5 Inversion

Contents

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Data Redundancy and Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Composite Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6L2 / L1L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Zackhem Stromberg and Dunn (ZSD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Noise Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Paradigm Rock & Fluid Canvas 2009 | Epos 4.0 Inversion 5-1

Overview

See also Notes on Inversion on page 1-30.

Principles

T2 relaxation measured by the CPMG echo train represents a distribution of T2 values that reflect the multi-exponential nature of T2 relaxation in rocks. T2 relaxation can be attributed to 3 key processes:

1. Surface relaxation, which is related to pore size.2. Bulk relaxation, which is related to fluid properties.3. Molecular diffusion relaxation due to magnetic field gradients.

Accurate description of the underlying T2 distribution furnishes information about pore size, fluid type and hydrocarbon volume.Inversion is used to determine the underlying T2 distribution that produces the observed CPMG decay. The underlying T2 distribution is most accurately described as a continuous function. However, for the purposes of petrophysics, the CPMG decay modeled as a multi-exponential function that assumes that the distribution has a discrete set of T2 relaxation components. These components are associated with pore size and the non-wetting fluids.A typical T2 distribution is composed of 30 bins spread between 0.3 and 3000ms. The bins are equally spaced in logarithmic space.The mathematical model that describes inversion is discussed in Notes on Inversion on page 1-30.

Data Redundancy and Compression

Generally, the results of inversion show that the CPMG T2 decay contains only a few independent pieces of information. Consequently, the CPMG can be reduced by compressing the data into a few predefined bins. The compressed bins of data consist of a series of windows that average parts of the CPMG distribution to a single value. Usually the windows are equally spaced logarithmically such that the first window will contain the fewest echoes and the last window will contain the most. The standard deviation of noise (sigma) is calculated independently for each window, and passed to the inversion routine.Data compression has the advantage in that it greatly reduces the computational time for inversion.Inversion 5-2Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Fig. 1: Binning the CPMG data allows data compression of CPMG data prior to inversion

Fit

The inversion algorithm attempts to find a best fit of the CPMG data that is closest to the underlying trend in the CPMG T2 relaxation curve. The deviation of the measurement data (deviation due to the noise) should be random and evenly spaced over the fitted curve. Any systematic deviation indicates that the data has been incorrectly fitted by the inversion procedure.Inversion 5-3Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Smoothing

Mathematically, inversion is an "ill posed" problem in that there is more than one solution to the problem. To find a solution to the inversion of the data, smoothing is usually imposed on the T2 distribution. Increased smoothing improves the stability of solution, but reduces the number of peaks resolved in the T2 distribution. If the smoothing parameter is set too low, spurious peaks may appear in the T2 distribution. The smoothing parameter is usually selected so that the T2 distribution contains 1, 2 or 3 peaks.

Smoothing also depends on the S:N ratio of the CPMG data. Noisy data usually requires a higher level of smoothing.The optimum smoothing parameter is often calibrated by comparing the log data with NMR laboratory data, or pore size distributions derived from capillary pressure data. Laboratory NMR data and capillary pressure data can be loaded into Geolog, and the distributions can be compared with results of inverting the CPMG data.

Composite Data Processing

Both the MRIL Prime and CMR Plus provide high resolution data acquisition modes. This data is referred to as C/TP (MRIL) or EPM (CMR) data. To collect this data, the tool excites regular pulse sequences (e.g. TE = 1.2 ms, collecting 400-800 echoes with a wait time of 8 sec). The tool then acquires a short-burst CPMG sequence with a short recovery time (e.g. TE = 0.6ms collecting 10 echoes with a wait time 0.02 sec).Inversion 5-4Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Normally, the short-burst data and the regular data are inverted separately, and the resultant

T2 distribution is spliced together to form a high-resolution T2 distribution. However, Geolog allows for both sets of data to be included in the inversion process at the same time. This is termed "simultaneous inversion", where matrix manipulation allows both sets of data to be solved simultaneously to generate single continuous T2 distribution. The exact mechanism of the matrix manipulation is detailed in:

