In situ measurement of the dispersion coefficient of liquid/supercritical CO₂–CH₄ in a sandpack using CT

Abstract

Dispersion exists in many scientific and engineering applications, especially for CO₂ enhanced gas recovery, which is a vital factor for controlling the contamination of remaining natural gas and gas recovery. In this paper, an in situ method for the dispersion coefficient measurement of liquid/supercritical CO₂–CH₄ in a sandpack using CT was proposed. The dispersion coefficient in the sandpack was obtained directly from the CO₂ mole fraction profiles translated from a CT greyvalue image, which eliminate the deviation caused by the entry/exit effect. The finite difference method, Crank–Nicolson method, was applied to solve the advection dispersion equation for obtaining the dispersion coefficient. The breakthrough profile of the effluent gas was also analyzed and the apparent dispersion coefficient containing the entry/exit effect was measured using the dynamic column breakthrough method. The entry/exit effect enlarged the dispersion coefficient in the range of 14–23% under a water-free experiment according to the deviation between the two methods. And the dispersion coefficient with the sandpack containing residual water was smaller than that of the water-free condition, which was probably caused by the dissolution of CO₂ in the displacing frontier into residual water. The dissolution stabilized the dispersion in the displacing frontier and resulted in the reduction of the dispersion coefficient.

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1. Introduction

Dispersion in porous media is a physical blending process, which occurs in many natural and anthropogenic processes, such as oil and gas engineering, chemical engineers, water pollution control engineering and CO₂ geo-sequestration. Research on dispersion has been performed for decades, mainly through measuring and analysing the dispersion coefficient.^1–4 The dispersion is a fundamental issue of multi-flow and mass transfer, especially in the field of CO₂ geo-sequestration. CO₂ geo-sequestration is widely considered as a CO₂ mitigation strategy,⁵ involving the new hybrid disciplines of enhanced oil recovery by CO₂ (CO₂-EOR), enhanced natural gas recovery by CO₂ (CO₂-EGR) and so on. In CO₂-EGR, as the remaining natural gas may be affected and even contaminated by the injected CO₂, gas recovery efficiency and the purity of gas production will be seriously influenced by the dispersion of CO₂. Simulation results^6–9 and the first field scale test (Rotliegend gas field's K12-B platform at the Dutch North Sea)^10–12 showed that the dispersion between CO₂ and natural gas have an important influence on the gas recovery efficiency.

To measure the dispersion coefficient, the dynamic column breakthrough (DCB) method is commonly used, by detecting the displacing fluid in the effluent fluids.¹³ The dispersion coefficients are calculated by fitting the breakthrough profiles of displacing fluid in the effluent fluids, based on advection dispersion equation (ADE).

In recent decades, researchers conducted some investigations of the dispersion of CO₂–natural gas using DCB method. Seo and Mamora^14,15 carried out CO₂–CH₄ dispersion experiments through a dry carbonate core, the apparent dispersion coefficient was obtained in the range of 0.01–0.12 cm² min⁻¹. Nogueira de Mago¹⁶ found that the dispersion coefficient increased when the displacing fluid CO₂ contain impurity N₂. Zhang et al.¹⁷ discovered that CO₂–CH₄ dispersion coefficients in sandpack decreased with pressure during 10–14 MPa at 40 °C. The obtained dispersion coefficients were measured without considering the effect of experimental pipelines, valves, the entry/exit, etc. To obtain the dispersion coefficient in porous media, Hughes et al.¹⁸ proposed a method to correct the effect of tubing and entry/exit, and measured the dispersion coefficients of CO₂–CH₄. Whereafter, Honari et al.,¹⁹ Liu et al.²⁰ and Honari et al.²¹ conducted experiments of CO₂–CH₄ dispersion using the correction method of Hughes et al.¹⁸ Honari et al. focused on the measurement of dispersivity,¹⁹ Liu et al.²⁰ analysed the influence of temperature, pressure, flow rate and particle size on the dispersion, respectively, and Honari et al.²¹ evaluated the effect of medium heterogeneity on the dispersion and found the dispersion in heterogeneous carbonate rocks exhibiting higher than that in homogeneous sandstone cores.

