From molecular dynamics to lattice Boltzmann: a new approach for pore-scale modeling of multi-phase flow

Xuan Liu; Yong-Feng Zhu; Bin Gong; Jia-Peng Yu; Shi-Ti Cui

doi:10.1007/s12182-015-0018-9

From molecular dynamics to lattice Boltzmann: a new approach for pore-scale modeling of multi-phase flow

Liu, Xuan; Zhu, Yong-Feng; Gong, Bin; Yu, Jia-Peng; Cui, Shi-Ti 2015-03-28 00:00:00 Pet. Sci. (2015) 12:282–292 DOI 10.1007/s12182-015-0018-9 ORIGINAL PAPER From molecular dynamics to lattice Boltzmann: a new approach for pore-scale modeling of multi-phase ﬂow 1,2 3 1 1 3 • • • • Xuan Liu Yong-Feng Zhu Bin Gong Jia-Peng Yu Shi-Ti Cui Received: 19 September 2014 / Published online: 28 March 2015 The Author(s) 2015. This article is published with open access at Springerlink.com Abstract Most current lattice Boltzmann (LBM) models 1 Introduction suffer from the deﬁciency that their parameters have to be obtained by ﬁtting experimental results. In this paper, we Multi-phase ﬂow in porous media is a common process in propose a new method that integrates the molecular dy- production of oil, natural gas, and geothermal ﬂuids from namics (MD) simulation and LBM to avoid such defect. natural reservoirs and in environmental applications such The basic idea is to ﬁrst construct a molecular model based as waste disposal, groundwater contamination monitoring, on the actual components of the rock–ﬂuid system, then to and geological sequestration of greenhouse gases. Con- compute the interaction force between the rock and the ventional computational ﬂuid dynamics (CFD) methods are ﬂuid of different densities through the MD simulation. This not adequate for simulating these problems as they have calculated rock–ﬂuid interaction force, combined with the difﬁculty in dealing with multi-component multi-phase ﬂuid–ﬂuid force determined from the equation of state, is ﬂow systems, especially phase transitions. In addition, the then used in LBM modeling. Without parameter ﬁtting, this complex pore structure of rocks is a big challenge for study presents a new systematic approach for pore-scale conventional grid generation and computational efﬁciency modeling of multi-phase ﬂow. We have validated this ap- (Guo and Zheng 2009). proach by simulating a two-phase separation process and In hydrology and the petroleum industry, it is common gas–liquid–solid three-phase contact angle. Based on an practice to model multi-phase ﬂow using Darcy’s law and actual X-ray CT image of a reservoir core, we applied our relative permeability theory. The relative permeability workﬂow to calculate the absolute permeability of the core, curve, usually obtained from laboratory experiments, is the vapor–liquid H O relative permeability, and capillary key to calculate ﬂow rates of different phases. Although pressure curves. measuring approaches have been widely used and results have been largely accepted for many years in most cases, laboratory experiments are usually expensive, not robust Keywords Molecular dynamics Lattice Boltzmann Multi-phase ﬂow Core simulation especially for low and ultra-low permeability core mea- surements, can damage the cores, and cannot always be repeated for different ﬂuids or under different ﬂow scenarios. & Bin Gong It is desirable to obtain the core properties through nu- gongbin@pku.edu.cn merical modeling based on actual pore structure charac- College of Engineering, Peking University, Beijing 100871, terizations. Recently, the lattice Boltzmann (LB) method, China which is based on a molecular velocity distribution func- Sinopec Petroleum Exploration and Production Research tion, has been proposed as a feasible tool for simulation of Institute, Beijing 100083, China multi-component multi-phase ﬂow in porous media (Huang Petrochina Tarim Oilﬁeld Exploration and Production et al. 2009, 2011; Huang and Lu 2009). In 1991, Chen et al. Research Institute, Korla 841000, Xinjiang, China proposed the ﬁrst immiscible LB model that uses red and blue-colored particles to represent two types of ﬂuids Edited by Yan-Hua Sun 123 Pet. Sci. (2015) 12:282–292 283 (Chen et al. 1991). The phase separation is produced by the Previous work in combining LB and MD methods to- repulsive interaction based on the color gradient. In 1993, gether can be classiﬁed into two types: one is conducted by Shan and Chen proposed to impose nonlocal interactions Succi, Horbach, and Sbragalia (Chibbaro et al. 2008; between ﬂuid particles at neighboring lattice sites by adding Horbach and Succi 2006; Sbragaglia et al. 2006; Succi an additional force term to the velocity ﬁeld (Shan and Chen et al. 2007). They applied MD and LB methods for the 1993, 1994; Shan and Doolen 1995). The potentials of the same problem and then compared the results. The second interaction control the form of the equation of state (EOS) type is conducted by Duenweg, Ahlrichs, Horbach, and of the ﬂuid, and phase separation occurs naturally once the Succi (Ahlrichs and Du¨nweg 1998, 1999; Fyta et al. 2006). interaction potentials are properly chosen. In 1995, Swift They applied the two methods to the motion simulation of et al. (1995, 1996) proposed a free-energy model, in which polymer, DNA, or other macromolecules in water. The the description of non-equilibrium dynamics, such as the coupling of the MD calculation for the macromolecule part Cahn–Hilliard approach, is incorporated into the LB model and the LB modeling for the solvent is achieved via a using the concept of the free-energy function. However, the friction ansatz, in which they assumed the force exerted by free-energy model does not satisfy