Permeability Prediction for Low Porosity Rocks by Mobile NMR

Permeability prediction by NMR 1

Permeability prediction for low porosity rocks by mobile NMR

H. Papea,d,*, J. Arnolda,c, R. Pechniga,c, C. Clausera,

E. Talnishnikhb, S. Anferovab, B. Blümichb

aApplied Geophysics and Geothermal Energy E.ON Research Center of Energy, RWTH Aachen

University, Germany

bInstitute of Technical and Macromolecular Chemistry, RWTH Aachen University, D-52056 Aachen,

Germany

cnow at: Geophysica Beratungsgesellschaft mbH, D-52064 Aachen, Germany

dnow at: Eichendorffstr. 10, D-30952 Ronnenberg, Germany

submitted for publication in

International Journal of Rock Mechanics and Mining Sciences

corresponding author: Dr. Hansgeorg Pape Eichendorffstr. 10 D-30952 Ronnenberg, Germany Tel.: +49511 1227451 E-mail address: [email protected] [email protected]

Permeability prediction by NMR 2

Abstract

Permeability prediction from NMR well logs or mobile NMR core-scanner data is attractive as

the measurements can be performed directly in the formation and on fresh cores right after drilling,

respectively. Besides, the method is fast and non-destructive. Compared to T1 relaxation times,

commonly measured T2 distributions are influenced by external and internal magnetic field gradients.

We performed two-dimensional T1 and T2 relaxation experiments on Rhaetian sandstone samples with

low porosity and small pore radii using a mobile NMR core scanner which provides a nearly

homogeneous static magnetic field. Because small pore sizes are associated with high internal magnetic

field gradients standard methods from NMR logging in the oil industry can not be applied for an

accurate permeability prediction. Therefore, a new model theory was developed which describes the

pore radius dependence of the surface relaxivity ρ 2 by both an analytical and a more practical empirical

equation. Regarding corrected ρ 2 values, permeability can be predicted accurately from the logarithmic

mean of the T2 distribution from the physically based Kozeny-Carman equation. Additional core plug

measurements of structural parameters such as porosity, permeability, specific inner surface area and

pore radius distributions provide confidence in the NMR results.

1. Introduction

Storage and transport properties such as porosity and permeability are essential for

characterizing reservoir rocks and different petrophysical measurements such as resistivity and NMR

are used to estimate them. The common approach to porosity in NMR logging is based on polarizing

the magnetic spin moments of hydrogen nuclei of fluids in porous rocks. NMR logs are considered the

most reliable logs for estimating the pore size distribution and to infer the formation permeability [1].

Commonly the build-up of nuclear magnetization and the irreversible signal decay are measured in

terms of relaxation curves following an excitation with suitable radio-frequency (rf) pulses. The

characteristic times for build-up of longitudinal magnetization and the loss of transverse magnetization

are denoted as T1 (longitudinal relaxation time) and T2 (transverse relaxation time), respectively.

Permeability prediction by NMR 3

Distributions of T1 and T2 times are obtained from experimental relaxation curves by inverse Laplace

transformation or by inversion with suitable fitting functions. Assuming the fast diffusion regime,

which conforms to the majority of rocks, the rate-limiting step is relaxation at the surface [2]. As a

consequence, the decay rate of magnetization decay depends mainly on the surface-to-volume ratio for

measurements in a homogeneous magnetic field. Thus the relaxation time distribution reflects the pore

radius distribution of the rock. Compared to T1, T2 measurements are preferred in NMR logging due to

shorter measurement times and correspondingly a higher possible logging speed. However, the

measured T2 values may be affected not only by the properties of interest, such as the pore size, but also

by the inhomogeneity of the static magnetic field and the internal magnetic field gradients [1]. This

causes a further reduction in relaxation times. Internal magnetic field gradients arise from magnetic

susceptibility contrasts between rock mineral surfaces and the pore fluid [3, 4]. The magnetic

susceptibility value of water is negative (-9.26 × 10-6 SI at 20 °C) like for diamagnetic minerals such as

rock salt, calcite, gypsum, quartz, and graphite. In contrast, most minerals and rocks have positive

values [5]. Compared to high-field NMR, measurements in low magnetic fields are preferred for rock

studies, because the internal field gradient is proportional to the applied magnetic field strength [2].

However, estimating permeability only from NMR T2 relaxation time measured in a low and

homogeneous static magnetic field is difficult. This is due to the unknown nature the surface relaxivity

and the internal magnetic field gradients of each sample. Hence, NMR data should be calibrated using

additional information from core analysis.

We studied the transport properties of a low-porosity sandstone formation in a hot water aquifer

drilled and logged in the Allermoehe borehole, which is located in the northern German sedimentary

basin. When we started, a commercial CMR Log (Combinable Magnetic Resonance tool recorded by

Schlumberger) was available which yielded logarithmic mean values of surface relaxivity, T2,LM and

additionally three fractions of porosity: clay bound porosity with T2 < 3 ms, capillary bound water with

3 ms < T2 < 33 ms and free water with T2 > 33 ms. From this data permeability was estimated in two

ways: One value was calculated with an empirical equation from porosity and T2 and the other value

with an empirical permeability-porosity relationship. Due to the unknown nature of the surface

relaxivity and the unidentified amount of relaxation caused by diffusion in zones with magnetic field

gradients, the pore radius-T2 relationship is biased resulting in an underestimation of permeability. A

comparison with laboratory measurements on core material indicated the need for more petrophysical

Permeability prediction by NMR 4

studies and a calibration with independent methods. To this end, we studied 20 plugs sampled from drill

cores which were recovered within an interval of 30 m in the same borehole. One-dimensional (1D) T2

measurements and two-dimensional (2D) T1- T2 experiments were performed in the laboratory on

several core plugs with a mobile, low-field NMR core-scanner. 2D NMR measurements allow to

compare T2 and T1 distributions where the latter ones are not influenced by diffusion within external

and internal magnetic field gradients. Using a new model based on the relation between surface

relaxivity ρ 2 and pore radius, permeabilities of the Allermoehe samples can be determined accurately.

Additionally, a set of petrophysical properties was measured including gas permeability, porosity,

formation factor, and specific inner surface area. Moreover, we obtained the pore size distribution on

identical core plugs by mercury injection.

Based on this data, this study evaluates how well permeability can be predicted from NMR

logging and core-scanner data within the sandstone formation of the Allermoehe borehole if additional

petrophysical data measured on core plugs are integrated in the analysis.

2. Samples

The Allermoehe borehole is located in the northern German sedimentary basin near Hamburg. It

was drilled into the Rhaetian hot water aquifer that is considered a geothermal resource. Rhaetian

sandstones were cored in the depth interval between 3220 m and 3250 m. From the same depth interval,

additional NMR logging data is available which was recorded by the CMR (Combinable Magnetic

Resonance) tool of Schlumberger. For laboratory NMR measurements and additional petrophysical

investigations we drilled core plugs with diameters of 20 mm at 20 positions from the Allermoehe

cores. The selected cores are fine-grained sandstones with high quartz content. They have low porosity

in the range between 2 % and 12 % and low permeability of up to 20 mD1. Mercury porosimetry yields

pore-sizes between 0.01 µm and 100 µm. Magnetic susceptibility values were recorded by a Geotek®

multi-sensor Core Logger [6] with range from 0 × 10-6 SI to 25 × 10-6 SI.

1 1 mD = Millidarcy; darcy = 1 D = 9.87 ×10-13 m²

Permeability prediction by NMR 5

3. Experimental procedures

3.1. Mobile NMR

Prior to NMR laboratory testing, the Allermoehe core plugs had been dried to a constant weight

in an oven at a temperature of 60° C to remove all water from interconnected pores. Then they were

saturated with distilled water in a vacuum exsiccator to constant weight.

