Optimization of CPMG sequences to measure NMR transverse relaxation time T 2 in borehole applications

Nuclear Magnetic Resonance (NMR) can provide key information such as porosity and permeability for hydrological characterization of geological material. In particular the NMR transverse relaxation time T2 is used to estimate permeability since it reflects a pore-size dependent relaxation process. The measurement sequence (CPMG) usually consists of several thousands of electromagnetic pulses to densely record the relaxation process and to avoid relaxation processes that are due to diffusion. These pulses are equidistantly spaced by a time constant tE. In NMR borehole applications the use of CPMG sequences for measuring the transverse relaxation time T2 is limited due to requirements on energy consumption. For measuringT2, it is state-of-the-art to conduct at least two sequences with different echo spacings ( tE) for recording fast and slow relaxing processes that correspond to different pore-sizes. We focus on conducting only a single CPMG sequence and reducing the amount of energy while obtaining both slow and fast decaying components and minimizing the influence of relaxation due to diffusion. Therefore, we tested the usage of CPMG sequences with an increasing tE and a decreasing number of pulses. A synthetic study as well as laboratory measurements on samples of glass beads and granulate material of different grain size spectra were conducted to evaluate the effects of an increasingtE. We show that T2 distributions are broadened if the number of pulses is decreasing and the mean grain size is increasing, which is mostly an effect of a significantly shortened acquisition time. The shift of T2 distributions to small decay times as a function of tE and the mean grain size distribution is observed. We found that it is possible to conduct CPMG sequences with an increasedtE. According to the acquisition time and increasing influence of relaxation due to diffusion, the sequence parameters need to be chosen carefully to avoid misinterpretations.


Introduction
The method of nuclear magnetic resonance (NMR), discovered in the forties of the last century (Bloch et al., 1946;Purcell et al., 1946), has found widespread use in scientific and daily life.It was not later than the sixties that a first borehole tool was developed (Brown and Gamson, 1960) to take advantages of the unique properties of NMR in geophysics.It allows a direct detection and quantification of water since the initial amplitude of an NMR signal corresponds to the amount of protons exited.Furthermore, the measured relaxation times (longitudinal relaxation time T 1 , transverse relaxation time T 2 ) depend on the pore geometry and can therefore be used to estimate hydraulic permeabilities (Seevers, 1966).
In laboratory-NMR, artificial magnetic fields and sequences to measure T 2 were early available.These CPMG sequences (Carr and Purcell, 1954;Meiboom and Gill, 1958) with several thousand pulses and a short equidistant pulse spacing (t E ) allow for densely recording the decaying signal, ensuring a sufficient acquisition time and minimize relaxation processes due to diffusion.Each of these pulses causes an NMR echo to be the measured signal (Hahn, 1950).Thus, t E is referred to as the echo spacing of a CPMG sequence.
In borehole-NMR, there has been significant changes in the design of borehole tools in the nineties (Miller et al., 1990) using artificial magnetic fields and measuring T 2 using a CPMG.This made NMR a frequently used tool in oil and gas exploration.However, due to limitations on available pulse power for conduction of several thousands of pulses, until now it is common to obtain T 2 distribution based on conducting at least two different CPMG sequences in NMR borehole measurements.While slow decaying components are acquired with a sequence consisting of 1000 pulses and a large echo spacing (t E ), fast decaying components can be measured with only a few dozen echos and short echo spacing (Kruspe et al., 2009).Recently and in addition to obtaining T 2 , there is an increasing interest in measuring multidimensional distributions involving diffusion and relaxation based on sequences, such as presented in Hürlimann and Venkataramanan (2002).
Beside applications in oil and gas exploration, NMR borehole measurements are needed in near-surface groundwater investigation.There are several reasons NMR logging has just started for this purpose.Standard NMR logging of groundwater monitoring wells is far too expensive with common oil and gas tools and service.Monitoring wells are of diameters as small as two inches, so a large diameter NMR tool cannot be used.A first commercially available small diameter NMR tool was introduced by Vista-Clara in 2011.This tool is designed for slimhole boreholes and hydrogeological investigations.An external surface station provides the necessary pulse power from the surface to the probe that is lowered into the well.This currently limits the tool to borehole depths of about 220 m (Vista Clara, 2011).The Larmor frequency for this tool can be switched between 245 and 290 kHz.This ensures a sufficient penetration depth (190 mm from the tool center), even for drilling in unconsolidated material.Nevertheless, NMR measurements conducted with a low Larmor frequency provide a low signal-to-noise ratio, which has to be compensated by a high number of stacks.This leads to logging usually performed as point measurements.In cases of shallow boreholes and unconsolidated material that need sufficient penetration depth to obtain reliable results, this is probably the best way to design an NMR borehole tool; however, it is not applicable for deeper boreholes and incompatible with standard logging equipment.
It is this huge energy needed for CPMG pulse sequences that limits the construction of a small diameter tool that is interchangeable with logging equipment usually used in nearsurface logging business.Additionally, if a borehole tool should operate in depths up to 1000 m, it is inevitable that all electronics for generating a NMR pulse sequence have to be implemented in the borehole tool.Therefore, research for developing an NMR borehole tool to characterize groundwater layers is still needed.
At the current state, we do not focus on a development of sequences that allow for estimating multi-dimensional relationships of diffusion and relaxation.To our knowledge a single CPMG measurement will not be sufficient for this purpose.We focus on conducting a single CPMG sequence that allows for obtaining slow and fast decaying components, minimizing the influence of relaxation due to diffusion and reducing the amount of energy.This can be reached by reducing the number of pulses and adjusting the echo spacing (t E ) between the pulses.We evaluated our optimized CPMG sequence with different parameter sets and various ways of increasing t E in a synthetic study.The synthetic study is based on simple T 2 relaxation processes.We are aware of complex spin dynamic that may occur due to changed echo spacing within one sequence.But numerical modeling based on the modified Bloch-Torrey equation (Torrey, 1956) would be beyond the scope of this paper.However, to prove our findings we conducted laboratory experiments.In order to have sufficient control on pore-size, laboratory measurements were conducted on samples of glass beads and granular material of different grain size spectra.This helps to cover a broad range of material that may be observed in field experiments.

