Journal of Environmental Quality 31:477-486 (2002)
© 2002 American Society of Agronomy, Crop Science Society of America, and Soil Science Society of America
Article
SYMPOSIUM PAPERS
Nuclear Magnetic Resonance Imaging for Studies of Flow and Transport in Porous Media
Quan Chen,
Wolfgang Kinzelbach* and
Sascha Oswald
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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Nuclear magnetic resonance imaging (NMRI) methods for visualization of fluid flow and transport in porous media are reviewed in this paper. They are illustrated with experiments showing applications of velocity imaging, NMRI measurements of multiphase flow, and NMRI measurements of density flow. The latter two are compared with numerical simulations. The examples show the capacity of NMRI to give structural information both of the medium and the fluid distributions as well as their temporal development. The resulting data can be used in a black box–white box comparison and as benchmarks for numerical models.
Abbreviations: NMR, nuclear magnetic resonance • NMRI, nuclear magnetic resonance imaging • TE, echo time • TR, repetition time
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INTRODUCTION
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TRADITIONALLY, CORE AND COLUMN analysts have been forced to assume that those objects were homogeneous black boxes. Volume and composition of fluids that were injected and recovered could be measured, but how the fluids were distributed and moving inside the core or column could only be inferred. With techniques such as nonintrusive neutron radiography (Brunner and Mardock, 1946), X-ray (Morgan et al., 1950; Laird and Putuan, 1951), gamma ray, and microwave absorption, it became possible to obtain one-dimensional fluid saturation and solute distribution data. The development of X-ray computerized tomography (CT) allowed for determination of two-dimensional and three-dimensional rock densities, saturations of fluids, and concentrations of solutes (Wellington and Vinegar, 1985, 1987; Hunt and Bajsarowicz, 1988) in the core or column. However, since this method is sensitive to mass density contrast, experiments usually have to be carried out with large tracer concentrations leading to unwanted fingering (Wooding, 1969; Perkins and Johnston, 1969).
About 10 years after the discovery of nuclear magnetic resonance (NMR) spectroscopy (Purcell et al., 1946; Bloch et al., 1946), the method was applied to porous media for the first time (Brown and Fatt, 1956). With the advent of NMR imaging (Lauterbur, 1973), a range of important new applications to porous media opened up and today many successful examples prove the value of this nondestructive detection method in porous and fractured media research.
Nuclear magnetic resonance imaging is similar to X-ray CT in that planar and three-dimensional images can be obtained from selected regions of an object. However, X-ray CT can only image electron density and atomic number, while NMRI, besides spin density of a variety of nuclei (1H, 19F, 23Na, 31P, etc.), also measures a number of other quantities such as relaxation times and chemical shifts (Vinegar, 1986; Blackband et al., 1986). It is inherently more informative than radiographic techniques. It is also much less hazardous. Special features of many NMRI techniques include the ability to measure the fluid flow velocities and to distinguish different liquids using their differences in intrinsic NMR properties. This type of information would be difficult to obtain by other methods. Besides three-dimensional imaging, a four-dimensional imaging in time is often possible. To avoid the blurring of images due to the fluid movement, the flow interruption technique can be used.
Multiphase flow and solute transport can be imaged by NMR through discrimination between fluids, for example on the basis of differences in relaxation times (Edelstein et al., 1988). For this application to be successful, it is necessary that the excited spins in different phases or components have different relaxation times. Relaxation times of fluids in porous media are dominated by fluid–surface interactions and pore geometry (D'Orazio et al., 1989), which results in a wide distribution of relaxation times as well as a decrease in the difference of relaxation times of fluids. Therefore, the feasibility of fluid identification based on relaxation times is considerably reduced. Consequently, experiments for both immiscible and miscible flooding in porous media were designed such that only a single fluid phase is imaged. This is done either by doping the fluid phase with small concentrations of paramagnetic ions such as Cu2+ or Mn2+ (Chen et al., 1996; Chen and Zhou, 1998; Baldwin and Yamanashi, 1991; Gaigalas et al., 1989; Guillot, et al., 1991; Pearl et al., 1993), or by using deuterated water (Chen, 1996; Chen et al., 1993) or fluorinated oil as a fluid. If one of the phases is gaseous the fluid phase has to be marked by the tracer. If the ranges of the paramagnetic ion concentrations are well chosen, an almost linear correspondence between signal and tracer concentration is found and a mass balance can be computed (Greiner et al., 1997; Oswald et al., 1997).
