NUMERICAL STUDY OF NON-AQUEOUS PHASE LIQUID TRANSPORT IN A SINGLE FILLED FRACTURE BY LATTICE BOLTZMANN METHOD*

DOU Zhi, ZHOU Zhi-fang, HUANG Yong

School of Earth Science and Engineering, Hohai University, Nanjing 210098, China,

E-mail: dz.uriah@gmail.com

WU Wei

School of Materials Science and Engineering, Southeast University, Nanjing 211189, China

NUMERICAL STUDY OF NON-AQUEOUS PHASE LIQUID TRANSPORT IN A SINGLE FILLED FRACTURE BY LATTICE BOLTZMANN METHOD*

DOU Zhi, ZHOU Zhi-fang, HUANG Yong

School of Earth Science and Engineering, Hohai University, Nanjing 210098, China,

E-mail: dz.uriah@gmail.com

WU Wei

School of Materials Science and Engineering, Southeast University, Nanjing 211189, China

(Received July 15, 2011, Revised October 18, 2011)

In this article, the Non-Aqueous Phase Liquid (NAPL) transport in the single filled fracture was studied with the Shan-Chen multi-component multiphase Lattice Boltzmann Method (LBM) with special consideration of wettability effects. With the help of the model, the contact angle of the non-wetting phase and wetting phase interface at a solid wall could be adjusted. By considering a set of appropriate boundary conditions, the fractured conductivity was investigated in condition that the NAPL blocks the channels in the single filled fracture. In order to study the wettability effects on the NAPL transport, a constant driving force was introduced in the Shan-Chen multi-component multiphase LBM. Flow regimes with different wettabilities were discussed. Simulated results show that the LBM is a very instrumental method for simulating and studying the immiscible multiphase flow problems in single filled fracture.

single filled fracture, Lattice Boltzmann Method (LBM), Non-Aqueous Phase Liquid (NAPL) transport, wettability, contact angle, immiscible two-phase flow

Introduction

Water is the basic source of life and an important indicator of the quality of our environment. However, the quality and availability of water are increasingly endangered. Non-Aqueous Phase Liquid (NAPL) contamination continues to persist as a problem in the industrialized nations, but this type of contamination has received little attention in the developing world[1]. Especially, in single filled fractures, compared with solute transport[2,3], the NAPL transport is very complex. The NAPL has very low solubility in water, which means that it remains as a separate phase, and hence, the interfacial area is of key importance for NAPL transport. The properties of the interfacial area are controlled by the interfacial tension resulting from the different wettabilities. Therefore, it is necessary to study the NAPL transport concerning with the wettability effects and the behavior of the multiphase flow system in single filled fractures. In single filled fractures, the effective fractured aperture is a very important parameter in the cubic law. The experiments have confirmed that the effective fractured aperture is the most significant influence upon the fractured conductivity[4]. To the best of our knowledge, the channels in the single filled fracture are easily blocked by the NAPL, which has effects on flow regimes and the fractured conductivity. Therefore, in the single filled fracture, it is necessary to study the effect of NAPL cloggings on the effective fracture aperture and the fractured conductivity.

Numerical methods can be very instrumental in enhancing understanding of fluid behavior in complex systems. The solute transportation in fractured rocks was simulated by Huang et al.[5]with the coupling model based on the Finite Volume Method (FVM), but the simulation was not applied in studying the immiscible two-phase flow. Simulation of the evolu-tion of fluid interface is a challenging work, which results from the fact that any phase interface boundary is mesoscopic by nature[6], however, in recent years, the Lattice Boltzmann Method (LBM) has won broad recognitions. Compared with conventional Computational Fluid Dynamics (CFD) methods, the LBM is a powerful technique for the numerical modeling of a wide variety of complex fluid flow problems included in single and multiphase flows in complex geometries[7]. It is a discrete computational method based upon the Boltzmann equation. It considers a typical volume element of fluid to be composed of a collection of particles that are represented by a particle velocity distribution function for each fluid component at each grid point. The time is counted with discrete time steps and the fluid particles can collide with each other as they move, possibly under applied forces. The rules governing the collisions are designed such that the time-average motion of the particles is consistent with the Navier-Stokes equations. This method naturally accomodates a variety of boundary conditions such as the pressure drop across the interface between two fluids and wetting effects at a fluid-solid interface. It is an approach that bridges microscopic phenomena with the continuum macroscopic equations. Du and Shi[8]reported that the multi-relaxationtime LBM is of better numerical stability and has attracted more and more research interests. Tan and Zhou[9]presented an LBM for studying on solute transportation in parallel single fracture. However, their method was unfavorable for studying the immiscible two-phase flow (NAPL transportation). There have been several popular lattice Boltzmann techniques for the analysis of multiphase flows. The Shan-Chen LBM (SC LBM) led to an LBM for multiphase flow by introducing a non-local interaction force between particles at neighboring lattice sites. The SC LBM is a more widely used multiphase model for simulating the immiscible two-phase flow due to its simplicity and remarkable versatility[10].

