Variations of Permeability and Pore Size Distribution of Porous Media with Pressure

Variations of Permeability and Pore Size Distribution of Porous Media with Pressure Quan Chen, Wolfgang Kinzelbach,* Chaohui Ye, and Yong Yue ABSTRACT ters have on reserves and productivity estimates as well Porosity and permeability of porous and fractured geological media as subsidence rates. Laboratory measurements of rock decrease with the exploitation of formation fluids such as petroleum, compressibility are applied to production forecasts and natural gas, or ground water. This may result in ground subsidence reservoir pressure maintenance evaluations, as well as and a decrease of recovery of petroleum, natural gas, or ground water. compaction and subsidence studies (Johnson et al., 1989; Therefore, an evaluation of the behavior of permeability and porosity Ruddy et al., 1989; Teufel et al., 1991; Rhett and Teufel, under formation fluid pressure changes is important to petroleum and 1992; Ruistuen et al., 1999). Permeability heavily influground water industries. This study for the first time establishes a ences reservoir productivity and injectivity and is essenmethod, which allows for the measurement of permeability, porosity, tial in performance forecasting (Rhett and Teufel, 1992). and pore size distribution of cores simultaneously. From the observation of the pore size distribution by low-field nuclear magnetic resoinvestigated the relationships between rock matrix per- Since the early 1950s a number of researchers have nance (NMR) relaxation time spectrometry the mechanisms of presmeability and applied external pressure. sure-dependent porosity and permeability change can be derived. This information cannot be obtained by traditional methods. As the largesize pores or fractures contribute significantly to the permeability, measure the variations of porosity, permeability, and However, traditional experimental methods cannot their change consequently leads to a large permeability change. The pore size distribution during formation pressure changes contribution of fractures to permeability is even larger than that of in the same experiment. In the studies presented here, pores. Thus, the permeability of the cores with fractures decreased the effect of pressure changes on porosity, permeability, more than that of cores without fractures during formation pressure and pore size distribution of sandstone were investidecrease. Furthermore, it did not recover during formation pressure gated simultaneously in the same sample, by adding increase. It can be concluded that in fractures, mainly plastic deformation takes place, while matrix pores mainly show elastic deformation. NMR to the classical measurement procedure. Therefore, it is very important to keep an appropriate formation fluid When talking about pressure in a rock formation we pressure during the exploitation of ground water and petroleum in a have to differentiate between the pressure of the liquid fractured formation. in the formation (i.e., formation pressure), and the over- burden pressure of the rock. In the laboratory setup the overburden pressure is simulated by the confining During the pumping of oil, natural gas, or ground pressure on the core holder. For the deformation of the water from fractured or porous formations, a proband overburden pressure is of interest. The difference rock only the difference between formation pressure lem occurs when, due to the drawdown created by the well, the pressure in the adjacent formation is not suffipressure and keeping the liquid (formation) pressure in the experiment is varied by changing the confining cient to keep fractures or pore spaces open. If fractures collapse, the permeability of the rock and with it the constant. In nature the difference varies due to changing productivity of the well decrease, and the sustainability formation pressure at constant overburden pressure. of production is endangered. Even if higher pressure The methods used here can also be applied to problevels in the well are restored, for example by decrease lems from soil science and contaminant hydrology when- in the pumping rate in the case of ground water, or by ever the variation of the pore size distribution is of interest. injection of brine in a neighboring well in the case of oil production, the damage to the fractures in the vicinity RELAXATION