Journal of Environmental Quality 31:500-505 (2002)
© 2002 American Society of Agronomy, Crop Science Society of America, and Soil Science Society of America
Article
SYMPOSIUM PAPERS
Variations of Permeability and Pore Size Distribution of Porous Media with Pressure
Quan Chena,
Wolfgang Kinzelbach*,b,
Chaohui Yea and
Yong Yuea
a Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, P.O. Box 71010, Wuhan, 430071, China
b Swiss Federal Institute of Technology Zurich, Institute of Hydromechanics and Water Resources Management, ETH-Hoenggerberg, CH-8093 Zurich, Switzerland
* Corresponding author (kinzelbach{at}ihw.baug.ethz.ch)
Received for publication June 2, 2000.
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ABSTRACT
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Porosity and permeability of porous and fractured geological media decrease with the exploitation of formation fluids such as petroleum, natural gas, or ground water. This may result in ground subsidence and a decrease of recovery of petroleum, natural gas, or ground water. Therefore, an evaluation of the behavior of permeability and porosity under formation fluid pressure changes is important to petroleum and ground water industries. This study for the first time establishes a method, which allows for the measurement of permeability, porosity, and pore size distribution of cores simultaneously. From the observation of the pore size distribution by low-field nuclear magnetic resonance (NMR) relaxation time spectrometry the mechanisms of pressure-dependent porosity and permeability change can be derived. This information cannot be obtained by traditional methods. As the large-size pores or fractures contribute significantly to the permeability, their change consequently leads to a large permeability change. The contribution of fractures to permeability is even larger than that of pores. Thus, the permeability of the cores with fractures decreased more than that of cores without fractures during formation pressure decrease. Furthermore, it did not recover during formation pressure increase. It can be concluded that in fractures, mainly plastic deformation takes place, while matrix pores mainly show elastic deformation. Therefore, it is very important to keep an appropriate formation fluid pressure during the exploitation of ground water and petroleum in a fractured formation.
Abbreviations: CPMG, pulse sequence proposed by Carr, Purcell, Meiboom, and Gill • ESEM, environmental scanning electron microscopy • NMR, nuclear magnetic resonance
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INTRODUCTION
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DURING THE PUMPING of oil, natural gas, or ground water from fractured or porous formations, a problem occurs when, due to the drawdown created by the well, the pressure in the adjacent formation is not sufficient to keep fractures or pore spaces open. If fractures collapse, the permeability of the rock and with it the productivity of the well decrease, and the sustainability of production is endangered. Even if higher pressure levels in the well are restored, for example by decrease in the pumping rate in the case of ground water, or by injection of brine in a neighboring well in the case of oil production, the damage to the fractures in the vicinity of the well may be irreversible. In the case of overpumping of ground water or natural gas, land subsidence may result, as is experienced on a large scale in places such as Mexico City and Bangkok, and the Po Delta (National Research Council, 1995; Gambolati et al., 1974; Srivardhana, 1994).
To estimate the consequences of overexploitation, reliable data on rock compressibility and permeability are essential due to the significant impact these parameters have on reserves and productivity estimates as well as subsidence rates. Laboratory measurements of rock compressibility are applied to production forecasts and reservoir pressure maintenance evaluations, as well as compaction and subsidence studies (Johnson et al., 1989; Ruddy et al., 1989; Teufel et al., 1991; Rhett and Teufel, 1992; Ruistuen et al., 1999). Permeability heavily influences reservoir productivity and injectivity and is essential in performance forecasting (Rhett and Teufel, 1992). Since the early 1950s a number of researchers have investigated the relationships between rock matrix permeability and applied external pressure.
However, traditional experimental methods cannot measure the variations of porosity, permeability, and pore size distribution during formation pressure changes in the same experiment. In the studies presented here, the effect of pressure changes on porosity, permeability, and pore size distribution of sandstone were investigated simultaneously in the same sample, by adding NMR to the classical measurement procedure.
When talking about pressure in a rock formation we have to differentiate between the pressure of the liquid in the formation (i.e., formation pressure), and the overburden pressure of the rock. In the laboratory setup the overburden pressure is simulated by the confining pressure on the core holder. For the deformation of the rock only the difference between formation pressure and overburden pressure is of interest. The difference in the experiment is varied by changing the confining pressure and keeping the liquid (formation) pressure constant. In nature the difference varies due to changing formation pressure at constant overburden pressure.