K. J. Dunn, G. A. La Torraca. The Inversion of NMR Log Data Sets with Different Measurement Errors. Journal of Magnetic Resonance. 140, 153-161(1999).Simultaneous inversion is included as an option in all of the inversion methods available in the Geolog NMR inversion modulesInversion 5-5Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Inversion

L2 / L1L2

This module uses zeroth order regularization to invert CPMG data to a distribution of exponentials (T2 distribution). The regularization parameter controls the smoothness of the T2 distribution. For noisy data (S:N < 2), regularization should be set between 1 and 10; for better quality data, a lower value can be used (e.g. 0.5). The smoothing parameter should be increased to remove spurious peaks from the resultant T2 distributions. The T2 distributions are corrected for incomplete polarization.

Operation1. Select Petrophysics > NMR > Inversion > L2/L1L2 to display nmr_inversion_compress.2. Set the required input Values as described in Table 5-1, "nmr_inversion_compress Parameters"

on page 5-7.Inversion 5-6Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

The nmr_inversion_compress ModuleTable 5-1: nmr_inversion_compress Parameters

Name Default Description

OPT_CTP NoDouble click in the field then click on the Dropdown List button to select either yes or no and thus turn composite data processing on or off.

OPT_COMPRESS YesDouble click in the field then click on the Dropdown List button to select either yes or no and thus turn data compression on or off.

FIRST_ECHO Available when OPT_COMPRESS is set to "yes"; first echo of the CPMG data to be included in the inversion.

LAST_ECHO Available when OPT_COMPRESS is set to "yes"; last echo of the CPMG data to be included in the inversion; all echoes lying between the FIRST_ECHO and LAST_ECHO contribute to the solution.Inversion 5-7Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Table 5-1: nmr_inversion_compress Parameters (Continued)COMPRESS_BINS Available when OPT_COMPRESS is set to "yes"; number of bins that the data is compressed to. Vary between 10 and 100. The minimum number of bins must exceed the number of bins in the T2 distribution.

OPT_COMPRESS_OUT NoDouble click in the field then click on the Dropdown List button to select either yes or no and thus decide whether or not to output an array of compressed data.

REG_PARAM Controls the smoothness of the T2 distribution. Larger values increase the smoothing component.

INVERSION_TYPE L2Double click in the field then click on the Dropdown List button to select either L2 or L1L2 inversion (see Notes on Inversion on page 1-30).

OPT_WAIT YesDouble click in the field then click on the Dropdown List button to select either on or off and thus turn wait time correction on or off.

Note: In data compression, only the regular (ECH_AMP_TP) data is compressed; the high resolution part data is not compressed. When the OPT_CTP is on, the FIRST_ECHO and LAST_ECHO refers to the compression of the regular data only (ECHO_AMP_TP).T1T2 1.5 T1:T2 ratio.NECH_V_C Available when OPT_CTP is set to "yes"; number of data points in the

short-burst CPMG echo.TE_C Available when OPT_CTP is set to "yes"; echo spacing of the short-burst

CPMG echo.TW_C Available when OPT_CTP is set to "yes"; wait time of the short-burst

CPMG echo.NECH_V_TP Number of data points in the CPMG echo train.TE_TP Echo spacing of the regular CPMG echo.TW_TP Wait time of the regular CPMG echo.T2_MIN Minimum T2 of the T2 distribution.T2_MAX Maximum T2 of the T2 distribution.T2_BIN Number of T2 bins.ECHO_AMP_C Available when OPT_CTP is set to "yes"; CPMG echo amplitude for the

short-burst data.ECHO_NOISE_C Available when OPT_CTP is set to "yes"; CPMG noise amplitude for the

short-burst data. If this data is not available run the "Noise Extraction" on page 5-15 module using the above echo amplitude log.

ECHO_AMP_TP CPMG echo amplitude for the regular data.ECHO_NOISE_TP CPMG noise amplitude for the regular data. If this data is not available

run the "Noise Extraction" on page 5-15 module using the above echo amplitude log.