The NMR measurements were performed by a mobile Halbach sensor (Fig. 1) as described by

Anferova et al. [7]. The magnet system weighs less than 30 kg and features a sufficiently homogeneous

magnetic field within the cylindrical volume of the sample. The static magnetic field BB0 is equal to 0.22

T, corresponding to a proton resonance frequency of 9.6 MHz. The device is constructed from two

identical arrays, which are stacked behind each other and separated by an axial gap of 10 mm for

improved homogeneity of the magnetic field. Each array consists of three magnet rings containing 16

magnet blocks of a volume of 30 mm³. The inner diameter of the magnet is 140 mm. The homogeneity

of the magnetic field in the central part of the magnet with a diameter of up to 80 mm is sufficient for

non-destructive transverse relaxation measurements. By applying exchangeable cylindrical rf-coils, T2

distributions can be studied on water-saturated drill cores and plugs of diameters between 20 mm and

80 mm. Additionally, 2 D T1-T2 correlation experiments can be performed in the central part of the

magnetic field with the highest homogeneity on plugs with diameters of 20 mm. Whereas 1D T2

distributions are measured by applying a CPMG pulse sequence [8], T1-T2 correlation experiments

require a relaxation-relaxation editing pulse sequence [9]. A multimodal log-normal distribution

(Matlab code by courtesy of Andreas Hartmann; RWTH Aachen University 2006; now at Baker

Hughes INTEQ GmbH, Celle) was used to obtain 1D T2 distribution curves. 2D T1-T2 distribution

curves are obtained from inverse Laplace transformations. Measurement times for T2 distributions and

T1-T2 correlation experiments on the Allermoehe core plugs depend on the sample porosity and on the

number of scans. They vary from a few minutes for CPMG experiments to 30 minutes for 2D

experiments. Experimental parameters of the Halbach sensor used for 1D T2 measurements and for 2D

T1-T2 correlation experiments are shown in Table 1.

Permeability prediction by NMR 6

3.2. Independent methods

To extend the interpretation of NMR core data with respect to permeability the following

measurements were performed on corresponding core plugs: porosity φ, permeability k, mercury

porosimetry, specific surface area. Additionally, electrical resistivity was measured to obtain the

formation factor F and thus tortuosity T.

All NMR porosities were correlated to porosity values measured on corresponding core plugs

with a helium gas pycnometer.

Permeability was determined from gas flow measurements. For low permeability rocks, the

effective gas permeability dependents on pressure and may therefore deviate from that for water.

Hence, the Klinkenberg correction [10] was applied to account for this effect.

To estimate pore size distributions from NMR relaxation measurements, additional information

was obtained by mercury porosimetry on selected core plugs. Mercury injection is a well established

and widely used method for obtaining pore size information [11]. It yields the pore throat sizes

weighted by the relative pore volumes to which the pore throats provide access [12]. Mercury is

injected into a core plug with incrementally increasing pressure up to a maximum of 60,000 psi2.

During each step, both the pressure and volume of mercury are measured after the pressure reaches

equilibrium. Applying the Washburn equation [13], each pressure step can be associated with a

particular pore throat size which decreases with pressure.

Precise measurements of the specific inner surface were performed based on nitrogen-adsorption

according to the Brunauer-Emmet-Teller (BET) method [14]. The specific surface Spor,BET is defined as

the ratio of the absolute surface area of a solid and its total volume. The surface area includes all parts

of accessible inner surfaces, i.e. mainly pore wall surfaces. The BET measurement determines the

amount of adsorbed gas required to cover the accessible internal pore surfaces of a solid with a

complete monolayer. The volume of the adsorbate at full monolayer can be calculated from the

adsorption isotherm by means of the BET equation [15].

2 1 psi (pound per square inch) = 6.98×10³ pascal

Permeability prediction by NMR 7

The formation factor F was obtained from electrical resistivity measurements. The inverse of the

formation factor describes the effective porosity with respect to transport processes such as fluid flow,

electrical conductance, and diffusion [16]. For the determination of F we used the equation of electrical

conductivity κ0 which is the inverse resistivity:

0 w w1

qelF T q

φκ κ κ κ= + = + κ , (1)

with electrical conductivity of the electrolyte κ0 and interlayer conductivity κq . Tel is the electrical

tortuosity.

F and κq are calculated from a series of resistivity measurements on a sample stepwise saturated with

saltwater with increasing salinity. Tel can be substituted by porosity according to Archie´s first law [17]:

mF Aφ −= . (2)

Table 2 presents NMR data and petrophysical parameters from independent methods, measured

on the Allermoehe core plugs.

4. Theory and integrated studies for permeability prediction

4.1. Porosity

NMR transverse relaxation time measurements can be calibrated directly to porosity. The initial

amplitude of the decay curve and the area under the T2 distribution curve are directly proportional to the

number of polarized hydrogen nuclei in the pore fluid. Hence, the T2 distribution can be directly

calibrated in terms of porosity [12]. NMR logs from the Allermoehe borehole were compared to mobile

NMR data measured on core plugs in the laboratory. Because these porosities are independent of the

matrix mineralogy, they can be compared directly with conventional laboratory porosity measurements.

The results show that NMR may predict porosity reasonably well. The correlation coefficient R²

between NMR core scanner and pycnometer measurements is 0.8 with a standard deviation of 1.1 %.

Moreover, NMR porosity from logging data agree well with individual porosity values measured with

the mobile NMR core-scanner on core plugs (Fig. 2).

Permeability prediction by NMR 8

4.2. Permeability

According to the Kozeny-Carman equation [19, 20], permeability is related to porosity φ,

tortuosity T and the effective hydraulic pore radius reff:

2

8 effk rTφ

= , ( )2effin µm ; in µmk r . (3)

The term (φ/T) can be replaced by the inverse formation factor F -1, and by φ m/A, according to the first

Archie equation (2) [17] .The so-called cementation or tortuosity factor m varies between 1 and 3 and

the coefficient A ranges from 0.6 to 2, depending on rock texture. The effective pore radius can be

substituted by Spor,hydr , the specific surface normalized by pore volume. The value of Spor,hydr is smaller

than the respective Spor,BET because the surface is smoothed by the physical process of fluid flow. For a

hydraulic model with smooth pores, reff is related to Spor,hydr by

effpor,hydr

arS

= , (4)

with a = 2 for cylindrical pores and a = 3 for spherical pores.

This equation is primary based on a pore model which consists of a bundles of smooth, cylindrical

capillaries of radius reff [16]. In contrast, the most simple model for NMR relaxation which relates

relaxation times T1 and T2 to Spor and reff, was developed for isolated smooth spherical pores [2] with:

1,2 por,NMR1,2

1= S

Tρ , (5)

where ρ1,2 is the surface relaxivity given in μm/s

and

effpor,NMR

3=r

S. (6)

Permeability prediction by NMR 9

Based on a fractal model for porous rocks, several geometrical relations were established by

Pape et al. [16, 21] in which specific surface, pore radius, tortuosity, and porosity are connected via the

fractal dimension D. This is the fundamental geometric parameter for the description of the structure of

pore-space. A standard value for northern German sandstones is D = 2.36 [22]. The principal idea of

the fractal concept is the dependence of the measure of geometrical parameters such as the area of a

rough surface on the power of resolution of the measuring method. The largest value is attributed to

Spor,BET measured with the highest resolution by the smallest yardstick, which is given by the size of

adsorbed nitrogen molecules. In contast, the dynamic process of fluid flow in a capillary is determined

by the smoothed surface of the hydraulic shear zone Spor,hydr. Pape et al. [22] thought for the pore radius

as a suitable yardstick and described the relationship between both specific surfaces by

(7) 0.36por,hydr por,BET eff0.1410 −=S S r

Then equations (7) and (4) yield:

10.64

effpor,BET

20.141

⎛ ⎞= ⎜⎜

⎝ ⎠r

S⎟⎟ . (8)

From equations (8) and (3) follows:

3.125Spor,BET por,BET497k S

Tφ −⎛ ⎞= ⎜ ⎟

⎝ ⎠

(kSpor,BET in μm2, Spor,BET in μm-1) . (9)

Table 3 presents petrophysical parameters which are derived from data of Table 2.

Permeabilities of the Allermoehe core plugs were calculated with equation (9) and Spor,BET. As tortuosity

value, we used the quotient of formation factor F and pycnometer porosity φpyc, which is strictly spoken

an electrical tortuosity Tel. Except for one outlier calculated permeabilities agree within one order of

magnitude with the measured gas permeabilities.