Free induction decay
As a property of matter, namely the spin, a hydrogen proton posses a microscopic magnetic moment.In a static magnetic field B 0 , a torque is applied that leads to an orientation of the magnetic moments along the stream lines of that field.According to the field strength of B 0 , the protons precess with the Larmor frequency ω L = γ • B 0 about the stream lines of B 0 , where γ [Hz T −1 ] is the gyromagnetic ratio.If an electromagnetic pulse (B 1 ) with the frequency ω L perpendicular to B 0 is applied, the protons can be flipped about an angle dependent on the pulse duration and amplitude.Immediately after termination of the excitation pulse, all affected protons return (relax) to the equilibrium state, i.e., the direction of the external static magnetic field (B 0 ).During relaxation the protons precess about the stream lines of B 0 .If the stream lines of B 0 denote the z-direction, the relaxation process is observed in the x-y plane.The decay in the x-y plane can be described by Eq. (1).
Here, M 0 is the initial amplitude, T 2 the transverse relaxation time and M xy the magnetisation in the x-y plane.For further information on basic NMR theory we refer to Dunn et al. (2002) or Coates et al. (1999).Basic parameters of interest are the initial amplitude M 0 at t = 0, due to its sensitivity to water content and the relaxation time T 2 as an indicator of pore space properties.Hence, the transverse relaxation time T 2 denotes how fast or slow the magnetisation decreases in the x-y plane.
However, the relaxation process in the x-y plane might be significantly influenced by inhomogeneities in the magnetic field, i.e., magnetic gradients.These inhomogeneities can be on macro scale like variations in the field strength of B 0 , or small scaled due to particles with a significant magnetic susceptibility in the porous medium.Variations of the magnetic field intensities lead to slightly different Larmor frequencies and therefore to a loss of phase coherence in the x-y plane.The precession of protons in the x-y-plane under the influence of magnetic gradients is illustrated schematically in Fig. 1.At t = t 0 all protons are coherent.The slightly different Larmor frequencies lead to a dephasing of the protons, which results in a reduced transverse magnetization.This relaxation time observed is mostly referred to as T * 2 (Dunn et al., 2002).This relaxation time T * 2 of water in porous media consists of three parts: the bulk relaxation T 2,B , the surface relaxation T 2,S , and the relaxation due to any kind of dephasing T 2,P .All parts are parallel processes and sum up to T * 2 , the relaxation time of the free induction decay (FID) as decribed in Eq. ( 2) (Mohnke and Yaramanci, 2008).
The bulk relaxation T 2,B denotes the relaxation time of free water, the surface relaxation T 2,S describes any influences of the pore surface to the protons in the vicinity of the surface, and T 2,P includes all kinds of magnetic gradients (macroscopic, microscopic).

