CO₂ diffusion in n-hexadecane investigated using magnetic resonance imaging and pressure decay measurements

Abstract

A method combining magnetic resonance imaging (MRI) and dual-chamber pressure decay (i.e., the pressure–volume–temperature method) was used to measure the diffusivity of CO₂ in bulk n-hexadecane, a representative oil, at 21 °C. Images of the proton density of n-hexadecane were obtained by MRI using a high magnetic field and high resolution, and the pressure decay was recorded. Overall diffusion coefficients of CO₂ were calculated from both pressure decay and MRI intensity data. The two results showed good agreement. MRI was successfully used to study the diffusivity of CO₂ in n-hexadecane. The change of image density (i.e., proton density) demonstrates that the oil density decreases as CO₂ diffuses into it. The finite volume method was applied to the images obtained by MRI, allowing the diffusion coefficient of CO₂ to be directly obtained. MRI methods can measure the unsteady-state local diffusion coefficient and overall diffusion coefficient of such systems.

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1 Introduction

Two miscible fluids will diffuse into each other when they come into contact. The molecular transport of one substance relative to another is known as diffusion. Diffusion plays an important role in the mixing of CO₂ and crude oil after injection of CO₂ into an oil reservoir.

CO₂-enhanced oil recovery (CO₂-EOR) has proven to be effective for improving domestic oil production. Moreover, use of CO₂ in oil recovery also provides considerable capacity for CO₂ mitigation.¹ The viscosity of oil can be lowered significantly by diffusion of CO₂ into it; thus, the mobility of oil in a reservoir can be improved.² Mass transfer between CO₂ and oil, which is mainly controlled by diffusivity, is the first mechanism that occurs during the CO₂-EOR process.^3,4 To describe this process quantitatively, the diffusion coefficient of CO₂ gas in oil needs to be determined.

Experimental methods used to measure the diffusivity of gas in oil can be broadly divided into two types: direct and indirect. The direct measurement of diffusivity is time consuming and involves measuring the change of concentration over time.⁵ Indirect methods involve measuring the change of a parameter related to diffusion over time, such as volume, density or pressure.^6,7 The pressure decay method is a typical indirect method used to measure diffusivity. Riaziet et al.⁸ reported a pressure decay method using a pressure–volume–temperature (PVT) cell. In their experiment, both the position of the gas/liquid interface and gas pressure decay in the PVT cell were monitored during the diffusion process. Zhang et al.⁹ measured the diffusion coefficient of CO₂ in heavy oils using a similar experimental method except that the position of the interface was considered constant. They also presented a nonlinear regression method that allowed the diffusion coefficient to be obtained directly from pressure decay data. Recently, non-intrusive detection methods such as X-ray computed tomography (CT) and magnetic resonance imaging (MRI) have been extended to the area of petroleum research. Liu et al.¹⁰ observed the miscible and immiscible CO₂ displacement of n-decane in glass-bead packed beds using an MRI scanner under high magnetic field. Residual oil saturation was quantitatively measured by intensity analysis of magnetic resonance (MR) images. Song et al.¹¹ determined the minimum miscible pressure of CO₂ and n-decane by the MRI method, and their results agreed well with previous data obtained with traditional methods. Meanwhile, X-ray CT has been used to measure density distribution in heavy oil and hydrocarbon solvent mixing experiments.^12–14 In these experiments, concentration profiles were obtained from CT images based on the relationship between bulk density and CT image intensity. Other researchers have used MRI to observe mass-transfer processes. Fisher et al.¹⁵ used MRI to observe vapor extraction processes. They performed bitumen and heavy-oil vapor extraction experiments using butane and propane as solvents in dry sand and sand with connate water. In their experiments, the MRI signal, which corresponded to the oil saturation, provided not only the position of the vapor chamber but also concentration gradients.