Name Default DescriptionInversion 5-8Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Table 5-1: nmr_inversion_compress Parameters (Continued)Singular Value Decomposition (SVD)

This module uses singular value decomposition (SVD) to invert the CPMG data to a T2 distribution (see Notes on Inversion on page 1-30 for further information). Included in the SVD module is the option to recalculate the standard deviation of noise (SIGMA_RECALC). The inversion is performed twice if the RMS error between the input and the theoretical fitted curve varies by more than 2%, in which case the inversion is repeated with the recalculated sigma.

Operation1. Select Petrophysics > NMR > Inversion > SVD to display the nmr_inversion_svd module.2. Set the required input Values as described in Table 5-2, "nmr_inversion_svd Parameters" on

page 5-10.

Note: If the echo data has been stacked to improve S:N ratio, the stacked data should be specified. Both the echo amplitude and the echo noise should be stacked using the same level of stacking. Stacked data should not be mixed with non-stacked data. The data should be rotated. MRIL data should have had the minimum RA applied during processing.PHI_T2_DIST The T2 porosity distribution.T2_TIME T2 time constants used in inversion. The T2 distribution is plotted using

T2_TIME on the x-axis and AMP_DIST on the y-axis. See Notes on Inversion on page 1-30.

QUALITY Quality of the inversion; 0 = good, 1 = bad.SIGMA_C Available when OPT_CTP is set to "yes"; standard deviation of the noise

calculated from the noise of the short burst dataSIGMA_TP Standard deviation of the noise calculated from the noise of the regular

data.INVERT_BAD No solution found for inversion. Normally, the solution will converge to

minima. When the solution does not converge, the INVERT_BAD flag is triggered.

ECHO_FIT_C Fitted CPMG for the short-burst data (i.e., the model used for inversion; see also "Fit" on page 5-3).

ECHO_FIT_TP Fitted CPMG for the regular data (i.e., the model used for inversion; see also "Fit" on page 5-3).

ECHO_AMP_COMP Compressed Echo Amplitude.

Name Default DescriptionInversion 5-9Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

The nmr_inversion_svd ModuleTable 5-2: nmr_inversion_svd Parameters

Name Default Description

OPT_CTP NoDouble click in the field then click on the Dropdown List button to select either yes or no and thus turn composite data processing on or off.

OPT_COMPRESS YesDouble click in the field then click on the Dropdown List button to select either yes or no and thus turn data compression on or off.

FIRST_ECHO Available when OPT_COMPRESS set to "yes"; first echo of the CPMG data to be included in the inversion.

LAST_ECHO Available when OPT_COMPRESS set to "yes"; last echo of the CPMG data to be included in the inversion; all echoes lying between the FIRST_ECHO and LAST_ECHO contribute to the solution.

COMPRESS_BINS Available when OPT_COMPRESS set to "yes"; number of bins that the data is compressed to. Vary between 10 and 100. The minimum number of bins must exceed the number of bins in the T2 distributionInversion 5-10Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Table 5-2: nmr_inversion_svd ParametersOPT_COMPRESS_OUT NoDouble click in the field then click on the Dropdown List button to select either yes or no and thus decide whether or not to output an array of compressed data.

OPT_RECALC_SIGMA no Recalculate signal-to-noise based on RMS error. Double click in the

field then click on the Dropdown List button to select "no", "recalc" or "weight".

SNR_FACTOR Controls the smoothness of the T 2 distribution. Smaller values increase the smoothing component.

OPT_WAIT YesDouble click in the field then click on the Dropdown List button to select either on or off and thus turn wait time correction on or off.

Note: In data compression, only the regular (ECH_AMP_TP) data is compressed, the high resolution part data is not compressed. When the OPT_CTP is on, the FIRST_ and LAST_ECHO refers to the compression of the regular data only (ECHO_AMP_TP).T1T2 1.5 T1:T2 ratio.NECH_V_C Available when OPT_CTP is set to "yes"; number of data points in

the short-burst CPMG echo.TW_C Available when OPT_CTP is set to "yes"; wait time of the short-burst