In the field of geoscience and oil & gas engineering, the non-destructive and non-invasive technologies, such as CT and NMR, attracted more and more attentions. Honari et al.¹³ measured the dispersion coefficient of CO₂–CH₄ in sandpack by in situ method using low field NMR, which is a revolutionary method avoiding the system error caused by tubing, valves, entry/exit, and so on. CT has higher spatial resolution to obtain the visual pore structure of porous media and the density of substance X-ray passed through.

In this work, an in situ method using CT was proposed to measure the dispersion coefficient of liquid/supercritical CO₂–CH₄ in porous media. The principle details of the in situ measurement using CT were described in Section 3.

2. Material, apparatus and procedure

The methane, carbon dioxide and helium used in the measurement were provided by Dalian Special Gases Co., LTD, China, with the purities higher than 0.9999 mol mol⁻¹, 0.99999 mol mol⁻¹ and 0.99999 mol mol⁻¹, respectively. And three standard CO₂–CH₄ mixtures (CO₂ mole fraction as 0.2012, 0.5012 and 0.7988) prepared gravimetrically by the same provider, were used to calibrate the gas chromatograph (Techecomp, GC7900), in which the carrier gas was helium. The spherical glass beads of BZ01 type and BZ04 type with the mean diameter of 0.12 mm and 0.40 mm, respectively, were provided by AS ONE Corporation. The sandpack was prepared with the cleaned and dried glass beads packed into the polyetheretherketone (PEEK) coreflooding cell. The PEEK cell is 120 mm long with the inner diameter of 15 mm, which can endure 12 MPa with light weight and has good transparency to X-ray.

A specialized coreflooding experiment system was improved from our previous experimental apparatus²⁰ for measuring the dispersion coefficients of CO₂–CH₄, shown as Fig. 1. Three syringe pumps (Teledyne ISCO, 260D) were used to inject CO₂, CH₄, and water, respectively. The water pump would work when experiment with the sandpack containing the residual water was conducted. In addition, the temperature of fluid in the syringe pumps was controlled by the heating circulator (JULABO, EH39, accuracy ±0.1 °C). The precision of the pressure and the flow rate of the ISCO pumps was ±0.5% at a steady temperature. The sandpack was fixed on the stage of the X-ray micro-CT scanner (Shimadzu, InspeXio SMX-225CT) vertically. And the temperature of the sandpack sample was controlled by a thermoregulator with a thermocouple provided the feedback signal. An automatic back pressure regulator (JASCO, BP-2080-M, accuracy ±3.5%) was connected to the sandpack maintaining the system pressure at a specified level. The effluent gas were injected into gas chromatograph by a six-port valve, separated by repacked column TDX-01 and determined by thermal conductivity detector (sensitivity: ≥10 000 mV ml mg⁻¹ n-hexadecane). And a branch pipe was used for vacuumizing the system. Furthermore, four pressure transducers (Rosemount, 3051TA, ±0.04%) and the thermocouple were connected to a data acquisition system (Advantech, ADAM-4019+) with a computer to collect data.

Fig. 1 (a) Schematic diagram of the coreflooding apparatus for the in situ measurement of CO₂–CH₄ dispersion coefficient using CT; (b) the detail of one-dimensional coordinate system along the sandpack with the scanned area lying in x = 40–78 mm.

Prior to commencing a coreflooding experiment, the sandpack was first scanned by CT to obtain the pore structure. If conducting the experiment containing the residual water, the sandpack was first saturated with deionized water by the water pump, and then the deionized water was displaced by CH₄ with the flow rate of 1 ml min⁻¹ until no water appeared in the vent and continuing injection about 2–3 PV (pore volume) CH₄. In the experiment, the sandpack sample was scanned by CT with the voxel size set as 0.084 mm. The effluent gas was analysed by the gas chromatograph. When the effluent gas was close to pure CO₂, the experiment ended.

The sandpack experiments were carried out with the mean interstitial velocity of CO₂ at 1.429 × 10⁻⁵ to 8.733 × 10⁻⁵ m s⁻¹ at a temperature of 25 °C and 40 °C and the pressure of 10 MPa in BZ01 sandpack and BZ04 sandpack with water-free or containing the residual water. And the detailed experiment condition was listed in Table 1.

