Method for obtaining NMR bound fluid volume using partial polarization

[0002] This invention relates to the field of well logging of earth boreholes and, more particularly, to methods for obtaining bound fluid volume from nuclear magnetic resonance measurements using partial polarization.

[0003] Well logging provides various parameters that may be used to determine the “quality” of a formation from a given borehole. These parameters include such factors as: resistivity, porosity, permeability and bound fluid volume (BFV). The parameters may be used to determine the amount of hydrocarbons present within the formation, as well as provide an indication as to the difficulty in extracting those hydrocarbons from the formation. BFV, in general, represents the volume of fluid that normally cannot be extracted from the formation (versus free fluid volume—the volume of fluid that will flow through the pores of the formation and thus, may be extracted). BFV is therefore, an important factor in determining whether a specific well site is commercially viable.

[0004] There are various known techniques for determining BFV. For example, it is known to apply nuclear magnetic resonance (NMR) techniques to measure BFV. NMR measurements, in general, are accomplished by causing the magnetic moments of nuclei in a formation to precess about an axis. The axis about which the nuclei precess may be established by applying a strong, polarizing, static magnetic field (B₀) to the formation, such as through the use of permanent magnets. This field causes the proton spins to align in a direction parallel to the applied field (this step, which is sometimes referred to as the creation of longitudinal magnetization, results in the nuclei being “polarized”). Polarization does not occur immediately, but instead grows in accordance with a time constant T₁, as described more fully below, and may take as long as several seconds to occur (even up to about eight seconds). After sufficient time, a thermal equilibrium polarization (M_∞) parallel to B₀has been established (i.e., M_∞ is proportional to B₀)

[0005] Next, a series of radio frequency (RF) pulses are produced so that an oscillating magnetic field, B₁, is applied. The first RF pulse (referred to as the 90° pulse) must be strong enough to rotate the magnetization from B₀substantially into the transverse plane (i.e., transverse magnetization). The rotation angle is given by:

[0006] α=γB₁t_p  (1)

[0007] (where t_pis the pulse length) and is adjusted to be 90°. Additional RF pulses, preferably a α=180° (referred to as 180° pulses), are applied to create a series of spin echos. The frequency of the RF pulses is chosen to excite specific nuclear spins of a particular region of the sample that is being investigated. The rotation angles of the RF pulses are adjusted to be 90° and 180° in the center of this region.

[0008] Two time constants are associated with the relaxation processes of the longitudinal and transverse magnetization. These time constants characterize the rate of return to thermal equilibrium of the magnetization components following the application of each 90° pulse. The spin-lattice relaxation time (T₁) is the time constant for longitudinal magnetization to return to its thermal equilibrium value M_∞ in the static magnetic field. The spin-spin relaxation time (T₂) is the time constant for the transverse magnetization to return to its thermal equilibrium value which is zero. In addition, B₀is typically inhomogeneous and the transverse magnetization decays with the shorter time constant T₂*, where: $\begin{array}{cc}\frac{1}{T_2^{*}}=\frac{1}{T_2}+\frac{1}{T^{\prime }}& \left(2\right)\end{array}$

[0009] but the part decaying with T, which is due to the inhomogeneous B₀, can be recovered by refocusing pulses that produce the echos.

[0010] The most common method for determining BFV involves determining the entire fully polarized T₂distribution and then computing results based on T₂values less than a fixed cutoff value (e.g., 33 milliseconds for sandstone). The portion of protons with T₂'s smaller than the fixed cutoff represents the BFV, while the portion of protons with T₂'s larger than the fixed cutoff represents the amount of free fluid in the formation.