Galilean invariance, and the ﬂuid on one monomer was proportional to the differ- the temperature dependence of the surface tension is in- ence between the monomer velocity and the ﬂuid velocity correct (Nourgaliev et al. 2003). In 2003, Zhang and Chen at the monomer’s position. proposed a new model, in which the body force term was To be exact, the ﬁrst type summarized above is not the directly incorporated in the evolution equation (Zhang and coupling of LB and MD. The second approach is consid- Chen 2003). Compared with the Shan and Chen (SC) ered as multi-scale coupling of LB and MD, where MD is model, the Zhang and Chen (ZC) model avoids negative used for the focus part such as the polymer or the ﬂuid–gas values of effective mass. However, simulation results from interface, while LB is used for other parts of the system, the Zhang and Chen model show that the spurious current such as the solvent or the ﬂuid ﬂow. LB and MD simula- gets worse and the temperature range that this model can tions are conducted at the same time step, and the variables deal with is much smaller than the SC model (Zeng et al. are exchanged between these two simulation domains un- 2009). In 2004, by introducing the explicit ﬁnite difference der certain boundary constraints. Such a synchronous cal- (EFD) method to calculate the volume force, Kupershtokh culation method is extremely time-consuming in porous developed a single-component Lattice Boltzmann model media ﬂow simulation because of the large amount of (LBM) (Kupershtokh and Medvedev 2006; Kupershtokh calculation of MD simulation on both gas–liquid and rock– et al. 2009; Kupershtokh 2010). Compared to previous ﬂuid interfaces. models, this model has a signiﬁcant improvement in pa- Our proposed method integrates, rather than couples si- rameter ranges of temperature and density ratio. Our work multaneously, the LB and MD models efﬁciently. In this ap- in this paper is partially based on this model. proach, the interaction forces between rock and ﬂuid of In conventional LB models, the force between ﬂuid and different density are ﬁrstly calculated by MD simulation. rock is supposed to be proportional to the ﬂuid density. Combined with the ﬂuid–ﬂuid force determined from the EOS, This assumption lacks theoretical support and cannot de- the two types of interaction forces are then accurately described scribe the true physical phenomena under certain circum- for LBM modeling. We validated our integrated model by stances. As an improvement, we propose to simulate the simulating a two-phase separation process and gas–liquid– force between the ﬂuid component and rock for different solid three-phase contact angle. The success of MD–LBM re- ﬂuid density using the molecular dynamics (MD) method. sults in agreement with published EOS solution, and ex- Molecular dynamics simulation is an effective method perimental results demonstrated a breakthrough in pore-scale, for investigating microscopic interactions and detailed multi-phase ﬂow modeling. Based on an actual X-ray CT im- governing forces that dominates the ﬂow. Among the MD age of a reservoir core, we applied our workﬂow to calculate studies, various issues in multi-phase processes were paid the absolute permeability of the core, the vapor–liquid H O close attention. Ten Wolde and Frenkel studied the ho- relative permeability, and capillary pressure curves. mogeneous nucleation of liquid phase from vapor (Ten Wolde and Frenkel 1998). Wang et al. studied thermody- namic properties in coexistent liquid–vapor systems with 2 Methodology liquid–vapor interfaces (Wang et al. 2001). A sharp peak and a small valley at the thin region outside the liquid– 2.1 The lattice Boltzmann model vapor interface were found to be evidence of a non-equi- librium state at the interface. In our work, we established a The Boltzmann equation describes the evolution with re- similar system to simulate forces between the ﬂuid and gard to a space–velocity distribution function from motions solid components for different ﬂuid densities. of microscopic ﬂuid particles (Atkins et al. 2006). 123 284 Pet. Sci. (2015) 12:282–292 equation implies two kinds of particle operations: stream- of þ n r f þ a r f ¼ Xðf Þ; ð1Þ x n ing and collision. The term on the left side of Eq. (2) de- ot scribes particles moving from the local site x to one of the where t is time, vector x is location, vector n is the ﬂuid neighbor sites x ? e dt within each time step. The ﬁrst term molecular velocity at time t and location x, and f is the on the right side of Eq. (2) describes the collisions con- velocity distribution function of the ﬂuid molecules, which tributing to loss or gain of the particles with a velocity of e . is equivalent to the density of the ﬂuid molecules whose After collision, the velocity distribution will relax to an velocity is n at time t on location x, a is the acceleration of eq equilibrium distribution, f . the ﬂuid molecules, and X(f) is the velocity distribution The ﬂuid density q and its velocity u at one node are function change caused by collision between ﬂuid calculated in Eqs. (3) and (4) (Qian et al. 1992). The re- molecules. lationship between s and ﬂuid kinematic viscosity m can be It is not realistic to solve the integral–differential ðÞ 2s 1 dx described as v ¼ , where dx is the lattice Boltzmann equation directly. An alternative approach is to ðÞ 6dt solve the discrete form of the Boltzmann equation. The constant and dt is the lattice time (Qian et al. 1992; Swift most widely used approach is the LBM. The key idea of et al. 1996). LBM is both the location and velocity