JOURNAL OF ELSEVIER Journal of Petroleum Science and Engineering 17 (1997) 247-264 ENGINEERING Stress-dependent permeability measurement using the oscillating pulse technique P. Suri a, M. Azeemuddin b, M. Zaman ~~, A.R. Kukreti a, J.-C. Roegiers b a School of Ciril Engineering and Encironmental Science, The Um) ersitv ~/'Oklahoma, Norman, OK 73019. USA h School of Petroleum and Geological Engineering, The Unicersity ~f Oklahoma, Norman, OK 73019. USA Accepted 28 October 1996 Abstract Permeability of Indiana Limestone samples undergoing deformation in a triaxial cell along several stress paths such as triaxial compression, hydrostatic compression and uniaxial strain was measured using the oscillating pulse technique. This technique consists of applying a sinusoidal pressure wave at the upstream end of the sample and recording the pressure-time behavior at the downstream end. The solution to the general one-dimensional diffusivity equation can be obtained by applying the appropriate boundary conditions to give a method of evaluating diffusivity and permeability in a relatively short time. The process of sending a pulse to evaluate permeability has been programmed in Visual Basic which can be incorporated into any regular triaxial testing automated routines to measure permeability at different stages of loading until failure. Stored waveforms of the upstream and downstream pressure responses with time are analyzed to evaluate the attenuation and phase shift in the waveform. For comparison purposes, permeability was also measured using steady-state methods for samples subjected to a hydrostatic compression stress path and good correlations were observed. During triaxial compression failure tests, in most tests the permeability decreased continuously until the end of the test. Permeability also decreased with increasing confining pressure, while under uniaxial strain (i.e. one-dimensional compaction (k 0) tests), there was a sudden reduction of permeability at a particular stress level. 1. Introduction Permeability of rocks is one of the important parameters of concern to petroleum engineers, hydrogeologists, seismologists and environmental engineers. Permeability is among the main factors responsible for: (1) reservoir production (Marek, 1979; Simon et al., 1982); (2) earthquake prediction (Healy et al., 1968: Raleigh et al., 1976); (3) selection of nuclear waste disposal sites (Bredehoeft et al., 1978), etc. Changes in permeability that take place in a reservoir due to an increase in effective stresses caused by fluid withdrawal are of direct concern to the petroleum engineers and hydrogeologists. Specifically, the accurate knowledge of permeability changes for a variety of production scenarios is important when dealing with pore collapse prone reservoirs. High-poros- * Corresponding author. 0920-4105/97/$17.(X1 Copyright ~5 1997 Elsevier Science B.V. All rights reserved. PII S0920-4105(96)00073-3 248 P. Suri et al. / Journal of Petroleum Science and Engineering 17 ~1 997) 247 264 ity formations, when subjected to effective stress increases due to productions, are often found to undergo sudden and drastic volume reductions through a micro-level mechanism called pore collapse. Reductions in permeability can result in a decrease of ultimate recovery, reservoir compaction, subsidence, etc. Consequently, the measurement of permeability in the laboratory under simulated conditions of pressure and temperature is important for the petroleum industry. Permeability measurements in a triaxial test setup have traditionally been carried out in one of two ways: (1) steady-state methods; and (2) transient methods (e.g., Brace et al., 1968). Rocks with high permeability can easily be measured by the former technique while it is more convenient to use transient methods for low-permeability rocks having permeability values of the order of < 1 - 10 14 cm 2 (l ~.LD) (Brace et al., 1968). For tight rocks, either due to the fact that the flow rate is too small to be measured accurately, or that unrealistic long times are required to reach stabilized flow conditions, the steady-state measurement methods become inconvenient and impractical. Moreover, pressure can usually be measured more accurately than flow rate. In such cases, transient techniques, originally pioneered by Brace et al. (1968) are resorted to. The method consists of applying a sudden pressure pulse upstream of a jacketed sample subjected to confining pressure and pore pressure and attached to upstream and downstream reservoirs of known volumes; and recording the pressure-time histories in both the reservoirs. The decay characteristics depend upon the permeability, the dimensions of the sample and the reservoirs, and on the physical characteristics of the fluid. The permeability