Lattice Boltzmann Simulations of Supercritical CO2âWater Drainage

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Article 2

Lattice Boltzmann Simulations of Supercritical CO-Water Drainage Displacement in Porous Media: CO Saturation and Displacement Mechanism 2

Hirotatsu Yamabe, Takeshi Tsuji, Yunfeng Liang, and Toshifumi Matsuoka Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/es504510y • Publication Date (Web): 26 Nov 2014 Downloaded from http://pubs.acs.org on December 2, 2014

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Environmental Science & Technology

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Lattice Boltzmann Simulations of Supercritical CO2-

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Water Drainage Displacement in Porous Media:

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CO2 Saturation and Displacement Mechanism

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Hirotatsu Yamabe†*, Takeshi Tsuji††, Yunfeng Liang†, and Toshifumi Matsuoka†

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†

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††

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University, Fukuoka 819-0395, Japan

Environment and Resource System Engineering, Kyoto University, Kyoto 615-8540, Japan International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu

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Conflicts of Interest The authors declare no competing financial interest.

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TOC/Abstract Graphic

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ACS Paragon Plus Environment

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ABSTRACT

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CO2 geosequestration in deep aquifers requires the displacement of water (wetting phase) from

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the porous media by supercritical CO2 (non-wetting phase). However, the interfacial instabilities,

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such as viscous and capillary fingerings, develop during the drainage displacement. Moreover,

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the burst-like Haines jump often occurs under conditions of low capillary number. To study these

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interfacial instabilities, we performed lattice Boltzmann simulations of CO2-water drainage

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displacement in a 3D synthetic granular rock model at a fixed viscous ratio and at various

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capillary numbers. The capillary numbers are varied by changing injection pressure, which

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induces changes in flow velocity. It was observed that the viscous fingering was dominant at

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high injection pressures, whereas the crossover of viscous and capillary fingerings was observed,

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accompanied by Haines jumps, at low injection pressures. The Haines jumps flowing forward

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caused significant drop of CO2 saturation, whereas Haines jumps flowing backward caused

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increase of CO2 saturation (per injection depth). We demonstrated that the pore-scale Haines

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jumps remarkably influenced the flow path and therefore equilibrium CO2 saturation in crossover

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domain, which is in turn related to the storage efficiency in the field-scale geosequestration. The

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results can improve our understandings of the storage efficiency by the effects of pore-scale

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displacement phenomena.

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INTRODUCTION

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CO2 geosequestration is one of the promising solutions for reducing carbon emissions

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and global warming.1-3 The estimation of geological CO2 storage capacity, evaluation of leakage

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risk, and enhancement of storage efficiency are current focuses and require understanding of

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microscopic CO2 flow in porous media, such as the fingering phenomenon.

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To examine CO2 flow in porous media, a number of experimental studies have been conducted.

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The preceding experiments can be divided into two approaches: core-flooding experiments using

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magnetic resonance imaging (MRI) or X-ray computed tomography (CT)4-9 and the observation

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of fluid displacement in fabricated micromodels.10-14 With X-ray CT scanners or MRI, we can

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measure the saturation and fluid distribution changes during core flood experiments with CO2.4-8

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However, the microscopic fluid state cannot be detected with medical CT scanners due to not

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high resolution (millimeter scale) compared with pore sizes. The resolution of microfuocus CTs

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are high, but it takes a long time to take one image unless fast synchrotron-based sources are

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used.9 Recent advances in microfabrication have enabled us to create arbitrary micropore

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network models. The experimental studies with two-dimensional microporous media have been

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conducted to reveal the mechanisms of immiscible fluid displacement. Despite the use of simple

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two-dimensional porous media, the contributions of these experimental studies to knowledge of

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fluid dynamics are considerable.10-14

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In general, understanding the flow of multiphase fluids in porous media has been a

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subject of great interest over a wide range of scientific and engineering disciplines.11,