Permeability prediction by NMR 10

Effective hydraulic pore radii were derived from equation (3) and the calculated permeabilties kSpor,BET

and the measured permeabilities kgas. We find that reff values calculated from F and kgas are generally

smaller than the logarithmic mean values reff,LM from pore size distribution curves, which we regard as

representative for the hydraulic radius. In order to get better reff values from equations (3) and (9), the

porosity and the tortuosity term have to be corrected. Firstly, the bulk porosity φ has to be reduced by

the clay bound porosity φclay. Secondly, the electric tortuosity Tel has to be replaced by the hydraulic

tortuosity Thydr which is by a factor larger. Numerical simulations on two-dimensional networks by

David [23] showed that the so-called ‘network tortuosity’ for hydraulic flow was 1.5 times larger than

the ‘network tortuosity’ for electrical current. The tortuosity with respect to processes of transport

depends mainly on path elongation and on the variation of pore radius, which is important in sandstone.

Interpretation of PFG-NMR measurements on Allermoehe sandstones by Pape et al. [24] showed that

the radius of pore bodies, about 12 µm, exceeds the derived pore throat radius of about 2 µm.

Tortuosity values for self-diffusion of water were determined in the range between 5 and 20. The

influence of path elongation is similar for electric tortuosity and for hydraulic tortuosity. However, the

impact of constrictions on tortuosity is larger for fluid flow, which was shown by modeling of Schopper

[25].

Therefore, equations (3) and (9) have to be extended to:

( )clay 2

ffhydr

18 ek

Tφ φ−⎛ ⎞= ⎜ ⎟

⎝ ⎠r (10)

and

por,BET

3.125clay por,BET

hydr

= 497S

- Sk

T qφ φ −⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

, (11)

where q is a factor of surface enlargement corresponding to the clay content, which is discussed in

detail in [21, 22].

The expression (φ-φclay) corresponds to the porosity which can be calculated from the pore radius

distribution from mercury intrusion as shown below. For further calculations we estimated Thydr by

Permeability prediction by NMR 11

0.6φ−1.2 , which corresponds to Archie’s first equation, but with a higher exponent as one gets from the

measured electrical formation factor and porosity.

The corrections in equation (11) balance in some way because of (φ-φclay)/Thydr < φ/Τel and

(Spor,BET/q)-3.125 > Spor,BET-3.125. This is the reason for suitable permeability values which we got from

equation (9).

Based on fractal theory [21], a power-law relationship can be established between permeability

and porosity. A fit to average sandstones yields:

2 10= 31 + 7463 +191(10 )kφ φ φ φ ( )2in nmk . (12)

From data of Allermoehe sandstone Pape et al. [26] derived the relationship:

( )4.85= 0.309 100kφ φ ( )2in nmk (13)

which was used to obtain permeability from NMR logging and NMR core-scanner data. The

permeability values from NMR data according to equation (13) and from gas flow measurements agree

well with the exception of a few outliers (Fig. 3).

The empirical Kenyon equation [27] equation is commonly used for estimating permeability

from porosity and NMR logging data, based on the logarithmic mean T2,LM of the T2 distribution:

(14) 4 22,LMk a Tφ=

Usually, for sandstones a value of a = 4 mD/(ms)² is used [28]. Permeabilities from the Schlumberger

CMR-log® recorded in the Allermoehe borehole were calculated from the free water fraction φfree (T2

cutoff = 33 ms) using the factor of a = 4 in equation (14). The permeability log is plotted in (Fig. 4).

The same plot presents the permeabilities of core plugs , calculated from equation (14) and laboratory

Permeability prediction by NMR 12

1D T2 measurements. For comparison, the measured gas permeabilities are shown. There is a large

disagreement between NMR and gas permeabilities. This can be explained by the low porosity values

of the Allermoehe samples varying between 2 % and 11 %. Due to the high exponent of porosity equal

to 4 in the T2,LM equation (14) which is suitable for formations of high porosity, permeability calculated

from Allermoehe logging and core data is too low. Therefore, the following more detailed analysis of

the low porosity Allermoehe sandstones is applied to predict permeability more accurately.

To develop a method for permeability prediction which is suitable for different formations,

information on the effective pore radius derived e.g. from mercury porosimetry should be used. The

following discussion of the radius distribution obtained from mercury injection compared to relaxation

time distributions is necessary to understand the structural and physical conditions in detail. This will

help us later to interpret T1 and T2 distributions for estimating permeability.

4.3. Mercury injection curves

The curves of mercury injection porosimetry were used firstly to obtain an estimate of the

effective hydraulic pore radius and secondly to determine the factor 3ρ 1,2 which relates pore radius r to

T1,2 of the NMR relaxation curves according to equations (5) and (6):

1,21,2 r,1,2 por,NMR

1,2 1,2

1 3= =3

3

TT cS

r = T

ρ

ρ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

r, (15)

where cr,T1,2 is the conversion factor in ms µm-1.

Fig. 5 presents the mercury injection curve of plug AC9 as a typical example, where the relative

volumes are plotted versus radius r. Additionally, we find four curves for different kinds of SiVΔ por

functions, which depend on the radius . In the following, the arguments are set in [ ]-brackets.

1) Spor,capillary[r] = 2/r . This is the specific surface divided by volume for each individual pore

class.

2) [ ] 0.36por,frac por,BET0.1410S r S r−= , (16)

Permeability prediction by NMR 13

which represents the fractal specific surface, derived from Spor,BET and adjusted to each radius

r of the distribution as resolution yardstick. Equation (16) corresponds to equation (7)

3)[ ] 1,

,

11 1

2

2

j

iipor frac n i

por mercury j jn n

ii iii ii

VS r r

S rVV V

r

=

== =

⎛ ⎞⎛ ⎛ ⎞Δ⎜ ⎟⎜ ⎜ ⎟

⎝ ⎠⎜ ⎟⎜⎡ ⎤ =⎣ ⎦ ⎜ ⎟⎜⎛ ⎞ ΔΔ Δ⎜ ⎟⎜⎜ ⎟⎜ ⎟⎜⎝ ⎠⎝ ⎠⎝

∑

∑∑ ∑

⎞⎟⎟⎟⎟⎟⎠

. (17)

This function describes the specific surface, calculated from the radius distribution from

mercury intrusion. The resolution power increases with decreasing radius. The nominator of

the second term contains the sum of the surfaces of all pore classes from the largest pore with

i = 1 to the pore class with index j ≤ n. The sum in the denominator is the total volume of the

same pore classes. Due to the first term, Spor,frac[rn] is equal to Spor,mercury[rn], where rn is the

smallest radius of the distribution.

4) [ ]-0.36

j relpor,frac,free j por,mercury border

borderi

1

j

i

r VS r S rr V

=

⎛ ⎞⎜ ⎟⎛ ⎞ ⎜⎡ ⎤ = ⎜ ⎟⎣ ⎦ ⎜ ⎟⎝ ⎠ Δ⎜ ⎟⎝ ⎠∑

⎟ , (18)

1

1n

rel ii

V V=

= Δ =∑ . The radius rborder shall mark the border between the clay volume part and

the volume part of the distribution which is relevant for fluid flow.. We derive rborder from the

shape of Spor,mercury and take the radius, where the slope changes from steep to less steep in the

direction of increasing radius (see Fig.5). We find rborder equal to 1 μm, which value was later

used for all samples. Besides, this value is consistent with the cutoff T2 < 3 ms used in the

interpretation of CMR data. Spor,frac,free is based on the value of Spor,mercury [rborder] . The function

Spor,frac,free describes a fractal specific surface which has nearly the same fractal dimension as the

specific surface described by Spor,frac due to the term (rj/rborder)-0.36. In the clay region of the radius

distribution, we find Spor,frac,free = Spor,frac / q, where q is the factor of surface enlargement due to

Permeability prediction by NMR 14

the clay content from equation (11). The last term in equation (18) causes a smaller slope of the

curve in the log-log plot (Fig. 5) compared to the slope of equation (16). Spor,frac,free is an

empirical function which shall describe the free water part of the pore distribution.

