Project Topics
Get Complete Project Material File(s) Now! »
Fluid viscosity
Since the fluid viscosity is widely introduced in literature, only a brief quantitative description will be mentioned in this section. Viscosity is a fluid property that is very important for fluid flow through pipes or porous media, and it could be define as a measure of fluid’s resistance to flow. For example, the water is easier to pass through a tube than the oil, because water has less viscosity. Fluid viscosity can be abstracted by quantifying the forces of frictional that appear between adjacent fluid layers that are in relative motion. Although this phenomenon becomes more complicated when it comes to the fluid flow in porous media, it is an essential parameter that controls the two-phase flow in porous media.
Consider a fluid trapped between two large parallel plates separated by a fluid layer (Fig. 2.2). The bottom one is fixed and the upper plate moves in parallel at a constant velocity 𝒰. If the top plate velocity is low (laminar flow case), then the fluid moves in parallel layers with different speeds. In many fluids, the fluid velocity varies from zero on the bottom plate to 𝒰 at the top. The force F applied to the top plate is proportional to the velocity 𝒰, and the area A of the plates. 𝐹=𝜇𝐴 𝒰𝑦 (2.2).
Interfacial tension
When two immiscible liquids come into contact with each other, the molecules of the two phases undergo unbalanced attraction forces. This imbalance creates interfacial tensions. It will be noted that the surface tension is the property of the liquid in contact with gaseous phase, usually air. The interfacial surface tension is measured in force per unit length (mN/m), and noted by the character σ. The interfacial tension occurs due to the imbalance of the molecular attraction forces experienced by the molecules on the surface. Figure 2.5 shows an example of interfacial tension in water droplet.
Leverett J-Function
Capillary pressure data are obtained on small core samples that represent an extremely small part of the reservoir and, therefore, it is necessary to combine all capillary data to classify а particular reservoir. The fact that the capillary pressure-saturation curves of nearly all naturally porous materials have many features in common has led to attempts to devise some general equation describing all such curves. Leverett (1941) approached the problem from the standpoint of dimensional analysis [58].
Realizing that capillary pressure should depend on the porosity, interfacial tension, and mean pore radius, Leverett defined the dimensionless function of saturation, which he called the J-function, as: 𝐽(𝑆𝑤 )=𝑃𝐶𝜎 𝑐𝑜𝑠 𝜃.√(𝐾/𝜑) (22.11).
where, K is the permeability, σ is the interfacial tension (dynes/cm), 𝜑 is the porosity.
Magnetic Resonance Phenomenon
The magnetic resonance phenomenon goes through three distinct stages:
• Polarization ⟹ under the effect of a static magnetic field.
• Resonance ⟹ disturbance using a radio-frequency field.
• Relaxation ⟹ return to balance condition.
Polarization [18, 72, 73]: Naturally, the protons are randomly arranged in the material to remove the influence of each other. For this reason, the vector sum of nuclear magnetic moments I for this material is zero (Σ I = 0), figure 3.2.a. In this case, all spins are distributed on the same level of energy. Nuclear Magnetic Resonance (NMR) studies the response of atomic nuclei when subjected to an external magnetic field. Here, we apply the field along the z-axis. The protons magnetic moment is therefore oriented in the direction of the magnetic field vector (Fig. 3.2.b). Nuclear magnetic moment z then takes the value of +/2 or -/2.
Transverse relaxation time T2
It is the time required for the transverse component of M to decay to 37% of its initial value via irreversible processes. It is also known as the spin-spin relaxation time or transverse relaxation time. Recall from previous that M0 is oriented only along the z (B0) axis at equilibrium and that no portion of M0 is in the XY plane. The coherence is entirely longitudinal. Absorption of energy from а 90o RF pulse, causes M0 to rotate entirely into the XY plane, so that the coherence is in the transverse plane at the end of the pulse. As time elapses, this coherence disappears, while at the same time the protons release their energy and reorient themselves along B0. This disappearing coherence produces the FID (Free Induction Decay). As this coherence disappears, the value of M in the XY plane decreases toward 0. T2 or T2* relaxation is the process by which this transverse magnetization is lost [18]. 𝑀𝑋𝑌(𝑡)=𝑀𝑋𝑌 𝑚𝑎𝑥 (𝑒(−𝑡𝑇2)) (3.7).
The frequency of precession is not perfectly homogeneous because of inhomogeneities of the field which leads to a loss of signal. This loss of signal is characterized by a T2* time, lower than T2, due to the loss of phase coherence between the different magnetizations during the precession. This signal decreases exponentially in T2* and not in T2 (Fig. 3.8). If the field was perfectly homogeneous, the shape of the signal would be an exponential decay in T2. The T2 exponential curve given by formula (3.6).
NMR signal recovery and images
To obtain an MR image, magnetic field gradients are applied in X, Y, and Z directions. For instance, a slice selection in a sample is achieved by applying a one-dimensional magnetic field gradient on Z-axis, during the period that RF pulse is applied (Fig. 3.9.). Therefore, only the magnetization of the spins belong to the selected slice will be tilted, and the conversely the spins outside the slice will be neglected. The placement of a volume-element (voxel) in a slice is encoded with a phase Y-axis and a frequency gradient in the X-axis direction. The resolution of obtained image relies mostly on the strength of the gradients: if the gradients are higher the resolution will be better.
• Fourier plane (k space):
The recovered signal is recorded in a frequency space called k-space or Fourier plane. The filling of this space is carried out following the application of gradients in X, Y, and Z directions. A classic spin echo sequence fills the k-space line by line as follows [78]:
1. Beginning of the sequence, 90° RF excitation + gradient of slice selection: position at the centre of the Fourier plane.