14-20

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Lenormand et al. have discussed the pore-scale displacement mechanism of the drainage process

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from the standpoint of viscous and capillary forces.21 The effects of these forces on drainage

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displacement processes can be characterized by two dimensionless numbers: the capillary


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number (Ca), which is defined as Ca = µinUin/σ, where µin, Uin, and σ are the viscosity of injected

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fluid, velocity of the injected fluid, and interfacial tension, respectively; and the viscosity ratio

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(M), defined as the ratio of viscosities of the non-wetting and wetting fluids. In subsurface rocks,

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CO2 phase usually behaves as non-wetting phase, thus the displacement process in CCS can be

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considered as drainage.. These phenomena have been studied in core-scale. When we consider

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the CO2 injection in subsurface rocks, the flow speed is slow (Ca > Scrossover. However, the displacement process in the crossover region has

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not been revealed in detail.


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In summary, CO2 saturation in subsurface rocks is a significant factor influencing storage

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efficiency. We need to understand CO2 displacement mechanisms and its effect on saturation in

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reservoir conditions. To achieve that, we conducted simulation studies with the lattice Boltzmann

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(LB) method, which enables us to simulate multiphase fluid flow in complex pore structures.

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Drainage displacement of a water phase by CO2 is simulated in a “granular model,” which is a

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3D rock model constructed by the packing of numerous spherical grains. In this paper, we

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discussed the displacement mechanisms in pore-scale and its effect on core-scale CO2 saturation

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which is related to storage efficiency.

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METHODOLOGIES

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Lattice Boltzmann Method

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The lattice Boltzmann (LB) method26 has been explored as a numerical method for simulating

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viscous fluid flow in complicated porous media because the LB method can easily handle

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complex solid boundaries and is suitable for parallel computation.27-28 For a more detailed

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description of the LB method, see reviews of the LB method by Succi (2001), Rothman and

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Zaleski (1997), and Wolf-Gladrow (2000).29-31For multiphase flow simulations, various LB

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models have been developed28, 36: the color-fluid,32 Shan-Chen,33 free-energy34 and mean-field35

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models. In this work we used a color-fluid model because the model requires fewer grids for

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representing interfaces and results in a reduction of computational cost. The color-fluid model

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has a potentially serious drawback; perturbations can cause spurious fluid velocities at an

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interface. Such a spurious velocity disturbs the fluid flow when the flow is very slow. In this

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work, we applied a modification suggested by Latva-Kokko and Rothmann (2005) to reduce the


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magnitude of spurious velocities.37 See the Supporting Information for a detailed LB method use

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in this paper.

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Rock Model and Simulation Conditions

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We used a synthetic granular rock model to provide the LB simulation with a pore-

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geometry. This model is constructed by a random packing process38 with a number of spherical

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grains. We assumed the grain-size distribution to be “well-sorted”.39 The most important

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parameter in the well-sorted distribution is the “sorting” parameter.40-41 Here, simulation studies

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were conducted for a granular rock model with a sorting parameter of 0.50 and dimensions of

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100 × 100 × 100 lattice units. Because the realistic scale of one lattice unit is configured as 10

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µm in the present study, the size of a rock model is 1 mm × 1 mm × 1 mm. See Supporting

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Information for the detailed description about the rock model.

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The fluid properties used in the study are shown in Table 1. The values in Table 1

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simulate the properties of supercritical CO2 and water at 13.8 MPa (2000 psi) and 50 °C,42-43

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which is close to the conditions at the pilot carbon capture and storage (CCS) site in Nagaoka,

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Japan.44 The surface of solid grains in the media was configured to be completely hydrophilic. It

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should be noted that the dissolution and mineralization is not considered in this study, thus the

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flow is under the condition of immiscible flow. We set identical densities for both fluids (994

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kg/m3), and the viscosities of both fluids were configured as 10 times the realistic viscosities

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because the multiphase LB model used in this study cannot deal with density contrasts and small

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viscosities. The effects of these assumptions are discussed in the last section in this paper.