The main objective of the present study is to use MRI technology to measure gas diffusion coefficients in oils, and investigate diffusion and mass transfer processes in situ. Diffusion coefficients of CO₂ in n-hexadecane are determined by measuring the pressure in a dual-chamber PVT system. MRI with high spatial resolution is used to observe the concentration distribution of CO₂ during diffusion. In addition, pressure data are recorded to calculate the molar mass of CO₂ diffusing into n-hexadecane. Pressure decay analysis is performed to calculate the steady-state diffusion coefficient of CO₂ in n-hexadecane. This study helps us to understand the diffusion behavior of CO₂ in a representative oil, so it will provide useful information for several oil recovery processes.

2 Experimental details

2.1. Apparatus and materials

An experimental system containing a gas cylinder and vessel for the MRI sample was constructed to measure the diffusion coefficient of CO₂ into a representative oil, n-hexadecane, using both the pressure decay method and MRI.¹⁰ A schematic diagram of the experimental apparatus is shown in Fig. 1. The gas cylinder is a constant-volume (500 mL) stainless steel cylinder that is wrapped with an electric heater. The self-designed MRI sample vessel (Fig. 2) is made of polyimide to hold the diffusion samples at high pressure (up to 12.0 MPa). The sample vessel contained three main types of components: an inner tube with an inner diameter of 15 mm and wall thickness of 4.5 mm, an outer tube with an outer diameter of 38 mm, and inlet and outlet connectors. The outlet was blocked in the experiments. The temperature of the sample in the vessel was controlled by a circulator (F-25ME, Fluorinert™, Julabo, Inc., Seelbach, Germany) with a temperature control range of −45 to 200 °C and precision of ±0.5 °C. Inert liquid Fluorinert™ FC-40 was used as the temperature-control medium of the imaging vessel because it does not contain hydrogen atoms and thus has no influence on ¹H nuclear imaging. The gas cylinder and sample vessel formed a dual-chamber system, which can improve the pressure stability and measurement precision relative to those of a single-chamber system.

Fig. 1 Schematic diagram of the experimental apparatus.

Fig. 2 Vertical-section diagram of the sample vessel.

Images were measured using a 400 MHz NMR spectrometer (Varian Inc., Palo Alto, CA, USA) with microimaging kits including a 55 mm i.d. gradient coil, which provided a maximum gradient magnetic field of 50 Gauss per cm, and a 40 mm i.d. 400 MHz ¹H Millipede vertical imaging probe. All data and images were processed with a data processor.

n-Hexadecane of 98% purity was contained in a glass vial (i.d. = 8.8 mm) placed at the bottom of the sample vessel. Gaseous CO₂ of 99.99% purity was injected via a syringe pump into the gas cylinder to attain the desired pressure.

2.2 Procedure

Prior to performing experiments, the diffusion cell and all connections were pressurized with nitrogen and tested for leakage at pressures up to 8 MPa for at least 24 h.

All experiments were performed at 21 °C. The glass vial was filled to a height of about 20 mm with n-hexadecane. The system was placed under vacuum to extract air. After careful tuning, shimming and setting of the pulse parameters, the oil sample was scanned to obtain an initial MR image. CO₂ was then injected into the gas cylinder with the syringe pump until the desired pressure was reached. The pressure in the gas cylinder was set between 3000 and 5000 kPa. The gas valves were then turned on to allow CO₂ to enter the sample vessel and diffuse into the oil. The pressure was automatically recorded at the same time. Proton density images of the longitudinal planes along the diffusion direction were obtained while the pressure was being plotted. Each experiment was stopped after the pressure maintained a constant value (equilibrium pressure) for 1 h. When the experiment had finished, the system was slowly depressurized to atmospheric pressure, dismantled and cleaned before starting a new experiment.

The steps of applying MRI into diffusion experiment imaging are as follows:

(1) Put the n-hexadecane-containing glass vials upright in the sample vessel, and keep the height consistent with the imaging probe.

(2) Debug the MRI analyzer; open Vnmrj in the image acquisition computer, and enter “tune” into the dialog box; tune by regulating the knobs “tune” and “match” below the probe; then start to shim: click “manual shim” on the “start” interface, entering the shimming interface; regulate by using the right and left mouse buttons, until the images are appropriate.