The construct with four different Spor curves allows to obtain three parameters which are needed

in the calculation of permeability from equations (10) or (11). We just mentioned the surface

enlargement factor q. Additionally we get free porosity (φ-φclay) from the volume Vfree = ib

ii 1

V=

Δ∑ with

the index ib of the class with rborder. For this, Vfree is firstly transformed into the respective volume of

mercury which intruded into the sample and secondly related to the volume of the sample. The third

parameter of interest is an estimate of reff. In Fig. 5 we find reff,BET at the radius, where the line of

Spor,frac crosses the line of Spor,capillary . This hydraulic radius is solely based on Spor,BET ; therefore it is

exactly the same as reff from equation (8). In analogous way we take the radius reff,frac at the crossing of

the curve of Spor,frac,free and the line of Spor,capillary. This radius is based on Spor,BET and on the radius

distribution as well and gives a good estimate for reff, which was proven by comparison of measured

gas permeability with calculated permeability from equation (10). The new parameters, derived from

radius distributions of several Allermoehe samples, and calculated permeability values are presented in

Table 4.

This method for obtaining reff values from individual pore radius distribution curves is time

consuming. However, we used it in order to compare reff,frac with the logarithmic mean values reff,LM

(Table 4), which are defined by

i i

ieff,LM 10

rel

Δ log( )= exp

⎡ ⎤⎢⎢⎢ ⎥⎣ ⎦

∑ V rr

V⎥⎥ (19)

with the integral volume Vrel equal to one.

In general, the calculated reff,LM values were close to the reff,frac. For sample AC23 with a

considerable high clay content, the effective pore radius reff,free,LM is calculated from the upper part of

the pore radius distribution with radius > 0.05 μm. Permeabilties were calculated from equation (10)

using reff,frac and reff,LM values as effective pore radii. In the case of sample AC23 the term reff,LM is

Permeability prediction by NMR 15

substituted by reff,free,LM. Table 4 presents correction factors cφ with ( )claycφφ φ φ= − , which are used

to calculate permeabilities from reff,frac. For the permeability prediction from reff,LM and reff,free,LM we

used a medium value values cφ = 0.7. In all cases tortuosity was set to Thydr = 0.6 φ -1.2. The

corresponding results are presented in Table 4. Fig. 6 shows the calculated permeabilities from reff,frac,

reff,LM, and reff,free,LM plotted versus depth in comparison to the measured permeabilties kgas. The fit of

permeabilities calculated in both ways is considerable good.

4.4. Surface relaxation and “de-phasing”’

As equation (15) indicates a linear relationship between NMR-relaxation times and pore radii,

the T1,2 distribution curves can be processed in a similar way as the mercury injection curves by

calculating the logarithmic mean values T1,2,LM. In the case of longitudinal relaxation the linearity is

sufficiently fulfilled, i.e. relaxivity ρ1 is constant with respect to pore radius although ρ1 may vary for

different samples. The physical process related to ρ1 is called surface relaxation. It acts directly at the

mineral surface of the pore walls where the magnetization of the hydrogen atoms is lost. For the same

sample, the T2 distribution curves are shifted towards smaller values. This indicates a second effective

relaxation process. It is caused by de-phasing of the transverse magnetization vectors due to diffusion

of water molecules within an internal, inhomogeneous magnetic field. Large field gradients are caused

by paramagnetic minerals at the pore walls. They are effective within a certain distance from the

minerals. For simplification, this will be represented by a thickness d of this field gradient interlayer

(Fig. 7).

In this situation, transverse relaxation can be described by:

2 1 de-phase

1 1 1= + ,T T T

(20)

where Tde-phase is the relaxation time due to de-phasing with corresponding “de-phasing

relaxation” ρde-phase as shown in Appendix A:

Permeability prediction by NMR 16

de-phasede-phase

-= 3

r dTρ

. (21)

Combining equations (20), (21), (4) and (5) we obtain:

de-phase2 1= +

1- dr

ρρ ρ

⎛ ⎞⎜ ⎟⎝ ⎠

, (22)

dephase1

21

= 1+1-

TdTr

ρ

ρ ⎛ ⎞⎜ ⎟⎝ ⎠

. (23)

Equations (22) and (23) show that ρ2 and T1/T2 decrease with pore radius r. As the external field

used by the NMR core-scanner used for NMR relaxation measurements is very homogeneous, surface

relaxivity and interlayer relaxivity can be separated in an analysis of a two-dimensional T1-T2 (2D)

correlation experiment.

4.5. 2D T1-T2 correlation experiments

We performed 2D T1-T2 relaxation correlation experiments with the NMR core-scanner in order

to study the influence of diffusion on the shape of the T2 distribution function for low porosity rocks,

such as those from the Rhaetian sandstone at Allermoehe. Initial results are published in Anferova et al.

[7]. The T1/T2 ratio, determined from 2D T1-T2 correlation experiments performed on the Allermoehe

samples is not constant even for short echo spacing. It decreases with pore size from values of 10 to 4.

Fig. 8 shows the 2D T1-T2 map based on measurements on the Allermoehe core plug AC15.

The T1-T2 correlation experiments were used to determine the relationship between ρ 1 and ρ 2.

In analogy to the mercury injection curves, the differential magnetizations Δm of each T1,2 fraction of

the distribution curves are plotted as relative volumes, with the integral volume mrel equal to one. First,

all relaxation curves were correlated with the radius distribution curves from mercury injection by

Permeability prediction by NMR 17

shifting the relative volume distribution relative to the pore size distribution until the maxima of both

curves coincide for large radii and large T1,2 values. Then, ρ 1 and ρ 2,eff are determined from equation

(15) for each sample (Table 5).

As core plug AC9 is considered representative for all Allermoehe samples studied in this work

due to

r at

T1 = 749

its petrophysical properties (porosity, range of pore sizes, and permeability) it will also be used

to demonstrate the pore radius dependence of ρ 2 and to calculate the thickness of the interlayer d.

For large T1,2 values, the first maxima of the relative volume distribution occu

.9 ms and T2 = 133.4 ms. The second maxima for the next largest T1,2 values occur at

T1 = 177.8 ms and T2 = 17.78 ms. Correlating the T1 curve with the radius distribution curve at the first

maximum yields a conversion factor of cr,T1 = 1/3ρ1 = 78.64 ms µm-1 according to equation (15).

Consequently, the first maximum corresponds to r1 = 9.53 μm and the second maximum corresponds to

r2 = 2.26 μm. Inserting the appropriate T1 and T2 values corresponding to both maxima in equation (22)

and dividing the two equations yields:

( )( )( )( )

1 1

2 1 2

1 2

12 2

1 1-= h =

1-1

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝⎛ ⎞ ⎛

−⎜ ⎟ ⎜⎜ ⎟ ⎝⎝ ⎠

T r dT r rT r d

rT r

⎠⎞⎟⎠

. (24)

From th one obtains: h (1 - d/r1) = (1 - d/r2) or h×(r2/r1)×(r1 - d) = (r2 - d), yielding the thickness of the is

interlayer:

( )

22

1

1- h= .

1- h⎛ ⎞⎜ ⎟⎝ ⎠

d rrr

(25)

Inserting the appropriate values of the parameters on the right-hand side of equation (25) yields

d = 1.25 μm. The surface relaxivities of AC9 obtained from the comparison of the T1,2 distribution

Permeability prediction by NMR 18

curves with the mercury injection curve are ρ 1 = 4.24 μm/s and ρ 2,eff [r1] = 23.82 μm/s. Thus, equation

(22) yields the “de-phasing relaxation” ρ de-phase = (ρ 2,eff [r1] - ρ 1) (1 – d/r1) = 17.01 μm/s.

The relationship between pore radius r, surface relaxivity ρ 2, and transverse relaxation time T

is described by equations (15) and (22). We calculated a tabl with ρ and T values calc

2

e 2 2 ulated for a

column

of r values. Because the function ρ 2 according to equation (22) has a pole for r = d, the values

for radii near the pole make no physical sense. Therefore we truncated the function at a pore radius r =

3 μm which is about three times larger than d and corresponds to T2 = 40 ms. To estimate correct ρ 2

values for T2 times shorter than 40 ms, the ρ 2 curve which has been calculated for large pore radii was

extrapolated by a straight line in the plot of log ρ 2 versus log T2 with the slope at r = 3 μm.

Data measured on Allermoehe sample AC9 yielded a slope of -0.3 which yields:

[ ] [ ] [ ][ ]

-0.3

2 i2 i 2,eff 1 1

2 1

= f i

T rr r r

T rρ ρ

⎛ ⎞or r<⎜ ⎟⎜ ⎟

⎝ ⎠ , (26)

whith r1 = 9.54.