[0011] Another known method for calculating BFV determines the entire T₂distribution for a given sample, and then computes the amplitude of the signal components with T₂values less than the “free fluid” cutoff value (or relaxation cutoff time). Instead of the traditional fixed cutoff, a tapered cutoff is theoretically determined that accounts for bound fluid volume lining large pores that would otherwise be considered to be free or extractable (see, R. L. Kleinberg et al., “Tapered Cutoffs for Magnetic Resonance Bound Fluid Volume,” Society of Petroleum Engineers, Doc. No. SPE 38737, 1997) (the “Kleinberg theoretical analysis”). For example, previous determinations of bound fluid assume that the bound fluid occupies small pores and free fluid occupies large pores. It was then assumed that the large pores would empty so that any fluid therein was free fluid, not bound fluid. The Kleinberg theoretical analysis, however, found that under certain circumstances, such as in clean well-sorted sands and carbonates where fixed cutoff computations provide low bound fluid results, the tapered cutoff provides a more accurate measure of BFV.

[0012] Another way to determine BFV using a tapered cutoff is based on empirically derived tapered cutoffs rather than the theoretically determine d tapered cutoff described above. The derived tapered cutoff relates each relaxation time to a specific fraction of capillary bound water, assuming that each pore size has an inherent irreducible water saturation (see, G. R. Coates et al., “A New Characterization of Bulk-Volume Irreducible Using Magnetic Resonance,” SPWLA38th Annual Logging Symposium, Jun. 15-18, 1997) (the “Coates empirical analysis”). The Coates empirical analysis utilized permeability models that do not rely on a specific model of pore geometry t o relate irreducible water saturation (S_WIRR) to the T₂distribution as follows: $\begin{array}{cc}\frac{1}{S_{\mathrm{WIRR}}}=m{T}_2+b& \left(3\right)\end{array}$

[0013] (where m and b a re empirical factors used for calibration of empirical data sets) which was then applied to different values of m with b=1, and different values of b with m=0.0618. In both the Kleinberg theoretical analysis and the Coates empirical analysis, the entire T₂distribution is used (rather than the cutoff method) to determine BFV, which is a direct output of the inversion of the echo data.

[0014] Another known method for calculating BFV using NMR is the fixed cutoff method which is described in commonly-assigned, Sezginer et al. U.S. Pat. No. 5,389,877 (Sezginer). Sezginer describes using NMR techniques in which a short train of spin echos (i.e., j echoes) are used to obtain a sharp cutoff (i.e, the fixed cutoff relaxation time T_C) that can be used to determine the producible volume in a borehole by measuring bound fluid and subtracting it from total porosity. These NMR techniques apply a weighted sum of the echos to determine BFV as follows: $\begin{array}{cc}\stackrel{\_}{\mathrm{BFV}}=\sum _{j=1}^J{w}_j{\mathrm{echo}}_j& \left(4\right)\end{array}$

[0015] (where w_jis a weighting factor chosen to weight different T₂components differently to sharpen (i.e., make steeper) the produced cutoff curve, and the overbar represents an estimate of BFV). The estimator of BFV is a linear function that acts on the relaxation-time distribution: $\begin{array}{cc}\stackrel{\_}{\mathrm{BFV}}={\int }_0^{\infty }f\left(T_1\right)\alpha \left(T_1\right)d{T}_1& \left(5\right)\end{array}$

[0016] where 𝑓(T₁) is a weighting function as follows: $\begin{array}{cc}f\left(T_1\right)=\left[1-e^{-T_r/T_1}\right]\sum _{j=1}^J{w}_j\exp \left(\frac{-2j{t}_{\mathrm{cp}}\left(T_1/T_2\right)}{T_1}\right)& \left(6\right)\end{array}$

[0017] (where T_ris the recovery time before a CPMG sequence, and t_cpis the Carr-Parcell spacing). 𝑓(T₁) is approximately equal to 0 if T₁is greater than T_C, and approximately equal to 1 if T₁is less than T_C. Sezginer assumes that the T₁/T₂ratio is approximately constant, being about unity for bulk water samples and about 1.5 for water saturated sandstones. A potential problem with the sharp cutoff of Sezginer may occur if the echo decays faster than predicted, for example, if motion of the measuring probe occurs during measurements. Under these conditions, the resultant data may be degraded.