T 2 -relaxation
By conducting a CPMG sequence (Carr and Purcell, 1954;Meiboom and Gill, 1958), influences of magnetic gradients can be minimized.Due to the CPMG sequence, T 2,P can be simplified to a diffusion term T 2,D .The relaxation due to diffusion T 2,D occurs if a proton moves through magnetic gradients either macroscopic or microscopic.While moving, the proton passes through regions with varying magnetic field strengths, which results in changing Larmor frequencies.According to different pathways of single protons within a set of protons and therefore differences in the Larmor frequencies, the set of protons lose their phase coherence.This diffusion effect is non-removable.The diffusion term can be expressed via Eq.(3) (e.g., Kleinberg and Horsfield, 1990): with the gradient G [G cm −1 ] of the magnetic field, γ [Hz T −1 ] the gyromagnetic ratio, D [m 2 s −1 ] the diffusion coefficient and t E [s] the spacing between an echo and the inducing pulse.As T 2,P can be reduced to T 2,D , Eq. (2) transforms to Eq. ( 4), which now describes all effects that influences the T 2 decay time.
According to Eq. (3), diffusion effects can be minimized by choosing a small t E while conducting a CPMG sequence.Equation (3) also indicates that diffusion effects take more effect if the echo spacing t E is too large.If the diffusion term can be assumed to be negligible, only T 2,B and T 2,S affect the decay time in Eq. (4).

Estimation of pore size by means of T 2
According to Eq. ( 4), the transverse relaxation time T 2 measured by a CPMG sequence consists of T 2,B and T 2,S .The bulk relaxation T 2,B denotes the relaxation of free water.Here the protons are only affected by other protons during relaxation.The surface relaxation T 2,S includes all effects of the grain surface that lead to a faster relaxation.These effects imply magnetic properties and the structure of the inner surface of the material.The surface relaxation, assuming a fast diffusion regime (Brownstein and Tarr, 1979), can be expressed by Eq. ( 5): whereas ρ [m s −1 ] denotes the surface relaxivity, S [m 2 ] the surface area and V [m 3 ] the volume of a pore.In a porous media, V and S refer to the volume and the surface area of the whole sample.In this case the surface relaxivity is a mean value of all water-filled pores.According to Eq. ( 5), the influence of surface relaxation increases for small pores and/or large surface relaxivities.Due to these dependencies fine material with small pores will create a faster decaying signal compared to coarse material.Thus, by knowing the surface relaxivity (ρ) of a sample it is possible to calculate a mean pore radius.If, for example, a spherical pore geometry is assumed, the surface to volume ratio ( S V ) equals 6 d ; or if a cylindrical geometry is assumed, S V = 1 d (without top and bottom).

Classic CPMG
The CPMG sequence consists of an excitation pulse (P90), which flips the orientation of the macroscopic magnetisation www.geosci-instrum-method-data-syst.net/1/197/2012/Geosci.Instrum.Method.Data Syst., 1, 197-208, 2012 about 90 • from the z-direction into the x-y plane and several refocusing pulses (P180), which flip the protons about 180 • in the x-y plane.After the excitation pulse (P90), the protons lose their phase coherence due to slightly different Larmor frequencies caused by magnetic gradients.After a time span τ , a refocusing pulse (P180) is applied, which flips the protons in a manner that those with a higher ω L lag behind those with a lower one.Due to refocusing, an echo is build up and reaches its maximum after t E = 2 τ .The maximum of the echo marks the point of maximal coherence of the protons.
In general, τ is constant over the whole CPMG sequence and has to be very small to ensure that the diffusion term is negligible, otherwise the protons are not refocused completely, which leads to a reduced amplitude of the echo.In laboratory measurements, τ is typically set between 100-300 µs.In order to acquire the whole decay process a large number of echos (n echo ) is needed.Commonly in laboratory measurements several thousand echos are performed within one CPMG sequence.The maximum of all echos detected gives the relaxation curve for estimating the T 2 relaxation time.A classical CPMG sequence is shown in Fig. 2a.Note that, if for a small τ diffusion can be neglected, then one can conduct a number of CPMG with different τ to measure diffusion processes.