On the base of equation (26) it is possible to get corrected pore radii from radii rρ2,eff which are

e constant ρ2,eff and equation (15). In order to calculate permeability later on, we

applied

calculated with th

this correction to the logarithmic mean values rρ2,eff,LM derived from the logarithmic mean T2,LM

of the T2 distribution. We got the correction term by multiplying equation (26) with 3T2[ri] and

deviding ing T2[ri] and T2[r1] in equation (26) by 3ρ2,eff :

0.3

1corr 2,LM 2, 2,LM =

effrr T r T

r Tρ⎡ ⎤ ⎡ ⎤ 2, 12, 2,LM

for effeff

r rρρ

⎛ ⎞<⎜ ⎟⎣ ⎦ ⎣ ⎦ ⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

, (27)

with r1 = 9.45.

Permeability prediction by NMR 19

Using the exponent of 0.3 yields a corrected pore radius rcorr [T2,LM ]which is similar to the pore

radius rρ 1 [T1,LM]. The pore radius rρ 1 [T1,LM] is obtained with equation (14) and ρ 1 from the T1

distribution curve which is measured simultaneously on sample AC9.

In Fig. 10a, the shape of the T1 and T2 distribution curves of sample AC9 are compared. The

relaxation curves are correlated by shifting the relative volume distributions until the maxima of both

curves for large T1,2 values coincide. This is achieved by multiplying the T2 values by the term (ρ 2,eff /

ρ 1). In a next step the T2 distribution curve is renormalized in two ways. Firstly, it is corrected using

the approach based on the thickness of the field gradient interlayer. Therefore, each measured T2 value

of the T2 distribution curve is multiplied by the term (ρ 2 / ρ 1). The individual ρ 2 values are determined

from equations (15) and (25). Secondly, the T2 distribution curve was renormalized empirically.

Therefore, T2 values were multiplied by the correction term (9.53 µm / rρ 2,eff)0.3 and with the term

(ρ 2,eff / ρ 1). Both approaches yield a nearly perfect match with the T1 distribution (Fig. 10b).

For all samples for which a pore radius distribution curve was available, values of rρ 1 [T1,LM]

and rρ 2,eff [T2,LM] were calculated from the T1 and T2 distribution curves. Mean values of ρ1 and ρ 2,eff

(Table 5) of these samples were used to determine values of rρ 1 [T1,LM] and rρ 2,eff [T2,LM] for the rest of

samples. From the rρ 2,eff [T2,LM] values also rcorr [T2,LM] values were calculated according to equation

(27). Permeability was then predicted using the following equations:

21 1,

hydr

1= 8 LM

ck r T

Tφ

ρφ⎛ ⎞ ⎡⎜ ⎟ ⎣⎝ ⎠

⎤⎦ (28)

and

22,

hydr

1= 8

corr LM

ck r T

Tφ φ⎛ ⎞ ⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

( , (29) )21 1,LM corr 2,LM in µm ; and in µmk r T r Tρ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

Permeability prediction by NMR 20

with tortuosity Thydr = 0.6 φ -1.2 (cf. Table 3). The factor cφ = 0.7 was chosen due to the

considerable high clay-content of the Allermoehe samples. Table 5 shows the resulting pore radii and

permeabilities.

In Fig. 11, permeabilities calculated from 2D T1 and T2 distributions according to equations (28)

and (29), are plotted versus depth and compared to gas permeabilities. Additionally, Fig. 12 compares

calculated permeability according to the standard method in oil industry (equation (14)) from 2D T2

distributions using a factor of a = 4 which is generally accepted for sandstones. Whereas permeabilities

derived from equations (28) and (29) agree well with gas flow measurements, results from the standard

method used in oil industry based on equation (14) seem to underestimate permeability systematically

for the low porosity Rhaetian sandstones from Allermoehe.

4.6. Permeability from 1D T2 relaxation

Permeabilities of the Allermoehe core plugs were also derived from 1D transverse relaxation

measurements performed with the NMR core-scanner. For the samples with available pore size

distribution, individual relaxivities ρ 2,eff were determined by coinciding the maxima of both frequency

distributions. A mean value of ρ 2,eff was calculated and used for the other samples. Pore radii rρ 2,eff

(T2,LM) were calculated from logarithmic mean relaxation times T2,LM according to equation (15). In the

case of samples AC1 and AC3 with very small pore radii, T2,LM values were calculated using a lower T2

cutoff of T2 = 0.9 ms. The pore radii were corrected according to equation (27) by multiplication with

the term (9.53/rρ 2,eff)0.3 resulting in rcorr (T2,LM).

Permeability was calculated from equation (29) with cφ = 0.7 and tortuosity Thydr = 0.6 φ -1.2 for

φ ≥ 0.05, and Thydr = 20 for φ < 0.05 (Pape et al., 2005b).

Equation (29) is consistent with:

( )2

1,2,eff 2,LM,corrhydr

1= 38

ck T

Tφ φ

ρ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

, (30) ( 21,2,eff in µm ; in µm/sk ρ )

Permeability prediction by NMR 21

where T2,LM,corr is the corrected logarithmic mean:

0.3

2,LM,corr 2,LM2,eff 2,LM

3.18 µm= ⎛ ⎞⎜⎜⎝ ⎠

T TTρ

⎟⎟ . (31)

The corrected pore radii rcorr [T2,LM] and permeabilities are presented in Table 6. To obtain

permeability values according to equation (30) the mean value of ρ 2,eff of the set of samples was used.

In Fig. 13, the permeabilities calculated from 1D T2 distributions according to equation (30) are plotted

versus depth and compared to gas permeabilities. Additionally, Fig. 14 shows the comparison to

permeability values calculated by the standard method in oil industry from 1D T2 distributions

according to equation (14) using a factor of a = 4 which is generally accepted for sandstones. Whereas

permeabilities derived from equation (30) agree well with gas flow measurements, results from the

standard method used in oil industry based underestimate permeability systematically. Particularly,

permeability is underestimated for the samples AC1 and AC3 which contain a considerable amount of

clay by four orders of magnitude.

5. Conclusion

NMR logging data recorded in a sandstone formation or NMR measurements performed with

a mobile NMR core scanner on corresponding core plugs are useful to determine porosity

directly and accurately. The empirical T2,LM equation with T2,LM, which is commonly used in

the oil industry for estimating permeability from NMR logging data, is acceptable for typical

reservoir rocks with porosity > 10%. A high exponent to porosity φ in this equation restricts

the applicability to a small range of porosities and the effective hydraulic pore radius has to

be larger than about 10 μm for using a constant surface relaxivity ρ2. In the case of the

Rhaetian sandstone formation at Allermoehe with low porosity ( ≤ 11%) and permeability,

this calculation scheme yields large errors. Therefore, the relationship between ρ2 and pore

radius r has been studied. From T1-T2 correlation experiments, the relationship between the

Permeability prediction by NMR 22

surface relaxivities ρ 1 and ρ 2 can be determined. A constant value for ρ 1 can be obtained

from the comparison of T1 distributions with mercury injection curves. Based on a model,

where an interlayer with large magnetic field gradients envelopes the inner side of the pore

walls, the thickness of this interlayer and a relaxation term ρ de-phase , describing de-phasing of

magnetic spins, can be calculated. Surface relaxivity ρ 1 and ρ de-phase compose ρ 2 which

increases with decreasing pore radius. Based on this analysis it is possible to calculate an

individual surface relaxivity ρ 2 for each T2 value. On this base, accurate permeability

prediction from transverse relaxation measurements is possible. A hydraulic effective pore

radius can be calculated for each sample from the logarithmic mean value of the T2

distribution curves. Consequently, permeability can be predicted accurately from the

effective pore radii with the physically based Kozeny-Carman equation [19, 20]. Moreover,

an empirical equation which is calibrated for T2 distributions measured on the Allermoehe

samples can be used to calculate permeability with the T2 distribution from measured decay

curves of transverse magnetization..

Appendix

Here, we give an explanation for equations (15) and (21) which relate T1 and Tde-phase to the pore

radius r and a reduced radius (r-d) respectively. We use a spherical pore model.