[0018] While Sezginer assumes that the T₁/T₂ratio is approximately constant, the T₁/T₂ratio must be considered when a shortened wait time between NMR experiments (e.g., as set forth in Sezginer) is used in determining BFV. For example, it is known that the NMR signal from a fluid is proportional to the hydrogen index (HI) and T₁Effect (TOE Factor) as described by Kleinberg & Vinegar in “NMR Properties for Reservoir Fluids,” The Log Analyst, Nov.-Dec. 1996. Kleinberg and Vinegar provide that the TOE Factor is defined as:

[0019] TOE Factor=[1−exp(−t_r/T₁)]  (7)

[0020] (where t_ris the wait time between NMR measurements and T₁is the longitudinal relaxation time, as described above). Kleinberg and Vinegar also showed that the signal for a given T₂may be determined as:

[0021] S(T₂)=V_water(T₂)×HI_water×[1−exp (−t_r/T_1water)]

[0022] +V_oil(T₂)×HI_oil×[1−exp(−t_r/T_1oil)]

[0023] +V_gas(T₂)×HI_gas×[1−exp(−t_r/T_1gas)]  (8)

[0024] where T₂is the transverse relaxation time as described above. T₁is a different function of T₂for each fluid. By factoring the ratio of T₁/T₂, the TOE Factor can be calculated based on T₂instead of T₁as follows:

[0025] TOE Factor =[1−exp(−t_r/(Ratio*T₂)]  (9)

[0026] where Ratio is the ratio of T₁/T₂which is expected to be about 1.5 for clay and capillary bound fluid formations, but may be equal to 1 under other conditions. Applying this ratio results in polarization expressions (i.e., equations 7 and 9) that are based on either T₁or T₂.

[0027] Most of the above-described methods for determining BFV suffer from at least the common problem that the entire T₂distribution must be obtained (for example, Sezginer, on the other hand, utilizes only part of the T₂distribution). Under some circumstances, for example, in logging-while-drilling, it simply may not be possible to acquire the necessary T₂distribution, or the motion of logging-while-drilling may degrade the measured data (e.g., such as when the techniques of Sezginer are applied). Under other circumstances, such as when high logging speed is required, it may not be practical to obtain the entire T₂distribution. Under still other circumstances, it only may be practical to obtain the initial amplitude of the NMR signal, in which case, known methods for determining BFV are not available. Moreover, well logging typically is a time intensive task, requiring large expenditures of effort to acquire well data.

[0028] For at least the foregoing reasons, it is an object of the present invention to provide methods for determining BFV utilizing NMR techniques when less than a substantial part of the T₂distribution is available.

[0029] It is also an object of the present invention to provide methods for determining BFV utilizing NMR techniques when high logging speed is required.

[0030] It is a further object of the present invention to provide methods for determining BFV utilizing NMR techniques in which only the initial amplitude of the NMR signal is obtained.

[0031] It is a still further object of the present invention to provide methods for determining BFV utilizing NMR techniques in which well logging time is reduced.

[0032] These and other objects of the invention are accomplished in accordance with the principles of the invention by providing methods for determining the bound fluid volume (BFV) of a formation utilizing nuclear magnetic resonance (NMR) techniques in which less than full polarization occurs and in which less than a substantial part of the NMR distribution is acquired. The preferred embodiments include methods in which a shortened wait time is utilized so that only part of the formation is polarized and a limited number of echo pulses are utilized to determine BFV. By only polarizing the bound fluid of the formation and by only requiring a limited number of echo pulses, the NMR methods of the present invention are capable of providing BFV at high logging speeds, such as those in excess of 1,800 feet per hour without a substantial degradation of vertical response. Moreover, the techniques of the present invention also may be used during logging-while-drilling operations, at least because of the shortened necessary duration of the echo train that results from measurements of partial polarization instead of T₂decay.