(3) Setting of parameters. All images were acquired with a standard spin-echo multi-slice (SEMS) pulse sequence, because this sequence is free from the impacts of main magnetic field inhomogeneity, and produces high-quality images. The specific imaging parameters are as follows: echo time (T_E) = 4.9 ms, repetition time (T_R) = 5 s, image data matrix = 96 pixels × 96 pixels, field of view (FOV) = 40 × 40 mm², slice thickness = 2 mm.

(4) Image acquisition. After gas injection was started, the MRIs were continuously acquired, until the end of the experiment.

2.3 Mathematical analysis

The diffusion process may be analyzed as shown in Fig. 3. This analysis can be derived based on the following assumptions:^9,16,17

Fig. 3 Schematic diagram of the diffusion model.

(1) The swelling height of the oil is negligible, that is, z₀ remains constant during the experiments.

(2) The temperature remains constant.

(3) The volatilization of n-hexadecane is negligible.

(4) The diffusion coefficient, D, does not change markedly with concentration during the experiments.

(5) The gas phase is a pure (single-component) gas.

The diffusion process can be expressed by Fick's law:⁹

where x is the molar concentration of the gas at position z and time t, D is the diffusion coefficient.

Based on this hypothesis and combined with Fick's law, the relationship between pressure P and time t can be acquired using semi-infinite boundary conditions and the matter balance principle. When the higher-order terms were ignored, a 2-order analytical expression was obtained as follows:⁹

The first exponential term in eqn (2) fits the main part of the pressure–time curve (large k) and the second term fits the initial part of the pressure–time curve during the incubation period (small k). Numerical fitting is required to obtain the values of m₁, m₂, k₁, k₂ and P_eq. Because the incubation period is relatively short, the first exponential term is more suitable to calculate the integral diffusion coefficient. The diffusion coefficient D is then given by:⁹

where k* is the greater of k₁ and k₂, z₀ is the liquid surface height of n-hexadecane.

2.4 MR image analysis

An MR image of the proton density of n-hexadecane was obtained after one MRI scan. A region of interest (ROI) in the image was selected to analyze the diffusion process. Each ROI was 20 mm high, 7 mm wide and divided into ten segments of equal height. The image intensity of each segment at different times (defined as I_z,t) was then calculated. The initial I₀ (at t = 0) and final I_eq (at t = t_eq) values are constant at all positions.

CO₂ has no signal in MRI. A previous study¹⁸ confirmed that the MR signal intensity at any location is proportional to the content of oil. When CO₂ diffuses into the oil, the content of oil decreases. Thus, the signal intensity of the MR images will weaken as the concentration of CO₂ increases. Therefore, the distribution of dimensionless CO₂ concentration can be calculated based on the measured signal intensity distribution of the MR images at each time period. According to Fick's law, the diffusion coefficients related to time and displacement can be computed based on such dimensionless concentrations. The concentration of CO₂ is related to the MR signal intensity. The normalized concentration of CO₂ in oil at different positions, z, and times, t, can be written as:

where x_z,t represents the CO₂ concentration at position z and time t. I₀ represents the intensity at t = 0, I_z,t represents the intensity at time t and position z, and I_eq represents the intensity at t = t_eq. According to Zhao et al.,¹⁸ the concentration of CO₂ is related to MRI intensity. So we can then obtain the following:

x_z,t = k(I₀ − I_z,t) (5a)

x₀ = 0 (5b)

x_eq = k(I₀ − I_eq) = const (5c)

where k is a constant.

Here, a non-iterative, finite-volume analytical technique was used to calculate the diffusion coefficient of CO₂ in oil (n-hexadecane). It was assumed that there was a small composition gradient in the oil along the diffusion direction, and that the diffusion front did not move appreciably. Thus, Fick's Law (eqn (1)) can be used to relate the concentration and diffusion coefficient.

As shown in Fig. 4, the medium domain was discretized with mesh size Δz, time step Δt and grid points z_i = i × Δz (i = 1, 2, …10) and t_n = n × Δt (n = 0, 1, 2, …).^19,20

Fig. 4 Model of the finite-volume analytical method used in this work.