CPMG with variable τ spacing
For the purpose of using NMR as a borehole tool, a significant reduction of energy is needed for conducting a CPMG due to logging requirements for transport and storage of energy.This is equivalent to a reduction of pulses used (accordingly n echo ) for a sequence.As a result of simply reducing the number of echos, the acquisition time is significantly shortened.This leads to errors in the estimated relaxation time if the decay process is not recorded completely.For compensation, i.e., to extent the acquisition time, the echo spacing t E = 2 τ can be increased.As mentioned before, this would increase the diffusion effects according to Eq. ( 3) from the beginning of the sequence, which is not intended.Additionally, the density of data points that forms the T 2 decay curve is reduced.Thus, it can happen that too few data points in the beginning of the decay curve are not sufficient for detecting fast decaying components.However, a CPMG with an increased constant τ causes insufficient sampling of fast decaying components in the beginning of the decay curve.Thus, it can be expected that a wrong decay time could be estimated.Additionally, diffusion effects will increase too, according to Eq. (3).
We investigated the effect of different variable τ spacings on the estimated decay time.We designed CPMG sequences with a variable τ , as schematically depicted in Fig. 2b.The important parameters for constructing a CPMG sequences with a variable τ spacing are the number of echos (n echo ) and a minimum and maximum τ .For generating a sequence with an increasing τ we fixed the values of τ min and τ max .Between these, an amount of n echo linearly increased; exponentially equidistant and logarithmically equidistant τ is calculated.Starting with the lowest possible τ min (τ 1 in Fig. 2b), the fast decaying components are sampled as accurately as possible, while the components with a slow decay are already appropriately sampled.
For the sake of clarity, the development of single CPMG sequence with a variable echo spacing does not allow for estimating diffusion or internal gradient at pore scale like experiments as described in Hürlimann and Venkataramanan (2002) or Song (2003).The intension of this optimized CPMG is to measure T 2 as appropriate as possible under the scope of reducing the number of pulses.

Synthetic study
We conducted a synthetic study to test the evolution of errors for the estimated transverse relaxation time T 2 from CPMG sequences with different approaches of increasing τ .First we generated synthetic data of a simple mono-exponential signal to get an overview of the different sequences.Secondly a synthetic multi-exponential signal was generated to evaluate if fast and slow decaying parts are distinguishable and can be determined correctly.Therefore, we used a bi-exponential signal with the transverse relaxation times T 2,1 and T 2,2 and fitted the data via a decay time distribution.Finally, to examine influences of diffusion on the data due to an increased τ , a diffusive part was added to the synthetic data.
We should note that forward modeling based on Eq. ( 4) is not the most rigorous approach to evaluate the influence of increasing τ .Changing the symmetry, i.e., the constant echo spacing, may cause complex spin dynamics that can be studied based on the Bloch-Torrey equation (Torrey, 1956).However, this would be beyond the scope of this paper.We decided to support our finding of the synthetic study by conducting laboratory-NMR experiments.

Mono-exponential signal
A mono-exponential signal can be generated using Eq. ( 1).The set of model parameters are the initial transverse magnetisation M 0 and the transverse decay time T 2 .Here, M 0 is set to 1 and the decay time for generating the synthetic data is named T true 2 and is varied between 1 ms and 1 s.Standard deviations between the true transverse decay time T true 2 (fixed model parameter) and the estimated T 2 of CPMG sequences with a variable τ spacing were calculated for three different noise levels and are plotted in Fig. 3, each as a function of the true decay time (T true 2 ) and n echo .All deviations above 20 % are red coloured.
The results for sequences conducted with a small and large constant τ are displayed in the first two rows of Fig. 3.As expected, the error of the estimated decay times increases if n echo decreases or the true decay time increases, i.e., slow decaying signals.This is a result of an insufficient acquisition time.If the noise increases, the range of high errors is extended to higher n echo and smaller T true 2 (see first row in Fig. 3).Additional errors for small decay times occur if the echo spacing (τ ) is increased (second row of Fig. 3), which is probably caused by a bad sampling of fast decaying signals.
Thus, increasing a constant τ to compensate for the reduction of n echo is not an appropriate approach.
All three sequences with a variable τ spacing, illustrated in the last three rows of Fig. 3, exhibit more or less the same behaviour for the range of errors.Comparing CPMG-sequences with a variable τ spacing with the one using a constant small τ (Fig. 3, first row), smaller errors occur for decreasing n echo due to the elongated acquisition time.Compared to the sequence with a constant large τ (Fig. 3, second row), all optimized sequences show smaller errors for fast decaying signals, which is a result of a tighter sampling of the decay curve at early times.As the noise level increases, the errors over 20 % expand to sequences with larger n echo and smaller decay times too.