CPMG echo.TE_C Available when OPT_CTP is set to "yes"; echo spacing of the short-

burst CPMG echo.NECH_V_TP Number of data points in the regular CPMG echo.TW_TP Wait time of the regular CPMG echo.TE_TP Echo spacing of the regular CPMG echo.T2_MIN Minimum T2 of the T2 distribution.T2_MAX Maximum T2 of the T2 distribution.T2_BIN Number of T2 bins.ECHO_AMP_C Available when OPT_CTP is set to "yes"; CPMG echo amplitude for

short-burst data.ECHO_NOISE_C Available when OPT_CTP is set to "yes"; CPMG noise amplitude for

the short-burst data.ECHO_AMP_TP CPMG echo amplitude for the regular data.ECHO_NOISE_TP CPMG noise amplitude for the regular data.Note: If the echo data has been stacked to improve S:N ratio, the stacked data should be specified. Both the echo amplitude and the echo noise should be stacked using the same level of stacking. Stacked data should not be mixed with non-stacked data. The data should be rotated. MRIL data should have had the minimum RA applied during processing.PHI_T2_DIST The T2 porosity distribution.PHI_T2_DIST_ERROR The error of the T2 distribution.

Name Default DescriptionInversion 5-11Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Table 5-2: nmr_inversion_svd ParametersZackhem Stromberg and Dunn (ZSD)

This module uses the ZSD method to invert the NMR data. For more information on this method, see Notes on Inversion on page 1-30.

Operation1. Select Petrophysics > NMR > Inversion > ZSD to display the nmr_inversion_zsd module.2. Set the required input Values as described in Table 5-3, "nmr_inversion_zsd Parameters" on

page 5-13.

T2_TIME T2 time constants used in inversion. The T2 distribution is plotted using T2_TIME on the x-axis and AMP_DIST on the y-axis. See Notes on Inversion on page 1-30.

EPSILON Noise/Error; general indicator of error in inversion.UXP Data span in solution matrix; 0 = good inversion.SIGMA_C Available when OPT_CTP is set to "yes"; standard deviation of the

noise calculated from the noise of the short-burst data.SIGMA_TP Standard deviation of the noise calculated from the noise of the

regular data.SIGMA_RECALC Recalculated sigma if OPT_ RECALC_SIGMA = yes.INVERT_BAD No solution found for inversion. Normally, the solution will converge

to minima. When the solution does not converge, the INVERT_BAD flag is triggered.

ECHO_FIT_C Available when OPT_CTP is set to "yes"; fitted CPMG for the short-burst data (i.e., the model used for inversion; see also "Fit" on page 5-3).

ECHO_FIT_TP Fitted CPMG for the regular data (i.e., the model used for inversion; see also "Fit" on page 5-3).

ECHO_AMP_COMP Compressed Echo Amplitude.

Name Default DescriptionInversion 5-12Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

The nmr_inversion_zsd ModuleTable 5-3: nmr_inversion_zsd Parameters

Name Default Description

OPT_CTP NoDouble click in the field then click on the Dropdown List button to select either yes or no and thus turn composite data processing on or off.

OPT_COMPRESS YesDouble click in the field then click on the Dropdown List button to select either yes or no and thus turn data compression on or off.

FIRST_ECHO Available when OPT_COMPRESS set to "yes"; first echo of the CPMG data to be included in the inversion.

LAST_ECHO Available when OPT_COMPRESS set to "yes"; last echo of the CPMG data to be included in the inversion; all echoes lying between the FIRST_ECHO and LAST_ECHO contribute to the solution.

COMPRESS_BINS Available when OPT_COMPRESS set to "yes"; number of bins that the data is compressed to. Vary between 10 and 100. The minimum number of bins must exceed the number of bins in the T2 distributionInversion 5-13Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Table 5-3: nmr_inversion_zsd Parameters (Continued)OPT_COMPRESS_OUT NoDouble click in the field then click on the Dropdown List button to select either yes or no and thus decide whether or not to output an array of compressed data.

ALPHA_PARAM 1 Starting value for alpha smoothing.BETA_PARAM 1 Starting value for beta smoothing.N_ITERATIONS 30 Number of iterations for inversion.OPT_WAIT Yes

Double click in the field then lick on the Dropdown List button to select either on or off and thus turn wait time correction on or off.