Assuming that the concentration in boundary A is constant and there is no flux in boundary B, the discretized equations are as follows:

Internal points

Boundary A

Boundary B

By arranging the discretized equations within the medium domain and at the boundary surfaces, eqn (6) can be written in matrix form as Ax = b. Vector b is composed of the concentration measurement at specified grid locations along the sample in the experiment. The components of vector x are the unknown diffusion coefficients.

3 Results and discussion

The pressure attenuation curves obtained with an initial pressure of 4377 kPa are shown in Fig. 5. A biexponential decrease in pressure is observed. Initially, the decrease was more pronounced, and became stable after an extended period. The decrease of pressure resulted from the dissolution of CO₂, which demonstrates that the mass transfer rate during the initial stage of the experiment was relatively high. This suggests that high concentration and density gradients existed along the internal diffusion direction for n-hexadecane, which led to considerable convection and mass transfer. The initial stage of this high-mass-transfer diffusion is called the “incubation period”.²¹ With lengthening diffusion time, the internal concentration gradient and density gradient of hexadecane were gradually reduced. At the same time, the rates of mass transfer and pressure decay decreased. This result also indicates that after a long time, normal diffusion behavior occurs, and that mass transfer after the incubation period is dominated by molecular diffusion. Furthermore, it was found that a higher initial pressure resulted in a higher pressure decay rate and equilibrium being reached in less time than a lower initial pressure. It is evident that the incubation period has a considerable influence on the diffusion of CO₂, and a high initial pressure increases this influence.

Fig. 5 Pressure decay during diffusion of CO₂ in n-hexadecane starting with an initial pressure of 4377 kPa.

The collected pressure–time data were numerically fitted in origin with an accuracy of 0.974–0.993. The overall diffusion coefficient of CO₂ in n-hexadecane can be obtained by the fitted k* using eqn (3). The overall diffusion data obtained under different pressures are listed in Table 1. We calculated that the overall diffusion coefficient of CO₂ in n-hexadecane to be about 3.21–4.30 × 10⁻⁹ m² s⁻¹. Even though the exact experimental conditions used in the present work have not been used before, Wang et al.²² and Grogan²³ obtained diffusion coefficients of 3.32–5.71 × 10⁻⁹ m² s⁻¹ and 1.8–3.21 × 10⁻⁹ m² s⁻¹, respectively, under similar conditions. Our diffusion coefficients are of the same order of magnitude as these.

Table 1 Pressure decay data for CO₂ diffusion in n-hexadecane at 21 °C

NO.	Oil height (mm)	Pressure of gas cylinder (kPa)	Initial pressure (kPa)	k* (s)	D (10^−9/m^−2 s^−1)
1	20	3000	2568	50 537.3	3.21
2	20	3500	3093	49 201.1	3.71
3	20	4000	3420	41 170.2	3.94
4	20	4600	3980	39 020.2	4.15
5	20	5000	4377	37 680.5	4.30

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The ROIs of MR images were cropped with image processing software. As an example, Fig. 6 shows a time series of ROIs of the diffusion process of CO₂ in n-hexadecane starting with an initial pressure of 4377 kPa. As CO₂ diffuses into n-hexadecane, a decrease in image intensity from top to bottom can be clearly observed because CO₂ is invisible in MR images. The CO₂ concentration distribution can be calculated from the image intensity data.

Fig. 6 Series of ROI images at different times during the CO₂ diffusion process for an initial pressure of 4377 kPa.

The normalized concentration distribution of CO₂ along the diffusion direction at different times calculated by eqn (4) for an initial pressure of 4377 kPa is shown in Fig. 7. Along the diffusion direction, the concentration of CO₂ decreases gradually. For a given diffusion distance, the concentration of CO₂ increases with time, which is consistent with the experimental results of Islas et al.²⁴

Fig. 7 Normalized CO₂ concentration distributions calculated from MR images with an initial pressure of 4377 kPa.