When the peaks of oil and water in porous media can be spectroscopically resolved, separate oil and water saturations can be obtained using various chemical shift imaging techniques. The use of the chemical shift technique in the study of the displacement of oil and water in cores is reported, for example, by Majors et al. (1990) and Chang and Edwards (1993).
The study of flow through porous media has been a topic of active research in many fields including ground water hydrology, petroleum engineering, and soil science. Nuclear magnetic resonance imaging can be used to directly observe the movement of molecules associated with diffusion or flow. This provides for an excellent opportunity to determine noninvasively local velocities within porous media (Robinson et al., 1992; Guilfoyle et al., 1992; Mansfield and Issa, 1994, 1996; Kutsovsky et al., 1996; Sederman et al., 1997, 1998).
The most important visualization methods are illustrated by the following examples.
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DIRECT FLOW VELOCITY IMAGING IN POROUS MEDIA
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The motion of excited spin produces changes in the amplitude and the phase of the magnetization in NMR. Flow, therefore, acts like an intrinsic contrast medium in NMRI. The use of phase shifts of the transverse nuclear magnetization to measure molecular velocity has been investigated for decades, and its potential application to the spatially resolved measurement of flow velocity was first suggested by Moran (1982). For an ensemble of molecules moving with a uniform velocity, the phase shift is expected to be proportional to the first moment of the gradient, and the flow velocity can be determined by a linear regression of phase vs. first moment (Caprihan et al., 1990).
We used NMRI to determine both porosity and velocity distributions in a column of packed glass beads. A mixture of epoxy resin and glass beads was applied to the inner wall of the column in order to avoid bypass flow at the column wall. The diameters of the glass beads were from 0.13 to 0.32 mm. The diameter of the column was 2.5 cm with an average porosity of 0.3.
The experiments were carried out using the Bruker (Faellanden, Switzerland) BIOSPEC 47/40 instrument with a 40-cm-bore superconductive magnet of 4.7-Tesla field strength and a B-G 15/A gradient system with maximum gradient field strength 12 G/cm.
A standard series of single-echo images at increasing echo times (TE) was used to gain the spin-density image. The echo time is defined as the time elapsed between the center of the 90° pulse and the echo maximum. In the experiment, the minimum echo time was 2 ms. The repetition time of the sequence (TR), which is the waiting time until recovery of the longitudinal magnetization before the next scan, was 2 s. Repeated scans are averaged to increase signal to noise ratio. The number of scans averaged here was two. The two-dimensional porosity image is shown in Fig. 1
. The grayscale bar on the righthand side of the figure was scaled with the average porosity obtained gravimetrically. The image matrix has 128 x 128 pixels and the spatial resolution is 0.35 x 0.35 x 10 mm3.
We used a modified spin–echo sequence to obtain the spatial distribution of flow velocity. A diagram of the sequence used is shown in Fig. 2
. The method (Young, 1984) employs a conventional three-dimensional Fourier transform (FT) spin–echo sequence, with the addition of flow-encoding gradients for the measurement of the flow velocity. The 90° radio frequency (RF) pulse was employed to generate transverse magnetization, supplying a read gradient (GR) for frequency encoding during the evolution time and for acquisition of spatial information during detection time. The phase-encoding gradients (GP and GS) are stepped independently during the evolution time to encode space information. The 180° RF pulse was used to refocus transverse magnetization and produce echo. The flow-encoding gradients (GF) were positioned before and after the 180° pulse for the measurement of the flow velocity.