The methods used here can also be applied to problems from soil science and contaminant hydrology whenever the variation of the pore size distribution is of interest.
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RELAXATION THEORY OF NUCLEAR MAGNETIC RESONANCE IN POROUS MEDIA
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The essential information on rock or sedimentary samples, which can be provided by low-field NMR, is the size distribution of fluid-filled pores. It is shown in the following how this distribution and the NMR signal are related to each other. The advantage of NMR is that the pore size distribution is determined nonintrusively and can be observed simultaneously under varying pressure with other experiments such as the permeability measurement. Hydrogen nuclei have a magnetic moment and behave like small bar magnets. When subjected to a magnetic field, such nuclei tend to align their magnetic moments parallel to the field, producing a net nuclear magnetization. In the NMR method their angle with respect to the magnetic field is changed by a radio frequency pulse. Once the pulse stops they regain their original orientation by relaxation. In a saturated porous medium the relaxation time depends not only on the fluid but also on the medium and the interaction between them. Thus, the study of relaxation time can provide information on the structure of a porous medium.
Pulsed NMR measures the magnetization (M) and transverse relaxation time (T2) of hydrogen nuclei contained in the pore fluids. The term M is proportional to the number of hydrogen nuclei in the sensitive region and can be scaled to give a NMR porosity (Timur, 1969; Kenyon, 1992). For fluids confined in pores, the T2 value can be shorter than that of the bulk fluid if the fluid interacts with the rock surface, which promotes NMR relaxation (Korringa et al., 1962; Kenyon, 1992). Three different mechanisms, which operate in parallel, contribute to the overall apparent relaxation rate 1/T2A of fluid in porous media (Kleinberg and Horsfield, 1990; Kleinberg et al., 1993):
 | [1] |
where the subscripts A, B, S, and D denote apparent, bulk, surface-induced, and diffusion-induced mechanisms, respectively. The bulk relaxation time is a property of the fluid only. Because the relaxation time of liquid in rocks is much shorter than the relaxation time of bulk liquid, the bulk terms in Eq. [1] can be neglected. The surface-induced relaxation is due to interaction between fluid and the solid surface while the diffusion-related relaxation is caused by diffusion in the inhomogeneous magnetic field arising from the magnetic susceptibility contrast between the grains and pore fluid. The surface and diffusion-induced relaxation rates are given by (Fukushima and Roeder, 1981; Cohen and Mendelson, 1982):
 | [2] |
 | [3] |
where 
2 is the surface relaxivity, S/V is the surface to volume ratio, 
is the gyromagnetic ratio, G is the "background" magnetic field gradient, TE is the echo time, and Do is the self-diffusion coefficient of the liquid.
The majority of rocks conform with the "fast-diffusion" (Brownstein and Tarr, 1979) or "surface limited" (Belton et al., 1988) relaxation regime in which the relaxation at the surface is slower than the transport of the hydrogen nuclei to the surface. Thus, the spins experience a rapid exchange of environments so that the local fields in each region of a pore are averaged to their mean value. As a consequence, a single exponential decay is observed for a given pore, and the rate of magnetization decay depends on surface to volume ratio only (Kleinberg et al., 1994). Under the conditions of low magnetic field strength (i.e., G is also small) and short TE (Kleinberg and Horsfield, 1990; Kleinberg et al., 1993), the enhancement in T2 decay coming from diffusion in the inhomogeneous local magnetic fields is negligible compared with the surface relaxation mechanism. Therefore, the measured T2 values are given by:
 | [4] |
Equation [4] forms the basis of NMR core analysis and log interpretation: T2 is proportional to V/S, which in turn is proportional to pore size. This means that in small pores relaxation is faster than in large pores.
In the measurement, the CPMG pulse sequence (proposed by Carr, Purcell, Meiboom, and Gill [Carr and Purcell, 1954; Meiboom and Gill, 1958]) was used. It consists of one 90° pulse followed by a series of 180° pulses. The time interval between two 180° pulses is the echo time TE, the time between two sequences is the recovery time TR. The term TR must be long enough to make sure that magnetization recovery to equilibrium is efficient. When hydrogen nuclei are tipped 90° from the direction of magnetic field, they precess and dephase due to the inhomogeneity of magnetic field. The nuclei can be refocused after a 180° pulse is transmitted. As the nuclei rephase, they generate a signal in a receive coil—a spin echo (Hahn, 1950). The 180° pulses can be applied repeatedly to produce a series of echo trains.