2. Intense negative phase gradient: downward movement of the Fourier plane.
3. Positive frequency gradient: move to the right of the plane.
4. RF wave of 180° + gradient of slice selection: symmetry with respect to the centre of the plane, position in top on the left.
5. Positive Frequency Gradient + echo recovery of the signal: move and fill to the right in the Fourier plane.
6. The sequence is then repeated, until the total filling of the Fourier plane, with phase gradients of increasing intensity. The K space is filled from bottom to top, line by line. It should be noted that the centre points of the Fourier space encode the low frequencies (contrast of the image) while the peripheral points encode the high frequencies (details of the image).
MRI Resolution and field of view
The MRI signal is acquired in a volume resonator with a 4cm diameter and a comparable height. It means that we cannot detect signal outside this limited volume. Therefore, we have to carefully select the imaging area, also called field of view (FOV). As we are looking at a macroscopic phenomenon, we should image the biggest possible area. We decided to select a FOV of 5cm*5cm, a little bigger than the detection volume. This allows us to limit aliasing from signal outside the FOV. Furthermore, we choose to place the read direction in the flow direction as there is no aliasing in this direction due to the digital filtering (contrary to the phase direction).
We acquired a matrix of 256 pixels × 256 pixels, which gives us a resolution of 0.19mm per pixel. It is possible to increase the size of the matrix, but the longer acquisition time leads to a decrease in signal due to T2* relaxation.
We had to design our study model according to the FOV constraints given above. To do so, our model was filled with glass beads of 3 mm diameter (Fig. 4.3.b), then installed inside a sample holder designed to fit in the imaging probe (Fig. 4.4). We acquired multiple images to precisely define the observation area in the sample. (Fig. 4.3.a).
Imaging protocol and methods
The magnetic resonance imaging technique is an effective, non-invasive method for determining fluid saturation in porous media by detecting 1H density. The imaging method used a classical multi-slice spin echo sequence. The MRI signal is explained in chapter 3 and can be described more as per S. Chen et al. and J. Yan et al. [5, 79]: 𝐼(𝑓,TE)=𝐼0(𝑓)(1−𝑒𝑥𝑝(−TRT1))exp(−TET2) (4.2). in which I is the observed intensity of the magnetic signal at resonance frequency f, I0 is the intrinsic magnetic intensity, TE is the echo time, TR is the repetition time, T1 is the longitudinal relaxation time and T2 is the transverse relaxation time. The intrinsic magnetic intensity I0 is in principle proportional to the number of protons 1H. Taking TR > 5T1 and TE < T2/5, the observed magnetic intensity is close to the intrinsic one, i.e. 𝐼(𝑓,𝑇𝐸)≈𝐼0(𝑓) which makes it possible to build an image in 1H density almost independent of the relaxation times.
Table of contents :
1. CONTEXT
1.1 Introduction
1.2 Importance of two phase flow in the oil industry
1.3 Applicati ons of MRI to porous media
1.4 MRI comparison to other visualisation techniques
1.5 Conclusions
2. TWO-PHASE FLOW IN POROUS MEDIA
2.1 Introduction
2.2 Basic elements
2.2.1 Porous medium
2.2.2 Fluid viscosity
2.2.3 Wettability
2.2.4 Interfacial tension
2.3 Flow equations in permeable media
2.3.1 Capillary pressure
2.3.2 Leverett J-Function
2.3.3 Governing equations for two-phase displacement
2.3.4 The competition between forces
2.4 Conclusions
3. PRINCIPLES OF MAGNETIC RESONANCE IMAGING (MR
3.1 Introduction and brief history
3.2 Nuclear magnetic moment
3.3 Magnetic Resonance Phenomenon
3.4 Acquisition of NMR signal
3.4.1 Longitudinal relaxation time T1
3.4.2 Transverse relaxation time T2
3.4.3 Spin-Echo sequence
3.5 NMR signal recovery and images
3.6 Conclusions
4. EXPERIMENTAL SETUP, INSTRUMENTATION AND METHODS
4.1 IntroductionIntroduction
4.2 ExperimentsExperiments
4.2.1 Experimental system
4.2.2 MRI Resolution and field of view
4.3 Sample materialSample material
4.3.1 Porous medium
4.3.2 Fluids
4.4 Experimental processExperimental process
4.5 Imaging protocol and methodsImaging protocol and methods
4.5.1 MSME protocol
4.5.2 Imaging methods
4.5.2.1 Selective method
4.5.2.2 Nonselective method
4.6 ConclusionConclusion
5. EXPERIMENTAL RESULTS AND DISCUSSIONS
5.1 IntroductionIntroduction
5.2 Experiments performedExperiments performed
5.2.1 Experiment with a packed glass-beads model, selective method
5.2.2 Experiment with a packed polystyrene-beads model
5.2.3 Experiments with packed sand column
5.3 Relative phase permeability calculationRelative phase permeability calculation
5.4 DiscussionDiscussion
5.4.1 Dimensionless groups
5.4.2 Displacement mechanism and the wettability effect
5.4.3 Additional observations
5.5 Conclusions
6. NUMERICAL SIMULATION AND COMPARISON WITH EXPERIMENTAL RESULTS
6.1 IntroductionIntroduction
6.2 Numerical modellingNumerical modelling
6.2.1 Case 1 classical model, viscous force effect
6.2.2 Case 2 gravity and viscous forces effect
6.2.3 Case 3 gravity, capillary and viscous forces effect
6.3 Comparison of the influence of three main Comparison of the influence of three main forcesforces
6.4 Comparison of experimental results with the numerical modelComparison of experimental results with the numerical model
6.5 ConclusionsConclusions
7. General conclusions and General conclusions and perspectivesperspectives
CONCLUSIONS AND PERSPECTIVES (ENGLISH VERSION)
BIBLIOGRAPHY