Table 1 The experiment condition and the dispersion coefficient calculated from in situ method using CT, D_{L-in situ}, and calculated from DCB method, D_L-DCB; and u is the interstitial velocity of CO₂ in the porous media

Sand type	Residual water	T (°C)	u (×10^−5 m s^−1)	DL-in situ (×10^−7 m^2 s^−1)	DL-DCB (×10^−7 m^2 s^−1)	Deviation (DL-DCB − DL-in situ)/DL-DCB
BZ01	Yes	25	8.733	0.855	1.020	0.162
		25	5.822	0.647	0.828	0.219
		25	2.911	0.550	0.723	0.239
		25	2.911	0.555	0.726	0.236
	Yes	40	8.733	1.728	2.348	0.264
		40	5.822	1.342	1.763	0.239
		40	2.911	1.027	1.436	0.285
	No	25	8.733	2.135	2.549	0.162
		25	5.822	1.664	1.941	0.143
		25	4.366	1.453	1.743	0.166
		25	2.911	1.181	1.447	0.184
	No	40	5.822	2.231	2.810	0.206
		40	4.366	1.923	2.429	0.208
		40	4.366	1.935	2.474	0.218
		40	2.911	1.754	2.266	0.226
BZ04	Yes	40	8.574	2.025	2.303	0.121
		40	5.716	1.243	1.496	0.169
		40	2.858	0.766	1.110	0.310
		40	1.429	0.445	0.651	0.316

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3. Principle of in situ measurement of dispersion coefficient of CO₂–CH₄ using CT

3.1 Dispersion theory

The dispersion in porous media occurs between two fluids, mainly due to different flow velocities and molecular diffusion.²² To quantitatively estimate the macroscopic dispersion effects, the ADE is the fundamental theory to obtain the dispersion coefficient. The dispersion is usually studied from two directions, the longitudinal dispersion coefficient D_L along the flow direction, and the transverse dispersion coefficient D_T perpendicular to the flow direction.²³ The fluid properties on dispersion coefficients in the two different directions were reported by researchers.^{1,3,4,24–28} In this article, attention was given to the longitudinal dispersion, and the transverse dispersion coefficients are normally much lower than the longitudinal dispersion coefficient, therefore the transverse dispersion is ignored during the analysis of longitudinal dispersion coefficient.²² The longitudinal dispersion was described by the one-dimensional differential ADE, with an assumption that the interstitial velocity u and the dispersion coefficients D_L independent to the concentration C.where x is the length along the sandpack in the direction of flow, and t is the time from experiment starting.

In CO₂–CH₄ displacement process, the solution of ADE is analytically shown as eqn (2) with the boundary conditions C(t = 0, x = 0) = 1, C(t = 0, x > 0) = 0 and the initial condition C(t, ∞) = 0.¹⁴

where x_D = x/L and t_D = ut/L are the dimensionless distance and the dimensionless time, respectively, L is the length of sandpack, and Pe_exp = uL/D_L is experiment Peclet number, showing the ratio of convection to dispersion on the experimental length scale, which is analogous to the definition of Peclet number. Peclet number is a key parameter related to the dominant factor to dispersion, diffusive or mechanical (advective) dispersion.where D is the diffusion coefficient and d_p is the characteristic length scale of the medium, which is the mean particle diameter of sand for an unconsolidated sandpack. At low fluid velocities, Pe < 0.01, the diffusive processes dominate and the dispersion coefficient is proportional to the diffusion coefficient divided by the tortuosity τ, which is a crucial parameter for flow channel of the porous media; at higher fluid velocities, Pe > 10, the advective dispersion will dominate; and a transition regime is sandwiched in the middle¹ with the dispersion coefficient as a function of Pe.where b is an adjustable constant. In previous experiment studies,^14,20,27 the apparent longitudinal dispersion coefficients were calculated by fitting the effluent gas concentration curves to the eqn (2) as a function of time.