The total measured magnetization signal is a superposition of the signals coming from all pores within the measurement volume. It can be expressed as a sum of exponentials:
 | [5] |
which shows that the overall decay is the sum of the individual decays and reflects pore size distribution. By using proper fitting routines Eq. [5] can be inverted into a T2 relaxation time distribution, where the T2i belong to a preselected basis set of relaxation time constants and Ai (0) are the signal amplitudes (Dunn and Latorraca, 1994; Bulter and Dawson, 1981). Because T2 depends linearly upon pore size, the T2 distribution corresponds to a pore-size distribution.
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MATERIALS AND METHODS
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Siltstone samples of Daqing oilfield, located in Heilongjiang Province of China, were cored with their axis approximately parallel to the bedding planes of the formation. The cores had a diameter of 3.8 cm and a length of 7.6 cm. Core 20 had a fracture along the axial direction of the core extending over its whole length, Core 39 had a fracture along the axial direction of the core extending over half the length, and Core 52 was a matrix core. The average clay content of the samples is 10.3%, while the relative content of clay is about 50 to 60% illite, 20 to 30% chlorite, and 26% illite–smectite mixture.
Cores were cleaned with ethanol and benzene by an extraction method. Cores were saturated with kerosene under vacuum conditions to minimize the interaction with the rock. The core porosity was measured by Archimedes' principle. The NMR images were produced on a Bruker (Faellanden, Switzerland) Biospec 47/40 superconductive nuclear magnetic resonance imaging (NMRI) instrument. The strength of the magnetic field is 4.7 Tesla, which corresponds to 200 MHz for the hydrogen nucleus (1H) resonance frequency. The two-dimensional proton density image was calculated from T2 images (CPMG sequence with TE = 2 ms and recovery time TR = 3 s), while the porosity distribution was scaled according to the proportionality between proton density and volume of liquid in the core. The image matrix was 128 x 128.
Nuclear magnetic resonance relaxation measurements were carried out on a homemade NMR spectrometer with a 1175-Gauss magnetic field strength, which corresponds to 5 MHz for the hydrogen nucleus (1H) resonance frequency. The permanent magnet has a 13-cm bore in the horizontal direction. A nonmagnetic core holder made of fiberglass material was put inside the probe and magnet. The profile schematic map of the nonmagnetic core holder was indicated in Fig. 1
. In Fig. 1, the core sample (1) was sealed in a fiberglass core holder (2) by two nylatron distributors (3) and a teflon tube (4). Different confining pressures can be controlled by a cylinder pump with injection or exsuction of deuterated water from the inlet (5). Kerosene was injected to the core from the inlet (6) and produced from the outlet (7) for the measurement of permeability.
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Fig. 1. Profile schematic map of the nonmagnetic core holder. (1) Core sample, (2) fiberglass, (3) nylatron distributor, (4) teflon tube, (5) inlet for confining pressure, (6) inlet for injection, (7) outlet.
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The CPMG pulse sequence was employed for the measurement of T2. The number of echoes was 1024, TE was 0.15 ms, and recovery time was 3 s.
Experimental Procedure
The experiments were carried out according to the following procedure. The matrix permeability of Core 20 was measured before fracturing of the core. The core samples were analyzed by environmental scanning electron microscopy (ESEM). The cores were saturated with kerosene under vacuum conditions for more than 24 h. The use of kerosene instead of water minimized the interaction between the rock and liquid (e.g., swelling). The total porosity of the rocks was measured using Archimedes' principle. A proton density image was taken by NMRI (using the CPMG pulse sequence) in order to determine the voxel porosity distribution. Then, the relaxation time distribution of the samples was measured by low field NMR. Simultaneously, two other measurements were carried out. Total porosity change was measured by volumetric determination of the displaced fluid, and permeability was measured by flooding of the system at different overburden pressures.
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RESULTS AND DISCUSSION
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The initial total porosities and permeabilities of cores are listed in Table 1. The matrix permeability of Core 20 amounted to 0.043 x 10-3 µm2.
Figures 2 through 4
show typical ESEM scans of material from the three cores. Core 20 has few resident pores between particles in the matrix, resulting in the measured low matrix porosity and permeability. In Core 39, microfractures can be seen at the edge of some particles and some pores between grains can be observed. Core 52 shows developed pores with good connectivity between grains.