According to Seevers [29], inverse longitudinal relaxation time Τ1-1 is described as the

arithmetic mean of the free fluid relaxivity ρFF for the free water volume and the so called interlayer

relaxivity ρinterlayer1 for the interlayer1 at the pore wall. The interlayer1 with thickness δ1 is different from

the interlayer with large field gradients in Fig. 7 with thickness d.

1 11

1

11 interlayer interlayerFF interlayer

total total

V VT V V

ρ ρ−

= + (32)

Permeability prediction by NMR 23

The free fluid relaxivity results from the interaction between water molecules themselves,

whereas the interlayer1 relaxivity is explained by the interaction of water molecules with the crystal

lattice of the pore wall. As ρFF is much smaller than ρ interlayer1, the first term can be neglected for small

pores. Vinterlayer1/Vtotal can be expressed by Spor,NMR and the thickness δ1 of interlayer1 which is in the

range of nanometers:

1 1 , 11

1interlayer por NMRS

T r3ρ δ= ρ= . (33)

The value of longitudinal magnetization Mlongitudinal starts with zero for t = 0 and

approximates M∞ for long time. The function (M∞-M)longitudinal would be equal to

transverse magnetization if de-phasing would not appear. Fig. 15 presents a schematic

plot of (M∞-M)longitudinal versus diametric extension from the pore wall to the center

(dashed line). The local values of (M∞-M)longitudinal correspond to the concentration of

protons in the activated magnetic state, which is arbitrarily set to 80% at the center.

The protons loose their energy at the pore wall. The shape of the curve results from

diffusion of activated water molecules towards the pore wall. The thickness of

interlayer δ1 is constructed with help of a step curve which confines the same area as

the original curve.

The geometrical conditions of transverse magnetization Mtransverse are illustrated

in the same figure. The local values of Mtransverse correspond to the concentration of

protons in the activated magnetic state, which are not irreversibly de-phased and

therefore can be measured by the spin-echo-method. This fraction can be brought

back into phase during echo time in contrast to those that changed their precession

velocity by diffusion in an inhomogeneous magnetic field which is caused by

paramagnetic minerals at the pore walls. An interlayer of thickness d enveloping the

pore wall contains this internal, inhomogeneous magnetic field (see Fig. 7). The

second curve (continuous line in Fig. 15) represents the transverse magnetization due

to irreversible de-phasing without surface relaxation at the pore wall. The value in the

pore center is set to 60% in order to show that the relaxation due to de-phasing goes

faster than pure surface relaxation fore (M∞-M)longitudinal. The curve decreases with a

steep slope at the beginning of the interlayer of large magnetic field gradients. This

interlayer is about one order of magnitude thicker than interlayer1. The thickness d is

Permeability prediction by NMR 24

again constructed with help of a step curve which confines the same area as the

original curve. Below this curve lays the third curve (dash dotted line) which

represents the measured transverse magnetization due to surface relaxation coupled

with irreversible de-phasing. How these two processes work together is explained by

equation (20). Here, we want to give a justification for equation (21), which relates the

relaxation time Tde-phase to the reduced radius (r-d).

The curve in the middle of Fig. 14 shows that it is allowed to neglect the

portion of activated protons within the interlayer with thickness d. Therefore the

relaxation process of de-phasing can be described by the decrease of activated protons

in the inner sphere with volume V[r-d] = (4/3)(r-d)3. Some activated protons pass the

surface with area S[r-d] = 4(r-d)2 by diffusion and become de-phased. As the amount

of activated protons is proportional to M, we get:

[ ]de-phase[ ]

r dr d

dM MSdt V

ρ −−

= − . (34)

Integration yields:

[ ]

[ ]

de-phase 0[ ]

[ ]0 de-phase

de-phase de-phase

log log

1exp with

r d

r d

r d

r d

SM t M

V

VtM M TT S

ρ

ρ

−

−

−

−

= − +

⎛⎛ ⎞⎜ ⎟= − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎞. (35)

V[r-d] and S[r-d] are expressed by the radius (r-d):

( )de-phase

de-phase

1T r

ρ=3

d−. (36)

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Permeability prediction by NMR 25

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Permeability prediction by NMR 26

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Permeability prediction by NMR 27

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List of Table captions

Table 1

Experimental parameters and measurement times of the mobile NMR sensor

Permeability prediction by NMR 28

Table 2

Petrophysical parameters measured on the Allermoehe core plugs from various depths: pycnometer

porosity φpyc, formation factor F, specific inner surface area Spor,BET measured with the BET method, gas

permeability kgas, logarithmic mean values from 1D T2 distributions T2,LM and from 2D T1 and T2

distributions T1,2LM

Table 3

Petrophysical parameters of Allermoehe samples: Tortuosity Tel from electrical measurements and

hydraulic tortuosity Thydr estimated from porosity; permeabilities kgas from gas flow measurements and

kSpor,BET derived from Tel and Spor,BET; effective pore radii reff calculated from kgas , reff from kSpor,BET , and

reff,LM from mercury injection curves

Table 4

Results based on the mercury intrusion measurements

Table 5

Results based on mercury injection curves and on 2D T1-T2 correlation experiments performed on the

Allermoehe samples: pore radii rρ 1 [T1,LM] and rρ 2,eff [T2,LM] are calculated according to equation (15)

using surface relaxivities ρ 1 and ρ 2,eff ; the rcorr [T2,LM] values are calculated according to the empirical

equation (27). Permeabilities k follow from equations (28) and (29). For comparison gas permeabilities

(kgas) are also shown.

Permeability prediction by NMR 29

Table 6

Results based on 1D T2 relaxation measurements with the improved Halbach core-scanner: ρ2,eff values

follow from equation (15); rcorr [T2,LM] values result from equation (27); and permeabilities “k from rcorr

[T2,LM]” are calculated with equation (29) and “k from T2,LM,corr” result from equations (30) and (31)

using the mean value of ρ2,eff.

Permeability prediction by NMR 30

List of figure captions

Fig. 1. Mobile Halbach NMR sensor with an inner diameter of 140 mm and combined with a special

insert for core plugs with a diameter of 20 mm

Fig. 2. Comparison of NMR logging data recorded in the Allermoehe borehole and mobile NMR core-

scanner results: the Schlumberger CMR porosity log (full line) correlates well with core plug porosities

measured with the gas pycnometer (circles) and with the NMR core-scanner (diamonds).

Fig. 3. Correlation of core plug permeabilities: results from gas flow measurements are compared to

calculated permeabilities from a power-law relationship between permeability and porosity according to

equation (13). Porosity data from NMR logging and mobile NMR sensor measurements were used.

Fig. 4. Comparison of NMR logging and mobile NMR sensor data: the CMR permeability (full line) is

compared to NMR core permeability (diamonds) based on equation (14). Permeability values derived

from gas flow measurements on core plugs (circles) are shown for comparison.

Fig. 5. Radius distribution curve of sample AC9, which serves to demonstrate a method to deduce

several parameters of pore structure bases on the theory of fractal dimensions

Fig. 6. Variation of Permeability of Allermoehe sandstone samples with depth calculated from pore

radius distributions according to equation (10) for different definitions of pore radius (crosses and

triangles see legend) and compared to gas permeability (circles)

Permeability prediction by NMR 31

Fig. 7. Schematic model of a pore in a constant, homogeneous static magnetic field of amplitude

B = 0.2 T. Strong local magnetic field gradients caused by paramagnetic centers at the pore wall affect a

layer of thickness d.

Fig. 8. Two dimensional T1-T2 map of measurements recorded on Allermoehe core plug AC15. Due to

high internal gradients, the T1/T2 ratio is not constant.

Fig. 9. Dependence of surface relaxivity ρ2 on pore radius demonstrated for sample AC9.

a) The surface relaxivity ρ2, calculated with equation (22) and the thickness d of the interlayer with

large magnetic field gradients is plotted for radius > 2μm. The other surface relaxivities (spheres from

the empirical equation (26) and triangles for ρ2 =ρ1(T2/T1) represent the values at the maxima of the T1

and T2 distributions of the two-dimensional measurement (see Fig. 8). They are plotted versus the

maxima of the pore radius distribution from mercury intrusion.

b) The surface relaxivities ρ2 (triangles) calculated for T2 with the empirical equation (26) is compared

with values (dashed line) calculated with aid of a table based on eqs. (22) and (15).