3.2 Theory of CT

X-ray CT is widely applied to obtain the inner structure information of substance and the flow processes of fluids in porous media. X-rays attenuate when penetrating substance, and the attenuation follows the Lambert–Beer's law.²⁹ In the previous studies, CT number is more commonly used, which is a linear function of the attenuation coefficient. Salama and Kantzas³⁰ and Luo³¹ conducted experiments and confirmed that the CT number was proportional to the density of the substance X-ray passed through. The CT greyvalue obtained from CT scanner in the experiment was analogous to CT number, to reflect the attenuation of X-ray. Shen³² verified that the CT greyvalue was also proportional to the substance density. Simultaneously, in the research of Uemura et al.³³ and Uehara and Takahashi,³⁴ CT greyvalue was used to calculate physical parameters instead of CT number, such as porosity (eqn (5)) and saturation (eqn (6)) in porous media, following the rule of CT number.where φ is the porous media porosity, and CT^sat_water, CT_dry, CT_water and CT_air are the CT greyvalue for the porous media saturated with water, dry porous media, pure water and air, respectively. And according to the CO₂ saturation of CO₂–water displacement process,^33,35 a specific CO₂ saturation in CO₂–CH₄ displacement process was proposed.where S_CO2 is the CO₂ saturation, and , and CT_exp are the CT greyvalue for the porous media saturated with CH₄, saturated with CO₂ and filled with CO₂–CH₄ mixture fluid, respectively.

3.3 Principle of in situ measurement of dispersion coefficient using CT

According to the fact that the CT greyvalue is proportional to the matter density, the density at different positions in substance can be represented by the CT greyvalue. In the coreflooding experiment, the structure of the porous media is definite, so the CT greyvalue of every voxel can represent the fixed porous media structure without considering noise signal. On the calculation of CO₂–CH₄ mixture density, the sandpack skeleton has no influence on the conversion of CT greyvalue to mixture density due to the subtraction in eqn (6) which offset of CT greyvalue representing the sandpack skeleton. Therefore, the density change of fluid in the pores of the porous media can be revealed by the CT greyvalue. In this paper, the density of CO₂–CH₄ mixture can be obtained from the conversion of the CT greyvalue, so the detail of CO₂–CH₄ dispersion process can be described by the CT image, as the mixture density changes along with CO₂–CH₄ displacement.

To reduce the noise signal and the effect of the porous media heterogeneity caused by sand packing, the specific CO₂ saturation (eqn (6)) in CO₂–CH₄ displacement process was applied in this paper. As the sandpack structure and its CT greyvalue were stationary, eqn (6) was also expressed as follows in the longitudinal direction, only related to the density of fluid in the pores of porous media.

where ρ_exp, ρ_CH4 and ρ_CO2 are the density of CO₂–CH₄ mixture, CH₄ and CO₂ in the dispersion process, respectively. Consequently, the density of CO₂–CH₄ mixture in the pores of sandpack during CO₂–CH₄ displacement process can be calculated from the CT greyvalue through eqn (6) and (7).

Considering the case of residual water in the sandpack, CO₂ will dissolve in water when CO₂ flow passes through the sandpack in the displacement. And then a weak carbonic acid is generated, which subsequently dissociates into HCO₃⁻ and CO₃²⁻ due to the chemical reactions. However, according to the equilibrium constants for their chemical reaction, the proportion of HCO₃⁻ and CO₃²⁻ is trivially small in the CO₂ solution, and majorities of the inorganic carbon exists as the CO₂ solute and carbonic acid.³⁶ In addition, the solubility of CO₂ in water is about 5% (ref. 36) on the experimental condition in this paper and the solubility of CO₂ in brine is 1% on the geological formation condition,³⁷ while the solubility of CH₄ is much smaller at all pressures and temperatures related to the reservoir condition. The density of the residual water after dissolving CO₂ increases only slightly (about 1%)^36,38,39 while the density increase of residual water after dissolving CH₄ is much less than that of CO₂.

On the experiment of containing residual water, the saturated sandpack with deionized water was displaced by CH₄ until no water appeared in the vent and continuing injection about 2–3 PV CH₄. Therefore, the amount of residual water in the pores is very little and the residual water was considered as immovably attached to the surface of the sand glass beads. In a sense, the residual water was identified as parts of the ‘core skeleton’ which contains the sandpack and the attached residual water. As a result, the CT greyvalue representing the ‘core skeleton’ is regarded as identical before and after the CO₂–CH₄ displacement. Therefore the density of CO₂–CH₄ mixture on the condition of residual water can also be calculated by eqn (6) and (7) sufficiently.

The density of CO₂–CH₄ mixture is strongly dependent on the component concentration, which can be described by equation of state (EOS). Among many EOS for mixture fluids, the GERG-2008 EOS⁴⁴ was proposed to predict the thermodynamic properties of natural gas. It had a good accuracy for mixtures with more than 21 natural gas components, containing methane and carbon dioxide.