However, there seems no report on applying the SC LBM to study the NAPL transportation concerning with the wettability effects in single fillled fractures. In this article, the Shan-Chen multi-component multiphase LBM will be reviewed briefly, and then applied to examine the contact angles under the different solid properties. Moreover, as the NAPL blocks the channels, the wettability effects on the NAPL transport in the single filled fracture are simulated and analyzed.

1. Method (Shan-Chen multi-component multiphase LBM)

Here we apply the SC LBM to a 2-D multi-component multiphase flow system. Two distribution functions are employed to represent wetting phase and non-wetting phase, respectively. The evolution equation of each distribution function satisfies the following lattice Boltzmann equation

where fi(x,t ) is the fluid particle distribution function, σ denotes either the wetting phase or the nonwetting phase, τσis the relaxation time which is related to the kinematic viscosity asis the local equilibrium distribution and defined as

where csis the lattice sound speed, and cs=c/3, where c=Δx/Δt is the ratio of lattice spacing Δx and time step Δt. In our simulations, one lattice unit (Δx) is defined as 1 lattice unit (l.u.). In this article, all the variables are in lattice units. In Eq.(2), the weights ωi, for the D2Q9 model, are given by

In Eqs.(1) and (2), theie’s are the discrete velocities. For the D2Q9 model, they are given by

The local mass densitiesσρ for each phase are obtained from

The macroscopic velocity uσ(eq)in Eq.(2) is given by

In Eqs.(6) and (7), u is the common averaged velocity of all the fluid components in the absence of any additional forces. Δu is a change in velocity. In the SC LBM model, the effect of body force is incorporated through adding an acceleration into the velocity field. In our simulations, the body force includes the fluid-fluid cohesionand the fluid-solid adhe-

In the SC LBM model, the fluid-fluid interaction is defined in Eq.(8) by only considering the coupling interaction between the nearest neighbours and the next-nearest neighbours[11]where σ and σ denote the wetting phase and the non-wetting phase, respectively, and Gcis a parameter that controls the strength of fluid-fluid interaction. The surface force between a fluid component and a solid phase,is defined as[12]

The pressure ()P x of the whole fluid is given as[14]

In our simulations, any lattice node in the computational domain represents either a solid node or a fluid node. For the solid node, before the streaming, the bounce-back algorithm instead of the collision step is implemented to non-slip wall boundary condition.

2. Results and discussion

2.1 contact angle simulation

Young’s equation for computing the contact angle contains interfacial tensions between the two fluids (12σ) and between each fluid and the surface (1sσ and2sσ). The contact angle θ is measured in Fluid 1.

Fig.1 Contact angle

From Fig.1, Young’s equation is simply

Huang et al.[12]proposed a straightforward application of Young’s equation with substitution of the LBM cohesion parameter and a density factorfor the fluid-fluid interfacial tension, and the adhesion parametersandfrom Eq.(11) for the corresponding fluid-solid interfacial tension

It is simple to determine the contact angle with Eq.(12). According to Huang et al.[12],Each node in the computational domain is occupied by every σ component, though one is dominant under most conditions. The minor dominant components can be thought of as dissolved within the dominant component. For instance, Fluid 1 with the densityρσis a droplet in Fluid 2 with the density ρσ. In thedroplet, Fluid 1 with the density ρσis a dominant component, which means that fluid 2 with the density ρσis a minor component. In Eq.(12),andare the dominant value and the minor value, respectively. In our simulation, the dominant valuerepresents 95% of the component σ of the density ρσ, and the minor valuerepresents 5% of the component σ of the densityρσ.