Fig. 10. Relationships between T1 and T2 distribution curves of the T1-T2 correlation experiment,

explained for sample AC9.

(a) The T2 distribution curve is shifted so that the first maxima for large T1,2 values of both distribution

curves coincide.

(b) Two correction procedures applied for the T2 distribution curve are demonstrated which

compensate the effect of enhanced relaxation due to magnetic field gradients in the layer at the pore

walls.

Fig. 11. Permeabilities calculated from 2D T1 and T2 distributions according to equations (28) and (29)

(Table 4) plotted versus depth and compared to gas permeabilities for several Allermoehe samples

Permeability prediction by NMR 32

Fig. 12. Correlations of Allermoehe core plug permeabilities: results from gas flow measurements are

compared to calculated permeabilities based on 2D T2 measurements:

1.) permeability calculated from T2,LM according to the standard method in oil industry with equation

(14) using a factor a = 4,

2.) calculated from rcorr (T2,LM) according to equation (29).

Fig. 13. Permeabilities calculated from 1D T2 distributions according to equation (30) are plotted versus

depth and compared to gas permeabilities.

Fig. 14. Correlations of Allermoehe core plug permeabilities: results from gas flow measurements are

compared calculated permeabilities based on 1D T2 distributions measured with NMR core-scanner:

1.) permeability calculated from T2,LM according to the standard method in oil industry with equation

(14) using a factor a = 4,

2.) calculated with equation (30).

Fig. 15. Schematic drawing for explaining the relationship between T1 and r1 and the relationship

between Tde-phase and (r-d). See text for more details.

Table 1

Experimental parameters and measurement times of the mobile NMR sensor

Parameter Modified Halbach

core-scanner

Frequency 9.6 MHz

Number of scans 32-80

Pulse length 9.7 µs

Inter-echo time 0.06 ms - 0.15 ms

Number of echoes 1500-6400

Saturation delay time 1 s

Saturation recovery time 1 ms – 3.5 s

Measurement time

1D measurements: 30 s - 5 min

2D measurements: 30 min

1

Table 2

Petrophysical parameters measured on the Allermoehe core plugs from various depths: pycnometer

porosity φpyc, formation factor F, specific inner surface area Spor,BET measured with the BET method, gas

permeability kgas, logarithmic mean values from 1D T2 distributions T2,LM and from 2D T1 and T2

distributions T1,2LM

Sample Depth

[m]

φpyc

[%]

F Spor,BET

[μm-1]

kgas

[mD]

T2,LM_1D

[ms]

T1,LM_2D

[ms]

T2,LM_2D

[ms]

AC1 3224.45 2 53 40.42 0.16 2.95 - -

AC3 3225.64 2 54 39.79 0.42 7.95 - -

AC4 3235.34 9 56 2.38 11.6 82.17 510 81.84

AC6 3236.67 6 72 10.59 1.85 14.23 250 30.11

AC7 3236.79 8 67 9.40 3.59 32.93 220 29.07

AC9 3239.88 8 70 8.45 5.26 24.37 300 37.04

AC10 3240.69 11 38 5.24 20.7 30.37 330 41.34

AC12 3241.44 9 68 10.89 3.13 32.78 330 28.17

AC13 3241.75 9 58 9.21 4.42 31.79 190 24.34

AC14 3242.56 9 44 5.63 10.1 30.60 320 35.76

AC15 3243.16 9 50 4.34 19.6 34.02 440 35.03

AC16 3243.42 8 55 6.60 8.99 26.01 330 33.29

AC18 3243.78 8 56 10.89 6.16 21.95 320 33.73

AC19 3243.92 10 47 6.50 13.9 32.01 370 39.90

AC20 3244.79 8 60 5.17 7.2 29.77 320 35.35

AC21 3245.10 7 79 6.71 3.1 19.49 - -

AC22 3245.54 7 89 9.34 0.39 17.47 150 7.68

AC23 3246.15 8 92 8.37 0.4 18.88 130 7.47

AC24 3246.65 3 65 38.67 0.91 25.65 - -

AC25 3247.31 6 66 12.52 1.71 21.57 240 23.89

2

Table 3

Petrophysical parameters of Allermoehe samples: Tortuosity Tel from electrical measurements and

hydraulic tortuosity Thydr estimated from porosity; permeabilities kgas from gas flow measurements and

kSpor,BET derived from Tel and Spor,BET; effective pore radii reff calculated from kgas , reff from kSpor,BET , and

reff,LM from mercury injection curves

Sample Tel Thydr

= 0.6φ -1.2kgas

[mD] kSpor,BET

[mD] eq. (9)

reff [μm] from kgas

eq. (3)

reff [μm]

from kSpor,BET

eq. (3)

reff,LM [μm] from

mercury intrusion

AC1 1.06 65.60 0.16 0.09 0.26 0.19 - AC3 1.08 65.60 0.42 0.09 0.42 0.20 - AC4 5.04 10.79 11.6 598.38 2.26 16.27 - AC6 4.32 17.55 1.85 4.38 1.02 1.58 - AC7 5.36 12.43 3.59 6.84 1.37 1.90 3.14 AC9 5.6 12.43 5.26 9.13 1.70 2.25 3.78 AC10 3.8 8.48 20.7 82.35 2.37 4.74 5.27 AC12 6.12 10.79 3.13 4.25 1.29 1.51 2.76 AC13 5.22 10.79 4.42 8.42 1.42 1.96 2.82 AC14 3.96 10.79 10.1 51.66 1.87 4.24 4.90 AC15 4.5 10.79 19.6 102.53 2.77 6.36 9.23 AC16 4.4 12.43 8.99 25.15 1.97 3.31 4.45 AC18 4.48 12.43 6.16 5.16 1.64 1.51 4.63 AC19 4.7 9.51 13.9 30.87 2.26 3.38 - AC20 4.8 12.43 7.2 49.45 1.84 4.84 4.87 AC21 5.53 14.59 3.1 16.63 1.39 3.22 - AC22 6.23 14.59 0.39 5.25 0.52 1.92 - AC23 7.36 12.43 0.4 7.16 0.54 2.28 0.29 AC24 1.95 40.33 0.91 0.08 0.68 0.21 - AC25 3.96 17.55 1.71 2.83 0.94 1.22 -

3

Table 4

Results based on the mercury intrusion measurements

Sample reff,frac

[µm]

q cφ

k from

reff,frac

[mD]

eq. (10)

reff,LM

[µm]

reff,LM,free

[µm]

k from

reff,LM

[mD]

eq. (10)

k from

reff,LM,free

[mD]

eq. (10)

AC7 3.62 1.15 0.77 8.10 3.14 5.62

AC9 3.18 1.79 0.82 6.69 3.78 8.17

AC10 3.32 1.02 0.87 12.56 5.27 31.97

AC12 2.58 2.07 0.79 5.46 2.76 5.64

AC13 2.60 1.72 0.80 5.59 2.82 5.89

AC14 2.38 1.00 0.84 4.93 4.90 17.73

AC15 4.21 1.00 0.88 16.26 9.23 63.01

AC16 1.76 1.00 0.78 1.95 4.45 11.3

AC18 2.98 1.00 0.82 5.85 4.63 12.22

AC20 3.59 1.00 0.82 8.60 4.87 13.53

AC23 1.61 3.51 0.33 0.68 0.29 0.9 0.46

4

Table 5

Results based on mercury injection curves and on 2D T1-T2 correlation experiments performed on the

Allermoehe samples: pore radii rρ 1 [T1,LM] and rρ 2,eff [T2,LM] are calculated according to equation (15)

using surface relaxivities ρ 1 and ρ 2,eff ; the rcorr [T2,LM] values are calculated according to the empirical

equation (27). Permeabilities k follow from equations (28) and (29). For comparison gas permeabilities

(kgas) are also shown.

sample ρ1

[μm/s]

ρ2,eff

[μm/s]

rρ 1

[T1,LM]

[µm]

eq. (15)

rρ 2,eff

[T2,LM]

[µm]

eq. (15)

rcorr

[T2,LM]

[µm]

eq. (27)

k [mD]

from rρ 1

(T1,LM)

eq. (28)

k [mD]

from rcorr

(T2,LM)

eq. (29)

kgas [mD]