The experimental data of CO₂–CH₄ mixture in literature, whose experimental temperature was close to our experiment, and the absolute average deviation (AAD) of density and CO₂ mole fraction between GERG-2008 calculation result and experiment data, were listed in Table 2. The detailed deviation of density and CO₂ mole fraction between calculated result and experiment data were shown in Fig. 2. As the pressure lied in 9.0–11.0 MPa, close to the pressure of our experiment 10 MPa, the deviations of density were in the range of −0.02 to 0.01 and the deviations of CO₂ mole fraction were in the range of −0.02 to 0.02. As a result, GERG-2008 had a good accuracy for calculating the concentration based on the density of CO₂–CH₄ mixture in this paper. Therefore, the density of CO₂–CH₄ mixture at the different position in the pores of the sandpack during CO₂–CH₄ displacement process, which was translated from the CT greyvalue through eqn (6) and (7), can be converted to CO₂ mole fraction. And the CO₂ mole fraction profiles at different positions in the sandpack can be obtained from the CT image data.

Table 2 The T–P–C_CO2 ranges of chosen experiment data as the experiment temperature close to 25 °C or 40 °C

Data resource	T/°C	Range CCO2	Chosen range P/MPa
Reamer et al.^40	37.78	0.1528–0.7961	1.38–17.24
Brugge et al.^41	26.85	0.09990–0.90112	0.20–9.82
	46.85	0.09990–0.90112	0.19–9.47
Hwang et al.^42	26.85	0.09826–0.90112	2.11–19.37
NIST^43	25	0, 1	1–20
	40	0, 1	1–20

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Fig. 2 (a) The density deviation between GERG-2008 calculation results and experiment data with the temperature near 25 °C and 40 °C under 20 MPa; (b) the mole fraction deviation between GERG-2008 calculation results and experiment data with the temperature near 25 °C and 40 °C under 20 MPa.

For verification of this method, four types of CO₂–CH₄ mixture gas (CO₂ mole fraction of 0.2012, 0.5012, 0.7988, 0.8988) in the sandpack on the condition of 40 °C and 10 MPa were measured, and every type of mixture gas was scanned 4 times by CT. The averaged CO₂ mole fraction of the different positions in one scan was plotted in Fig. 3 to compare with the actual mole fraction. It was shown that the measured CO₂ mole fractions were all around the actual CO₂ mole fraction value. And the maximum deviation of 0.0456 appeared in the measurement of mixture gas with CO₂ mole fraction of 0.5012 while the average deviation of all scans was 0.0035. As a result, this method has a sufficient precision to measure the CO₂ mole fraction.

Fig. 3 The averaged CO₂ mole fraction of different positions along the sandpack for four types of CO₂–CH₄ mixture gas (CO₂ mole fraction of 0.2012, 0.5012, 0.7988, 0.8988) measured by CT methods was compared with the actual CO₂ mole fraction.

Generally, the dispersion coefficient was calculated according to ADE using eqn (2), containing the effect of pipes, entry/exit, etc. In simulation, the ADE can be solved easily using the finite difference methods (FDM), in which Crank–Nicolson (C–N) method is convenient and unconditionally stable. The finite difference equation eqn (8) based on eqn (1) is obtained using the C–N discretization scheme.

where, C_jⁿ is the concentration at time n and space node j with time step Δt and space step Δx; c and s are the dimensionless numbers, termed as Courant number and diffusion number, respectively.

In this paper, the C–N numerical method was applied to calculate the dispersion coefficient. The CO₂ mole fraction profile at the first position was set as the initial boundary condition, and the CO₂ mole fraction profile at another position was fitted to calculate dispersion coefficient. Accordingly, a Matlab program using C–N discretization method was written, and the dispersion coefficients were figured out.

4. Result and discussion

4.1 Pore structure and porosity of sandpacks

As shown in Fig. 1, the experiment was conducted with CO₂ displacing CH₄ vertically from the bottom to the top in the sandpack. In order to avoid the entry/exit effect, the CT scanned area was set in the middle part of the sandpack, x = 40–78 mm, as the bottom of sandpack was set as x = 0 mm.