Fig.2 Simulations of different contact angles (light gray represents Fluid 1, black represents Fluid 2 and dark gray represents solid wall)

In the SC model, the parameter Gccontrols the fluid-fluid interfacial tension, which means that Gcdoes not change, if the ρσandρσare identified.andcontrol the fluid-solid interfacial ten

sion. For the identified Fluid 1 and Fluid 2, the different values ofandcan be used to modify the contact angle of the interface at the solid surface.

In our simulation of contact angles, ρσ=1, ρσ=0.82, and any lattice node in the computational domain represents either a solid node or a fluid node. For the solid node, before the streaming, the bounceback algorithm instead of the collision step is implemented to non-slip wall boundary condition. We place a pure droplet of Fluid 2 (ρσ) inside a 210×110 domain of Fluid 1(ρσ) with periodic boundaries. After 60 000ts, Figure 2 demonstrates that different contact angles can be obtained by adjustingand. 2.2 Validation of model: Immiscible two-phase flows

in a single fracture

For immiscible two-phase flow in single fracture, the wetting phase typical covers and moves along the solid surface, while the non-wetting phase is not in direct contact with the solid surface and it flows in the central part of fracture (see Fig.3). The flow velocity of the non-wetting phase at the center of fracture is affected by the viscosity ratio of the non-wetting phase and wetting phase, M=μnw/μw(where μnwand μware the dynamic viscosities in non-wetting phase and wetting phase, respectively). In the SC LBM, the kinematic viscosity is introduced by v= cs

2(τ?0.5). If both fluids have the same kinematic viscosity, the viscosity ratio is only dependent of the ratio of fluids densities. In this simulation, M=0.1. Both fluids have the same kinematic viscosity.

Fig.3 Schematic of immiscible two-phase flow in single fracture (The wetting phase moves along the solid surface while the non-wetting phase flows in the center of fracture)

For a given value of the wetting phase saturation Sw, we take the wetting phase flowing along the fracture in the region a＜x＜b, and where Sw=(b?a)/b and Snw=a/ b. Assuming a poiseuille-type flow in the fracture, then the analytical solutions for two-phase flow can be derived by solving the appropriate Navier-Stokes equations[15],

wherewν,nwν,wρ,nwρ are the kinematic viscosities and densities of the wetting and non-wetting phases, respectively. The length of fracture in the direction of the flow is L. In Eqs.(14) and (15), the pressure gradient in the direction of the flow is taken equal to F,established. Figure 4 shows that the results of LBM simulation (with the SC LBM) and analytical solution with =0.1M. The LBM simulation result is consistent with the analytical solution.

所謂的定向鉆探，就是在同一個位置多個方位都存在有斜孔，與此同時在斜孔內(nèi)可以進(jìn)行定向鉆探技術(shù)的使用。這一項技術(shù)主要應(yīng)用于深度地質(zhì)的相關(guān)勘查工作，并且主要是在這三個生態(tài)環(huán)境中：首先是地勢較高并且比較陡峭，需要先進(jìn)行大規(guī)模修路工程的地域；其次是在地質(zhì)的勘查過程中，鉆探的深度大于5000米；最后是鉆孔直徑為65mm，且地質(zhì)中的巖石中心直徑為43mm，可以通過一個鉆機場地完成多個方向的鉆探工作。所以說，此項技術(shù)的應(yīng)用不僅大幅減少了勘探工程在地表自然環(huán)境方面的占地面積，還具有十分顯著的綠色地質(zhì)勘查效果。

Fig.4 Velocity profile u( x) in the middle of the 2-D channel. M=0.1

2.3 Immiscible two-phase flow through the single filled fracture

In this section, the clogging resulting from the NAPL transport concerning with the wettability and flow regimes with different wettabilities were simulated in single filled fracture with the SC LBM. Here we briefly review the basic law about the single filled fracture. Based on the cubic law, we can obtain the flow rate q in the single filled fracture,

where J is the hydraulic gradient, ν the viscosity, g the gravitational acceleration, and b the effective fractured aperture. Comparing with the Darcy law, we can rewrite the cubic law as[16]

wherefV is average velocity in the whole fracture and the fractured conductivityfK is given by

Fig.5 Model for single filled fracture

Fig.6 Simulation NAPL transport (light gray represents wetting phase, black represents non-wetting phase (NAPL) and dark gray represents fillers)

In our simulation, the filled structure is not a topic we are concerned with here. For simplicity, the isotropic filler is represented by the square of 10×10 in single fracture (see Fig.5). The left side and theright side are inlet and outlet, respectively. The whole computational domain is 410×100. For the fracture and the filler nodes, the non-slip bounce-back algorithm is employed for the fluid-solid interfaces. In the inlet and outlet directions, constant pressure inlet boundary condition is implemented for the periodic boundary conditions. The densities of the NAPL ρσand the water ρσare 0.82 and 1.00, respectively.