of the particles q1 which are discretely characterized (Fig. 1). A typical LB q ¼ f ð3Þ equation can be written as (Shan and Chen 1993; Qian i¼0 et al. 1992) q1 qu ¼ f e ð4Þ i i fðÞ x þ e dt; t þ dt fðÞ x; t i i i i¼0 ð2Þ eq ¼ðÞ f ðÞ x; t fðÞ x; tþ wðÞ x; t ; eq i i We use the equilibrium distribution function f in the standard form (Yuan and Schaefer 2006) where x denotes the position vector, e (i = 0, 1,…, q - 1) is the particle velocity vector to the neighbor sites, q is the eq eq f ðx; tÞ¼ f ðq; uÞ i i number of neighbors, which depends on the lattice ge- ; ð5Þ e u ðÞ e u u i i ometry, f is the particle velocity distribution function along ¼ qx 1 þ þ eq 2 h 2h 2h the ith direction, f is the corresponding local equilibrium distribution function satisfying the Maxwell distribution, s where q and u are for q(x, t) and u(x, t), respectively, is the collision relaxation time, and w is the change in the corresponding to local ﬂuid density and macro velocity. distribution function due to the body force. The LB With this equilibrium distribution function, the ‘‘kinetic temperature’’ in standard LB models, such as ‘‘D2Q9’’ and dx ‘‘D3Q19,’’ is equal to h ¼ (Yuan and Schaefer ðÞ 3dt 2006). In this work, we use the D2Q9 model for the 2D simulations. The weighting factor and discrete velocity for this model are given below: 6 2 5 ðÞ 0; 0 ; i ¼ 0 e e e < 6 2 5 e ¼ ðÞ 1; 0ðÞ 0; 1 ; i ¼ 1; 2; 3; 4 3 ðÞ 1; 1 ; i ¼ 5; 6; 7; 8 3 0 1 < 4=9; i ¼ 0 x ¼ 1=9; i ¼ 1; 2; 3; 4 e e e 7 4 8 1=36; i ¼ 5; 6; 7; 8 7 4 8 For the implementation of the body force term, w,itis appropriate to use the exact difference method (Kupersh- tokh and Medvedev 2006; Kupershtokh 2010) eq eq wðÞ ~ x; t ¼ fðÞ q; ~ u þ D~ u f ðÞ q; ~ u ; ð6Þ i i where the change of the velocity is determined by the force F acting on the node Fig. 1 Illustration of the lattice Boltzmann model 123 Pet. Sci. (2015) 12:282–292 285 the following equations for the P–R equation to be used in Du ¼ FDt=q ð7Þ the LB model (Yuan and Schaefer 2006) F is identiﬁed in three categories: attractive (or repulsive) l r l r l r T T p p q q m m force between local ﬂuid and the ﬂuid of their neighbor ¼ ¼ ¼ ; ð14Þ l r l r l r T T p p q q lattices F , attractive force between the local lattice ﬂuid c c c c m;c m;c ff and the boundary wall F , and the macro body force, e.g., sf where the superscript l and r stand for the lattice and real the gravity, F . systems, respectively, and c means the critical state. With the body force F, the real ﬂuid velocity v should be evaluated at half the time step (Kupershtokh 2010) 2.3 Application of the molecular dynamics q1 simulation to obtain rock–ﬂuid interaction qv ¼ f e þ FDt ð8Þ i i forces F 2 sf i¼0 Thus, the most important part for LB modeling is the de- It is clear that the force acting between ﬂuid components termination of the interaction forces between neighboring and the rock boundary wall has a signiﬁcant effect on ﬂow ﬂuids F and the ﬂuid and rock wall F . ff sf modeling with LBM. In fact, this force decides the capil- lary force and relative permeability curve for multi-phase systems. 2.2 Application of the equation of state to obtain In previous studies (Huang et al. 2009, 2011; Huang and ﬂuid–ﬂuid interaction forces F ff Lu 2009; Martys and Chen 1996; Hatiboglu and Babadagli 2007, 2008), the force between the ﬂuid and boundary wall It is suggested that the ﬂuid–ﬂuid interacting force can be is simpliﬁed as obtained by solving the state equations (Shan and Chen 1993, 1994; Kupershtokh and Medvedev 2009; Yuan and F ðÞ x; t ¼uqðÞ ðÞ x; t G x dðÞ x þ e e ; ð15Þ Schaefer 2006) sf sf i i i i¼0 F ðÞ x; t ¼rUðÞ x; t ; ð9Þ ff where q is the ﬂuid component density, boolean d = 0, 1 to where U(x, t) is a function of state that can be expressed as indicate if it is the ﬂuid or solid lattice, respectively. G is sf (Kupershtokh 2010) the force strength factor between ﬂuid and solid, and it is usually determined by ﬁtting the macroscopic ﬂuid–solid UðÞ x; t ¼ pðÞ q ðÞ x; t ; TðÞ x; t q ðÞ x; t h; ð10Þ m m contact angle. where q is the speciﬁc density of the ﬂuid, which can be There are two disadvantages of this method. One is that expressed as q = q M where M is the standard molar G cannot always be obtained because of the lack of sf quality of ﬂuid component. T is temperature. p(q , T)isa contact angle data for some common ﬂuids such as certain state equation. Replacing U with the effective mass methane, nitrogen, carbon dioxide, and polymers for in- density u, we get stance. Another disadvantage is that the assumption that F sf is directly proportional to the ﬂuid effective mass density uðÞ x; t ¼jj UðÞ x; t ð11Þ yields signiﬁcant error under actual conditions. For ex- Then the interacting force between ﬂuids can be written as ample, this assumption cannot represent the adsorption force for coal gas and shale gas to the formation rock. F ðÞ x; t ¼ 2uðÞ x; truðÞ x; t ð12Þ ff To overcome these limitations, we propose to obtain the In our work, the Peng–Robinson (P–R) state equation is force between ﬂuid components and the rock through used in the expression molecular dynamic simulation. 