Bi-exponential signal
In the next step a bi-exponential signal was generated to examine if both the fast and the slow decaying parts of the signal can be estimated while reducing n echo .For generating the synthetic data, Eq. ( 1) was extended to a sum of two exponentials with the transverse decay times T 2,1 and T 2,2 .The decay times were chosen in a manner that they are far apart from each other.Therefore, the synthetic decay curve is composed of the decay times T 2,1 = 0.01 s and T 2,2 = 1 s with 5 % noise added.For all sequences with different variable τ spacing, the recorded n echo were reduced successively from 1000 to 100 echos.The decay curves were fitted with a multi-exponential approach using an inverse Laplace transformation.This calculates corresponding amplitudes to a predefined amount of decay times.For smoothness constraints a Tikhonov regularisation of first kind was used.The regularisation parameter was determined via the L-curve criterion that gives a solution closest to the corner of the L-curve (Aster et al., 2005).The T 2 distributions of the fitted signals are illustrated in Fig. 4 and consist of 100 logarithmically-equidistant-spaced decay times within a range of T 2 ∈ [10 −3 , 10 1 ].
In Fig. 4a the signal is sampled with n echo = 1000.In all curves a bi-exponential behaviour is observable.Sequences with a small constant τ (blue curve in Fig. 4) cannot resolve the slow decaying parts of the signal due to a short acquisition time (about 0.2 s).As a result the T 2 distribution is broadened and decay times above 3 s are fitted, which exceeds the maximal possible transverse decay time (T 2 ) of water.Thus, a logarithmic mean decay time of this T 2 distribution will be overestimated.
In contrast, fast decaying components of the signal cannot be resolved by a CPMG sequence with a large constant τ spacing (see the black curve in Fig. 4a).This can also be seen in Kleinberg (1994).
Both parts of the signal can be more or less resolved by all sequences with a variable τ spacing.The sequence with an exponentially equidistant τ spacing shows a broadened T 2 distribution for fast decaying components, which is a result of a fast increasing τ that lead to an insufficient sampling of the decay curve for early times.
In general, if n echo decreases, the bi-exponential behaviour vanishes in conjunction with a broadened T 2 distribution due to the shortened acquisition time.If n echo is reduced to 200 echos (see Fig. 4c), only sequences with a linearly increased and exponentially-equidistant τ spacing exhibit a maximum for slow decaying components.The fast decaying component is resolvable by all sequences.

Bi-exponential signal with diffusion
According to Eq. ( 4), the τ spacing is the only sequence parameter that influences the diffusion term of an NMR signal.Thus, different kinds of increasing τ lead to different impacts of the diffusion on the measured signal.To examine this effect we added a diffusion component to the signal.
According to Coates et al. (1999), the magnetic gradient is influenced by three factors.One depends on the tool design (i.e., tool size, frequency and shape of the used magnet).The second one handles conditions of the formation, such as the temperature.The third factor is described by gradients induced by the applied magnetic field, which are the result of differences in the magnetic susceptibility of grains and pore fluid and are referred to as internal gradients.The magnetic gradient of an MRIL tool (Coates et al., 1999) is denoted with 17 G cm −1 , but Kenyon (1997) stated that a common magnetic gradient from commercial NMR logging tools are about 20 G cm −1 .Thereby, we set G to 18 G cm −1 as a realistic magnetic gradient of the B 0 field.Furthermore, the diffusion constant of water D = 2.5 × 10 −9 m 2 s −1 was used.
The different impacts of G are included by the increased τ .The T 2 distributions for the CPMG sequences with different variable τ spacings and different n echo are plotted in Fig. 5. Compared to the non-diffusive signal (Fig. 4), all T 2 distributions in Fig. 5 are shifted to smaller decay times, except the sequence with constant τ = 0.1 ms (blue curve in Fig. 5).As mentioned before, there are only small influences of diffusion relaxation on the T 2 relaxation time for sequences with a short τ spacing.Nevertheless, the short acquisition time leads to a broadened decay spectrum, which leads to an overestimated log mean decay time.This is more pronounced if n echo is reduced (see Fig. 5b-d).
Thus, if a variable τ spacing is applied, T 2,D increases for increasing τ max and/or a fast increasing τ .In addition it can be seen that slow decaying components are more influenced by diffusion than the faster ones.This is already visible in Eq. ( 4).Relatively small τ spacings in the beginning of the sequences lead to small influences of diffusion on the signal, but increases successively to larger τ spacing in the end of the sequence.
The largest shifts to short decay times are observable at CPMG τ,exp (see orange curve in Fig. 5), which has the fastest increasing τ spacing.The sequence with a logarithmically equidistant τ spacing (CPMG τ,log , red curve in Fig. 5) has the slowest increasing τ and therefore the smallest shift to short decay times.However, it also exhibits the shortest acquisition time, and thus leads to the most broadened distribution within the CPMG sequences with a variable τ spacing.
Considering the trade-off between effects due to short acquisition times (broadening) and diffusion (shift to smaller decay times), the CPMG sequence with a linearly increasing τ (Fig. 5, green curve) is superior compared to sequences with an exponentially and logarithmically equidistant τ .Therefore, the CPMG sequence with a linearly increased τ was used for laboratory measurements.