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We conducted 6 case studies, designated “Case 1” to “Case 6,” by changing the injection

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pressure. Differences in injection pressure alter the fluid velocity and capillary number. In the

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present study, the capillary number was calculated by using the average fluid velocity until the

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CO2 front reached the outlet (breakthrough). The fluid velocity (UCO2) was calculated from the

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CO2 flux (QCO2), which can be defined as the rate of CO2 volume change; QCO2 = dVCO2/dt,

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where VCO2 is the volume of CO2 at each time step. The fluid velocity is obtained by dividing the

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CO2 flux by the cross-sectional area of the rock model (A), i.e. UCO2 = QCO2/A. The calculated

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capillary numbers for the case studies were 1.21×10−3, 6.35×10−4, 3.42×10−4, 1.07×10−4,

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5.13×10−4, and 2.90×10−5 for Cases 1–6. The conditions simulated in the present study are

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plotted on the phase diagram in Figure 1. The capillary numbers in this study varied by two

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orders of magnitude, this is the limitation of current lattice Boltzmann model. When the capillary

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number is lower than the limit, i.e. the flow speed is slow, fluids cannot flow due to the

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prevention by spurious velocity.

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NUMERICAL RESULTS AND DISCUSSION

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In the simulation studies, CO2 was injected into a synthetic granular medium saturated

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with water. The injection direction was defined as the x-direction. The simulation continued until

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the distributions and saturation of fluids reached an equilibrium state. Simulated CO2

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distributions in six case studies are shown in Figure 2.

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Displacement Mechanism


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For the purpose of discussing the displacement mechanism, we focused on Cases 2 and 5.

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Figure 3 shows the time series of CO2 flow in Cases 2 and 5. Clear differences in migration

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patterns between the two cases are associated with altered displacement mechanisms imposed by

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different capillary numbers (6.35×10−4 in Case 2, 5.13×10−5 in Case 5). The simulation results

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shown in Figure 3 indicate characteristics of viscous and capillary fingering. In Case 2, the

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invading CO2 flows via many fingers toward the outlet (clearly observed in Case 2-b, c and d),

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passing through many pore spaces despite the narrow size of some pores (i.e., high capillarity).

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This is the characteristic of viscous fingering and we can conclude that viscous fingering is the

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dominant displacement mechanism in Case 2. In contrast, the invading CO2 in Case 5 had only a

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few fingers passing through large pores selectively. The front of the fingers in Case 5 moved not

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only in the direction of injection (along the x-axis) but also in the vertical direction and even

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toward the inlet (backwards), as shown in Case 5-d in Figure 3. The movement of fingers toward

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the inlet is characteristic of capillary fingering. Nevertheless, the front moved toward the outlet

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(shown in the change of CO2 distribution from Case 5-d to Case 5-e) in the same time, despite

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narrower pores. We can conclude that both viscous and capillary fingering occurred in Case 5

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and that the crossover of viscous and capillary fingering is the dominant displacement

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mechanism in this case.

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Pore-Scale Displacement Event: Observation of Haines Jump

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Figure 4 gives the plots of CO2 saturation versus the normalized position of CO2 front

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from the inlet at each time step. Note that the plotted points show the data from injection through

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to breakthrough. In Cases 1–3, CO2 saturation increased smoothly with increasing front position,

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whereas the curves for Cases 4–6 are not smooth. To study the different flow characteristics and

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the pore-scale displacement event in detail, we have investigated three particular stages of the


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flow, as shown in Figure 4 by the vertical dashed lines. The first typical behavior is observed

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when the normalized distance from the inlet is between 0.3 and 0.55 (labeled (i) in Figure 4). In

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this range, the gradients of plots for low capillary numbers (Cases 4–6) are smaller than those

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with high capillary numbers (Cases 1–3). A small gradient indicates that CO2 saturation does not

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increase rapidly during CO2 migration toward the outlet. The small incremental increase in CO2

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saturation is due to the presence of only a few fingers. Because the fingers of CO2 selectively

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pass through large pores when the capillary number is low, only a few fingers grow, which cause

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a small increment in CO2 saturation in Cases 4–6.