2 There are a variety of minerals in the reservoir rock. But it is q RT aaðÞ T q m m p ¼ ð13Þ mainly composed of minerals including quartz (SiO ), calcite 2 2 1 bq m 1 þ 2bq ðÞ bq m m (CaCO ), dolomite (CaMg(CO ) ), feldspar (KAlSi O , 3 3 2 3 8 2 2 where a = 0.458R T /p , b = 0.0778RT /p , a(T) = NaAlSi O ,CaAl Si O mixture), and clay (Al Si O (OH) ) c c c c 3 8 2 2 8 3 2 5 4 0.5 2 [1 ? f(x)(1-(T/T ) )] where T and p are the critical with a mixing ratio. In our preliminary investigation, we chose c c c temperature and pressure of ﬂuid components, x is the monocrystalline silicon to represent the rock solid, although acentric factor which depends on the ﬂuid composition itself. elemental silicon does not occur in reservoir rocks. It is According to the correspondence principle, the molar however convenient for model validation against theoretical density q , viscosity l, etc. are the same when the two and experimental results. In our on-going research, we now ﬂuids have the same values of T/T and p/p . Following this use the more realistic assumption that the rock grains are c c principle, the lattice ﬂuid and the real ﬂuid should satisfy composed of SiO and other components. 123 286 Pet. Sci. (2015) 12:282–292 The SPC/E model (Jorgensen et al. 1983) was used to y directions. The length along these two directions is both model water molecules. This model speciﬁes a 3-site rigid 43.45 Angstrom, and the length along the z direction is 60.14 water molecule with charges and Lennard-Jones pa- Angstroms. The solid walls were represented by layers of rameters assigned to each of the 3 atoms. The bond length DM (face-centered cubic) silicon atoms (1152 9 2). The of O–H is 1.0 Angstrom, and the bond angle of H–O–H is speed and the force applied on the Si atoms were both set as 109.47. zero to make them represent boundary walls and to keep a The well-known LJ potential and the standard constant volume of the system. Coulombic interaction potential were applied to calculate For different cases, at the beginning of simulation, water the intermolecular forces between all the molecules. The molecules of different numbers were sandwiched by the parameters are listed in Table 1. The cross parameters of two solid walls. LJ potential were obtained by Lorentz Berthelot combining Firstly, using the constant NVT (certain number, vol- rules. As originally proposed, this model was run with a 9 ume, and temperature–canonical ensemble) time integra- Angstrom cut-off for both LJ and Coulombic terms. tion via the Nose/Hoover method, the whole system was set LAMMPS code was used to construct the simulation at a uniform temperature of 373.15 K (110 C). system. Then an annealing schedule of 340,000 steps was ap- As shown in Fig. 2, the simulation domain was a box with plied to allow the system to reach equilibrium (there were periodic boundary conditions applied in the x and four cycles altogether in the annealing schedule. In one single cycle, the temperature is raised from 373.15 to 423.15 K in 20,000 steps, to 473.15 K in 20,000 steps, and Table 1 Parameters for LJ potential and Coulombic potential then reduced to 423.15 K in 20,000 steps, to 373.15 K in Parameters r, Angstroms e, kcal/mole q,e 20,000 steps). After that, the equilibrium state of the sys- tem was reached at 373.15 K (110 C). Then the NVT Silicon 3.826 0.4030 0 ensemble was changed into NVE (certain number, volume, Oxygen 3.166 0.1553 -0.8476 and energy– micro-canonical ensemble) and run for 20,000 Hydrogen 0.0 0.0 0.4238 steps. In these two processes, the equivalent length scale for the simulations was 0.5 feets to ensure energy conservation. With the equilibrium system, the force between Si and H O can be calculated. As Fig. 2 shows, since the distance between domain 1 and the solid wall is larger than the cut- off radius (9 Angstroms), the water molecules in domain 1 can be considered as free molecules. So the mass densities of free water in different cases were calculated based on the molecules in this domain. For the water molecules in do- main 2, the force F applied by the silicon solid wall was directly calculated according to the potential ﬁeld in the simulation. Then we can get the stress r = F/A, where A is the area of the wall. 2.4 Integrated workﬂow from MD to LBM simulation For the ﬁrst step, we build a MD model as presented in Sect. 2.3 to calculate the rock–ﬂuid interaction forces F sf between any solid component of the wall and the ﬂuid of different density. Input parameters for this step include molecular species of the ﬂuid and boundary solid and the potential parameters. The result of this step is the rela- tionship between the solid–ﬂuid interface and the ﬂuid Fig. 2 The molecular dynamics model of the Si–H O system. density. L = L = 43.45 Angstrom, L = 60.14 Angstrom. Green atom is x y z Then we calculate u(q) of different ﬂuid density q from for Si, pink for O, and white for H. The density of water phase is P–R EOS calculations as presented in Sect. 2.2. The ﬂuid– determined by H O molecules in domain 1. Force between water and ﬂuid interaction force F is then determined from Eq. (12). the boundary is determined by H O molecules in domain 2 2 sf 123 Pet. Sci. (2015) 12:282–292 287 Input parameters for this step include ﬂuid density, critical (a) (b) pressure and temperature, and acentric factor of the ﬂuid 1000 component. With an actual core X-ray scan image for lattice grid construction and the interaction forces being determined from MD and EOS calculation, we use the LB method to simulate the ﬂuid ﬂow in porous media and to obtain the absolute permeability, relative permeability, and capillary (c) (d) pressure curves. The integrated workﬂow is illustrated in Fig. 3. 