Note: In data compression, only the regular (ECH_AMP_TP) data is compressed, the high resolution part data is not compressed. When the OPT_CTP is on, the FIRST_ and LAST_ECHO refers to the compression of the regular data only (ECHO_AMP_TP).T1T2 1.5 T1:T2 ratio.NECH_V_C Available when OPT_CTP is set to "yes"; number of data points in

the short-burst CPMG echo.TE_C Available when OPT_CTP is set to "yes"; echo spacing of the short-

burst CPMG echo.TW_C Available when OPT_CTP is set to "yes"; wait time of the short-

burst CPMG echo.NECH_V_TP Number of data points in the regular CPMG echo.TE_TP Echo spacing of the regular CPMG echo.TW_TP Wait time of the regular CPMG echo.T2_MIN Minimum T2 of the T2 distribution.T2_MAX Maximum T2 of the T2 distribution.T2_BIN Number of T2 bins.ECHO_AMP_C Available when OPT_CTP is set to "yes"; CPMG echo amplitude

for short-burst data.ECHO_NOISE_C Available when OPT_CTP is set to "yes"; CPMG noise amplitude

for the short-burst data.ECHO_AMP_TP CPMG echo amplitude for the regular data.ECHO_NOISE_TP CPMG noise amplitude for the regular data.Note: If the echo data has been stacked to improve S:N ratio, the stacked data should be specified. Both the echo amplitude and the echo noise should be stacked using the same level of stacking. Stacked data should not be mixed with non-stacked data. The data should be rotated. MRIL data should have had the minimum RA applied during processing.PHI_T2_DIST The T2 porosity distribution.T2_TIME T2 time constants used in inversion. The T2 distribution is plotted

using T2_TIME on the x-axis and AMP_DIST on the y-axis. See Notes on Inversion on page 1-30.

ALPHA_BETA Smoothing applied to T2 distribution.ERROR Error of inversion related to porosity estimate.

Name Default DescriptionInversion 5-14Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Table 5-3: nmr_inversion_zsd Parameters (Continued)Noise Extraction

This module is used to extract the noise from an input signal. You would run this module to extract a clay or TP noise amplitude log which is required in the "L2 / L1L2" on page 5-6 inversion processing.Two methods are provided to extract the noise; HIGH for applying a second derivative operator and LOW for a low-pass operator.

Operation1. Select Petrophysics > NMR > Inversion > Noise extraction to display the

tp_nmr_noise_extraction module.2. Set the required input Values as described in Table 5-4, "nmr_noise_extraction Parameters" on

page 5-16.

The tp_nmr_noise_extraction Module

SIGMA_C Available when OPT_CTP is set to "yes"; standard deviation of the noise calculated from the noise of the short-burst data.

SIGMA_TP Standard deviation of the noise calculated from the noise of the regular data.

INVERT_BAD No solution found for inversion. Normally, the solution will converge to minima. When the solution does not converge, the INVERT_BAD flag is triggered.

ECHO_FIT_C Available when OPT_CTP is set to "yes"; fitted CPMG for the short-burst data (i.e., the model used for inversion; see also "Fit" on page 5-3).

ECHO_FIT_TP Fitted CPMG for the regular data (i.e., the model used for inversion; see also "Fit" on page 5-3).

ECHO_AMP_COMP Compressed Echo Amplitude.

Name Default DescriptionInversion 5-15Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Table 5-4: nmr_noise_extraction Parameters

Name Name Description

METHOD HIGHDouble click in the field, then click on the Dropdown List button to select the noise extraction method to use, either HIGH or LOW.

WINLEN 2 Available when the extraction METHOD is set to LOW; specify the required window length.

ECHO_AMP Enter or select input log filename.ECHO_NOISE If required, change default output filename.Inversion 5-16Paradigm Rock & Fluid Canvas 2009 | Epos 4.0

Index

Bbest fit of the CPMG data 3

Ccompressing CPMG data 2CPMG

compressing the data 2

Iinversion

and smoothing 4description of use in NMR 2L1/L2 module 6

Kkey processes for T2 relaxation 2

TT2

key relaxation processes 2Inversion 5-17Paradigm Rock & Fluid Canvas 2009 | Epos 4.0