The diffusion coefficients calculated from eqn (6) for initial pressures of 2568, 3093, 3420, 3980 and 4377 kPa are depicted in Fig. 8. All of the diffusion coefficients decreased as the diffusion distance increased. Moreover, the diffusion coefficients decreased dramatically near the gas/oil interface and gradually decreased to a stable value as the diffusion distance increased. The maximum diffusion coefficient was observed near the interface, indicating that concentration and density gradients exist in n-hexadecane during dissolution and diffusion of CO₂. In addition, because the dissolution of CO₂ is much higher at the gas/oil interface, the concentration gradient at this interface is relatively high, which causes convection and results in increased diffusion. Furthermore, Fig. 8 also reveals that the diffusion coefficient in a given region decreases over time, which suggests that the influence of the density gradient on convection and the incubation period decreases. In addition, diffusion coefficients increase with increasing initial pressure, which is consistent with the findings of the pressure decay method.

Fig. 8 Diffusion coefficients measured at different diffusion distances and times for initial pressures of (a) 2568 kPa, (b) 3093 kPa, (c) 3420 kPa, (d) 3980 kPa and (e) 4377 kPa.

MRI can be used to measure the overall diffusion coefficient of CO₂ in n-hexadecane. The overall diffusion coefficient is the average of diffusion coefficients measured during the time it takes for diffusion to reach equilibrium. First, the computational time step was determined, and then the distribution of dimensionless CO₂ concentration was calculated using eqn (4). The position dependence of diffusion coefficients was computed using eqn (6). According to the relationship between diffusion coefficient and position, the trapezoidal rule was used to obtain the averages that represent time-dependent diffusion coefficients. Likewise, considering the relationship between diffusion coefficient and time, the trapezoidal rule was then used to average values to calculate the overall diffusion coefficient for each initial pressure. Fig. 9 shows the overall diffusion coefficients for different initial pressures calculated from the experimental MRI data and by the PVT method. According to eqn (3), the error between the MRI and PVT is about 1.6–4.3%. The two results showed good agreement. MRI can measure both unsteady-state local diffusion coefficients and overall diffusion coefficients. In addition, according to the experimental results, the overall diffusion coefficients increase along with the increase of the initial pressure.

Fig. 9 Overall diffusion coefficients obtained by MRI and the PVT method.

4 Conclusions

This study demonstrates that visualization and qualitative analysis of CO₂ diffusion in n-hexadecane can be achieved by combining MRI with the dual-chamber pressure decay method. The non-iterative finite-volume analytical technique can be used to calculate CO₂ diffusion coefficients from concentration profiles obtained from MR images. The results agree very well with the conventional pressure decay method in binary mixtures. MRI can determine both unsteady-state local diffusion coefficients and overall diffusion coefficients. Based on our experimental results, the following conclusions can be drawn:

(1) The concentration of CO₂ decreases along the diffusion direction and is higher during the later stages of the reaction than during the initial stages.

(2) The diffusion coefficient of CO₂ is a function of both diffusion time and diffusion distance. The diffusion coefficient decreases as diffusion distance and time increase until an equilibrium state is reached.

(3) The initial pressure has an obvious effect on the diffusion coefficient of CO₂, with pressure being directly proportional to diffusion coefficient.

(4) The influence of the incubation period decreases with increasing diffusion time. Mass transfer after the incubation period is mainly governed by molecular diffusion.

Nomenclature

D	Diffusion coefficient (m^2 s^−1)
P(t), Peq	Measured and equilibrium pressure, respectively (kPa)
t	Time (s)
xeq(P)	Oil/gas interface molar concentration at a given time (mol cm^−3)
z0	Height of the oil in the vessel (cm)
xz,t	CO2 concentration at position z and time t from MRI images
xeq	CO2 concentration at equilibrium state
I0	Initial image intensity at time t = 0
Iz,t	Image intensity at position z and time t
Ieq	Image intensity at equilibrium state
x	Molar concentration of CO2 gas

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant no. 51206018, and 51106019), the Fundamental Research Funds for Central Universities, the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20120041120057), and the Ph. D. Startup Foundation of Liaoning Province (Grant no. 20121022).

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