3 Results 3.1 LBM simulation on liquid–vapor phase transition process of water Fig. 4 Mass density (kg/m ) distribution of water vapor and liquid at different time steps t. a t = 250. b t = 500. c t = 1000. d t = 2000 To validate the EOS model, the single-component ﬂuid phase transition process was simulated using the model low (about 0.76 kg/m ), which indicates it is saturated with described in Sect. 2.2. water vapor. It is clearly shown in this ﬁgure that the small Water is the ﬂuid we choose in our simulation. Its cri- liquid droplets aggregate to form bigger ones as time in- tical temperature is 373.99 C, critical pressure is creases. Eventually, all these small droplets coalesce to 22.06 MPa, critical density is 322.0 kg/m , and its acentric form bigger droplets and the rest of the space of the factor is 0.344. computational domain is occupied by vapor only. All these A 2D 200 9 200 square lattice was used in this simulation results for the single-component phase transi- simulation, and the periodic boundary conditions were tion process are consistent with those reported by Zeng applied on the four boundaries. The average mass density et al. (2009) and Qin (2006). of the computational domain was set to be 500 kg/m .At We then repeated the simulation at different tem- the beginning of simulation, the mass density was ho- peratures for model validation. Figure 5 shows the density mogenously initialized with a small (0.1 %) random perturbation. Figure 4 shows the mass density contour in the com- putation domain at different time steps (t = 250, 500, 1000 and 2000) when the temperature is 110 C. The density in the pink area is high (793 kg/m ), which indicates it is saturated with water liquid. The density in the green area is Single component multi-phase flow in porous media Interaction between Interaction between fluid component fluid and rock media Liquid density, Exp. P–R EOS MD simulation Vapor density, Exp. Liquid density, EOS Lattice Boltzmann model Pore network structure Vapor density, EOS Liquid density, LBM Vapor density, LBM Absolute Relative Capillary permeability permeability pressure 0 200 400 600 800 1000 ρ, kg/m Fig. 3 Schematic illustrations of the workﬂow. First, the interaction forces between ﬂuids are determined from P–R EOS calculation, and Fig. 5 Saturated water (in blue) and vapor (in red) densities versus the rock–ﬂuid interaction force is determined from MD simulation. temperature. ‘‘o’’ is the result of LB simulation. The solid line is the Then we run the lattice Boltzmann simulation based on the grid solution of the P–R equation using the Maxwell equal-area construc- constructed from the core X-ray scan image to obtain the absolute tion. asterisk Denotes the experimental data from Handbook of permeability, relative permeability, and capillary pressure curves Chemical Engineering (Liu et al. 2001) T, ºC 288 Pet. Sci. (2015) 12:282–292 distribution curves of both vapor and liquid when they vapor density is smaller than 0.787 kg/m or the liquid reach phase equilibrium at different temperatures. This density is greater than 793 kg/m . The relationship be- ﬁgure indicates that the density curve calculated with LBM tween density and stress is plotted in Fig. 7. It clearly is nearly identical to that obtained from P–R EOS using the shows that the stress varies with the ﬂuid density nonlin- Maxwell equal-area construction (Yuan and Schaefer early, thus demonstrates the assumption that ‘‘the force 2006). It suggests that the LBM, combined with EOS, between ﬂuid and rock is supposed to be proportional to the provides an accurate method to model single-component ﬂuid density’’ used by former studies (Martys and Chen gas–ﬂuid two-phase ﬂow. 1996; Hatiboglu and Babadagli 2007, 2008) is not It is noted that there are some differences between the appropriate. theoretical solution using EOS and the experimental results for liquid density calculations, and this is originated from 3.3 Calculation of monocrystalline silicon–water the inadequacy of P–R EOS and is in agreement with contact angle (Atkins and William 2006). Contact angle reﬂects the interaction forces at the phase 3.2 MD simulation in determination of force interfaces and the wettability. Thus, it is a signiﬁcant pa- between ﬂuid components and the boundary rameter to describe the interaction between oil, gas, water, wall and rock in petroleum reservoirs (Wolf et al. 2009). We simulated the contact angle of the monocrystalline In the MD simulation, 42 sets of systems with different Si–water liquid–water vapor system by applying our inte- H O densities were generated. We select 4 cases for il- grated MD–LBM approach and then compared the value lustration in Fig. 6. with literature for validation. As shown in Fig. 5, at the simulation temperature Figure 8 demonstrates the equilibrium state of the sys- (110 C), the force calculation is only meaningful when the tem after LBM simulation. In the simulation, the compu- tational domain is 100 9 50, where the upper and lower boundaries are solid walls, and the east and west boundary is periodic. For the solid node, before the streaming step, a bounce–back algorithm was implemented to mimic the non-slip wall boundary condition. In Fig. 8, the gray grid on the top and bottom is for Si, the red is for water liquid, and the blue is for water vapor. The interaction force between water ﬂuids was calculated by EOS, and the interaction force between water and Si was obtained by MD. This ﬁgure shows that the macro contact angel is approximate 101, which is consistent with the result given by (Williams and Goodman 1974) in which it states that the contact angle is near 90. Therefore, the method of MD used in ﬂuid–solid interaction force calcu- lation incorporated into LBM is reasonable in porous me- dia ﬂow simulation. 