can be evaluated by comparing the observed pressure decay in the upstream reservoir with the behavior predicted analytically. A more general approach was used by Trimmer et al. (1980) and Lin (1982), in which no experimental restrictions on the reservoir volumes were imposed. The measurements were performed in the same manner as described by Brace et al. (1968), and permeabilities were evaluated numerically using a one-dimensional finite-difference model. Hsieh et al. (1981) presented a general analytical solution which accounts for the compressive storage in the rock sample, and is applicable to all sizes of the upstream and downstream reservoirs. Based on the techniques developed by Hsieh et al. (1981) and Neuzil et al. (1981), one can graphically estimate the permeability and storage capacity of the rock samples. The transient step method requires measurements for long periods of time and utilizes graphical curve-matching procedures. Kranz et al. (1990), based on the literature on thermal diffusivity measurements (Cowan, 1961; Cerceo and Childers, 1963), applied oscillating boundary conditions to study the hydraulic properties of tight rocks. The method was also used by Fischer (1992) and Fischer and Paterson (1992) to determine the permeability and storage capacity of rocks under high temperature and pressure. The method consists of applying a sinusoidal pore pressure oscillation at the upstream end of a sample and recording the pressure-time response at the downstream end. In this paper, the oscillating pulse technique has been modified to be applicable to a wider range of rocks. The technique has been programmed in Visual Basic language and is used to determine the permeability changes in Indiana Limestone subjected to increasing effective stresses under different stress paths in a triaxial cell. The results are found to be consistent with those obtained using other techniques (steady-state methods). Drastic changes in permeability were observed during uniaxial consolidation at the time of pore collapse. The objectives of this study was to develop a fast and efficient means of measuring stress-dependent permeability in rocks undergoing given loading paths, and to determine the applicability of the oscillating pulse technique. It was also intended to evaluate the sensitivity of the frequency of the oscillating pulse on the values determined using this technique. 2. Oscillating pulse technique 2.1. General The oscillating pulse technique consists of applying a sinusoidal pressure wave of known amplitude and frequency to the upstream end of a sample, subjected to known confining and pore pressures. The downstream P. Suri el al./ Journal c~ Petroleum Science and Engineering 17 (1997) 247-264 249 pressure response is also, after an initial transient response, a phase-shifted sinusoidal wave with an attenuated amplitude; the ratio of which are a function of the physical properties of the rock sample (length, diameter, pore volume or storage capacity, permeability), fluid (viscosity, compressibility) and other experimental parameters (frequency of the pressure wave, size of the downstream reservoir, etc.). Among the many factors responsible, the permeability of the sample and its storage capacity are the only unknowns; and the remaining parameters can be either measured or evaluated through calibration procedures. 2.2. Methodology The general one-dimensionless equation of fluid flow for a compressible fluid is as follows: ax k /3 at (l) where P is the pressure; x the distance along the axis of the sample (flow direction); /z the fluid viscosity; /3 the fluid compressibility; k the sample permeability, /3m the effective compressibility of fluid saturated rock; /3~ the compressibility of the minerals; ch the connected porosity; and t the elapsed time. The above equation is also called the diffusivity equation, with diffusivity D, given by the expression: D ' =--~ [4,/3+/3~,-/3~(1 +4))] k (2) A simplified form of Eq. (1) was used by Brace et al. (1968). Their experimental configuration and the material used allowed them to treat the right-hand side as zero, resulting in: 02p ~tx 2 =0 (3) The initial and boundary conditions used in the present method can be mathematically represented at the upstream end as: P(x,0)=0 for 0
