A new method for multi-exponential inversion of NMR relaxation measurements

Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.3 265 276 265 A new method for multi-exponential inversion of NMR relaxation measurements WANG Zhongdong 1, 2, XIAO Lizhi 1 & LIU Tangyan 1 1. Petroleum University, Beijing 102249, China; 2. Liaohe Petroleum Administration, Panjin 124011, China Correspondence should be addressed to Xiao Lizhi (xiaolizhi@bjpeu.edu.cn) Received June 21, 2003 Abstract A new method for multi-exponential inversion to NMR T 1 and T 2 relaxation time distributions is suggested and tested. Inversion results are compared with MAP-II which is based on SVD algorithm and widely accepted in the industry. Inversed NMR relaxation spectra that have different pre-assigned relaxation times from echo trains with different SNR confirm that the new method with 16 to 64 equally spaced time constants in logarithm scale will ensure the relaxation distribution. Testing results show that the new inversion algorithm is a valuable tool for rock core NMR experimental analysis and NMR logging data process and interpretation. Keywords: NMR, inversion, relaxation, core analysis, well logging. DOI: 10.1360/ 03yw0111 The NMR technique has been widely applied to petroleum well logging and rock core analysis since the 1990s when NUMAR introduced a reliable NMR logging tool to the oil industry. It has been playing an important role for prospecting and exploiting resource of oil and gas for the last ten years. In an oil well, NMR can provide parameters of reservoir and fluid properties, such as porosity, pore size distribution, bound water volume, bulk volume of free water, permeability, in-situ fluid diffusion factor and viscosity at reservoir conditions. To obtain these parameters, an essential step is the inversion process that is called NMR multi-exponential fit for T 1 or T 2 distributions. Prammer [1,2] and Xiao [3,4] have used SVD (singular value decomposition) and NNLS (no-negative least square) to carry out the multi-exponential fit. The MAP-II method based on SVD technique and developed by Prammer is now a well-known and widely used method for T 2 relaxation inversion. Wang et al. [5], Wang et al. [6], and Weng et al. [7] have studied and discussed related issues of multi-exponential inversion, but how to build a fast and practical inversion algorithm with higher resolution to relaxation times and less dependence on measured SNR is still an important subject. A new multi-exponential inversion method for NMR relaxation signals is presented in this paper, which is based on a solid iteration rebuild technique (SIRT). The

266 Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.3 265 276 implementation process and application results are discussed in details. The T 2 spectra inversed by a new method are compared with MAP-II results. The T 1 and T 2 inversion results with different pre-assigned relaxation times and different SNR show that 16 to 64 logarithmically equally spaced time constants are obviously better than MAP-II. And in particular, they can ensure the relaxation time distribution when the SNR of the measured signal is very low. The new algorithm has been applied in rock core NMR analysis and NMR logging data process and interpretation. 1 Unified description for NMR T 1 and T 2 relaxation The NMR relaxation measurements, either longitudinal (T 1 ) or transverse (T 2 ) relaxation, in porous media can be expressed as a multi-exponential function. The T 2 relaxation signal measured by CPMG spin echo method, for example, can be written as t M () t = Pi exp. i T2i (1) The T 1 relaxation signal measured by inverse-recovery method may be expressed as t MII () t = Pi 1 2exp. i T1 i (2) If we unify the description, the expression can be yt () = P c + c exp t, i 1 2 i T (3) 1,2i where for the T 1 relaxation signal, c 1 = 1 c 2 = 2; and T 1, 2i is T 1i and P i is the contribution of the ith relaxation unit to the total signal. And for the T 2 relaxation measurements, c 1 = 0 c 2 = 1; T 1, 2i is T 2i and P i is the contribution of the ith relaxation unit to the total signal. The objective of inversion is to extract P i vs. T 1i or T 2i (i = 1,, n) from eqs. (1), (2) or (3), so that a so-called T 1 or T 2 distribution can be determined. 2 New multi-exponential inversion method for relaxation measurements Eq. (3) can be rewritten as a matrix format: Y = A P, (4) where the matrix Y n 1 = (y 1, y 2, y n ) T is the measured relaxation signal (T 1 or T 2 ); and the matrix P m 1 = ( p 1, p 2, p m ) T is the amplitude to be inversed with T 1j time (for T 1 relaxation inversion) or T 2j time (for T 2 relaxation inversion). The T 1j or T 2j ( j = 1,, m) are pre-assigned relaxation time constants in T 1 or T 2 distribution range, which are m bins chosen with logarithmically equally spaced between the minimal and maximal