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Fig. 2. The flow imaging pulse sequence. RF, radio frequency; GR, read gradient; GP and GS, phase-encoding gradients; GF, flow-encoding gradient; TE, echo time; rt, ramp time; Gi, magnitude of the gradient pulse.
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The flow velocity can be measured by the phase of the transverse spin magnetization. The phase shift (
) in a voxel due to fluid moving with a constant velocity component Vi in the ith coordinate direction is given by the Hahn equation (Hahn, 1960):
 | [1] |
where 
is the gyromagnetic ratio and m1 is the first moment of the gradient:
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where Gi is the magnitude of the gradient pulse.
Due to bipolar, trapezoidal gradients used in the sequence, Eq. [1] simplifies to:
 | [3] |
where 
is the separation time of two gradient pulses, 
is the duration of the gradient pulse, and rt is the ramp time of the gradient pulse.
Differentiating Eq. [3] with respect to Gi yields:
 | [4] |
Therefore, the velocity in a voxel is directly proportional to the slope of the line resulting from a graph of versus Gi. In the experiments, this slope was determined from the observed phase at three different values of Gi: -10.8 G/cm, 0 G/cm, and +10.8 G/cm. The echo time was 14 ms, the repetition time 0.5 s, and the ramp time of the gradient pulse 0.125 ms. The number of scans averaged was 32. The image matrix was 128 x 128, and spatial resolution 0.35 x 0.35 x 10 mm3.
Since the phase of the NMR signal is very sensitive to the inhomogeneity in the magnetic field, a reference phase map at zero velocity is required to correct for any phase errors caused by the field inhomogeneity (Casprihan et al., 1990). Such phase errors may arise from magnet shimming, susceptibility effects, and eddy currents due to gradient switching.
The two-dimensional velocity image is shown in Fig. 3
. The grayscale bar on the righthand side of the figure was scaled with average velocity. The average velocity of the experiment was 0.25 mm/s, measured by a conventional method. The average velocity measured by NMRI was 0.23 mm/s. Thus, the error in velocity is estimated at 8%.
In our experiment an empirical linear equation relating local velocity (V) and porosity (
) (Fig. 4)
was obtained from maps of velocity (Fig. 3) and porosity (Fig. 1). The single points in Fig. 4 correspond to averages over a region of interest (ROI) of approximately 16 by 16 voxels. Note that velocity in this context is a Darcy velocity. The error bars show the standard deviation within the ROI. The correlation coefficient was 0.9603 and the regression line is given by:
 | [5] |
Assuming an average head gradient, this relation indicates that permeability increases with the increase of porosity.
The successful use of the phase shift method to obtain a velocity map depends on the shape and range of the velocity distribution within each voxel, which, in turn, is influenced by the pore structure. There exists a fundamental limitation to the estimation of flow velocity using the phase shift of a NMR signal in a voxel for heterogeneous samples. The error of phase (Bevington, 1969) due to uncertainties in the measurements of an NMR signal is about the reciprocal of the signal to noise ratio if both channels of the quadrature detection cause the same signal error. Consequently, experimental results by Chang and Watson (1999) indicated that velocity imaging using the phase-shift method for a rock sample can be more difficult than for a bead pack due to the respective signal and more significant loss in phase coherence.
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FLOW IMAGING BY PARAMAGNETIC TRACERS
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Traditionally, porous flow experiments are interpreted on the basis of the homogeneous black box assumption. Volume and composition of fluids that are injected and recovered are measured, but distribution of the fluids inside the porous medium can only be inferred. The loss of information results in a series of problems.
One- and two-phase porous flow are imaged by NMR using paramagnetic ions as tracers. The inspection of internal distributions in addition to input and breakthrough curves allows showing the differences between black box and white box model interpretations.