0) (6) t /3 v2 (7) in which A is the cross-sectional area of the sample, V, is the volume of the downstream reservoir, and /z and /3 have been defined earlier. The solution to the diffusivity equation (Eq. (1)) with the above initial and boundary conditions is given in terms of R, the ratio of the amplitudes of the downstream to the upstream ends; and 6, the phase difference between the downstream and the upstream waves (Kranz et al., 1990). The permeability can be evaluated from the following equation, once R and 6 are obtained from experiments: I~V2h k - --~BV2 a~oL A y (8) A 250 P. Suri et al./ Journal of Petroleum Science and Engineering 17 (1997) 247-264 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 10 , . ........... 2.~O-2o0_¢~--~ ~~,:~- - ~-15:-f -16, ~, _. ....... .. ......... 0o gE1 70. -.Jl:O ~------~ d \".. .. 01\" 140-----Z%1/- ~, .~\" _J~o~ \" ~~_.~0-----~_--- //. \"t_.=D.',~ ~_~- ~- ~o o~ Fig. I. Nomograph for the evaluation of ot and y from the experimental evaluated R and ~. where c~ and ,/ are dimensionless numbers given by: A OL (2~oD) I/2 ~oL (2 o)D) 1/2 (9) and Y = (10) The expressions for 6 and R are given as follows: 4o~ 2 ] i/'2 R = (2a2 + 1)c°sh2y+ (2°t 2 _ 1)cos2y+ 2o~(sinh2y- sin2y) ] 6 = arctan (11) (12) tanh,/(2a tan 7 + 1)_+ tan,i] tan ?' - tanh 3, - 2 a ] The procedure to obtain permeability and diffusivity values from the above equations is to solve Eqs. (11) and (12) simultaneously for a and \",/, after having evaluated R and 6 from the experimental data (procedure is described later). Once a and 3/are evaluated, Eq. (6) can be used to evaluate the permeability. The procedure to solve the simultaneous equations is tedious; hence, Eqs. (11) and (12) are solved numerically for different values of o~ and Y. A nomograph (Fig. 1) was generated from which one can easily obtain the values of a and \",I, for given values of R and 6. This procedure has been programmed in Visual Basic 1 to obtain the corresponding exact values of a and T; hence, the permeability, based on R and 6 input. The values of R and 6 are evaluated by fitting sinusoidal functions to the upstream and downstream wave pulses. Fig. 2 shows a typical applied upstream pulse and the resulting downstream pressure response for an / Visual Basic is an object-oriented programming technique u, hich uses the Basic Language. P. Suri et al./ Journal of Petroleum Science and Engineering 17 (1997) 247 264 251 1150 -~ .... 11C]0 1050 @ 1000 950 ~- ---~-- upstream 1 ~ downstream 900 2505 2510 2515 2520 2525 2530 2535 2540 2545 2550 time 2555 (soc) Fig. 2. Typical upstream and downstream pressure waveforms. Indiana Limestone specimen. The entire procedure, from conducting the test to analyzing data and obtaining permeability values, is programmed under the Microsoft windows operating environment. The oscillating pulse technique is programmed as a separate module in a manner that it is easy to be incorporated into most triaxial testing procedures. 3. Experimental procedure and material used 3.1. Introduction The experimental results reported here were obtained from a series of tests conducted in the Halliburton Rock Mechanics Laboratory at The University of Oklahoma, under hydrostatic, triaxial, and k 0 (uniaxial strain) deformation paths on Indiana Limestone. The testing program is described in Table 1. At any stage of loading where permeability was to be measured, the axial stress or confining pressure was held and a sinusoidal pressure wave was applied at the upstream end. The amplitude of the applied sinusoidal pressure wave should be kept sufficiently small, to avoid any changes in the rock properties during the permeability measurements. Both 252 Table 1 Testing Program Type of test Hydrostatic compression Triaxial compression P. Suri et al./ Journal of Petroleum Science and Engineering 17 (l 997) 247-264 Confining pressure (psi) up to 9000 2000 4000 6000 8000 2000 Pore pressure (psi) 1000 2000 1000 1000 1000 1000 500 Number of tests 2 1 2 2 1 1 2 Uniaxial strain upstream and downstream pressure data were recorded with time. The two waves were compared to evaluate the attenuation and the phase lag, as described later in Section 3.6, and permeability was evaluated. 3.2. Material used For the objectives of this study, the material chosen should be homogeneous and isotropic such as Indiana Limestone. This material has high porosity and low permeability, and is obtained from a quarry outside Cleveland, Ohio. Its chemical composition analysis showed 99.2% calcite and 0.8% magnesium. Microscopic analyses from two different places from the block show that this rock is quite homogeneous. Using oil injection on 1-in (2.54 cm) plugs, the porosity of the material was found to be ~ 18%. Its permeability evaluated by flowing oil under constant pressure head was found to be ~ 6 mD. 3.3. Sample preparation and saturation Rock samples were drilled from rectangular blocks with a thin-walled core barrel. The samples were cut into circular cylinders of height-to-diameter ratio of at least 2. The diameter of the samples was kept at ~ 2.125 in (54.0 mm). The ends of the samples were surface ground parallel to within 0.01 mm using an 80-grit aluminum oxide wheel. The samples were saturated via a vacuum pump; calculations based on the dry and wet weights as well as porosity indicated that the samples were saturated to ~ 95-100%. 