Laboratory measurements
As it can be seen in Fig. 5, two main effects influence the estimated decay time distributions and thus the T 2 decay time: (i) a shortened acquisition time due to a reduction of n echo , and (ii) an increased diffusion term caused by an increased inter-echo spacing t E = 2 τ .On the one hand, these have to be minimized for the widest possible grain size spectrum.On the other hand, the energy consumption for conducting a CPMG sequence has to be minimized in order to not exceed the possibilities of the borehole tool.
The requirement on estimating the T 2 decay time is a minimal or acceptable deviation to a reference sequence.Thus, optimal settings for n echo and τ max have to be found.For this purpose laboratory measurements with the Maran Ultra 8 (Resonance Instruments) were conducted.Unfortunately, our instrument does not allow for constant gradients of the magnetic field to simulate gradients as expected from borehole tools.Thus, the influence of diffusion that is observed belongs to internal gradients at pore scale and inhomogeneities of the artificial magnetic field.

Sample preparation and parameter sets of CPMG sequences
Samples of glass beads and granular material with grain size spectra varying from coarse silt (gU) to coarse sand (gS) were used.The grain size spectra of the used material are given in Table 1.
For each grain size distribution, three samples were prepared.The material was filled in small cylindrical vessels with a height of h = 2.6 cm and a diameter of d = 3.6 cm.The sample material was trickled into the vessel, which was filled with distilled water.To assure comparability of porosity between the three samples of the same material, an ultrasonic bath was used to raise the packing density to a maximum.Afterwards the samples were filled up and sealed with a foil to minimize evaporation effects during NMR measurements.To avoid further heating during NMR measurements and thus temperature effects on the initial amplitude and the decay time (Godefroy et al., 2001), the samples were heated up to 25-30 • C. The samples were weighed before and after NMR measurements to determine if evaporation occurred during the NMR measurements.Variations of the mass less than 1 % were observed.
If a sample is completely water saturated, the porosity can be calculated using the initial amplitude.A comparison between the NMR-and gravimetric-porosity is illustrated in Fig. 6, which is an easy way to determine if any faults were made during the preparation or measurement.
We performed 9 CPMG sequences per sample with different values of τ max and n echo .The logarithmic mean of the T 2 distribution was compared with the log mean decay time (T 2,lg ) of the reference sequence, which consists of 5000 pulses and a constant τ of 300 µs.The pulse-echo spacing increased in a linear manner, staring with τ min = 100 µs, while τ max was set between 400 and 4000 µs.The minimal echo spacing (τ min ) was the lowest possible value that could be realized with the Maran Ultra 8.The number of echos was decreased, beginning with n echo = 500 down to n echo = 50.By choosing these sequence parameters, an outline of the effect on the acquisition time together with diffusion effects for Table 2. Sequence parameter for the tested CPMG sequences.Starting from a fixed τ min = 100 µs, the echo spacing was increased up to τ max , which equals to the total acquisition time t tot given in column 3. different grain sizes, i.e., signals with different decay times, could be obtained.A full list of the main sequence parameters n echo and τ max together with the resulting acquisition time t tot is given in Table 2.

