Magnetic Resonance Imaging 19 (2001) 379 –383
Contributed Papers
Direct measurement of porous media local hydrodynamical permeability using gas MRI Martin Bencsik,* Chandrasekhar Ramanathan School of Physics and Astronomy, University of Nottingham, GB-NG7 2RD Nottingham, United Kingdom
Abstract The concept of hydraulic permeability is at the core of modeling single phase or multi-phase flow in heterogeneous porous media, as it is the spatial distribution of the permeability that primarily governs the behavior of fluid flow in the medium. To date, the modeling of fluid flow in porous media has been hampered by poor estimates of local permeability. Magnetic Resonance Imaging is well known for its ability to measure non-invasively the local density and flow rate of different fluids saturating porous media [1,2]. In this paper we demonstrate the first non-invasive method for the direct measurement of a single projection of the local permeability tensor of a porous medium using gas-phase MRI. The potential for three-dimensional imaging of the medium permeability is also discussed. The limitations of the method are listed and results are presented in a model porous medium as well as in a real oil reservoir rock. © 2001 Elsevier Science Inc. All rights reserved. Keywords: Hydraulic permeability; Porous medium; MRI.
1. Introduction Hydraulic permeability is a parameter that is characteristic of the porous medium only. It has a complex dependence on other properties of the medium such as porosity, connectivity, geometry of the pore structure etc. It is defined via a phenomenological relationship usually referred to as Darcy’s law:
⫽
Q L ⌬P A
(1)
where is the hydraulic permeability of the medium (in m2), Q is the volume flow rate (in m3/s) of the fluid going through the medium, is the viscosity of the fluid (in Ns/m2), L (in m) is the length of the medium, A (in m2) its transectional area, and ⌬P (in N/m2) is the pressure drop experienced by the fluid between the inlet and the outlet of the medium. The relationship in Eq. (1) is applicable for laminar flow of an incompressible Newtonian fluid, on a mesoscopic or macroscopic scale. Some porous media may be homogeneous over their
* Corresponding author. Tel.: ⫹44-115-846-60-03; fax: ⫹44-115-95151-66. E-mail address:
[email protected] (M. Bencsik).
entire volume, like a packed bed of spheres of the same size for example. In this case, the permeability of an elementary volume unit of the medium remains constant everywhere. In many natural porous systems of interest, like porous rocks and coals, strong variations of the value of the local permeability can be observed [3]. In order to measure these variations, invasive measurement of the fluid pressure may be performed using pressure sensors placed inside the medium. Alternatively, the medium may be physically sectioned, and the permeability of the individual elements measured. We propose and demonstrate here the first non-invasive and non-destructive method to achieve spatially resolved permeability measurements.
2. Principle of the method It is necessary to measure the local fluid pressure to infer local permeability (see Appendix). If a compressible gas is used as the fluid experiencing flow in the porous system, local fluid pressure will be revealed by the local NMR signal strength, which is very sensitive to the volume density of the nuclear spins. This local pressure information, P(rជ ), in combination with the local fluid flow vector, qជ (rជ ), will yield one projection of the local permeability tensor, (rជ ), according to the differential form of Eq. (1):
0730-725X/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved. PII: S 0 7 3 0 - 7 2 5 X ( 0 1 ) 0 0 2 5 2 - 1
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qជ 共rជ 兲 ⫽ ⫺
共rជ 兲 ជ P共rជ 兲 ⵜ 共rជ 兲
(2)
2.1. Measurement of local fluid pressure The local dependence of the NMR signal strength on gas density is complex, because of the variable parameters affecting the signal: the longitudinal and transverse relaxation rates, the diffusion coefficient of the gas molecules and the internal distribution of magnetic susceptibilities of the medium. The longer the signal is allowed to evolve in the medium, the greater the weighting due to relaxation and diffusion. As it is not usually possible to make a short enough NMR experiment so that the signal strength is simply proportional to the spin density, the protocol we suggest starts with a signal strength calibration procedure using the gas without a pressure gradient. The assumption is then made that this calibration remains valid in the presence of the pressure gradient. The protocol we adopted is the following: 1. Saturate the porous medium with a pure NMR visible gas; 2. Obtain MR images of the gas saturated medium at various static pressures in order to calibrate local signal strength with local pressure; 3. Apply a steady state pressure gradient to the system and make an MR image; and 4. Reconstruct the fluid pressure map of the flowing gas image by using the calibration.