3.4. Experimental setup The experimental setup is shown in Fig. 3 and consists of: (1) a loading flame, MTS Model 315.02, having a capability of applying 600-kips (2670 kN) force; (2) a control panel, MTS Model 939.42; (3) a 30-ksi (207 MPa) SBEL HI-3000 intensifier for confining pressure; (4) a 10-ksi (69 MPa) SBEL HI-1000 intensifier for pore pressure; (5) SBEL Model 604-1 Dynamic Test System console for controlling both confining and pore pressure intensifiers; (6) a triaxial cell, SBEL Model 10, with a 10-ksi (69 MPa) confining pressure rating; and (7) a 486 personal computer for test control and data acquisition, equipped with Data Translation boards, DT2801A and DT2809, for analog-to-digital and digital-to-analog (I/O) conversions. The sample was first jacketed while mounted on the bottom platen of the triaxial cell. Porous plates were introduced, for even spreading of the pore pressure, between the sample and the top and bottom platens. Each porous plate holds six porous stones of 0.25-in (64 mm) diameter and 40-1~m pore size placed in a circular manner around the circumference of the plate. It was found that the clogging of porous stones affects the downstream pressure response; hence these stones were periodically ultrasonically cleaned. A Teflon jacket was P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247 264 Console for confining and pore pressure ccntroi 253 Console for load frame contr31 Computer for test control and data acquisition J j , \\ \\ Downstreamp orep ressure Reservoir transducer I frame n Confiningp ressurein tensifier- - Porep ressurein tensifier ]-~ ~~ ~ ! cell t Upstreamp orep ressure transducer Fig. 3. Experimental setup for load deformation testing including permeability measurement. heat glued to the sample with its ends properly secured to the top and bottom platens, with steel wires, for complete isolation from the confining pressure. Proper connections were made for the pore pressure inlet and outlet. The measurement devices, two LVDTs (Linear Variable Differential Transformers) and a circumferential extensometer, were mounted on the sample. The extensometer was placed at the center of the sample while the LVDTs were placed at two diametrically-opposite positions on the top platen. The LVDTs measure the deformations of the sample alone and do not require any corrections for the elastic deformations of the frame. A pressure transducer was located at the downstream end of the sample, between the sample and a downstream reservoir of 150-cm 3 capacity. The capacity of the downstream reservoir could be varied, depending upon the size of the sample being tested, by introducing a number of steel ball bearings of known volume. This may be required to increase the accuracy of the amplitude ratio measurement or to change it in order for it to fall within the nomograph presented in Fig. 1. The upstream pore pressure to the sample was supplied by the servo-con- trolled intensifier using oil (PG-1 Multitherm oil) as pore fluid. 3.5. Test procedure Initial confining and pore pressures were applied to the sample. Permeability was measured by the oscillation technique applied between two successive increments of confining pressure or axial load, depending upon the test. At the end of any confining pressure application, the sample was allowed to stabilize. A pore pressure sinusoidal oscillation of given frequency and amplitude was induced at the upstream end. The magnitude of the oscillation amplitude was always maintained sufficiently low, at ~ 10% of the equilibrium pore pressure, to 254 P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247-264 ensure that the rock and fluid properties were essentially constant during a particular measurement interval. During the oscillation interval, only the upstream and downstream pressures were stored as a function of time. The upstream and downstream waves were then analyzed to obtain R and 6. The two parameters, along with other fluid and rock properties, were used to evaluate the permeability of the sample. However, the analysis of the waveforms and the calculation were deferred until the test was completed. 