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TWO-PHASE FLOW
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Using NMRI, an experiment on the basis of a quarter five spot pattern was carried out and visualized. The model consists of artificial consolidated sandstone sealed by epoxy resin with inlet and outlet at opposite corners of the model. The artificial consolidated sandstone was made of 92% silica sand with diameters from 0.25 to 0.42 mm and 8% epoxy resin as cementing agent. The size of the cube-shaped model is 6.4 x 6.4 x 1.5 cm, the average porosity is 0.31, and the average permeability is 1.6 µm2.
The NMRI experiments were carried out using the Bruker BIOSPEC 47/40 instrument with a 40-cm-bore superconductive magnet of 4.7-Tesla field strength and a B-G 15/A gradient system with maximum gradient field strength of 120 mT/m.
The model was saturated with oil under vacuum conditions. The heterogeneity of the artificial sandstone was assessed on the basis of a two-dimensional porosity distribution derived from proton spin-density imaging with the same experimental parameters as described above. The two-dimensional proton spin-density image is shown in Fig. 5
, the corresponding histogram of voxel porosity is given in Fig. 6
. The y axis shows the distribution frequency while the x axis corresponds to the grayscale bar of Fig. 5, scaled to average porosity.
Oil in the model was flooded by an aqueous paramagnetic ion solution (500 mg/L of MnCl2) at a constant flux of 0.25 mL/min. The NMR kinetic imaging was conducted by the spin–echo method during flooding using the following experimental parameters: echo time 2 ms, repetition time 0.5 s, image matrix 128 x 128, number of images for averaging 2, spatial resolution 0.63 x 0.63 x 15 mm3. Every image was gathered over a time of 2.2 min. The distribution of the two phases at different stages of the water flooding is shown in the right column of Fig. 7
. The time is expressed in units of injected pore volumes. The fluid distribution is characterized by fingers, which are caused by heterogeneity of the material.
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Fig. 7. Comparison of fluid distributions (black box simulation at left, nuclear magnetic resonance imaging [NMRI] experiment at right) during water flooding.
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A numerical calculation using a standard two-phase flow code was made under the following hypotheses. The porous medium is assumed to be homogeneous with a two-dimensional flow in the plane of the two larger dimensions. It is further assumed that Darcy's law is valid and that the compressibilities of fluid and porous medium are very small. Capillary pressure is neglected.
The results are shown for the same times as the experimental results in the left column of Fig. 7. The comparison of experimental and numerical results illustrates the important effect of heterogeneity. The distributions of the fluids are distinctly different in both cases. While in the beginning the oil recovery rate in both cases is identical for continuity reasons, at a later stage the sweep efficiency of the experiment decreases due to early water breakthrough in fingers. The porosity distribution of the sandstone slab was mapped with NMR via spin-density imaging. The permeability distribution can be derived from the NMRI velocity distribution using Darcy law. These data can serve for purposes of benchmarking in the following way. The distributions of porosity and permeability as measured by NMRI serve as input for the numerical simulation. The capillary pressure curve of each voxel can be calculated from the distributions of porosity and permeability using the "J-function" (Leverett, 1941) obtained by conventional methods. At the same time, the relative permeability of each voxel can be calculated from the capillary pressure curve (Purcell, 1949; Fatt and Dyksta, 1951; Burdine, 1953). Both the capillary pressure and relative permeability function of every voxel can be provided as input of the numerical simulation. The saturation distribution calculated at different times of displacement can then be matched with saturation distributions measured by NMRI for the same times.