A new method for multi-exponential inversion of NMR relaxation measurements 267 relaxation times (T 1, 2min and T 1, 2max ). There are different ways to pre-assign the relaxation time constants, such as 2-to-the-nth-power spaced time constants in MAP-II introduced by NUMAR. t i In the coefficient matrix An m= aij = c1 + c2 exp, n m T 1,2 j n m for T 1 inversion, c 1 =1, c 2 = 2, T 1,2j = T 1j, and t i ( i = 1,, n) is the ith inverse-recovery measurement time; for T 2 inversion, c 1 = 0 c 2 = 1 T 1,2j = T 2j and t i ( i = 1,, n) is the time when the ith echo is recorded, t i = i*te, TE is a time interval between two echoes, or called echo space. Eq. (4) can be over-determined, determined or under-determined, depending on which is bigger, n or m, where n is the number of measured relaxation signals, and m is the number of pre-assigned relaxation time constants. The solution of eq. (4) determines the T 1 or T 2 time distribution. This is a typical ill-posed problem in mathematics, and usual very difficult to have analytical solution, so iteration solution has been employed. For example, Prammer, Xiao et al. and currently widely used commercial software MAP-II all applied SVD and NNLS to the iteration process. These methods, however, all have disadvantages such as being too sensitive to SNR and un-flexible to bin selection. Dines and Lyttle introduced an improved ART algorithm in their early work [8]. Making use of their algorithm, a new method is developed for NMR multi-exponential inversion. The implementation steps for the new method are as follows: First, given an initial model P, compute an error Y from the forecasted relaxation signal Y and the measured relaxation signal Y, that is: Y = Y Y, m a p = y. ij j i j= 1 (5) (6) For matrix A, all the elements a ij are greater than or equal to zero; y i, therefore, can be attributed to each p j according to the magnitude of a ij. It is essential for the new method to compute the correct magnitude p j from y i, so as to keep the continuity of NMR relaxation time distribution and ensure the inversed results physics meaningful. Second, letting p j = λa ij y i, and taking it into eq. (6), we have λ = 1. m 2 aij j= 1 (7) This is the coefficient for attributing y i to p j. The correction values for all rows (i.e. the correction value for each relaxation signal) are estimated respectively. Then, these www.scichina.com

268 Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.3 265 276 corrections are averaged according to the attributing coefficients and the number of non-zero elements at the jth column. The correction for p j is n ( q+ 1) 1 ( q) j λa IA( j) ij i i= 1 p = y, (8) where LA( j ) is not a fixed value m as in the ART method, but the number of non-zero elements of the jth column in matrix A. ( q) y i is the residual for the ith relaxation signal computed at the qth iteration. Because matrix A has many zero elements, LA( j ) is normally less than m; therefore the iteration of the new method is faster than that of the ART method. The expression for the corrected relaxation signal components is P ( q+ 1) ( q) = P + P. (9) Eqs. (8) and 9 are the basic iteration formulae for NMR T 1 and T 2 relaxation inversion. The new algorithm is called simultaneous iterative reconstruction technique (SIRT) because the approximate solutions to eq. (4) are not calculated until all equations are processed. In the physical sense, the amplitude in the NMR T 1 and T 2 spectra cannot be less than zero. Therefore p j ( j = 1,, m) must be non-negative in the process. When is negative, give it a very small positive number, for example, 1.0E-30. Then, the iteration computation is restarted till the variation of the correction for the equation solutions ( q) ( P/ p j ) satisfies the requirements or the pre-set iteration time finishes. Restricting p j to no-negative avoids the situation in which we are unable to calculate ( q) ( q) P / p j when the negative p j is set to zero. When we use adjacent relaxation spectrum data to counterpoise the negative p j, relaxation time distribution will keep smooth and have physical meanings. The SIRT method is superior to the traditional SVD method because it possesses the merits of simplicity, easy programming, no need for user s interaction, and no need for many complex pre-set inversion control parameters in the process. So the artificial errors for the inverted results are reduced. In addition it is faster than the SVD method when all the measured relaxation signals take part in the inversion process or the number of pre-assigned relaxation time constants is big. 3 Numerical results and discussion Using echo data for T 2 and inverse-recovery data for T 1 with different SNR or different pre-assigned bins, respectively, we test the applicability and feasibility of the new method. ( q) p j