Fig. 1. Cylindrical porous medium experiencing fluid flow. Pi and Po are the fluid pressures at the inlet and at the outlet respectively.
As the gas flow we use is isothermal and the viscosity dependence on density is negligible [4], (z) can be fully determined by the knowledge of P(z). If three dimensional mapping of is required, 3D imaging of the fluid flow vector, qជ (rជ ), is required in addition to 3D mapping of the pressure field, P(rជ ). This is possible thanks to “q-space imaging” techniques [5].
2.2. Measurement of local permeability The measurement of the pressure drop provides enough information for deriving a one dimensional profile of the medium permeability. Assume a pressure drop measurement along the main flow direction as shown in Fig. 1. The total fluid volume flow rate, q(z), at position z is given by the projection of Eq. (1) along this direction q共 z兲 ⫽ ⫺
共 z兲 dP共 z兲 共 z兲 dz
(3)
In the steady state regime of fluid flow, the gas mass flow ˙ ( z), is rate through the transverse section of the medium, M constant along the core length and is given by ( z)q( z) ⫽ ˙ where (z) is the gas volume density. M Assuming an ideal gas, we have:
共 z兲 ⫽
Mw P共 z兲 RT共 z兲
(4)
where Mw is the molecular mass, R the ideal gas constant, and T(z) is the local temperature of the gas. Substituting Eq. (4) into Eq. (3) we obtain
共 z兲 ⫽ ⫺
冉
˙ RT共 z兲 共 z兲M dP共 z兲 P共 z兲 Mw dz
冊
⫺1
.
(5)
3. Limitations of the method (i) During MR imaging of the flowing gas, caution has to be taken that molecular displacements due to the pressure gradient are negligible compared with the spatial resolution of the experiment. Only permeabilities below a certain threshold can be measured, otherwise the fluid flow will be too rapid. The shortest signal acquisition duration is limited by the hardware performance and the best spatial resolution is set primarily by the acceptable SNR for the desired accuracy on . A typical echo time of 400 s limited our studies to systems with permeabilities lower than 10 milliDarcy (mD). Acceptable SNR was achieved in experiments typically lasting two hours. (ii) Only dry porous media can be used. (iii) Only one projection of (rជ ) is determined by the method, that dictated by the local fluid flow vector as seen from Eq. (2). (iv) In some media the permeability can be affected by the gas presence, such as coal, for example, which can adsorb large amounts of gas. There is no simple means available at the moment to account for this process.
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381
Fig. 2.a. Results in the model porous medium. NMR 1D profiles of the static gas at 2 atm (upper curve) and flowing gas (lower gas) acquired at 2.35 T. Positions of the low permeability walls is indicated with dotted lines. Their presence as well as the imperfection of the plastic cylinder we used, caused mild signal changes. Spatial resolution ⫽ 0.5 mm. No. of averages ⫽ 8 000. Gradient echo sequence with hard 90° excitation pulse, TE ⫽ 1.36 ms. TR ⫽ 160 ms. Imaging time (for one curve) ⫽ 21 min. In the flowing gas experiment, outlet flow rate is 9.6 mL/min. 2b. Curve corresponding to the division of the flowing gas profile by the static gas profile of Fig. 2a. Walls are indicated as before. Signal strength is now only a function of gas pressure. The first, second, fifth and sixth walls caused the greatest pressure drops.
(v) Only single phase “bulk” local permeability can be measured at present. (vi) The porous medium has to be “NMR-friendly,” that is, its internal local magnetic field gradients must be as low as possible. Typically, T2* values shorter than 1 ms are not acceptable. (vii) Local permeability heterogeneities must be negligible on the length scale of the spatial resolution of the experiment. Pixel intensity will otherwise not necessarily reveal the mean local fluid pressure during imaging of the flowing gas.