The dry sandpack was scanned by CT before experiment to obtain the pore structure. Through the reconstruction of the CT data, 2D views of BZ01 sandpack and BZ04 sandpack along the axis of sandpack were obtained, shown in Fig. 4. It was observed that the pore distribution along the axis of BZ01 sandpack and BZ04 sandpack were basically uniform. Simultaneously, the pores of BZ01 sandpack were smaller than that of BZ04 sandpack.

Fig. 4 The 2D views of pore structure along the axis of BZ01 sandpack and BZ04 sandpack from 40 to 78 mm.

According to the CT data, the porosities of BZ01 sandpack and BZ04 sandpack were obtained. The slice-averaged porosity along the axis of sandpack was calculated using eqn (5) considering the sandpack as one-dimensional, and the results were shown in Fig. 5. The slice-averaged porosities of BZ01 sandpack changed little along the axis of sandpack, and the average porosity was 0.324. The slice-averaged porosities of BZ04 sandpack had comparatively bigger fluctuation than that of BZ01 sandpack. And the average porosity of BZ04 sandpack was 0.330. However, on the whole, the porosity fluctuations of the two kinds of sandpack were slight which was consistent with the visual result of the 2D views. As a result, it was illustrated that BZ01 sandpack and BZ04 sandpack could be regarded as homogenous in 1D vertical direction along the axis of the sandpack, and the average porosities were taken as their porosities in this paper.

Fig. 5 The slice-porosity of BZ01 sandpack and BZ04 sandpack along the axis.

4.2 Dispersion coefficient obtained from in situ method using CT

Due to the scanned area lying in x = 40–78 mm and the voxel size of 0.084 mm in CT image, the CT image of the scanned area of sandpack contained 450 slices. And the CT greyvalues of 5 successive slices were averaged as that of the middle slice to reduce the effect of inhomogeneity and noise signal. Moreover, considering the property of CT radiation source, the data near the edges of the scanned area was not used, and only data at the position with x = 45–75 mm can be applied to calculate the dispersion coefficient.

The experiment of 25 °C, 10 MPa and 2.911 × 10⁻⁵ m s⁻¹ with BZ01 sandpack containing the residual water was taken as a calculation example. The CO₂ saturation and mole fraction of 6 different moments along the axis of the sandpack were obtained from the CT images using eqn (6) and (7) and GERG-2008 EOS and illustrated in Fig. 6. The CO₂ saturation and mole fraction were around 0 at t = 0 min, which qualitatively agreed with the actual condition. When the displacement appeared into the view of the scanned area, CO₂ occurred from the bottom to the top as CO₂ saturation and mole fraction decreased progressively with x increased. As time passed, the CO₂ saturation and mole fraction in different position of sandpack went up until to 1. And the CO₂ saturation and mole fraction at the bottom position reached 1 earlier. When the CO₂ saturation and mole fraction at the top position reached 1, it can be considered that the CH₄ in the scanned area were entirely displaced by CO₂.

Fig. 6 Experiment condition: 25 °C, 10 MPa and 2.858 × 10⁻⁵ m s⁻¹ at 6 moments (t = 0, 80, 100, 120, 140 and 180 min) along the axis of BZ04 sandpack (a) CO₂ saturation; (b) CO₂ mole fraction.

According to the CO₂ mole fraction along the axis of the sandpack at different time, the CO₂ mole fraction profiles with time at different positions were obtained. Two repeated experiments were conducted in the BZ04 sandpack with the same condition of 25 °C, 10 MPa and 2.858 × 10⁻⁵ m s⁻¹. Two typical CO₂ mole fraction profiles at positions of x = 52.80 mm and x = 73.18 mm in the repeated experiments were illustrated in Fig. 7(a). The CO₂ mole fraction profiles of the repeated experiments at the two positions agreed with each other. Therefore, it was demonstrated that the experiments had good repeatability.

Fig. 7 (a) The CO₂ mole fraction profiles at 52.80 mm and 73.18 mm of repeated experiments in BZ04 sandpack with the condition of 25 °C, 10 MPa and 2.858 × 10⁻⁵ m s⁻¹; (b) the CO₂ mole fraction profiles of example experiment data of 25 °C, 10 MPa and 2.858 × 10⁻⁵ m s⁻¹ at 61.84 mm, 67.30 mm and 73.18 mm were presented, and the corresponding fit lines were theoretical calculated results based on the in situ method using CT with the initial boundary condition as the CO₂ mole fraction at x₀ = 52.80 mm.