3.4 Rock permeability determination based on an actual X-ray CT image of a reservoir core Similar to the methodology in previous literature (Guo and Zheng 2009; Huang et al. 2009, 2011; Huang and Lu 2009; Hatiboglu and Babadagli 2007, 2008; Jorgensen et al. 1983), we calculated the relative permeability curves by simulating a two-phase ﬂow in the porous media. Figure 9 is a post-processed 2D X-ray image of a reservoir core in Tarim Basin conglomerate with a length L = 2.7 mm and width A = 1.35 mm. The domain is gridded into Fig. 6 Si–H O system in equilibrium with 4 different water densities. 3 3 540 9 270 = 145,800 cells with each one being a In vapor phase: a q = 0.43 kg/m and b q = 0.66 kg/m . In liquid 3 3 5 9 5 lm square. In this image, the part in black phase: c q = 842 kg/m and d q = 911 kg/m 123 Pet. Sci. (2015) 12:282–292 289 0.8 5.0 (b) (a) 0.6 4.5 0.4 4.0 0.2 3.5 0 0.2 0.4 0.6 0.8 700 800 900 1000 1100 ρ, kg/m ρ, kg/m Fig. 7 The stress–density relationship. a in the vapor region, linear ﬁtted by the least squares method and b in the liquid region, quadratic ﬁtted by the least squares method represents pores and white for the rock grains. Porosity / the image in Fig. 9. The system is at temperature = 43.5 %. T = 110 C. At the initial state, each cell was saturated We applied a single-vapor phase ﬂow of water to cal- with H O steam with a density of 0.754 kg/m . In order to culate the absolute permeability of the core represented by use the MD simulation results on interaction forces be- tween H O and Si consistently, we assumed that the rock grain is made of Si (in our on-going research, we use the more realistic assumption that the rock grains are com- posed of quartz (SiO ) and other minerals). As in Fig. 10, we applied a virtual body force g and periodic boundary conditions in calculating permeability based on LBM solution of the water ﬂow problem. The body force is equivalent of adding a pressure drop Dp ¼ qgL (q ¼ q =n is the average density of water in the i¼1 Body force g Fig. 8 Equilibrium state simulated from LBM for a water liquid drop in contact with the Si surface. The gray grid on the top and bottom is for Si, the red for water liquid, and the blue for water vapor L = 2.7 mm Periodic boundary condition Fig. 10 The ﬂow problem in calculating absolute permeability. Closed boundaries on the top and bottom. A virtual body force g towards right is applied to the simulation domain, and periodic Fig. 9 A 2D CT image of a reservoir rock in Tarim Basin boundary condition is set on left and right sides of the simulation conglomerate domain σ, MPa A = 1.35 mm σ, MPa 290 Pet. Sci. (2015) 12:282–292 simulation domain, n is the total number of the grids ﬁlled with water) between the left and right side of the core. By applying different values of g, we can calculate ﬂow ve- locity under different pressure drops. For a given pressure drop Dp, the ﬂow velocity of H O steam reaches equilibrium after some steps of simulation. The volumetric velocity of H O steam ﬂow is calculated as v ¼ / v =n (Cihan et al. 2009). Figure 11 depicts the g i i¼0 calculated relationship between v and Dp. The linearity between v and Dp shown in Fig. 11 is in Fig. 12 Distribution of vapor (in pink) and liquid (in blue) phases of H O in rock pores agreement with Darcy’s law v = KDp/(lL), where l is the viscosity of water vapor, L is the length of the core, and K is permeability of the core. -6 represent different saturations (Fig. 12). At the simulation Plug L = 2.7 mm and l = 12.4 9 10 Pa s into temperature, 110 C, saturated water vapor density is Darcy’s law and use least square linear ﬁtting of calculated 0.787 kg/m , and saturated water liquid density is 793 kg/ points in Fig. 11, we calculate the permeability of the rock m . Then the average density of water in simulation do- as 636 mD. main, q, was calculated for different saturations. Given that Dp ¼ qgL, the pressure drop for any values of g can be 3.5 Vapor–liquid two-phase relative permeability obtained. determination based on an actual X-ray CT At the simulation temperature, the viscosities of image of a reservoir core -6 saturated water vapor and liquid are l = 12.4 9 10 Pa -6 s and l = 252 9 10 Pa s. Relative permeability, as a function of phase saturation, We calculated the permeability of the vapor and liquid could be modeled by multi-phase ﬂow simulation using the phases using Darcy’s law for different phases, K = v l L/ proposed workﬂow. Similar to the conditions set for ab- g g g Dp, K = v l L/Dp. K and K at different saturations solute permeability simulation, virtual body force g and w w w g w were obtained from repeated simulations and calculations periodic boundary condition were applied to the simulation of volumetric velocity of each phase. Finally, the relative domain, and temperature was set at 110 C. permeability K = K /K and K = K /K is obtained as At initial state, the core was randomly ﬁlled with vapor– rg g rw w shown in Fig. 13. liquid two-phase water at different mixing ratios to -3 1.0 x 10 rw 0.9 rg 0.8 0.7 0.6 0.5 0.4 0.3 2 0.2 0.1 1 Simulated result Least-squares method fitted result 0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 200 250 300 Δ p, Pa Fig. 13 Calculated vapor–liquid two-phase water relative perme- Fig. 11 Calculated water ﬂow velocity versus pressure drop ability curves v , m/s r Pet. Sci. (2015) 12:282–292 291 0.20 Different from the simulation of absolute permeability, the multi-phase ﬂow modeling results using a statistical method such as LBM could be unstable, thus causing the volumetric velocity obtained at different times be slightly 0.15 different. This means that the relative permeability of each phase at a given saturation could be a distribution as shown in Fig. 13. The relative permeability curves were con- structed by taking the average values of K at each r 0.10 saturation. 