[0033] The methods of the present invention utilize a significantly shortened wait time t_r, where t_ris the time from the last 180° pulse of one NMR experiment until the application of the 90° pulse to start the next NMR experiment—persons skilled in the art will appreciate that, in the circumstance where only single pulse sequences are utilized (e.g., while the drill pipe is moving), the shortened wait time is defined from the time the spins enter the static magnetic field. The shortened wait time is selected so that the static magnetic field B₀is only applied for a short period of time before the next NMR experiment begins. This results in B₀polarizing only the bound fluid in the formation, rather than the bound and free fluid, so that the measured amplitude represents the actual BFV of the part of the formation being analyzed.

[0034] The length of time it takes to run any NMR experiment is mainly determined by the wait time, which is greater than the time required to measure the echos. Therefore, because of the shortened wait time, the partial polarization NMR experiments of the present invention always take less time to run than conventional NMR experiments. This results in the ability to stack more experiments—i.e., perform and add up more experiments in a given period of time—resulting in an improved signal-to-noise ratio (SN) (because SN is proportional to the square root of the number of acquisitions (n)). The NMR experiments of the present invention provide information so that a partial polarization calculation curve in T₂lies almost equally between the Kleinberg and Coates tapered cutoff curves.

[0035] Further features of the invention, its nature and various advantages will be more apparent from the accompanying drawings and the following detailed description of the preferred embodiments.

[0036] FIG. 1 is a flow chart illustrating steps for determining bound fluid volume (BFV) in accordance with the principles of the present invention;

[0037] FIG. 2 is a flow chart illustrating a preferred embodiment of the method of determining BFV of FIG. 1 in accordance with the principles of the present invention;

[0038] FIG. 3 is a graph showing a comparison of polarization versus T₂as determined for two theoretical tapered cutoff analyses (i.e., Kleinberg theoretical analyses) and several partial polarization determinations in accordance with the present invention;

[0039] FIG. 4 is a graph showing a comparison of T₂distributions for a low permeability sample as determined for a theoretical tapered cutoff analysis (i.e., Kleinberg theoretical analysis) and for a partial polarization analysis in accordance with the present invention;

[0040] FIG. 5 is a graph showing a comparison of T₂distributions for a high permeability sample as determined for a theoretical tapered cutoff analysis (i.e., Kleinberg theoretical analysis) and for a partial polarization analysis in accordance with the present invention;

[0041] FIG. 6 is a graph showing another comparison of T₂distributions for a low permeability sample as determined for an empirical tapered cutoff analysis (i.e., Coates empirical analysis) and for a partial polarization analysis in accordance with the present invention;

[0042] FIG. 7 is a graph showing another comparison of T₂distributions for a high permeability sample as determined for an empirical tapered cutoff analysis (i.e., Coates empirical analysis) and for a partial polarization analysis in accordance with the present invention;

[0043] FIG. 8 is a graph showing a comparison of the cutoff that is produced by the partial polarization techniques of the present invention versus theoretical (i.e., Kleinberg theoretical analysis) and empirical (i.e., Coates empirical analysis) tapered cutoff; and

[0044] FIG. 9 is a flow chart illustrating steps for another embodiment of determining bound fluid volume (BFV) in accordance with the principles of the present invention.

[0045] Various well known techniques exist for determining many of the parameters associated with the production of hydrocarbons from ground formations using nuclear magnetic resonance (NMR) tools. These parameters include porosity φ (the fraction of the formation per unit volume occupied by pore spaces), permeability k (the ease with which fluid passes through the formation), free fluid volume FFV (the fluid within the formation that is capable of being extracted through the pores), and bound fluid volume BFV (the fluid within the formation that is trapped inside pores and will not flow out during the extraction process). The present invention describes methods for determining BFV based on a limited quantity of acquired data.