The CO₂ mole fraction profiles of randomly selected 4 positions at 52.80 mm, 61.84 mm, 67.30 mm and 73.18 mm were shown in Fig. 7(b). With the CO₂ mole fraction profile at 52.80 mm set as the initial boundary condition, the dispersion coefficients were calculated by fitting the CO₂ mole fraction profiles at 61.84 mm, 67.30 mm and 73.18 mm, respectively, using the in situ method. The corresponding fit lines were figured in Fig. 7(b) and they had good agreements with the CO₂ mole fraction profiles at the three positions. The dispersion coefficients calculated from the three positions were similar but not identical to each other, as 0.559 × 10⁻⁷ m² s⁻¹, 0.512 × 10⁻⁷ m² s⁻¹ and 0.579 × 10⁻⁷ m² s⁻¹, respectively. The possible reason was the porosities variation at different position, which caused the interstitial velocity of CO₂ changing along the axis of sandpack. Therefore the dispersion coefficients calculated from different position maybe have diminutive fluctuation. However, the porosity and CO₂ interstitial velocity were supposed as fixed values in the in situ method. For this reason, the dispersion coefficient was obtained through fitting the CO₂ mole fraction profiles at the three positions with the CO₂ mole fraction profile at x₀ = 52.8 mm as initial boundary condition and considering the average value as the dispersion coefficient in sandpack. And the subsequent experiments were also processed in this way.

4.3 Effect of the entry/exit on dispersion coefficient

Apart from in situ measurement using CT, the CO₂ breakthrough profiles of the effluent gas were also analysed with the DCB method and the method of Hughes et al.¹⁸ to correct the tubing effect. The dispersion coefficients obtained from the two methods were shown in Table 1. It was obvious that the dispersion coefficients of different experiment conditions obtained from DCB method were bigger than that calculated from in situ method using CT in BZ01 sandpack shown in Fig. 8. However the dispersion coefficient obtained from different methods had the similar rising tendency with the mean interstitial velocity of CO₂, which was caused by the advection. The dispersion coefficient calculated from the in situ method only described the dispersion contribution in the sandpack. While the dispersion coefficient calculated from DCB method also contained the entry/exit effect. Therefore, the deviation of the dispersion coefficient between the two methods reflects the contribution of the entry/exit effect to dispersion. Therefore the entry/exit effect enlarged the dispersion coefficient in the range of 14–23% on the condition of the water-free sandpack, shown in Table 1 and its magnitude was similar to the data of Honari et al.¹³ and the effect of entry/exit in Hughes et al.¹⁸

Fig. 8 The dispersion coefficient of different experiment conditions in BZ01 sandpack obtained from the DCB method and the in situ method using CT.

4.4 Effect of the residual water on dispersion

The dispersion coefficients of the experiments carried out with the sandpack containing the residual water or water-free were shown in Fig. 9 as a function of the mean interstitial velocity of CO₂. All series of the dispersion coefficients increased with mean interstitial velocity, which could attribute to the advection. The high flow velocity accelerated the dispersion of CO₂ into CH₄. And the phenomenon was still valid in the sandpack containing residual water.

Fig. 9 The dispersion coefficient calculated from the in situ method using CT at different experiment condition.

Compared with the experiment conducted in the water-free sandpack, the dispersion coefficient obtained from the experiment in the sandpack containing the residual water was lower. This phenomenon existed in BZ01 sandpack and BZ04 sandpack with CO₂ in liquid state (25 °C) and supercritical state (40 °C). It had been verified that CO₂–CH₄ mixing zone existed in the displacing frontier.²⁰ Therefore, during CO₂–CH₄ displacement, some CO₂ from the gas mixture of CO₂–CH₄ would dissolve into the residual water as the displacing frontier passed through the sandpack, causing a small reduction of CO₂ volume and CO₂ mole fraction in the displacing frontier. With CO₂ progressively dissolved into the residual water with time until saturation, the gradient of CO₂ mole fraction profiles would increase. And the breakthrough of CO₂ also delayed which was consistent with the simulation results of Al-Hasami et al.⁷ and Hussen et al.⁸ In other words, the displacing frontier was stabilized, and the length of CO₂–CH₄ mixing zone was shortened along the axis of the sandpack in the displacement direction compared to the case of the water-free condition. Therefore the dispersion caused by free diffusion and advection was reduced, and the dispersion coefficient was lower than that of the water-free experiment.