3.6 Capillary pressure curve determination based 0.05 on an actual X-ray CT image of a reservoir core We also applied our methodology to calculate the capillary pressure curve. As shown in Fig. 14, constant pressure 0.00 0 0.2 0.4 0.6 0.8 1.0 boundary conditions (p on the left and p on the right) and 1 2 closed boundary conditions on the top and bottom were applied to the simulation domain. Fig. 16 Calculated vapor–liquid H O capillary pressure curve In the initial state, the domain was ﬁlled with saturated water vapor. We injected saturated water liquid from the 4 Conclusions left side under constant pressure p . Since p [ p , water 1 1 2 vapor will ﬂow out of the domain from the right side. By In this work, we proposed a new systematic workﬂow to applying different pressure drops Dp = p -p , the injected 1 2 integrate MD simulation with the LB method to model liquid ﬂowed into the pores to yield different phase multi-phase ﬂow in porous media. As an improvement, this saturations. At an equilibrium state as shown in Fig. 15, the new approach avoids parameter ﬁtting or incorrectly as- capillary pressure is equal to the pressure drop, i.e., suming a linear relationship between the rock–ﬂuid inter- p (S ) = Dp. The calculated capillary pressure curve is c g action force and ﬂuid density. We have validated this shown in Fig. 16. approach by simulating a two-phase separation process and a gas–liquid–solid three-phase contact angle. The success of MD–LBM results in agreement with published EOS solution, and experimental results demonstrated a break- Liquid Vapor through in pore-scale, multi-phase ﬂow modeling. Based inflow outflow on an actual X-ray CT image of a reservoir core, we ap- plied our workﬂow to calculate absolute permeability of p p 1 2 the core, vapor–liquid water relative permeability, and capillary pressure curves. With the application of this workﬂow to a more realistic model considering actual Fig. 14 The ﬂow problem in calculating capillary pressure curve. A reservoir rock and ﬂuid parameters, the ultimate goal is to constant pressure boundary is set on both sides of the simulation develop an accurate method for prediction of permeability domain tensor, relative permeability, and capillary curves based on 3D CT image of the rock, actual ﬂuid, and rock components. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, dis- tribution, and reproduction in any medium, provided the original author(s) and the source are credited. References Ahlrichs P, Du¨nweg B. 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Publisher: Springer Journals
Copyright: Copyright © 2015 by The Author(s)
Subject: Earth Sciences; Mineral Resources; Industrial Chemistry/Chemical Engineering; Industrial and Production Engineering; Energy Economics
ISSN: 1672-5107
eISSN: 1995-8226
DOI: 10.1007/s12182-015-0018-9
Publisher site: See Article on Publisher Site

Abstract

Pet. Sci. (2015) 12:282–292 DOI 10.1007/s12182-015-0018-9 ORIGINAL PAPER From molecular dynamics to lattice Boltzmann: a new approach for pore-scale modeling of multi-phase ﬂow 1,2 3 1 1 3 • • • • Xuan Liu Yong-Feng Zhu Bin Gong Jia-Peng Yu Shi-Ti Cui Received: 19 September 2014 / Published online: 28 March 2015 The Author(s) 2015. This article is published with open access at Springerlink.com Abstract Most current lattice Boltzmann (LBM) models 1 Introduction suffer from the deﬁciency that their parameters have to be obtained by ﬁtting experimental results. In this paper, we Multi-phase ﬂow in porous media is a common process in propose a new method that integrates the molecular dy- production of oil, natural gas, and geothermal ﬂuids from namics (MD) simulation and LBM to avoid such defect. natural reservoirs and in environmental applications such The basic idea is to ﬁrst construct a molecular model based as waste disposal, groundwater contamination monitoring, on the actual components of the rock–ﬂuid system, then to and geological sequestration of greenhouse gases. Con- compute the interaction force between the rock and the ventional computational ﬂuid dynamics (CFD) methods are ﬂuid of different densities through the MD simulation. This not adequate for simulating these problems as they have calculated rock–ﬂuid interaction force, combined with the difﬁculty in dealing with multi-component multi-phase ﬂuid–ﬂuid force determined from the equation of state, is ﬂow systems, especially phase transitions. In addition, the then used in LBM modeling. Without parameter ﬁtting, this complex pore structure of rocks is a big challenge for study presents a new systematic approach for pore-scale conventional grid generation and computational efﬁciency modeling of multi-phase ﬂow. We have validated this ap- (Guo and Zheng 2009). proach by simulating a two-phase separation process and In hydrology and the petroleum industry, it is common gas–liquid–solid three-phase contact angle. Based on an practice to model multi-phase ﬂow using Darcy’s law and actual X-ray CT image of a reservoir core, we applied our relative permeability theory. The relative permeability workﬂow to calculate the absolute permeability of the core, curve, usually obtained from laboratory experiments, is the vapor–liquid H O relative permeability, and capillary key to calculate ﬂow rates of different phases. Although pressure curves. measuring approaches have been widely used and results have been largely accepted for many years in most cases, laboratory experiments are usually expensive, not robust Keywords Molecular dynamics