Fig.7 Effective fracture apertures

In order to simulate the wettability effects, we adjust the contact angels of the filler and the fracture, which is easily achieved by adjusting the parameter. In our simulation, all the variables are in lattice

units, which can be related to physical units by dimensionless conversion. In Eq.(16), for the gravitational acceleration, by setting the characteristic length Lras 10–3m, the characteristic time tras 0.63×10–3s, the gravitational acceleration in the lattice unit, g, is obtained asg=9.8×tr×tr×Lr=0.00024.

2.3.1 NAPL blocks the channels in single filled fracture

In this section, the fractured conductivity concerning with the NAPL clogging in the single filled fracture was obtained by the cubic law. The NAPL at the inlet was injected instantaneously. By using the inlet boundary conditions under different pressures, different average velocity fields of the fluid were thus given. In the present simulations, under the appropriate inlet pressure, no NAPL can be driven out the single filled fracture, which keeps the NAPL content constant. According to Eqs.(16) and (17), the fractured conductivities and the effective fractured apertures are calculated. In order to display the NAPL transport better, the filler in fracture is introduced in Fig.6.

In Fig.6, the injected NAPL occupies 60×100 in the single filled fracture, and the body force parameters are Gc=1.5,=?0.2, which means that the contact angle is 116.6o. In this simulation, it is found that the smaller droplets/bubbles may disappear, however, the mass of non-wetting phase or wetting phase in the whole computational domain remains constant because the bigger droplet/bubbles grow at the same time. If the difference of average wetting phase velocities between every 3 000ts is smaller than 0.5%, it is assumed that the final steady state has arrived. At the final steady state, after 40 000ts, the NAPL does not move any more, and the average wetting phase velocity is obtained by Eq.(10). In Fig.6, it can be seen that the non-wetting phase is discontinuous and covers the filler surface resulting from the wettability. After 15 000ts, some droplets grow at filler surfaces and some vertical channels between the fillers are blocked by the discontinuous non-wetting phase. The flow regime of the non-wetting phase seems to be symmetrical, however, some droplets are drained into channels, and then the symmetrical flow regime is changed. In the horizontal direction, the non-wetting phases are distributed in the channels between the fracture and fillers, when the non-wetting phase flows in the horizontal direction, which can integrate some droplets in channels, so that the bigger droplets can not flow at the final steady state. In Fig.6, at the final steady state (t=40 000ts), the discontinuous non-wetting phases are distributed in the channels in the vertical direction, while the wetting phase flows in the fracture at horizontal direction. The wetting phase tends to keep its continuous distribution in the horizontal direction, which results from the higher pressure gradient in the horizontal direction. However, the non-wetting phase flows in the vertical direction so that it can not be driven out in the direction with lower pressure gradient. Under the circumstance of constant wettability, with the different injected NAPL contents and inlet pressure conditions, there are twenty-five cases of effective fractured apertures in Fig.7.

Figure 7 shows the effect of different injected NAPL contents on the effective fracture aperture, based on the Eq.(16), in which the horizontal and vertical axes represent g· J and 12νVf, respectively, and the slope of the fitting curve represents the square of effective fracture aperture, b2. The cases of N1, N2, N3, N4and N5are obtained under the NAPL contentsof 1×100, 20×100, 30×100, 40×100 and 60×100, respectively, and their correlation coefficients between the data and their linear fitting are 0.9986, 0.9994, 0.9989, 0.9991 and 0.9987, respectively. It is found that the effective fracture aperture is subject to the injected NAPL content. With the constant NAPL, although, the different inlet pressure conditions are introduced, smaller injected NAPL results in the bigger slope of the fitting curve and the bigger effective fracture aperture. The linear fitting of N1is obtained by the least injected NAPL content, which is considered to be very close to the effective fractured aperture obtained without injected NAPL. When the NAPL is injected, i.e., N2, N3, N4, N5in Fig.7, the slope of the fitting line is reduced obviously, namely the effective fractured aperture is smaller than the case without the injected NAPL. In our simulation, for the same injected NAPL content, under the different inlet pressure conditions, with the same wettability, the distributions of the NAPL is a little different, however, from Fig.7, the different distributions have little effects on the effective fracture aperture in the whole single filled fracture. According to Eq.(17), it can be found that the effective fracture aperture is a function of the fractured conductivity Kf. Depending on the five slopes in Fig.7 and Eq.(17), the fractured conductivities are given in Fig.8.