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The second behavior is the most typical one; the plots for lower capillary numbers have a

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CO2 increment without front migration in the x-direction at a normalized distance of 0.57 (range

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(ii)). The cause of this phenomenon may be explained by a Haines jump backward. In Figure 5,

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we focused on two sub-units (a) and (b) for Case 5 at different times. From (1) and (2), we can

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confirm the occurrence of CO2 flow back to the inlet, which is one of the characteristics of

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capillary fingering. By this phenomenon, two CO2 fingers coalesce. Figure 6 represents the CO2

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pressure inside each sub-unit in Case 5. It can be observed from the plot of sub-unit (a) that the

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back flow and coalescence of the two fingers occurred in a short time. After CO2 reached sub-

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unit (a) at a normalized time of around 0.11, the CO2 pressure increased because of the local

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pressure concentration. The pressure concentration caused a “burst-like flow” backward and the

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coalescence of two CO2 fingers. This flow caused a sudden pressure drop. The phenomena

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observed here for the flow and corresponding local pressure are characteristic of a Haines jump.9,

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in the sub-unit had locally reached equilibrium. The CO2 cluster in the sub-unit could not migrate

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further, and the CO2 front stopped momentarily, as shown in range (ii) in Figure 4. Note that

After the pressure drop, there was no further increase in the local pressure, indicating that CO2


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fingers elsewhere from the sub-unit grew and migrated toward the outlet while the CO2 front

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locally stopped in the sub-unit; these fingers are the reason for an incremental change in CO2

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saturation without advancement of the CO2 front.

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The third behavior is observed in range (iii) in Figure 4. As was observed in range (i), the

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slopes for Cases 4–6 are much smaller than those for Cases 1–3 in this range. The lower slopes

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are attributed to the fewer numbers of fingers, which is evident from the CO2 distribution at

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breakthrough in Figure 2. Fewer fingers in this range were a result of Haines jumps. These

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Haines jumps in the range (iii) can be observed in sub-unit (b). The flow from (3) to (4) in Figure

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5 occurred in a short time, which can be confirmed in the graph in Figure 6, and the jump

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decreased the CO2 pressure in sub-unit (b). In contrast to the backward Haines jump in sub-unit

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(a), this jump occurred in the forward direction. Because of this Haines jump, breakthrough of

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the CO2 phase occurred with only one finger around the outlet of our model.

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Equilibrium CO2 Saturation

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The saturation of CO2 at an equilibrium state is of great concern in the CCS project

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because equilibrium saturation is one of the most important factors affecting storage efficiency.

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Figure 7 shows plots of CO2 saturation versus capillary number for each case study. The highest

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CO2 saturation was observed in Case 1 (95.87%) and the lowest in Case 6 (40.91%). Since the

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rock model is homogeneous and isotropic, the CO2 phase can enter almost all the pore spaces

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when the capillary number is high, thus the saturation in Case 1 is extremely high. Figure 7

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shows that the saturation of CO2 drastically drops with decreasing capillary number. This is

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because of crossover of capillary and viscous fingering. The decreasing trend in CO2 saturation

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is considered reasonable when compared with preceding research.21


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To further characterize the different behaviors, we divided the rock model into two half

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sub-units: the inlet side of the sub-unit was denoted sub-unit 1 (x = 1–50 lattice units), and the

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outlet side was denoted sub-unit 2 (x = 51–100 lattice units). The saturations in sub-unit 1 were

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99.98% and 71.62% in Case 1 and Case 6, respectively; the corresponding saturations in sub-unit