T 2 distribution of glass beads
The T 2 distribution of different glass beads are illustrated in Fig. 7.The mean grain size spectra increases from top to bottom, in which samples of coarse silt (40-70 µm) are depicted in the first row (Fig. 7a-c), samples of medium sand (250-500 µm) in the second row (Fig. 7d-f) and samples of coarse sand (1250-1650 µm) in the third row.The echos decrease from left to right, starting with n echo = 500 down to  n echo = 50.Within each subplot of Fig. 7, the blue coloured T 2 distribution is associated with the reference sequence.The other curves denote CPMG sequences with a linearly increasing τ spacing and different τ max values.The corresponding log mean decay times of the distribution are given in Table 3.
For n echo = 500 (Fig. 7a) the T 2 distribution for finegrained glass beads are displayed.Compared to the reference sequence a shift of the main peaks to smaller decay times is observed, which is more distinctive for an increasing τ max .For a constant n echo the τ spacing increases faster for larger τ max .Thus, considering Eq. ( 3), a larger τ leads to a higher attenuation due to diffusion and results in a larger shift to smaller decay times.
For n echo = 500 and increasing grain size (from top to bottom), it shows that a shift to small decay times is observed for all grain sizes.For the fine-grained sample (Fig. 7a) the difference of the main peak to the reference signal is about 55.65 ms considering the sequence with τ max = 4000 ms, while the shift for the medium-grained sample (Fig. 7d) is (with a difference of 164 ms) increased.In contrast, for coarse sand samples the shift decreases with a larger τ max , which is due to the short acquisition time.material, NMR signals decay more slowly due to a decrease of S V of the surface relaxation T 2,S .Thus, maintaining the sequence parameter (n echo , τ max ), the acquisition time becomes insufficiently short for slow decaying signals and small τ max .Furthermore, the shorter acquisition time leads to a broadening of the distribution.
For decreasing n echo (from left to right) and fine-grained material, the same effect of broadening is observed; however, in addition to fully acquiring the signal, the sampling of the decay curve is the key factor.If the decay is completely acquired by the CPMG sequence, as it is the case for the fine-grained material and sequences with τ max = 4000 µs, the reduction of data points due to decreased n echo leads to a broadening of the T 2 distributions.If the signal is not completely acquired, as it is the case for τ max = 400 µs, a further broadening is observed.
For decreasing n echo and increasing the grain size, it shows that the effect of acquiring the complete signal dominates the data sampling effect.Theoretically a shortened acquisition time due to a reduction of n echo can be compensated by a larger τ max ; however, this leads to higher diffusion.

Log mean decay time of glass beads
The corresponding log mean decay times to the T 2 distributions of glass beads are given in Table 3.For each sample and amount of echos (n echo ), a systematic decrease of T 2,lg for an increasing τ max is observable.On the one hand, this can be explained by an attenuated diffusion for fast decaying signals, which are fully acquired.On the other hand, for slower decaying signals the decrease of T 2,lg results from the broadening of the T 2 distribution due to an insufficient acquisition time.
If n echo is reduced T 2,lg decreases for silt and medium sand samples with τ max = 1000/4000 µs, which is also an effect of acquisition time.In contrast, the sequence with τ max = 400 µs of medium sand and all sequences of coarse sand samples show an increasing T 2,lg if n echo decreases.This is a result of a flattened T 2 distribution with large decay times (beyond 3 s) fitted to explain the data.
Sequences conducted for example with n echo = 500 and τ max = 400 µs seems to overestimate T 2,lg compared to the reference sequence (silt and medium-grained glass beads in Table 3), although the decay process is recorded completely.Compared to the reference sequence, fast decaying components vanish in the T 2 distributions (see Fig. 7).A cut-off time for calculating T 2,lg could possibly solve this problem.

Comparison of T 2 distributions between samples of granular material and glass beads
The T 2 distributions for samples of granular material are illustrated in Fig. 8. Sequences with n echo = 50 are neglected because the T 2 distribution of glass beads (see Fig. 7c, f and i) showed that the acquisition time is too short for getting appropriate results.Also, the results for fine gravel are neglected because these large grained materials are assumed to have already good hydraulic properties, whose estimation was one of the major focuses.
In comparison to the glass beads, the T 2 distribution of granular material exhibits a bi-exponential behaviour that is more distinctive at coarse sand (Fig. 8c and d).Consistent with the glass beads, the shift of the T 2 distribution to shorter decay times is observed.In contrast to the fine material of glass beads (see Fig. 7a), the shift becomes more distinctive for decreasing τ max .It can be assumed that an insufficient acquisition time caused this effect, since an increased diffusion term leads to an increasing shift for larger τ max .
The absolute shift also increases with grain size, which could be an interaction of attenuated diffusion for increasing grain size and a short acquisition time.According to Fig. 8a  and c, a flattening of the T 2 distribution for coarse-grained material is observed.
A reduction of n echo shortens the record time and thus leads to broadened T 2 distributions.Additionally, as a result, the bi-exponential behaviour vanishes.

Log mean decay time for samples of granular material
In agreement with the results of glass beads, T 2,lg of granular material (see Table 4) decreases if τ max increases and is explainable in the same way.Due to an insufficient acquisition time, T 2,lg increases for sequences with τ max = 400 and 1000 µs of coarse sand and CPMG sequence with τ max = 400 µs of medium sand.Apparently, T 2,lg of the medium-grained granular material (first row in Table 4) is quite similar to the log mean decay time of fine-grained glass beads (Table 3, first row).This similarity between these measurements could be explained by a difference in the surface relaxivity and/or internal magnetic field gradients.Moreover, even though the mean grain size is different, the S/V may not be that different since glass beads have rather smooth surface and are spherical while the granulate material has a rough surface and grains are more angular.