[0046] Typical NMR techniques for determining various geological parameters include a lengthy, time consuming process in which T₂distributions with full polarization of almost all T₁components are obtained and analyzed. The present invention provides methods for determining BFV utilizing NMR techniques in which only partial polarization is performed so that recording fully polarized, complete, T₂distributions are unnecessary.

[0047] In accordance with the principles of the present invention, a substantially short wait time is utilized between NMR experiments so that the static magnetic field B₀only polarizes the bound fluid of the formation (as compared to traditional NMR experiments where the wait time is extensive to insure that the applied static magnetic field B₀has sufficient time to fully polarize the part of the formation being analyzed before the application of the oscillating magnetic field—a time typically up to five times T_1MAX(i.e., the time required to insure that all of the fluid—both bound fluid and free fluid—in the formation is polarized), where T_1MAXis the largest T₁value present in the sample).

[0048] In addition, because only the initial amplitude of the echo train is measured, a substantially limited number of echos are required to measure induced magnetic signals (conventional NMR techniques typically apply hundreds of echos to produce a complete T₂distribution). Moreover, the short wait time together with the reduced number of echos causes the NMR experiment to take less time than traditional NMR experiments. The short experiment time enables the operator to conduct more experiments in a given time period resulting in logging data having a significantly improved signal-to-noise ratio. The improvement in signal-to-noise ratio is often on the order of at least three times over conventional techniques, at least because the length of time to run one partial polarization NMR measurement is typically at least ten times shorter than a conventional measurement. Thus, at least ten times more partial polarization NMR experiments may be run for a given period of time than conventional NMR experiments and the signal-to-noise ratio scales with the square root of n, where n is the number fo experiments.

[0049] FIG. 1 shows a flow diagram that generally illustrates methods for determining BFV in accordance with the principles of the present invention. Initial preparations may occur in step 102 to determine the wait time t_rthat is used in the NMR measurement. As described above, the methods of the present invention require that t_rbe substantially shorter than t_rin conventional measurements. For example, it is known that t_rfor conventional NMR measurements in rocks is on the order of about one to eight seconds (1-8 sec.). In accordance with the present invention, however, t_ris always less than one-tenth of one second, and, for experiments in sandstones, is preferably on the order of ten to sixty milliseconds (10-60 msec.). The determination of t_rdepends on the petrophysical properties of the formation being sampled. For example, wait time t_ris less for sandstones than for carbonates. Thus, in general, the determination of t_ris empirical, based on known information about the formation being sampled.

[0050] Initial preparations may also occur in step 104 to determine the logging speed of the instant NMR measurement. For example, while it may be preferred to acquire full T₂distributions (because, for example, a full distribution is more useful for post-collection data analysis), circumstances may be such that it is not possible to obtain full distributions. Additionally, if only BFV is required, the partial polarization techniques of the present invention enable the logging operator to run the NMR measurement at high speeds of approximately 1800 feet per hour, versus traditional NMR well logging speeds, in order to save time. Moreover, the same techniques enable the operator to conduct BFV NMR experiments during logging-while drilling operations without data degradation even though it is likely that the NMR tool will experience lateral motion during NMR experiments.

[0051] Once the operational parameters have been determined, they may be entered into the NMR tool sequence program in step 106. The logging operator, in step 108, places the tool in the borehole. Persons skilled in the art will appreciate that the partial polarization NMR experiments of the present invention simply may begin at step 108, instead of step 102. Prior to the introduction of the NMR tool into the borehole, the spins of the nuclei will generally be aligned with the earth's naturally occurring magnetic field. The presence of the NMR tool in step 110, and its static magnetic field B₀, however, causes the spins to begin to be aligned with B₀. In accordance with the present invention, the wait time t_rbetween NMR experiments is substantially short so that the application of B₀is limited to polarizing the bound fluid of the formation, as compared to traditional NMR borehole measurements (as described above).