4. Results
figure is superimposed the profile of the gas when it is flowing (from right to left) with an inlet pressure of 2 atm. The outlet, which is left open, is at 1 atm. Fig. 2b shows the same flowing gas profile as in Fig. 2a, after calibration of the signal strength has been performed. For this system, signal strength turned out to be directly proportional to gas pressure. The constant pressure within the compartments, the steep pressure drops at the walls and the correct inlet and outlet pressures provide strong validation for the technique. Further validation is published elsewhere [6]. Three of the seven walls appear to exhibit very high permeability compared with the remaining four. This variation is due to the unpredictable aperture areas resulting from our manual drilling procedure.
4.1. Model porous medium 4.2. Oil reservoir rock The method was tested in a model porous medium in which the spatial distribution of the fluid pressure is well known. The phantom consists of a hollow plastic cylinder (80 mm in length, 22 mm in internal diameter). Seven thin (180 m) plastic sheets, each separated by 7 mm, were placed inside, perpendicular to the cylinder axis (see shaded areas in Fig. 1). A single hole (approx. 200 m in diameter) was made in each sheet. These holes form the only apertures for gas flow, and the low permeability at the walls therefore governs the distribution of pressure drops in the sample. Fig. 2a shows the 1D NMR profile of the system saturated with gas (in our case, heptafluoropropane, CF3-CHFCF3) at 2 atm. The presence of the seven plastic walls can be seen according to the sharp signal drops. On the same
Oil reservoir rocks with permeability of the order of 10 mD can often be seen in transition regions between shale and clean sand intervals. We used such a North Sea oil reservoir sandstone sample, from a shoreface parasequence, which has a small percentage of clay (mostly illite) particles interleaved between the quartz grains. This clay is partly the cause of the low permeability (8 mD) of this sample while its porosity is still quite high (20%) due to the clay distribution forming microporous rims around the quartz grains (D. Potter, personal communication). The cylindrical sample (diameter ⫽ 2.54 cm, length ⫽ 8.4 cm) was placed in a nitrile sleeve experiencing an inward radial pressure of 15 bar to prevent by-pass flow
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Fig. 3.a. Results in the oil reservoir rock. Gas pressure drop curves calculated from the gradient echo profiles for the North Sea sandstone. The curves are not mirror images of one another (about the center of the system) as would be the case in a homogeneous system. The profile exhibiting the negative slope was measured with gas flowing from the right to the left, with inlet pressure, Pi ⫽ 2.6 atm (flow rate at the outlet is 47 mL/min). The profile exhibiting the positive slope was obtained by reversing the direction of the flow, with the same Pi. Profiles were reconstructed from 25 000 acquisitions with TR ⫽ 127 ms. Spatial resolution ⫽ 417 m. 3b. Results in the oil reservoir rock. Spatial distribution of the reciprocal of the permeability, derived from both curves of Fig. 3a. The white curve corresponds to the gas flowing to the right, the black curve corresponds to the gas flowing to the left. At very high permeability values the thermal noise error (thickness of the curve) is of the order of the value of the data. Therefore ⫺1(z) is displayed instead of (z). Spatial resolution ⫽ 3 mm.
around the plug. We ran the NMR experiment (gradient echo sequence with hard 90° excitation pulse) with a shorter echo time (380 s) to minimize strong signal attenuation due to susceptibility effects (T2* ⬃ 1.5 ms). After static gas calibration profiles were acquired, flowing gas profiles were acquired for two opposite directions of flow. Fig. 3a illustrates the corresponding pressure drops. The fact that these two curves do not intercept in the middle of the sample illustrates the heterogeneity of the permeability of the system. The spatial distributions of the reciprocal of the permeability obtained from the two pressure profiles using Eq. (6) are displayed in Fig. 3b. Smoothing of the pressure curves (with a Gaussian window with standard deviation ⫽ 3 mm) was necessary prior to data processing to avoid unacceptable errors in the gradient calculation. The predicted permeability distributions should be the same irrespective of the direction of the gas flow. There is fairly good agreement with the permeabilities obtained from the two experiments. The predicted permeabilities at the extremities of the sample are lower than elsewhere, probably because of edge effects. These effects appear greater at the inlet, where the pressure is greater. They arise from the narrow pipe carrying the gas in and out, forcing the flow to probe an effective transverse area smaller than that of the core. The distribution of permeabilities is very broad, with values differing by one order of magnitude.