THEORY OF NUCLEAR of the well may be irreversible. In the case of overpump- MAGNETIC RESONANCE ing of ground water or natural gas, land subsidence may IN POROUS MEDIA result, as is experienced on a large scale in places such as The essential information on rock or sedimentary samples, Mexico City and Bangkok, and the Po Delta (National which can be provided by low-field NMR, is the size distribu- Research Council, 1995; Gambolati et al., 1974; Srivard- tion of fluid-filled pores. It is shown in the following how this hana, 1994). distribution and the NMR signal are related to each other. To estimate the consequences of overexploitation, The advantage of NMR is that the pore size distribution is reliable data on rock compressibility and permeability determined nonintrusively and can be observed simultane- are essential due to the significant impact these parameas the permeability measurement. Hydrogen nuclei have a ously under varying pressure with other experiments such magnetic moment and behave like small bar magnets. When Q. Chen, C. Ye, and Y. Yue, Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and subjected to a magnetic field, such nuclei tend to align their Mathematics, The Chinese Academy of Sciences, P.O. Box 71010, magnetic moments parallel to the field, producing a net nu- Wuhan, 430071, China. W. Kinzelbach, Swiss Federal Institute of clear magnetization. In the NMR method their angle with Technology Zurich, Institute of Hydromechanics and Water Resources Management, ETH-Hoenggerberg, CH-8093 Zurich, Switzer- respect to the magnetic field is changed by a radio frequency land. Received 2 June 2000. *Corresponding author (kinzelbach@ihw. baug.ethz.ch). Abbreviations: CPMG, pulse sequence proposed by Carr, Purcell, Meiboom, and Gill; ESEM, environmental scanning electron microscopy; NMR, nuclear magnetic Published in J. Environ. Qual. 31:500 505 (2002). resonance. 500

CHEN ET AL.: PERMEABILITY AND PORE SIZE DISTRIBUTION OF POROUS MEDIA 501 pulse. Once the pulse stops they regain their original orienta- two sequences is the recovery time T R. The term T R must be tion by relaxation. In a saturated porous medium the relaxation long enough to make sure that magnetization recovery to time depends not only on the fluid but also on the equilibrium is efficient. When hydrogen nuclei are tipped 90 medium and the interaction between them. Thus, the study from the direction of magnetic field, they precess and dephase of relaxation time can provide information on the structure due to the inhomogeneity of magnetic field. The nuclei can of a porous medium. be refocused after a 180 pulse is transmitted. As the nuclei Pulsed NMR measures the magnetization (M) and trans- rephase, they generate a signal in a receive coil a spin echo verse relaxation time (T 2 ) of hydrogen nuclei contained in the (Hahn, 1950). The 180 pulses can be applied repeatedly to pore fluids. The term M is proportional to the number of produce a series of echo trains. hydrogen nuclei in the sensitive region and can be scaled to The total measured magnetization signal is a superposition give a NMR porosity (Timur, 1969; Kenyon, 1992). For fluids of the signals coming from all pores within the measurement confined in pores, the T 2 value can be shorter than that of the volume. It can be expressed as a sum of exponentials: bulk fluid if the fluid interacts with the rock surface, which promotes NMR relaxation (Korringa et al., 1962; Kenyon, M(t) i A i (0)exp( t/t 2i ) [5] 1992). Three different mechanisms, which operate in parallel, which shows that the overall decay is the sum of the individual contribute to the overall apparent relaxation rate 1/T 2A of fluid decays and reflects pore size distribution. By using proper in porous media (Kleinberg and Horsfield, 1990; Kleinberg et fitting routines Eq. [5] can be inverted into a T 2 relaxation al., 1993): time distribution, where the T 2i belong to a preselected basis 1/T set of relaxation time constants and A i (0) are the signal ampli- 2A 1/T 2B 1/T 2S 1/T 2D [1] tudes (Dunn and Latorraca, 1994; Bulter and Dawson, 1981). where the subscripts A, B, S, and D denote apparent, bulk, Because T 2 depends linearly upon pore size, the T 2 distribution surface-induced, and diffusion-induced mechanisms, respec- corresponds to a pore-size distribution. tively. 