AC4 6.51 6.12 6.97 35.96 27.67 11.6

AC6 3.19 2.25 3.45 3.60 1.53 1.85

AC7 4.83 21.89 3.19 1.91 3.07 5.38 2.08 3.59

AC9 4.24 23.82 3.82 2.65 3.87 8.53 4.00 5.26

AC10 4.24 23.82 4.20 2.95 4.18 20.06 10.03 20.7

AC12 2.37 17.76 2.35 1.50 2.59 4.97 1.67 3.13

AC13 4.21 13.30 2.40 0.97 1.91 2.69 0.70 4.42

AC14 5.31 37.20 5.10 3.99 5.16 19.71 11.78 10.1

AC15 4.28 35.10 5.65 3.69 4.88 17.64 10.06 19.6

AC16 5.77 33.62 5.71 3.36 4.57 11.92 6.43 8.99

AC18 5.09 29.68 4.89 3.00 4.23 10.19 5.15 6.16

AC19 4.72 2.98 4.21 16.48 8.29 13.9

AC22 1.92 0.57 1.32 0.74 0.14 0.39

AC23 2.22 12.96 0.87 0.29 0.82 0.38 0.05 0.4

AC25 3.06 1.79 2.93 2.60 0.97 1.71

mean

value

= 4.256

mean

value

= 24.915

5

Table 6

Results based on 1D T2 relaxation measurements with the improved Halbach core-scanner: ρ2,eff values

follow from equation (15); rcorr [T2,LM] values result from equation (27); and permeabilities “k from rcorr

[T2,LM]” are calculated with equation (29) and “k from T2,LM,corr” result from equations (30) and (31) using

the mean value of ρ2,eff.

sample ρ2,eff [μm/s]

eq. (15) rcorr (T2,LM)

[μm]

eq. (27)

k [mD]

from rcorr [T2,LM]

eq. (30)

k [mD]

from T2,LM,corr

eqs. (30) and (31)

and mean ρ2,eff

kgas

[mD]

AC1 - 1.53 0.21 0.21 0.16 AC3 - 2.49 0.55 0.55 0.42 AC4 - 6.14 27.90 27.91 11.6 AC6 - 1.79 0.97 0.97 1.85 AC7 21.60 3.32 6.30 6.46 3.59 AC9 17.37 2.31 3.03 2.73 5.26

AC10 23.46 3.33 12.72 13.70 20.7 AC12 13.10 2.33 4.01 3.06 3.13 AC13 22.05 3.29 7.99 8.30 4.42 AC14 13.10 2.22 3.64 2.78 10.1 AC15 27.85 4.07 12.22 14.56 19.6 AC16 29.68 3.52 7.07 8.74 8.99 AC18 26.29 2.87 4.69 5.40 6.16 AC19 - 3.16 4.43 4.44 13.9 AC20 23.07 3.24 5.99 6.39 7.2 AC21 - 2.23 2.11 2.12 3.1 AC22 - 2.06 1.81 1.81 0.39 AC23 10.38 1.34 1.02 0.68 0.4 AC25 - 2.71 0.48 0.48 1.71

mean value 20.72

6

Fig. 1. Mobile Halbach NMR sensor with an inner diameter of 140 mm and combined with a special insert

for core plugs with a diameter of 20 mm

1

Fig. 2. Comparison of NMR logging data recorded in the Allermoehe borehole and mobile NMR core-

scanner results: the Schlumberger CMR porosity log (full line) correlates well with core plug porosities

measured with the gas pycnometer (circles) and with the NMR core-scanner (diamonds).

2

Fig. 3. Correlation of core plug permeabilities: results from gas flow measurements are compared to

calculated permeabilities from a power-law relationship between permeability and porosity according to

equation (13). Porosity data from NMR logging and mobile NMR sensor measurements were used.

3

Fig. 4. Comparison of NMR logging and mobile NMR sensor data: the CMR permeability (full line) is

compared to NMR core permeability (diamonds) based on equation (14). Permeability values derived

from gas flow measurements on core plugs (circles) are shown for comparison.

4

clay pore volume

q

surface enlargement q

reff,fracreff,BET

ΔVi

Spor,frac from Spor,BET (eq. 16)Spor=2/r (eq. 4)Spor,mercury [rn] (eq. 17) Spor,frac,free (eq. 18)

change of slopeS

por[μ

m-1

]

Fig. 5. Radius distribution curve of sample AC9, which serves to demonstrate a method to deduce several

parameters of pore structure bases on the theory of fractal dimensions

5

3250

3245

3240

3235de

pth

[m]

0.1 1.0 10.0 100.0

permeability [mD]

kgas

k from reff,LM and reff,LM,free

k from reff,frac,free

kgas

k from reff,LM and reff,LM,free

k from reff,frac,free

Fig. 6. Variation of Permeability of Allermoehe sandstone samples with depth. Permeability is calculated

from pore radius distributions according to equation (10) for different definitions of pore radius (crosses

and triangles see legend) and compared to gas permeability (circles)

6

Fig. 7. Schematic model of a pore in a constant, homogeneous static magnetic field of amplitude B = 0.2 T.

Strong local magnetic field gradients caused by paramagnetic centers at the pore wall affect a layer of

thickness d.

7

Fig. 8. Two dimensional T1-T2 map of measurements recorded on Allermoehe core plug AC15. Due to high

internal gradients, the T1/T2 ratio is not constant.

8

a)

b)

ρ1ρ2 from eq. (22)ρ2 from eq. (26)ρ2 =ρ1(T1/T2)

ρ2 from eqs. (22) and (15)ρ2 from eq. (26)

Fig. 9. Dependence of surface relaxivity ρ2 on pore radius demonstrated for sample AC9.

a) The surface relaxivity ρ2, calculated with equation (22) and the thickness d of the interlayer with large

magnetic field gradients is plotted for radius > 2μm. The other surface relaxivities (spheres from the

empirical equation (26) and triangles for ρ2 =ρ1(T2/T1) represent the values at the maxima of the T1 and

T2 distributions of the two-dimensional measurement (see Fig. 8). They are plotted versus the maxima of

the pore radius distribution from mercury intrusion.

b) The surface relaxivities ρ2 (triangles) calculated for T2 with the empirical equation (26) is compared

with values (dashed line) calculated with aid of a table based on eqs. (22) and (15).

9

Fig. 10. Relationships between T1 and T2 distribution curves of the T1-T2 correlation experiment, explained

for sample AC9.

(a) The T2 distribution curve is shifted so that the first maxima for large T1,2 values of both distribution

curves coincide.

(b) Two correction procedures applied for the T2 distribution curve are demonstrated which compensate

the effect of enhanced relaxation due to magnetic field gradients in the layer at the pore walls.

10

kgask from rρ1 [T1,LM]k from rcorr [T2,LM]

Fig. 11. Permeabilities calculated from 2D T1 and T2 distributions according to equations (28) and (29)

(Table 4) plotted versus depth and compared to gas permeabilities for several Allermoehe samples

11

Fig. 12. Correlations of Allermoehe core plug permeabilities: results from gas flow measurements are

compared to calculated permeabilities based on 2D T2 measurements:

1.) permeability calculated from T2,LM according to the standard method in oil industry with equation (14)

using a factor a = 4,

2.) calculated from rcorr (T2,LM) according to equation (29).

12

kgask from T2,LM (eq. 30)

Fig. 13. Permeabilities calculated from 1D T2 distributions according to equation (30) are plotted versus

depth and compared to gas permeabilities.

13

k based on eq. (14) k based on eq. (30)

Fig. 14. Correlations of Allermoehe core plug permeabilities: results from gas flow measurements are

compared calculated permeabilities based on 1D T2 distributions measured with NMR core-scanner:

1.) permeability calculated from T2,LM according to the standard method in oil industry with equation (14)

using a factor a = 4,

2.) calculated with equation (30).

14

100 101 102 103 1040

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

M∞-M rectangular functionM irreversible de-phasingM rectangular functionM de-phasing and surface relaxation

long

itudi

nal m

agne

tizat

ion

M∞

-M

trans

vers

e m

agne

tizat

ion

M

δ1

interlayer of field gradients

d

diametric extension [nm]

Fig. 15. Schematic drawing for explaining the relationship between T1 and r1 and the relationship between

Tde-phase and (r-d). See text for more details.

15