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DENSITY FLOW IMAGING WITH THE PARAMAGNETIC TRACER METHOD
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Density flow problems occur in aquifers where salt water and fresh water cause density gradients, which interact with the flow field. Prominent applications of density flow models are seawater intrusion, salt water upconing, and flow around salt domes. Models for computing density flow are available. In order to test these models benchmark problems are required. Unfortunately, few analytical solutions of density-driven flow are known and those often used such as the Henry problem are not a sensitive test of model properties. Therefore, a three-dimensional physical benchmark test was proposed (Oswald, 1999). The NMRI method was used to visualize three-dimensional density flow in an artificial porous medium. The medium consisted of more or less single-sized, uniform glass beads of spherical shape. The bead diameter was 1.2 mm on the average, varying between 1.0 and 1.3 mm. The size of the cube-shaped model was 20 x 20 x 20 cm3. The porosity was 0.37. A schematic view of the experimental array is shown in Fig. 8
.
The physical properties of a voxel, which determine the signal amplitude, are the proton density in the voxel, the spin–lattice relaxation time T1 and the spin–spin relaxation time T2. The measurements of T1 by an inversion recovery method and of T2 by the multiecho Carr–Purcell method showed that only T2 varies significantly for bead packings with bead diameters between 0.5 and 1.2 mm, whereas T1 is nearly constant. The time sequences chosen are characterized by echo times (TE) from 10.6 to 15 ms and repetition times (TR) from 90 to 300 ms so that in any case TR/T1 <
1 and TE/T2 << 1.
Measurements with Cu2+ solutions in a glass bead matrix with beads of a 1.2-mm diameter showed that the inverse relaxation times (1/T2) are nearly proportional to the Cu2+ concentration. The tested concentrations ranged between 0 and 12 mmol/L. This kind of relationship between ion concentration and inverse relaxation time has already been observed for other porous materials (e.g., Fischer et al., 1995). For small concentrations this linearity can lead to another linear relationship, namely between the Cu2+ concentration and the signal amplitude (Greiner et al., 1997), which allows computation of concentrations and mass balances from NMR images directly. The sensitivity of the method allowed significant measurement of concentrations down to values around 0.5 mmol/L of CuSO4.
The Cu2+ ions (i.e., CuSO4) were added to a NaCl solution of a density distinctly higher than freshwater to cause density flow. Since the Cu2+ ions do not segregate significantly from the Na+ ions during the relatively short duration of the experiments, the total salt concentration can be determined by measuring the signal amplitude as influenced by the concentration of paramagnetic ions. In fact, the CuSO4 is used as a NMR tracer to make the salt concentration visible. Using a low tracer concentration allows exploitation of the linear relation discussed above for mass balancing purposes. The errors of the method have been estimated in some of the experiments described in the following and will be given there.
In one experiment the dense fluid was initially positioned on top of the fresh water leading to fingering phenomena (similar to Pearl et al., 1993). To guarantee that the first finger always developed in the same place, a narrow pack of larger beads was built into the center of the box resulting in a more permeable vertical channel. Figure 9
shows the experimental results after some time as a three-dimensional image of the 50% tracer concentration contour. The four-pronged finger in the more pervious channel can be clearly seen together with the rising freshwater finger on top. Outside of the channel many smaller fingers have developed.
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Fig. 9. Three-dimensional nuclear magnetic resonance (NMR) image of finger distribution (50% contour, saltwater above, fresh water below).
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The density contrast had to be chosen carefully in order to keep the time scale of finger growth large enough compared with the imaging time, which was about three minutes for a single measurement period covering the full volume of the cube.
The situation was modeled using a standard numerical density flow model (SALTFLOW; Molson and Frind, 1994). The resulting distribution is comparable with the experimental results in Fig. 9 and is shown in Fig. 10
. The topology of the fingers is well reproduced by the numerical model. However, much less fingering is observed and the time for the main finger to develop is much larger in the numerical simulation than in reality. This is due to regularization of the numerical scheme, which tends to suppress numerical instability. But in doing this the real physical instability is also suppressed. Much finer grids than the one used here would be required to obtain more realistic finger dynamics.
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Fig. 10. Three-dimensional computation of finger distribution (50% contour, salt water above, freshwater below).