[0052] After the significantly shortened wait time t_rhas passed, in step 112, an oscillating magnetic field B₁is applied. The application of B₁causes the spins to tip into the transverse plane, at which point they begin to precess (at a frequency equal to the Larmor frequency). The application of B₁, as described in more detail with respect to FIG. 2, induces the protons to emit magnetic energy that may be measured by the NMR tool, in step 114. The measured signals are then analyzed, in step 116, to determine the initial amplitude (AMP_MAX) of the echo train at “time zero” (where “time zero” is the point of origin of the echo train whose position within the excitation pulse is understood by persons skilled in the art). The determination of AMP_MAXmay be performed by known techniques, such as extrapolation whereby AMP_MAX(the amplitude at time t=0) is extrapolated backwards from the measured NMR echos (e.g., the echos from time=1, 2, 3, 4, 5 and 6). The extrapolation should be insensitive to variations in the decay rate of the echo train (i.e., using a porosity estimator). Finally, in step 118, AMP_MAXis provided as an output that corresponds to the bound fluid volume (BFV).

[0053] The application of the oscillating field B₁is shown in more detail in FIG. 2. The first step in applying B₁is the application of an oscillating magnetic field perpendicular to the static magnetic field B₀(i.e., the 90° pulse), in step 202. This signifies the “beginning” of a single NMR measurement and the end of the significantly shortened wait time t_r. As the spins begin to precess, they generate a small magnetic field that can be detected by the NMR equipment (i.e., step 114 above). However, because the protons quickly lose synchronization, the net magnetization is quickly reduced. In order to obtain accurate NMR measurements, in step 204, one or more rephasing pulses (i.e., the 180° pulses) are applied that are perpendicular to static magnetic field B₀. The rephasing pulses induce additional magnetic signals from the protons (i.e., spin-echos) that are measured. The shortened wait time t_rbegins after the application of the last rephasing pulse for a given NMR experiment, and ends when the next 90° pulse is applied.

[0054] In accordance with the principles of the present invention, NMR measurements of BFV may be accomplished with a single echo pulse in step 204. It may be preferable, however, to obtain on the order of five or six echo pulses to improve signal quality. In any case, the present invention provides that BFV measurements may be obtained where less than ten echo pulses are generated, as compared to traditional NMR measurements in which hundreds or thousands of echo pulses are utilized to acquire a full T₂distribution.

[0055] Additional advantages of the present invention result from the shortened wait time t_rand the reduced number of echo pulses. These two factors combine to provide very short duration NMR experiments that provide the opportunity to stack significantly more NMR experiments in a given time period than conventional NMR experiments. The signal-to-noise ratio (SN) of the acquired data significantly improved by this process because SN is proportional to the square root of the number of acquisitions. Thus, BFV measurements obtained in accordance with this invention have superior signal-to-noise ratios as compared with conventional BFV measurements.

[0056] FIGS. 3-7 show various comparisons of partial polarization cutoffs versus the empirical and theoretical tapered cutoff calculations. FIG. 3 shows polarization versus T₂for a traditional fixed T₂cutoff (i.e., T_2cutoff) of 33 milliseconds (i.e., for sandstone) on curve 302. Curve 304 represents a tapered T₂cutoff derived theoretically based on T′=0.5*T_2cutoff, while Curve 306 represents a tapered T₂cutoff derived empirically based on T′=0.22*T_2cutoff. The dashed curves are partial polarization curves in T₂(according to equation 9) calculated for different wait times t_rand a ratio T₁/T₂of 1.5. For example, curve 308 shows the partial polarization curve in T₂for t_requal to 30 milliseconds; curve 310 shows the partial polarization curve in T₂for t_requal to 60 milliseconds; and curve 312 shows the partial polarization curve in T₂for t_requal to 100 milliseconds. FIG. 3 clearly shows the correlation between the theoretical tapered cutoff in T₂and the partial polarization cutoff according to the present invention expressed using equation 9 in T₂when t_requals 60 milliseconds; and also the correlation between the empirical tapered T₂cutoff and the partial polarization cutoff according to the present invention expressed using equation 9 in T₂when t_requals 30 milliseconds for a T₁/T₂ratio of 1.5.