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2 were 91.72% and 9.964%, respectively. The difference in saturation between the two cases is

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especially significant in sub-unit 2 (Figure 7). In sub-unit 1, CO2 was supplied to all pore spaces

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because the sub-unit 1 is located on the border with the inlet boundary. However, in sub-unit 2,

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CO2 was supplied only by the pore spaces where CO2 existed in sub-unit 1. Therefore, lower

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capillary numbers caused larger differences in CO2 saturation between sub-units 1 and 2 because

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CO2 selectively passed only through large pore spaces by capillary fingering.

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The Haines jump can have a strong impact on CO2 saturation, and the impact can be

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positive or negative. As discussed above, we observed two different types of Haines jumps,

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going backward and forward, when the crossover of viscous and capillary fingering was

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dominant. The “backward jump” caused the coalescence of two CO2 clusters and a decrease in

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pressure toward equilibrium. As a result of this phenomenon, fingers in other pore spaces can

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grow and effectively increase CO2 saturation (evident from range (ii) in Figure 4). Therefore, we

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can conclude that the “backward jump” can have a positive impact on CO2 saturation.

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Conversely, “forward jumps” posed fewer fingers than in cases with high capillary numbers.

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Once the CO2 phase had reached the outlet (breakthrough), CO2 saturation could not increase

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effectively when the capillary number was low because CO2 injected at low rates can only flow

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along paths that had already been formed prior to breakthrough. By comparing CO2 distributions

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at breakthrough and equilibrated states (Figure 2), we can confirm that there was little growth in

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the CO2 fingers. This is also evident from the differences in CO2 saturation at equilibrium and


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breakthrough. In Cases 1 and 2 (high capillary numbers), 131% and 108% increment of CO2

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saturation after breakthrough was observed; whereas only 27.6% and 7.38% increment were

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observed in Cases 5 and 6 (low capillary numbers), respectively. These data show that CO2

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saturation does not significantly increase when “forward” Haines jumps occur at low capillary

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numbers. For these reasons, the CO2 saturation in cases with lower capillary numbers was much

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lower than in those with higher capillary numbers.

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Implications

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Our simulation studies demonstrated that pore-scale displacement phenomena have

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strong effects on CO2 saturation. It was found that the existence of forward-going Haines jump

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decreases equilibrium CO2 saturation. The typical capillary number at reservoir conditions is

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generally believed to be at the order of 10-5.45 Our studies have shown that the CO2 saturation

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becomes low when the capillary number is on the order of 10-5. It should be noted that the

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relationship between the capillary number and displacement mechanisms is dependent on the

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geometry of porous media. The granular rock model that we employed, according to us, is

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relatively homogeneous. In addition, we have ignored the gravity effect intentionally. Even in

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such simple case, we demonstrated that a non-uniform CO2 distribution develops as a

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consequence of unstable displacement at pore-scale.

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The studies were conducted by assuming different absolute viscosities of water and CO2.

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However, the studies were conducted with fixed realistic viscosity ratio (M=1). Even though the

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absolute values of viscosities would have influences on displacement patterns, the key of the

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conclusion, the pore-scale phenomena (Haines jumps) have strong impact on core-scale CO2

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saturation in porous media, would be guaranteed.


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The results provided a fundamental understanding of CO2 behavior and its effect on

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storage efficiency in homogeneous region in CCS site. The effect of strong heterogeneity of CCS

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site (e.g. fractures) has not been explored in this study, however, it is important to consider the

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possibility that typical capillary number (the order of 10-5) under reservoir conditions can make

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the storage efficiency poor. Although the numerical studies were conducted with mm-scale, the

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results can be upscaled by following consideration: CO2 displacement process with viscous

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fingering will occur near an injection well where the capillary number is expected high.

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Whenever the crossover of capillary and viscous fingering occurs, CO2 storage efficiency

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becomes low. The latter will be observed far from an injection well where capillary number is

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expected low.