5. Conclusion We have presented a novel method for directly measuring the spatial distribution of the permeability of a dry porous medium using gas MRI. Its major advantage is that it is non-invasive and it offers the potential for 3D mapping of one projection of the permeability tensor. At the moment the technique is limited to systems exhibiting typical permeabilities lower than 10 mD. The use of a more viscous gas or of faster imaging techniques (PET for example) might overcome this limitation.
Acknowledgments M. Bencsik is especially grateful to Sir Peter Mansfield for his help and financial support. We thank Dr. David Potter (Petroleum Engineering Department, Heriot-Watt University, Edinburgh, U.K.) for providing us with the North Sea reservoir sandstone (from the Pegasus project), Prof. Ken Packer (Department of Chemistry, Nottingham University, UK) for providing us with the sample holder, and Dr. David Rourke for his help with the Appendix.
Appendix For a particular porous medium experiencing fluid flow, we assume that Darcy’s law holds and that we know the
M. Bencsik, C. Ramanathan / Magnetic Resonance Imaging 19 (2001) 379 –383
flow vector field, qជ (rជ ), and the inlet and outlet pressures, Pi and Po respectively. For simplicity, we take an incompressible fluid and the one dimensional case as in Fig. 1 so that the integrated z projection of qជ (rជ ), qz(z), is a constant. Assume that there are two different sets of pressure and permeability fields, [1(z); P1(z)] and [2(z); P2(z)], which may exist under the same flow conditions: qz ⫽ ⫺
1共 z兲 dP 1 dz
冊
z
⫽ ⫺
2共 z兲 dP 2 dz
冊
(6)
z
383
For the three dimensional case, we assume Darcy’s law as in Eq. (2) and the conservation of fluid mass flow over the ជ 䡠 qជ (rជ ) ⫽ 0, and that the fluid pressures at whole space, ⵜ the inlet and outlet plates are known. It is easy to show that the one dimensional case studied before is a particular case which fulfills all these conditions. This allows us to conclude that if Darcy’s law holds and if we know the fluid inlet and outlet pressures, there is an infinity of different sets of pressure and permeability fields which yield the same flow vector field.
The identical total pressure drop condition yields:
冕 L
共P o ⫺ P i兲 ⫽ q z
冕 L
dz ⫽ qz 1共 z兲
0
References
dz 2共 z兲
(7)
0
where L is the length of the medium. There is an infinity of sets of pressure and permeability fields which obey these conditions. For Pi ⫽ 2 atm and Po ⫽ 1 atm, we may find for example 关 1共z兲; P1共z兲兴 ⫽ and 关 2共z兲; P2共z兲兴 ⫽
冋 冑 冉 冊册 L z ; 2⫺ 2 L
冋冑
冉 冑 冊册
z ⫹ 1; 2 ⫺
z ⫹ 1 L
.
[1] Packer KJ. Diffusion and flow in fluids, in: John Wiley and Sons, editors. Encyclopædia of NMR, Chichester UK 1996; p. 1615–26. [2] Packer KJ. Oil reservoir rocks examined by MRI, in: John Wiley and Sons, editors. Encyclopædia of NMR, Chichester UK 1996; p. 3365– 76. [3] Collins RE. Flow of fluids through porous materials, Reinhold publishing corporation, New York, 1961. See in particular pages 17 and 18. [4] In the case of an ideal gas, viscosity is actually independent of density. See for example Glasstone S. and Lewis D. Elements of physical chemistry, London, MacMillan & Co. Ltd 1960. [5] Callaghan PT. Principles of Nuclear Magnetic Resonance Microscopy. Oxford: Oxford University Press, 1991. [6] Bencsik M, and Ramanathan C, Measurements of local hydraulic permeability using gas phase MRI. Phys Rev E Rapid Communications (in press).