[0057] For each of curves 308, 310 and 312, partial polarization is based on Equation (7) above, which shows that polarization corrections may be solely a function of T₁, however, because the curves in FIG. 3 are plotted against T₂, the T₁/T₂ratio is used as described above in accordance with Equation (9). Thus, only a knowledge of T₁is necessary to practice the partial polarization techniques of the present invention and the time required to obtain full T₂distributions may be avoided.

[0058] The tapered cutoff calculations are based on the “Kleinberg theoretical analysis,” which recites BFV as follows: $\begin{array}{cc}\mathrm{FV}=\sum _{T_{2i}=T_{2\min }}^{T^{\prime }}m\left(T_{2i}\right)+\sum _{T_{2i}=T^{\prime }}^{T_{2\max }}m\left(T_{2i}\right)\left(\frac{2T^{\prime }}{T_{2i}}-{\left(\frac{T^{\prime }}{T_{2i}}\right)}^2\right)& \left(10\right)\end{array}$

[0059] where m(T_2i) is the T₂distribution of fully saturated rock and T₂=0.5T_2cutoff, arid where the tapered cutoff data shown in FIG. 3 is plotted from the function within the outermost parenthesis of equation (10).

[0060] FIG. 4 shows an original T₂distribution for a low permeability sample compared to a tapered T₂cutoff calculation and a partial polarization calculation made using equation 9 in T₂. Curve 402 represents an actual laboratory T₂distribution obtained from a silty sandstone sample. NMR analysis of the sample showed it to have a porosity of 31.80 pu and a permeability of 6.39 md. Curve 404 represents a tapered cutoff calculation of T₂based on T′=0.22* T_2cutoff, where T_2cutoffis 33 milliseconds (i.e., the fixed cutoff). Curve 404 results in a tapered T₂cutoff calculation for BFV of 25.42 pu. Curve 406 represents a partial polarization curve in T₂according to equation 9 with a wait time t_rof 30 milliseconds. Curve 406 results in a partial polarization calculation in T₂for BFV of 25.42 pu—within 0.01 pu of the value for the tapered T₂cutoff.

[0061] FIG. 5 shows an original T₂distribution for a high permeability sample compared to a tapered T₂cutoff calculation and a partial polarization measurement in T₂. Curve 502 represents an actual laboratory T₂distribution obtained from a sandstone sample. NMR analysis of the sample showed it to have a porosity of 33.70 pu and a permeability of 154.36 md. Curve 504 represents a tapered cutoff calculation of T₂based on T′=0.22*T_2cutoff, where T_2cutoffis 33 milliseconds (i.e., the fixed cutoff). Curve 504 results in a tapered T₂cutoff calculation for BFV of 11.07 pu. Curve 506 represents a partial polarization curve in T₂according to equation 9 with a wait time t_rof 30 milliseconds. Curve 506 results in a partial polarization calculation in T₂for BFV of 12.06 pu—within 0.99 pu of the value for the tapered T₂cutoff. Thus, for both high and low permeability formations, the BFV results from tapered T₂cutoff and partial polarization in T₂are within 1 pu of each other.

[0062] FIG. 6 shows an original T₂distribution for a low permeability sample compared to an empirical tapered T₂cutoff calculation and a partial polarization distribution (in accordance with equation 9) in T₂. Curve 602 represents an actual laboratory T₂distribution obtained from a silty sandstone sample. NMR analysis of the sample showed it to have a porosity of 31.80 pu and a permeability of 6.39 md. Curve 604 represents an empirical tapered cutoff calculation of T₂based on m=0.0618 and b=1. Curve 604 results in an empirical tapered T₂cutoff calculation for BFV of 19.19 pu. Curve 606 represents a partial polarization distribution (in accordance with equation 9) in T₂with a wait time t_rof 15 milliseconds. Curve 606 results in a partial polarization calculation in T₂for BFV of 19.24 pu—within 0.05 pu of the value for the tapered T₂cutoff.

