is based on the following: 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 sandstonesIf 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: 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 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 destressed and restressed a number of times, there is some doubt that this cycle is truly reversible, so lab measurements may not represent insitu 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 insitu log results. A possible solution to this dilemma is described in later Sections. Errors in Poisson's ratio strongly affect calculation of insitu 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 minifracs, 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: A and B are constants that depend on porosity as shown below:
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:
Lacy proposed a
correlation to obtain static Young's Modulus, Yst, from Ydyn. First
convert Ydyn to English units: NOTE: Results are in GPa. Divide by 6.894 to get psi * 10^6.
Barba's
correlation for Yst is:
Examples of Mechanical Properties Logs


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