3.6. Data acquisition and analyses Experimental data were acquired with a 486 personal computer, which was also used to operate the test procedure. The following data were recorded by the computer program: time, axial load, axial stroke, axial deformation from two axial LVDTs, circumferential displacement from the radial extensometer, confining pressure, and upstream and downstream pore pressures. These data were acquired at a frequency of ~ 6 data points per minute, which could be varied depending upon the rate of loading and the test duration. During the application of the sinusoidal pressure wave, the wave data alone were acquired and stored in separate files corresponding to the stress level. Reading fewer channels increased the frequency of the data acquisition 60 7- 50 4O I >* 30 O laa c- O O 20 J • -- Axial Radial Volumetric 10 0 [3 0.1 0.2 [3.3 0.z~ 0,5 Strain (%) + 0,6 t 0.7 Fig. 4. Stress-strain response for the hydrostatic compression test (test 1 ) for Indiana Limestone. P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247-264 255 required to completely define a pressure cycle defined by at least 100 points, giving a resolution of ~ 4 °. The system had a capability of recording ~ 7 pressure-time data points per second. A new version of the programme is being developed in which the data acquisition frequency can be increased to ~ 200 data points per second. For the material used so far, the slower frequency of data acquisition did not pose any problem as the oscillation periods required were ~ 10-15 s, giving enough data points in a given cycle (70-105 data points, or a resolution of 5 ° to 3°). In order to evaluate the ratio of the amplitude of the downstream to the upstream pressure waves (attenuation) and the phase lag between them, the two curves are fitted with a sine function of the form (Fig. 2): (pressure) = a + b. sin[ w(t + c)] (13) where a, b and c are the unknowns; t represents time; and w = 2 rr,f, where f is the a known frequency. The parameter a represents the shift along the y-axis (pressure axis); b, the amplitude of the wave; and c, the shift along the x-axis (time axis). Two sets of values for a, b and c were evaluated by fitting to the upstream and downstream waves. Let them be represented by a u, b,, c u and ao, b d and Q, where the subscripts u and d refer to the upstream and downstream waves, respectively. The attenuation is given by (ba/b u) and the phase shift by (Q - cu). The value of the phase shift obtained this way may have to be adjusted to obtain the phase lag that will fall between zero and 360 ° . 60 50 ~40 ~3o o ,=., • 2O -- ~ • -- Test 1 Test 2 Static 10 I 0.1 0.2 0.3 04 I 0.6 ! 07 0.5 Volumetric Strain (%) Fig. 5. Comparison of stress-strain response for three hydrostatic compression test for Indiana Limestone. 256 P. Suri et al. / Journal o[ Petroleum Science and Engineering 17 (1997) 247-264 4. Discussion of results 4.1. Stress-strain behauior 4.1.1. Hydrostatic compression tests A total of three hydrostatic compression tests were conducted. Fig. 4 shows the behavior of three strains (axial, radial and volumetric) with increasing mean effective stress on the sample for Test 1. It can be observed from the figure that when the loading is stopped in order to transmit a pressure pulse, the sample deforms under constant effective stress. In all the tests, confining pressure was increased up to 9 ksi (62.06 MPa). In tests 1 and 2, pore pressure was kept at 1 ksi (6.9 MPa) resulting in a maximum mean effective stress of ~ 8 ksi (55.2 MPa). In these two tests, permeability was measured using the oscillating pulse technique. In the \"static\" test, the permeability was measured using the steady-state method involving measuring the steady flow in a given interval of time. For test 3, referred to as \"static\", the pore pressure was kept at 2 ksi (13.8 MPa) so the maximum mean effective stress was ~ 7 ksi (48.3 MPa). The confining pressure was increased manually in increments of 1 ksi (6.9 MPa), applied over a period of ~ 4 min. At the end of each increment, permeability was measured, after allowing the sample to stabilize under increasing load for ~ 5 rain. Fig. 4 shows that the 80 70 2 o 0 '/ 4C I I I i m m EffStress 6.9 MPQ Eft Stress 20.7 MPa Eff Stress 34.5 MPa Eft' Stress 48.3 MPa 30 / I 0 / t 10 I # t 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 18 Axial Strain (%) Fig. 6. Comparison of stress strain response for triaxial compression test at o-~ = 2, 4, 6 and 8 ksi (13.8, 20.7.41.4 and 55.2 MPa) for lndiana Limestone. P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247-264 257 response is linear and elastic in nature. Fig. 5 shows volumetric strain for these three tests against mean effective stress. It can be seen that volumetric strains for three tests at any particular mean effective stress is nearly the same, thus establishing the repeatability of the stress-strain behavior. 4.1.2. Triaxial compression tests Triaxial compression tests with permeability measurement were conducted at initial confining pressures of 2, 4, 6 and 8 ksi (13.8, 27.6, 41.4 and 55.2 MPa). Pore pressure in all the tests was kept at 1 ksi (6.9 MPa). Fig. 6 shows the deviatoric stress vs. axial strain for all the tests. It can be observed that the lower confining pressure sample fails at lower deviatoric stress and at lower axial strain in comparison to the tests at higher confining pressures. Material behavior changes from brittle to ductile for triaxial compression tests with increasing confining pressure. 