Conclusions
As part of a project for developing an NMR borehole tool to characterize near surface groundwater layer in unconsolidated material, optimized CPMG sequences for measuring T 2 decay time were examined.
A synthetic study of sequences with an exponentially-, logarithmically-equidistant and linearly-increased echo spacing showed that decay times of mono-and multiexponential signals can be resolved with respect to the sequence parameters.It can also be seen that the sequence with a linearly increasing τ yields the best results for slow and fast decaying components if diffusion is taken into account.
Laboratory measurements on glass beads and granular material of different grain size spectra showed that the main sequence parameters (n echo and τ max ) cannot be chosen arbitrarily.Regarding the usage of NMR data, for example to estimate the hydraulic conductivity, a sequence with 500-200 echos and 1000 µs ≤ τ max ≤ 2500 µs is an appropriate choice.Because coarse-grained material is of minor interest due to the high hydraulic conductivity, slow decaying signals may be estimated with higher uncertainty.Although, by adapting the sequence parameters it is possible to fit NMR data of coarse material with an acceptable accuracy.Therefore, an application of a CPMG sequence with a variable echo spacing to obtain slow and fast decaying components instead of using two CPMG sequences seems to be feasible, which results in faster measurements and thus an increased vertical resolution.A field test within an calibration pit is required to show the applicability under borehole conditions.Further investigation is needed to examine the behaviour on natural material.A closer look at the influence of different magnetic gradients on CPMG sequences with an increasing τ is of high interest.

Fig. 1 .
Fig. 1.Loss of phase coherence of protons due to differences in the Larmor frequency in the presence of magnetic gradients.

Fig. 2 .
Fig. 2. (a) Classical CPMG sequence with a P90 excitation pulse and several P180 pulses for refocusing.After each P180 pulse an echos appears.The envelope of all detected echos gives the T 2 decay curve.(b) First pulses and the resulting echos for a CPMG sequence with variable τ spacing.

Fig. 3 .
Fig. 3.The evolution of error for different signal-to-noise ratio.Standard deviations between the true transverse decay time T true 2 and the estimated T 2 depending on the number of echos for mono-exponential synthetic data is shown.All errors over 20 % are red coloured.

Fig. 4 .
Fig. 4. T 2 distributions of bi-exponential signals with different n echo beginning with 1000 echos (a) to 100 echos (d), all with 5 % noise added.Sequences with linearly increasing, exponentially equidistant and logarithmically equidistant τ spacing with τ max = 2 ms are plotted.Additionally two sequences with an constant τ spacing are shown.

Fig. 5 .
Fig.5.T 2 distribution of a bi-exponential signal with T 2,1 = 0.01 s and T 2,2 = 1 s with a diffusion component added and 5 % noise.CP-MGs with linearly increasing, exponentially equidistant and logarithmically equidistant τ spacing with τ max = 2 ms are plotted for different amounts of echos.Additionally, two sequences with an constant τ spacing are plotted.

Fig. 6 .
Fig.6.Cross-plot of porosities determined via NMR-and gravimetric-measurements.The NMR-porosity is in excellent agreement with the gravimetric ones, which is validated by the correlation coefficient of c = 0.982.

Fig. 7 .
Fig. 7. T 2 distribution of glass beads for CPMG sequences with a linearly increased τ .Columns indicate a decreasing n echo (500-200-50, also given in the legends).The grain size increases from top to bottom (first row -coarse silt; second row -fine sand; third row -coarse sand).

Fig. 8 .
Fig.8.T 2 distribution of granular material for pulse sequences with a linearly increased τ .n echo is decreasing from left to right beginning (n echo = 500-200).The grain size increases from top (medium sand) to bottom (coarse sand).

Table 1 .
Grain size spectra of different samples for glass beads and granular material used in laboratory measurements, with gU -coarse silt; mS, gS -medium, coarse sand.

Table 3 .
Logarithmic mean decay time T 2,lg for all CPMG sequences conducted on glass beads.The second column denotes T 2,lg of the reference sequence conducted with n echo = 5000 and a constant τ = 300 µs.

Table 4 .
Logarithmic mean decay time T 2,lg for all CPMG sequences conducted on samples granular material.The second column denotes T 2,lg of the reference sequence conducted with n echo = 5000 and a constant τ = 300 µs.