Simultaneously, it was also discovered from Table 1 that the residual water enlarged the entry/exit effect on the dispersion, as the deviation of the dispersion coefficient between the in situ method and DCB method was larger than that of the water-free condition. On the experiment of the sandpack containing the residual water, the sandpack was saturated by water first and then the water was displaced entirely with CH₄ until no water appeared in the vent. In this procedure, except for the residual water in the sandpack, some water may be stuck in the corner of the coreflooding cell entry deviate from the mainstream in the axis direction of sandpack. During CO₂–CH₄ displacement, CO₂ would dissolve into the water in the corner of the coreflooding cell which caused the slow growth of CO₂ breakthrough profiles of the effluent gas. For this reason, the dispersion coefficient calculated from DCB method relatively increased. Therefore the deviation of the dispersion coefficient in the experiment containing residual water was larger.

The ratio of dispersion coefficient to diffusion coefficient as a function of Pe was plotted in Fig. 10, containing the experiment data of Liu et al.²⁰ and Honari et al.¹³ as comparison. Pe was determined according to eqn (3) with the diffusion coefficient obtained from the formula of Hughes et al.¹⁸ fitting with the experiment data of Takahashi and Iwasaki.⁴⁵ The experiment condition of Liu et al.²⁰ was BZ04 sandpack, 40 °C, 10 MPa, and the experiment of Honari et al.¹³ was conducted in porous media packed with glass beads with mean diameter of 0.1 mm at 23 °C and 4.5 MPa. In Fig. 10, the dispersion coefficient lied in the transition regime from molecular diffusion domination to advective mixing domination. The ratio of dispersion coefficient to diffusion coefficient showed up the asymptotic growth trend with Pe in the range of 0.01–1, which was consistent with the conclusions of previous studies.^1,46,47 And in the experiment with sandpack containing residual water, the growth trend still existed. But the value of the ratio of the dispersion coefficient to diffusion coefficient was lower, and the effect was equivalent to enlarge the touristy of sandpack considering the relationship between the dispersion coefficient and Pe in eqn (4). As unconsolidated porous media, the sandpack has higher porosity than rock core, and the pores and throats are also bigger. And the vast majority of the residual water should be considered as the bound water and attached to the surface of sandpack particles due to the fact that 2–3 PV CH₄ displaces water before experiment. Consequently, with few obstructed flow paths, a majority of flow paths probably narrow down caused by the residual water. As a result, the tortuosity increases while the dispersion coefficient decreases on the condition of containing residual water due to the influence of CO₂ dissolution.

Fig. 10 The ratio of dispersion coefficient to diffusion coefficient change with Pe number; * is the experiment data of Liu et al.²⁰ and ** is the experiment data of Honari et al.¹⁹

5. Conclusions

An in situ method for dispersion coefficient measurement of liquid/supercritical CO₂–CH₄ in sandpack using CT was proposed. In this method, the effect of entry/exit was eliminated as the dispersion coefficient was calculated directly based on the CO₂ mole fraction profiles along the axis of the sandpack. The finite difference method, Crank–Nicolson method, was applied to solve the advection dispersion equation to obtain the dispersion coefficient. The apparent dispersion coefficient including entry/exit effect was also measured by analysing the breakthrough profiles of the effluent gas using DCB method. The entry/exit effect enlarged the dispersion coefficient in sandpack with the range of 14–23% under the water-free experiment. The effect of residual water on dispersion was also analysed. The dispersion coefficient was smaller under the condition of residual water, which probably due to the dissolution of CO₂ in the displacing frontier into residual water. The dissolution stabilized the mixing in the displacing frontier and shortened the length of mixing zone compared to the water-free condition. As a result, the dispersion coefficient reduced and the effect was similar to enlarging the tortuosity of the sandpack.

Acknowledgements

This paper was supported by National Natural Science Foundation of China (51576031, 51227005, and 51436003), Fundamental Research Funds for the Central Universities (DUT15LAB22).

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Footnote

† These authors contributed equally to this work and should be considered co-first authors.

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