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. MATERIALS AND METHODS [1] can be neglected. The surface-induced relaxation is due Siltstone samples of Daqing oilfield, located in Heilongjiang to interaction between fluid and the solid surface while the Province of China, were cored with their axis approximately diffusion-related relaxation is caused by diffusion in the inhoa parallel to the bedding planes of the formation. The cores had mogeneous magnetic field arising from the magnetic susceptifracture diameter of 3.8 cm and a length of 7.6 cm. Core 20 had a bility contrast between the grains and pore fluid. The surface along the axial direction of the core extending over and diffusion-induced relaxation rates are given by (Fukution its whole length, Core 39 had a fracture along the axial direc- shima and Roeder, 1981; Cohen and Mendelson, 1982): of the core extending over half the length, and Core 52 1/T 2S 2 S/V [2] 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% 1/T 2D [( GT E ) 2 D o ]/12 [3] illite, 20 to 30% chlorite, and 26% illite smectite mixture. Cores were cleaned with ethanol and benzene by an extracwhere 2 is the surface relaxivity, S/V is the surface to volume tion method. Cores were saturated with kerosene under vacratio, is the gyromagnetic ratio, G is the background uum conditions to minimize the interaction with the rock. The magnetic field gradient, T E is the echo time, and D o is the self- core porosity was measured by Archimedes principle. The diffusion coefficient of the liquid. NMR images were produced on a Bruker (Faellanden, Swit- The majority of rocks conform with the fast-diffusion zerland) Biospec 47/40 superconductive nuclear magnetic res- (Brownstein and Tarr, 1979) or surface limited (Belton et onance imaging (NMRI) instrument. The strength of the magal., 1988) relaxation regime in which the relaxation at the netic field is 4.7 Tesla, which corresponds to 200 MHz for surface is slower than the transport of the hydrogen nuclei to the hydrogen nucleus ( 1 H) resonance frequency. The twothe surface. Thus, the spins experience a rapid exchange of dimensional proton density image was calculated from T 2 imenvironments so that the local fields in each region of a pore ages (CPMG sequence with T E 2 ms and recovery time are averaged to their mean value. As a consequence, a single T R 3 s), while the porosity distribution was scaled according exponential decay is observed for a given pore, and the rate to the proportionality between proton density and volume of of magnetization decay depends on surface to volume ratio liquid in the core. The image matrix was 128 128. only (Kleinberg et al., 1994). Under the conditions of low Nuclear magnetic resonance relaxation measurements were magnetic field strength (i.e., G is also small) and short T E carried out on a homemade NMR spectrometer with a 1175- (Kleinberg and Horsfield, 1990; Kleinberg et al., 1993), the Gauss magnetic field strength, which corresponds to 5 MHz enhancement in T 2 decay coming from diffusion in the inhomo- for the hydrogen nucleus ( 1 H) resonance frequency. The pergeneous local magnetic fields is negligible compared with the manent magnet has a 13-cm bore in the horizontal direction. surface relaxation mechanism. Therefore, the measured T 2 A nonmagnetic core holder made of fiberglass material was values are given by: put inside the probe and magnet. The profile schematic map 1/T of the nonmagnetic core holder was indicated in Fig. 1. In 2A 2 S/V [4] Fig. 1, the core sample (1) was sealed in a fiberglass core Equation [4] forms the basis of NMR core analysis and holder (2) by two nylatron distributors (3) and a teflon tube log interpretation: T 2 is proportional to V/S, which in turn (4). Different confining pressures can be controlled by a cylinder is proportional to pore size. This means that in small pores pump with injection or exsuction of deuterated water from relaxation is faster than in large pores. the inlet (5). Kerosene was injected to the core from the inlet In the measurement, the CPMG pulse sequence (proposed (6) and produced from the outlet (7) for the measurement by Carr, Purcell, Meiboom, and Gill [Carr and Purcell, 1954; of permeability. 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 T E, the time between The CPMG pulse sequence was employed for the measurement of T 2. The number of echoes was 1024, T E was 0.15 ms, and recovery time was 3 s.