Fig.8 The fracture conductivity vs. the effective fracture aperture

In Fig.8, due to the NAPL clogging, the fractured conductivity is reduced obviously. As was mentioned above, the effective fractured aperture without the injected NAPL is obtained by the slopes of the case N1. Therefore, it can be seen that the NAPL clogging has prominent effects on the fracture conductivity from Fig.8.

2.3.2 Wettability effects on the NAPL transportation

In this section, the NAPL transport concerning with the different wettabilities were simulated. In the present simulations, the NAPLs are distributed randomly in the single filled fracture initially. In order to investigate that the effect of wettability on the NAPL transport in the single filled fracture, with the same random distributions of the NAPL, the contact angles were given to 63.4o, 90oand 132o, respectively. Thedirection and the periodic boundary conditions were applied in inlet and outlet directions. If the difference of average non-wetting phase velocities between every 3 000ts is smaller than 0.5%, it is assumed that the final steady state has arrived. At the final steady state, after 43 000ts, with the different contact angles, the initial NAPL distributions and the results of NAPL transport are displayed in Fig.9.

Fig.9 NAPL transport in single filled fracture

Fig.10 The average NAPL velocity with different contact angles

To the best of our knowledge, when θ＜90o, the NAPL is wetting phase for the fillers, however, when θ＞90o, the NAPL is non-wetting phase for the fillers. From Fig.9 it is found that, when the NAPL is wetting phase for the fillers (θ＜90o), it is difficult for the NAPL to enter the channels in the vertical direction, as opposed to the horizontal flow directions. It is alsofound that when the NAPL is non-wetting phase for the fillers(θ＞90o), it is easier for the NAPL to distribute in the whole single filler fracture, because there exists the strong interaction between the wetting phase and the fillers is favorable for the NAPL transportation. The wettability effects on the average NAPL velocity are displayed in Fig.10.

In Fig.10, before 10 000ts, the average NAPL velocity is unsteady obviously. After 10 000ts, there are few unsteady states in different cases, which results from the some droplets integrated into big droplets. From Fig.10, it is found that the contact angles have effects on the NAPL transport. For the same body force, the big contact angle improves the NAPL transport.

3. Conclusions

Our work demonstrates that the Shan-Chen multi-component multiphase LBM is a very instrumental tool to study the immiscible two-phase flow in single filled fracture due to its simplicity and capability of investigating wettability effects.

In this article, with the help of the Shan-Chen multi-component multiphase LBM, the contact angles of the non-wetting phase and wetting phase interface at a solid wall have been calculated accurately. By using this model, the clogging resulted from the NAPL transport concerning with the wettability and flow regimes with different wettabilities have been simulated in single filled fracture. It is found that if the injected NAPL is constant, even though with the different inlet pressure conditions, the effective fractured aperture and the fractured conductivity are constants, and the effective fractured aperture and the fractured conductivity is related to the injected NAPL content.

From the simulation it is found that, whether the NAPL is non-wetting phase or wetting phase depends on the contact angles for the filler. When the NAPL is wetting phase, it is distributed in the horizontal direction. When the NAPL is non-wetting phase, it is positive for the NAPL to be distributed in the whole single filler fracture. For the same body force, the big contact angle improves the NAPL transport.

Acknowledgement

The authors thank Prof. Huang Hai-bo and Dr. Qing C. for helpful discussions.

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10.1016/S1001-6058(11)60227-8

* Project supported by the National Natural Science Foundation of China (Grant Nos. 51079043, 41172204), the Program for Non-profit Industry Financial Program of Ministry of Water Resources (Grant Nos. 200901064, 201001020) and the Research Innovation Program for College Graduates of Jiangsu Province (Grant No.CXZZ11_0450).

Biography: DOU Zhi (1986-), Male, Ph. D. Candidate

ZHOU Zhi-fang,

E-mail: zhouzf@hhu.edu.cn

2012,24(1):130-137