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Figure 1. Phase diagram showing each displacement pattern plotted in the log Ca–log M plane.

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There exists a crossover zone among the displacement pattern zones. The boundaries described

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with blue lines are noted by Lenormand et al. (1988)21 whereas those described with red lines are

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results from Zhang et al. (2011).12 These boundaries are dependent on the system, especially on

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the geometry of the porous media, therefore should be understood only as a qualitative

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representation. The six plotted points indicated by circles represent the conditions simulated in

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Cases 1–6, from top down.


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Figure 2. CO2 distributions from the simulation results. Panels from top to bottom show

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results of Cases 1–6. Note that the water phase is not visualized in these figures and that CO2 is

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injected along the x-direction (from left to right). The six panels on the left show the CO2

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distribution (red-colored parts) in the granular rock model (gray-colored) when the CO2 front

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reached the outlet (breakthrough point); the right panels show the CO2 distribution in the

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granular rock model when fluid flow had reached the equilibrium state.

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Figure 3. Time series change of CO2 distribution in Case 2 (top) and Case 5 (bottom). From

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left to right, the five snapshots are ordered chronologically. The last snapshots (Cases 2-e and 5-

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e) show the equilibrated state (same as in the right panels in Figure 2).


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Figure 4. Plots of evolutional CO2 saturation in a porous rock model versus distance of CO2

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front from the inlet until breakthrough. Note that the distance is normalized by the size of the

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porous medium.

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Figure 5. CO2 distribution in Case 5 before (1) and immediately after (2) a burst-like flow

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backward occurred, and before (3) and immediately after (4) a burst-like flow forward occurred.

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(1-a) and (2-a) are focused CO2 distributions in sub-unit (a) shown in the white boxes identified

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in (1) and (2); (3-b) and (4-b) show CO2 distributions in sub-unit (b) which is shown in (3) and

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(4) as the white boxes.

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Figure 6. Time series plots of CO2 pressure in the focused sub-units (a) and (b). (1), (2), (3),

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and (4) correspond to the numbered images in Figure 5. Note that the simulation time is

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normalized by the time at which the CO2 phase reached the outlet (breakthrough).


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Figure 7. Plots of CO2 saturation versus capillary number on a logarithmic scale. The solid

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line shows the saturation in the whole rock model and two types of dashed lines show the

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saturation in sub-units 1 and 2. The exact value of CO2 saturation can be seen Table S2, in the

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Supporting Information.

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Table 1. Densities, viscosities, and interfacial tension used in simulations42-43. The values in

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parentheses are the values configured in the LB simulations; the densities of both fluids are set identical,

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and viscosities are 10 times the realistic ones.

Water (wetting)

CO2 (non-wetting)

Density (kg/m3)

994

994 (668)

Viscosity (cP)

5.51 (0.551)

0.521 (0.0521)

Interfacial Tension (mN/m)

35.0

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Supporting Information.

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The numerical method and synthetic rock model are detailed in the Supporting Information. The

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CO2 saturation and its profiles in each case study are also described there. This material is

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available free of charge via the Internet at http://pubs.acs.org.

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Corresponding Author

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*Hirotatsu Yamabe, E-mail: [email protected], TEL: +81-75-383-3206,

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FAX: +81-75-383-3203

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ACKNOWLEDGEMENTS

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The authors acknowledge the financial support of the Japan Society for the Promotion of Science

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(JSPS) through a Grant-in-Aid for Scientific Research A (no. 24246148) and JST/JICA-

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SATREPS. H. Yamabe is grateful for the support provided by a Grant-in-Aid for JSPS 246048

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Fellows. This research is partially supported by the Sumitomo Foundation. T. Tsuji gratefully

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acknowledges the support of the I2CNER, sponsored by the World Premier International

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Research Center Initiative (WPI), MEXT, Japan.

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