A new method for multi-exponential inversion of NMR relaxation measurements 269 First of all, we construct a standard relaxation time spectrum that has a typical double-peak feature as shown in fig. 1. It can be either a T 2 or T 1 relaxation time distribution. Fig. 1. Relaxation time distribution for simulation. Based on this relaxation time distribution, echo trains with different SNR are simulated at TE (echo space) = 1.2 ms and NE (number of echoes) = 512 for T 2 relaxation time inversion test; inverse-recovery data with different SNR for T 1 relaxation time inversion test are also simulated. 3.1 Inversion results for noise-free echo train Using the constructed ideal echo train (SNR = ), we test the effects of different pre-assigned time constants on T 2 spectra. The pre-assigned time constants are logarithmically equally spaced with 64, 32 and 16 bins in a range from 0.1 ms to 10 s. Because the inversion speed of SVD algorithm with a large number of bins is very slow, NUMAR s MAP-II only provides an inversed T 2 spectrum with 10 bins of 2 to the nth power. In order to compare with the result of MAP-II, a T 2 spectrum inversed by SIRT method is provided which has 10 bins with 2-to-the-nth-power from 2 ms to 1024 ms. The inversion results are shown in fig. 2, which includes SIRT results with 64 and 16 logarithmically spaced bins and 10 2-to-the-nth-power bins, and MAP-II result with 10 2-to-the-nth-power bins. It is clear that the inverted T 2 spectrum with a large number of bins (64 bins, for instance) is very similar to the constructed T 2 spectrum. The difference between inverted T 2 spectrum and the simulated T 2 spectrum increases with the reducing number of pre-set bins. But in all cases, the inversed T 2 spectra can ensure the double-peak feature of the constructed T 2 spectrum. The T 2 spectrum of SIRT with www.scichina.com

270 Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.3 265 276 Fig. 2. Inversion results for different bin sets. Noise-free echo, SNR =. 1, Constructed relaxation time distribution; 2, logarithmically equal spaced with 64 bins; 3, with 16 bins; 4, with 10 bins; 5, MAP-II with 10 bins. 10 2-to-the-nth-power bins is similar to that of MAP-II. In general, The T 2 spectrum with 16 logarithmically spaced bins ensures the facticity of rock T 2 spectrum. So the number of pre-set bins must be large enough, that is from 16 to 64 commonly, and the time range for the pre-set bins is from 0.1 ms to 10000 ms, spreading three orders of magnitude. 3.2 Inversion results for echo trains with different SNR The numerical testing shows that the higher the SNR of echo train, and the less the averaged standard deviation for echo fit, the faster the iteration. For a low SNR (< 20) echo train, the averaged standard deviation is relatively big and hardly changes with the increase in the iteration number, which indicates that the new method is able to find the general trend for a low SNR echo train and to invert quickly a T 2 spectrum. Fig. 3(a) and 3(b) demonstrate this feature by comparing the inverted T 2 spectra by SIRT and MAP-II for echo trains with different SNR. Fig. 3(a) shows the contrast of the T 2 spectra converted by SIRT; and fig. 3(b) shows the contrast of the T 2 spectra converted by MAP-II. It can be seen that the T 2 spectrum converted by SIRT still keeps the double-peak feature as the constructed T 2 spectrum, and they are similar to each other. The testing results indicate that the new method is suitable for different SNR NMR measurement, even when the SNR is very low, lower than five for example. However, results from MAP-II keep the double-peak feature only for the high SNR, but not for the low SNR ( 20) at which the inversed results deviate from the constructed T 2 spectrum. The T 2 spectrum from the echo train with SNR of five, for example, just has a single peak. So the MAP-II method is only suitable for the high SNR ( 40) NMR measurement, and

A new method for multi-exponential inversion of NMR relaxation measurements 271 has a lower resolution than the SIRT method. Fig. 3. Inversion results for different signal to noise ratio. (a) SIRT; (b) MAP-II. 3.3 Inversion results for inverse-recovery signals with different SNR A series of inverse-recovery signals with different SNR as, 100, 60, 40, 20, 10 and 5 are simulated based on the relaxation time spectrum as fig. 1. T 1 spectra are converted using the SIRT method with 64 logarithmically spaced bins from 0.1 ms to 15000 ms. The results are shown in fig. 4. It can be seen that the T 1 spectra keep the double-peak feature as the constructed T 1 spectrum, and different ways for pre-assign time constants have little influence on the inversion results. The right peaks of the T 1 www.scichina.com

272 Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.3 265 276 Fig. 4. T 1 inversion results. (a) Different bin pre-set; (b) different SNR. 1, Constructed relaxation time distribution; 2, logarithmically equal spaced with 64 bins; 3, with 16 bins; 4, 2-power spaced with 10 bins. spectra rise gradually with the noise level increasing, which indicates that long T 1 is sensitive to noise in this method. 4 Application NMR T 1 and T 2 relaxation data measured from a rock core and a well are processed and analyzed using the new method. Fig. 5 shows an example of echo data (a) and converted T 2 spectrum (b) for a rock core. In the measurement TE is 1.2 ms, NE is 512 and SNR is 15. Porosity of the rock is 17.67%. In the figure, the raw echo data (solid dots) is compared with the fitted echo train (solid line), and converted T 2 spectrum by the new method with 32 logarithmically