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In two further experiments the dense water was introduced from below, leading to a stably stratified situation. The configuration of the model is shown in Fig. 11
. Two density contrasts were chosen, with salt solutions of 1 and 10% salt mass fraction, respectively. Both experiments were run in three phases.
The box is initially filled with fresh water. In the first phase the saltwater is injected via an opening in the bottom of the model, displacing the fresh water radially. The excess fresh water can leave the box via openings in the four top corners of the box. In the second phase the saltwater body is allowed to equilibrate. In the third phase fresh water is injected at a fixed rate in one of the top corners while the diagonally opposing top corner valve is open.
The results are shown in the form of concentration profiles along the vertical diagonal plane of the cube (Fig. 12
and Fig. 13)
. In the low-density experiment (Fig. 12) the inflowing fresh water displaces the saltwater layer, pushing it gradually out of the box. The boundary streamlines of the inflowing water follow the vertical walls and bottom of the box. Comparison of outflow concentration and remaining concentration in the box proves that mass balancing on the basis of NMR images is feasible. At the end of the experiment 47% of the initial salt mass was found in the outflow, leaving 53% in the box. This is in close agreement with the value of 50% detected inside the box directly by the NMR images.
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Fig. 12. Nuclear magnetic resonance (NMR) image of saltwater distribution in a diagonal plane at different times for the low-concentration contrast experiment (height of section = 20 cm, width of section = 38 cm).
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Fig. 13. Nuclear magnetic resonance (NMR) image of salt water distribution in a diagonal plane at different times for the high-concentration contrast experiment (height of section = 20 cm, width of section = 38 cm).
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In the high-density contrast case (Fig. 13), the upconing is strongly reduced by gravity forces and the flow pattern changes significantly. The equilibration essentially already takes place in the injection of Phase 1 and nothing much happens in Phase 2. In Phase 3 the salt–freshwater interface is only slightly tilted. The boundary streamlines of the inflowing water now pass on top of the interface, only "shaving off" the saline water transported out of the saltwater body by lateral dispersion and molecular diffusion.
Due to the high salt concentration in the last experiment the electrical conductivity of the salt water becomes so large that it leads to shielding effects, which are shown in Fig. 14
. All images shown in Fig. 13 were corrected in the salt water region using the images of the second phase of the series. This is feasible as the shielding is only important for the inner region, which is at maximum concentration over the whole experiment. By using a nonpolar solute such as sugar to produce the density effect, the shielding effect could be avoided.
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Fig. 14. Image distortion by shielding effects (left) in the high-concentration experiment and correction (right) (height of section = 20 cm, width of section = 38 cm).
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The values of the experimental parameters of these laboratory experiments could be evaluated with high accuracy and the setup of the problem was quite simple in terms of its geometry and its homogeneity. Therefore, the experimental results are suited to be used for the verification of variable-density flow codes.
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CONCLUSIONS
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Nuclear magnetic resonance imaging can be used in the direct observation of the movement of molecules associated with flow. This provides for an opportunity to determine local velocities within porous media noninvasively. The information on flow structure inside porous media can be compared with results obtained on the basis of homogeneous black box models.
The examples show that predictions based on homogeneity assumptions may be risky. On the other hand, results from NMRI experiments can serve as benchmark for porous media flow models. In particular, differences between numerical calculation and physical experiments can be exploited for improving the physical and numerical basis of models.
Limitations of some techniques show for high paramagnetic content of the media and weak tracer signals. Therefore, high resolution imaging at the moment is restricted to natural materials with low contents of paramagnetic substances or artificial porous media.
Nuclear magnetic resonance imaging is a versatile noninvasive method, which has significant potential for further applications to flow processes in porous and fractured media.
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Q. Chen, F. R. Rack, and B. J. Balcom
Quantitative magnetic resonance imaging methods for core analysis
Geological Society, London, Special Publications,
January 1, 2006;
267(1):
193 - 207.
[Abstract]
[PDF]
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ABSTRACT
INTRODUCTION