4.1.3. Uniaxial strain tests The uniaxial strain (k 0) tests were conducted at 0.5-ksi (3.45 MPa) pore pressure and an initial confining pressure of 2 ksi (13.8 MPa) was maintained before starting the test under the strain control mode. The test was repeated at similar initial conditions. The variation of volumetric strain with axial stress for tests 1 and 2 are shown in Fig. 7. The slope of both curves changes at ~ 11.6-ksi (80 MPa) axial stress and becomes steeper, ,8 I 1,6 1.4 1.2 o 0,8 0.6 / 0.4 Test 1 0.2 ~ Test 2 o ~~1 0 20 I 40 I 60 I 80 Axial Stress (MPa) I 100 I 120 I 140 I 160 Fig. 7. Axial stress vs. volumetric strain response for the uniaxial strain test (test 1) for Indiana Limestone. 258 P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247-264 100 90 _ 80 \\ 70 613 50 40 30 ~-- Test 1 • Test 2 Static 29 A 10 a 0 I 10 I 20 I 30 I 40 ~ 50 60 Mean Effective Stress (MPa) Fig. 8. Comparison of permeability reduction during three hydrostatic compression tests for Indiana Limestone. suggesting that the sample undergoes pore collapse or increased volumetric compaction at ~ 11.6 ksi (80 Mpa). High-porosity formations are often found to undergo sudden and drastic volumetric reductions through micro-level mechanisms called \"pore collapse\" (Zaman et al., 1994). 4.2. Permeabilib\" L,a riation 4.2.1. Hydrostatic compression tests Two methods of permeability measurement were used, i.e. the dynamic (oscillating pulse technique) and the static (steady-state flow measurement technique). Tests 1 and 2 were dynamic and test 3 was static. The dynamic tests were conducted for two different time periods of the oscillating pulse: 10-s and 20-s, i.e. at oscillating pulse frequencies of 0.1 and 0.05 Hz. For 10-s time periods, the permeability of the samples decreases continuously with increasing mean effective stress, as expected (refer Fig. 8), but the variation for 20-s time period is nonuniform. From Fig. 8 it can be observed that for both the techniques the permeability reduction is nearly the same, ~ 25%. In Fig. 8, the y-axis represents (k/kb), where k b is the base permeability at a mean effective stress value of 6.9 MPa. It can thus be concluded that the oscillating pulse technique compares well with the steady-state method of permeability measurement. P. Suri et al. /Journal of Petroleum Science and Engineering 17 (1997) 247 264 259 9O 8O 70 60 50 40 30 _IiiI 20 t=lO secs 1 I • t=15 secs I 10 0 0 I 10 I 20 I 30 t 40 I 50 I 60 [ 70 Deviatoric Stress (MPa) Fig. 9. Permeability variation during triaxial compression test (test 2) at ~r~ = 2 ksi (13.8 MPa) for indiana Limestone. 4.2.2. Triaxial compression tests Triaxial compression tests were conducted for three different time periods, i.e. 5-s, 10-s and 15-s (frequencies of 0.2, 0.1 and 0.067 Hz) and for four different confining pressures. Fig, 9 shows the reduction in permeability with respect to the deviatoric stress for 10- and 15-s time periods for triaxial compression test at 2 ksi (13.8 MPa) confining pressure and 1 ksi (6.9 MPa) pore pressure, i.e. at 1 ksi (6.9 MPa) effective stress. It is observed that the permeability reduction is nearly the same (~ 27%). Similar behavior is observed for tests conducted at 3 ksi (20.7 MPa), 5 ksi (34.5 MPa) and 7 ksi (48.3 MPa) effective stress (not shown). It is observed that for 10- and 15-s time periods, the permeability variation is nearly the same whereas for 5 s it is quite different and erratic. A possible reason for such a behavior could be the fact that for a 5-s time period, the number of data points acquired were only 35 per cycle (at 7 data points per second); giving ~ 1 data point every 10 °, which is not sufficient to describe the wave precisely. It is recommended that at least 1 data point be recorded every 4 ° (Kranz et al., 1990). Figs. 10 and 11 show permeability reduction for all triaxial tests at different effective stresses for 10-s and 15-s, respectively. From these figures it can be observed that with increasing effective stress, the reduction in permeability is greater, as expected. It can be concluded that the oscillating pulse technique gives dependable results for Indiana Limestone when the time period is from 10-s to 15-s. If the data 260 P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247-264 100 90 80 11170 60 50 40 • 30 Eff Stress 6.9 MPa Eft Stress 20.7 MPa Eft Stress 34.5 MPa 20 Eft Stress 48.3 MPa 10 0 0 10 I 20 I 30 ~ 40 I 50 I 60 I 70 I 80 I 90 Deviatoric Stress (MPa) Fig. 10. Permeability reduction during triaxial compression tests at lO-s time period for Indiana Limestone. acquisition frequency is increased then 5-s time period could also be used. Also it was observed that changing the time period within a range does not cause a variation in the measured permeability values. The particular range of time periods (or frequencies) needs to be established for a given material. Generally speaking, the tighter the material, the higher will be the period (lower frequency) of the required oscillating pulse (Kranz et al., 1990). 