[0063] FIG. 7 shows an original T₂distribution for a high permeability sample compared to an empirical tapered T₂cutoff calculation and a partial polarization distribution (in accordance with equation 9) in T₂. Curve 702 represents an actual laboratory T₂distributions obtained from a sandstone sample. NMR analysis of the sample showed it to have a porosity of 33.70 pu and a permeability of 154.36 md. Curve 704 represents an empirical tapered cutoff calculation of T₂based on m=0.0618 and b=1. Curve 704 results in an empirical tapered T₂cutoff calculation for BFV of 9.14 pu. Curve 706 represents a partial polarization distribution (in accordance with equation 9) in T₂with a wait time t_rof 15 milliseconds. Curve 706 results in a partial polarization calculation in T₂for BFV of 8.18 pu—within 0.96 pu of the value for the tapered T₂cutoff. Thus, for both high and low permeability formations, the BFV results from an empirically determined tapered T₂cutoff and partial polarization in T₂still are within 1 pu of each other.

[0064] FIG. 8 shows polarization versus T₂for a partial polarization distribution (plotted in accordance with equation 9) on curve 802. Curve 804 represents a tapered T₂cutoff derived theoretically based on T′=0.22*33 (msec), while curve 806 represents a tapered T₂cutoff derived empirically based on m=0.0618 and b=1. FIG. 8 clearly shows the correlation between the theoretical and empirical tapered T₂cutoffs and the partial polarization cutoff according to the present invention in T₂when t_requals 20 milliseconds and the T₁/T₂ratio is equal to 1.5. In fact, it should be noted that the curve of the partial polarization calculation in T₂lies almost equally between the two tapered cutoff curves.

[0065] FIG. 9 is a flow chart illustrating steps for another embodiment of determining bound fluid volume (BFV) in accordance with the principles of the present invention. The inventive principles set forth with respect to FIG. 9 may be combined with the steps shown and described with respect to FIG. 1 above, or they may be carried out in a “single” NMR experiment that assumes the NMR tool is already in the borehole. Instead of choosing a single shortened wait time t_r, which may have been inappropriate for a given formation, a series of shortened wait times are selected and applied as a “multi-wait sequence NMR experiment” (if an inappropriate shortened wait time is chosen and full T₂distributions were not obtained, the well log cannot simply be rerun with a modified shortened wait time to produce the desired data). The shortened wait times in the multi-wait sequence are chosen to provide a smooth interpolation between them so that the redundancy of the acquired data provides adequate precision.

[0066] For example, assuming that the selected shortened wait time for an NMR experiment according to FIG. 1 is 20 milliseconds, a multi-wait sequence NMR experiment would apply three NMR “sub-experiments” at 10, 20 and 30 milliseconds as follows. In step 910, the static magnetic field is applied for 10 milliseconds, at which time, in step 912, the oscillating magnetic field is applied. The induced signals are detected in step 914 and analyzed in step 916 (which is substantially similar to step 116 described above) to produce AMP_MAX10. The result is stored in a step 920. A test is performed in step 922 to determine whether additional sub-experiments are to be run (in this instance, two more sub-experiments are to be run, one for 20 milliseconds and one for 30 milliseconds). Steps 910 through 920 are carried out again utilizing 20 milliseconds and 30 milliseconds as the shorted wait time to produce AMP_MAX20and AMP_MAX30, respectively. Finally, in step 924, AMP_MAX, AMP_MAX20and AMP_MAX30are interpolated and AMP_MAXis output. Persons skilled in the art will appreciate that the multi-wait sequence of FIG. 9 may be utilized using more or less than the three shortened wait times described herein (for example, four shortened wait times).

[0067] It will be understood that the foregoing is only illustrative of the principles of the invention, and that various modifications can be made by those skilled in the art without departing from the scope and spirit of the invention.