The voxel porosity distribution of Core 39 is shown in Fig. 5 . The map on the upper righthand side of Fig. 5 is a two-dimensional proton density image of the core in the cross section. The diagram on the lower part of Fig. 5 is the voxel porosity distribution; the x scale, running from 0 to 511, indicates a grayscale of the pixels in the cross section. The scale is linearly related to porosity and fixed by the average grayscale. The y scale is the distribution frequency. The minimum and maximum voxel porosities given in Table 1 were obtained from such measurements.
Figures 6 through 9
show the change of porosity 
relative to the initial porosity 0 with varying effective overburden pressure P for all three cores. The effective overburden pressure is the difference between overburden pressure and fluid pressure at the inlet. The decrease of formation pressure is simulated by increasing the effective overburden pressure.
In Fig. 6, the decrease of relative porosity with increasing effective overburden pressure (corresponding to the decrease of formation pressure) is shown for all three cores together. The porosity ratio decrease is larger for the fractured Cores 39 and 20 than for the unfractured Core 52. The functional relation between porosity ratio /0 and effective overburden pressure P was fitted. The functions (righthand side of Fig. 6) are linear for all three cores with a high correlation coefficient (R2).
Figures 7 through 9 depict the hysteretic behavior core by core. The upper branch corresponds to the curve of Fig. 6. Note the different scale. The porosity ratio of the fractured cores (20 and 39 in Fig. 7 and 8) recovers to more than 92%. The unfractured Core 52, however, (Fig. 9) recovers to more than 99%.
Similarly, Fig. 10 through 13
show the behavior of permeability K relative to initial permeability K0 under increase of effective overburden pressure P (corresponding to formation pressure decrease) and the corresponding hysteretic behavior. The relation of the permeability ratio (K/K0) and effective overburden pressure P was fitted. The functions are shown on the righthand side of Fig. 10. They are exponential with a high correlation coefficient (R2). The unfractured Core 52 shows a very small change in permeability ratio of less than 20% compared with the two fractured cores (20 and 39), which show a change of more than 80%.
Figures 11 through 13 depict the hysteretic behavior core by core. The upper branch corresponds to the curve of Fig. 10. Note the different scale. The permeability ratios of the fractured cores (20 and 39 in Fig. 11 and 12) show a recovery to less than 30% only. The unfractured Core 52 (Fig. 13), however, recovers to more than 92%.
The final three figures (Fig. 14–16)
show the distributions of relaxation times T2 for the three cores at initial effective overburden pressure (Pi = 2.5 MPa), under maximum effective overburden pressure (P = 20 MPa), and after relieving overburden pressure to the initial value (P = 2.5 MPa). The relaxation time is directly proportional to the pore size. The spectra show a bimodal shape. The changes in pore size distribution are restricted to the large pore size range of the spectrum, which contains large pores and the fractures. The large pore size distribution is shifted left toward smaller pores under pressure increase and does not recover to the original distribution when pressure is diminished again. The changes are smallest for the unfractured Core 52 (Fig. 16). They are much larger for the fractured Cores 20 and 39 (Fig. 14 and 15).
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Fig. 14. Changes in the distribution of relaxation time T2 with effective overburden pressure of Core 20.
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Fig. 16. Changes in the distribution of relaxation time T2 with effective overburden pressure of Core 52.
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Fig. 15. Changes in the distribution of relaxation time T2 with effective overburden pressure of Core 39.
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CONCLUSIONS
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This study for the first time establishes a method that allows for simultanous measurement of permeability, porosity, and pore size distribution of a core. From the observation of the pore size distribution the mechanisms of pressure-dependent porosity and permeability change can be derived.
With the decrease of formation pressure, porosity decreased slightly. The relation between the two quantities is approximately linear. At the same time, the distribution of pore sizes was shifted toward smaller radii in the large pore size range. As the large-size pores or fractures contribute significantly to the permeability, their change consequently leads to a large permeability change obeying an exponential relationship.
With the increase of formation pressure, porosity, pore size distribution, and permeability recovered gradually, but did not return to the original values. The recovery of porosity and permeability was less in the fractured cores compared with the unfractured one as the contribution of fractures to permeability is even larger than that of pores.
It can be concluded that in fractures mainly plastic deformation takes place, while matrix pores mainly show elastic deformation. The permeability of the formations with fractures can decrease seriously during formation pressure decrease and need not recover during formation pressure increase. Therefore, it is very important to keep an appropriate formation pressure during the exploitation of ground water and petroleum in a fractured formation.
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Geological Society, London, Special Publications,
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267(1):
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[Abstract]
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ABSTRACT
INTRODUCTION