502 J. ENVIRON. QUAL., VOL. 31, MARCH APRIL 2002 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. 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 Fig. 2. Environmental scanning electron microscopy (ESEM) scan of (ESEM). The cores were saturated with kerosene under vacsample from Core 20. uum 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 voxel porosities given in Table 1 were obtained from was measured using Archimedes principle. A proton density such measurements. image was taken by NMRI (using the CPMG pulse sequence) Figures 6 through 9 show the change of porosity in order to determine the voxel porosity distribution. Then, relative to the initial porosity 0 with varying effective the relaxation time distribution of the samples was measured by low field NMR. Simultaneously, two other measurements overburden pressure P for all three cores. The effective were carried out. Total porosity change was measured by overburden pressure is the difference between overbur- volumetric determination of the displaced fluid, and permeof formation pressure is simulated by increasing the den pressure and fluid pressure at the inlet. The decrease ability was measured by flooding of the system at different overburden pressures. effective overburden pressure. In Fig. 6, the decrease of relative porosity with increasing effective overburden pressure (corresponding RESULTS AND DISCUSSION to the decrease of formation pressure) is shown for all The initial total porosities and permeabilities of cores three cores together. The porosity ratio decrease is are listed in Table 1. The matrix permeability of Core larger for the fractured Cores 39 and 20 than for the 20 amounted to 0.043 10 3 m 2. unfractured Core 52. The functional relation between Figures 2 through 4 show typical ESEM scans of mateporosity ratio / 0 and effective overburden pressure rial 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 Table 1. Initial parameters of the soil cores. Average Minimum voxel Maximum voxel Core porosity K 0 porosity porosity % 10 3 m 2 % 20 6.31 199 0.26 23.8 39 7.38 1.35 0.62 23.9 Fig. 3. Environmental scanning electron microscopy (ESEM) scan of 52 9.33 0.85 0.31 27.9 sample from Core 39.

CHEN ET AL.: PERMEABILITY AND PORE SIZE DISTRIBUTION OF POROUS MEDIA 503 Fig. 5. Voxel porosity distribution of Core 39. Fig. 4. Environmental scanning electron microscopy (ESEM) scan of compared with the two fractured cores (20 and 39), sample from Core 52. which show a change of more than 80%. Figures 11 through 13 depict the hysteretic behavior P was fitted. The functions (righthand side of Fig. 6) core by core. The upper branch corresponds to the curve are linear for all three cores with a high correlation of Fig. 10. Note the different scale. The permeability coefficient (R 2 ). ratios of the fractured cores (20 and 39 in Fig. 11 and 12) Figures 7 through 9 depict the hysteretic behavior show a recovery to less than 30% only. The unfractured core by core. The upper branch corresponds to the curve Core 52 (Fig. 13), however, recovers to more than 92%. of Fig. 6. Note the different scale. The porosity ratio of The final three figures (Fig. 14 16) show the distributions the fractured cores (20 and 39 in Fig. 7 and 8) recovers of relaxation times T 2 for the three cores at initial to more than 92%. The unfractured Core 52, however, effective overburden pressure (P i 2.5 MPa), under (Fig. 9) recovers to more than 99%. maximum effective overburden pressure (P 20 MPa), Similarly, Fig. 10 through 13 show the behavior of and after relieving overburden pressure to the initial permeability K relative to initial permeability K 0 under value (P 2.5 MPa). The relaxation time is directly increase of effective overburden pressure P (corre- proportional to the pore size. The spectra show a bimodal sponding to formation pressure decrease) and the corresponding shape. The changes in pore size distribution are hysteretic behavior. The relation of the perme- restricted to the large pore size range of the spectrum, ability ratio (K/K 0 ) and effective overburden pressure which contains large pores and the fractures. The large P was fitted. The functions are shown on the righthand pore size distribution is shifted left toward smaller pores side of Fig. 10. They are exponential with a high correlation under pressure increase and does not recover to the coefficient (R 2 ). The unfractured Core 52 shows a original distribution when pressure is diminished again. very small change in permeability ratio of less than 20% The changes are smallest for the unfractured Core 52 Fig. 6. Relation between porosity ratio and effective overburden pressure.

504 J. ENVIRON. QUAL., VOL. 31, MARCH APRIL 2002 Fig. 7. Relation between porosity ratio and effective overburden pressure of Core 20. Fig. 11. Relation between permeability ratio K/K 0 and effective overburden pressure of Core 20. Fig. 8. Relation between porosity ratio and effective overburden pressure of Core 39. Fig. 12. Relation between permeability ratio K/K 0 and effective overburden pressure of Core 39. Fig. 9. Relation between porosity ratio and effective overburden pressure of Core 52. Fig. 13. Relation between permeability ratio K/K 0 and effective over- burden pressure of Core 52. Fig. 10. Relation between permeability ratio K/K 0 and effective over- Fig. 14. Changes in the distribution of relaxation time T 2 with effective burden pressure. overburden pressure of Core 20.

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