A new method for multi-exponential inversion of NMR relaxation measurements 273 Fig. 5. Measured echo train (a) and inversion results (b). 1, SIRT logarithmically spaced; 2, MAP-II 2-power spaced. spaced bins is compared with the converted T 2 spectrum by MAP-II with 10 2-to-thenth-power spaced bins. The porosities from both T 2 spectra are 18.03% and 17.05% respectively. It is clear that the porosities are identical, and the difference to core porosity is very small. But for the low SNR echo data (SNR < 20), the converted T 2 spectrum by MAP-II deviates from the core analysis result quite a lot, and not able to reflect the true pore size distribution. On the contrary, T 2 spectrum from SIRT is very close to the core NMR analysis result. Fig. 6 shows an NMR T 1 relaxation signal (a) measured from a saturated rock core and T 1 spectrum (b) by SIRT. The rock core has porosity of 23.78% and permeability of www.scichina.com

274 Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.3 265 276 357.31 md. Fig. 6. Measured inverse-recovery signal (a) and inversion results (b). Fig. 7 is an example of NMR well logging. It is from an exploration well in Liaohe oilfield shale-sands formation. The echo train is converted by MAP-II and SIRT separately. The T 2 spectra converted by MAP-II with 10 2-to-the-nth-power spaced bins from 4 ms to 2048 ms have a single peak in general, even if in beds have good reserve capability, for example, in the interval from 2910 m to 2915 m. The T 2 spectra do not tally with the aperture distribution feature of sand rocks that have two or more peaks. On the contrary, the T 2 spectra converted by the new method with 20 logarithmically spaced bins from 0.3 ms to 3000 ms have two or three peaks in sand rocks, and are identical with the core analysis results. The example indicates that the new method has a better resolution than MAP-II.

A new method for multi-exponential inversion of NMR relaxation measurements 275 Fig. 7. T 2 distribution of NMR logging data by MAP-II (track 2) and new method (track 3). 5 Conclusions A new method for multi-exponential inversion of NMR T 1 or T 2 relaxation time distribution from core analysis data and well logging data is presented and tested. The numerical results suggest that the reasonable number of pre-assigned relaxation time constants is 16 to 64 for T 1 or T 2 inversion, and the time regime should be from 0.1 ms to 10000 ms which spans five orders of magnitude. The new method is stable, and the converted relaxation time distributions have better resolution than the test from MAP-II especially when the SNR is low. The new method is applicable to both low SNR core analysis and well logging data process and interpretation. www.scichina.com

276 Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.3 265 276 References 1. Prammer, M. G., NMR pore size distributions and permeability at the well site, 1994, SPE 28368, in SPE annual technical conference and exhibition proceedings, v. omega, Formation evaluation and reservoir geology: Society of Petroleum Engineers, 55 64. 2. Prammer, M. G., Principles of signal processing NMR data and T 2 distributions, SPWLA 36th Annual Symposium, Paris, France; Chapter 4, in Nuclear Magnetic Resonance Logging Short Course Notes (ed. Georgi, D.T.), Paris, France: Society of Professional Well Log Analysts, variously paginated, 1995. 3. Xiao Lizhi, Shi Hongbin, Low field NMR core analysis and its applications to well logging interpretation, Well Logging Technology (in Chinese), 1998, 22(1): 42 49. 4. Xiao Lizhi, NMR Imaging Logging and Rock NMR (in Chinese), Beijing: Science Press, 1998. 5. Wang Weiming, Li Pei, Ye Chaohui, Multi-exponential inversions of nuclear magnetic resonance relaxation signal, Science in China, Ser. A, 2001, 44(11): 1477 1484.[Abstract] [PDF] 6. Wang Caizhi, Li Ning, On the method of analyzing and processing T 2 relaxation spectra from NMR log data and relative program design on Cif2000 platform, Well Logging Technology (in Chinese), 2002, 26(5): 360 363. 7. Weng Aihua, Li Zhoubo, On high resolution inversion of NMR logging data, Well Logging Technology (in Chinese), 2002, 26(6): 455 459. 8. Dines, K., Lyttle, J., Computerized geophysical tomography, Proc. IEEE, 1979, 67: 1065 1073.