4.2.3. Uniaxial strain tests The permeability reduction for tests 1 and 2 are shown in Fig. 12. It can be observed that there is a sudden reduction in the values of permeability at an axial stress of 80 MPa, corresponding to the yielding of the sample. Thus, during the post-yield or pore collapse deformation there is a drastic reduction in permeability, indicating decrease in pore connectivity due to pore collapse. It is known that pore collapse leads to reduction in permeability (Marek, 1979; Blanton, 1981). Thus the oscillating pulse can be used to measure the post-pore collapse permeability of a material. The permeability reduction at the end of the test is ~ 75%. Both tests give similar variations, establishing the reproducibility of the results and the reliability of the permeability measurement technique. P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247-264 261 5. Conclusions Permeability was measured using an oscillating pressure pulse technique which consists of sending a pressure pulse of known amplitude and frequency at the upstream of a sample. The downstream response is a function, among other experimental parameters and fluid properties, of the sample permeability. After an initial time delay, the downstream response comes to equilibrium with respect to the upstream pulse, though at an attenuated amplitude. Using the solution available in the literature for one-dimensional flow of an incompress- ible fluid with the appropriate boundary conditions, the permeability was evaluated. The method can also give the values of diffusivity and interconnected porosity. Under hydrostatic compression, it was found that the permeability decreases continuously with increasing confining pressure, matching with the observed pore space reduction and the volume change behavior. The results match those reported in the literature. The values were compared with the steady-state measured permeability and the results were found to correlate very well. The small discrepancy observed in the absolute values of permeability could be due to several factors such as inaccuracies in the measurement of flow volume, time, as well as the variation between the individual samples. Under triaxial conditions, the permeability decreased initially for a low confining pressure test and tended to 10o 90 80 70 6O 4O • 30 A Eft Stress 6.9 MPa Eft Stress 20.7 MPa Elf Stress 34.8 MPa Elf Stress 48.3 MPa 20+ I 0 I I 0 I 20 I 30 I 40 J 5(3 I 03 I 70 ~ 80 I 90 Deviatoric Stress (MPa) Fig. | 1. Permeability reduction during triaxial compression tests at 15-s time period for Indiana Limestone. 262 P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247-264 100 90 80 70 03 50 .Q 0 0 40 30 • 20 Test 1 Test 2 * 10 0 0 I 20 I 40 I 60 I 8O I 100 I 120 E 140 E 103 Axial Stress (MPa) Fig. 12. Permeability reduction during uniaxial strain tests tbr Indiana Limestone. increase with initiation of dilatancy, while for the test at higher confining pressures, where the volume change was totally compressive, the permeability decreased continuously. Under uniaxial strain tests there is a sudden decrease in the permeability values at a particular axial stress level, corresponding to the yield stress of the material. During pore collapse or increased volumetric compaction, the reduction in permeability was drastic, in accordance with the published literature. 6. Notation Off = /3= ~eff = /3s= 3,= 6= #= dimensionless variable fluid compressibility effective compressibility of the fluid saturated rock compressibility of minerals in the rock dimensionless variable phase difference between the upstream and downstream waves dynamic fluid viscosity angular frequency of oscillation P. Suri et al. / Journal of Petroleum Science and Engineering 17 (1997) 247-264 263 A= D= k= L = p= PA= R= t = V~= x= interconnected porosity sample cross-section perpendicular to flow direction diffusivity permeability sample length pressure amplitude of the generated oscillation amplitude ratio time downstream fluid reservoir distance along the axis of the sample Subscripts: u = d = s = upstream downstream rock minerals Acknowledgements The authors are thankful to Peter Lemmon for his assistance in providing the laboratory support at the Halliburton Rock Mechanics Laboratory, the University of Oklahoma. 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