Lattice Boltzmann Multi-phase ﬂow Core simulation especially for low and ultra-low permeability core mea- surements, can damage the cores, and cannot always be repeated for different ﬂuids or under different ﬂow scenarios. & Bin Gong It is desirable to obtain the core properties through nu- gongbin@pku.edu.cn merical modeling based on actual pore structure charac- College of Engineering, Peking University, Beijing 100871, terizations. Recently, the lattice Boltzmann (LB) method, China which is based on a molecular velocity distribution func- Sinopec Petroleum Exploration and Production Research tion, has been proposed as a feasible tool for simulation of Institute, Beijing 100083, China multi-component multi-phase ﬂow in porous media (Huang Petrochina Tarim Oilﬁeld Exploration and Production et al. 2009, 2011; Huang and Lu 2009). In 1991, Chen et al. Research Institute, Korla 841000, Xinjiang, China proposed the ﬁrst immiscible LB model that uses red and blue-colored particles to represent two types of ﬂuids Edited by Yan-Hua Sun 123 Pet. Sci. (2015) 12:282–292 283 (Chen et al. 1991). The phase separation is produced by the Previous work in combining LB and MD methods to- repulsive interaction based on the color gradient. In 1993, gether can be classiﬁed into two types: one is conducted by Shan and Chen proposed to impose nonlocal interactions Succi, Horbach, and Sbragalia (Chibbaro et al. 2008; between ﬂuid particles at neighboring lattice sites by adding Horbach and Succi 2006; Sbragaglia et al. 2006; Succi an additional force term to the velocity ﬁeld (Shan and Chen et al. 2007). They applied MD and LB methods for the 1993, 1994; Shan and Doolen 1995). The potentials of the same problem and then compared the results. The second interaction control the form of the equation of state (EOS) type is conducted by Duenweg, Ahlrichs, Horbach, and of the ﬂuid, and phase separation occurs naturally once the Succi (Ahlrichs and Du¨nweg 1998, 1999; Fyta et al. 2006). interaction potentials are properly chosen. In 1995, Swift They applied the two methods to the motion simulation of et al. (1995, 1996) proposed a free-energy model, in which polymer, DNA, or other macromolecules in water. The the description of non-equilibrium dynamics, such as the coupling of the MD calculation for the macromolecule part Cahn–Hilliard approach, is incorporated into the LB model and the LB modeling for the solvent is achieved via a using the concept of the free-energy function. However, the friction ansatz, in which they assumed the force exerted by free-energy model does not satisfy Galilean invariance, and the ﬂuid on one monomer was proportional to the differ- the temperature dependence of the surface tension is in- ence between the monomer velocity and the ﬂuid velocity correct (Nourgaliev et al. 2003). In 2003, Zhang and Chen at the monomer’s position. proposed a new model, in which the body force term was To be exact, the ﬁrst type summarized above is not the directly incorporated in the evolution equation (Zhang and coupling of LB and MD. The second approach is consid- Chen 2003). Compared with the Shan and Chen (SC) ered as multi-scale coupling of LB and MD, where MD is model, the Zhang and Chen (ZC) model avoids negative used for the focus part such as the polymer or the ﬂuid–gas values of effective mass. However, simulation results from interface, while LB is used for other parts of the system, the Zhang and Chen model show that the spurious current such as the solvent or the ﬂuid ﬂow. LB and MD simula- gets worse and the temperature range that this model can tions are conducted at the same time step, and the variables deal with is much smaller than the SC model (Zeng et al. are exchanged between these two simulation domains un- 2009). In 2004, by introducing the explicit ﬁnite difference der certain boundary constraints. Such a synchronous cal- (EFD) method to calculate the volume force, Kupershtokh culation method is extremely time-consuming in porous developed a single-component Lattice Boltzmann model media ﬂow simulation because of the large amount of (LBM) (Kupershtokh and Medvedev 2006; Kupershtokh calculation of MD simulation on both gas–liquid and rock– et al. 2009; Kupershtokh 2010). Compared to previous ﬂuid interfaces. models, this model has a signiﬁcant improvement in pa- Our proposed method integrates, rather than couples si- rameter ranges of temperature and density ratio. Our work multaneously, the LB and MD models efﬁciently. In this ap- in this paper is partially based on this model. proach, the interaction forces between rock and ﬂuid of In conventional LB models, the force between ﬂuid and different density are ﬁrstly calculated by MD simulation. rock is supposed to be proportional to the ﬂuid density. Combined with the ﬂuid–ﬂuid force determined from the EOS, This assumption lacks theoretical support and cannot de- the two types of interaction forces are then accurately described scribe the true physical phenomena under certain circum- for LBM modeling. We validated our integrated model by stances. As an improvement, we propose to simulate the simulating a two-phase separation process and gas–liquid– force between the ﬂuid component and rock for different solid three-phase contact angle. The success of MD–LBM re- ﬂuid density using the molecular dynamics (MD) method. sults in agreement with published EOS solution, and ex- Molecular dynamics simulation is an effective method perimental results demonstrated a breakthrough in pore-scale, for investigating microscopic interactions and detailed multi-phase ﬂow modeling. Based on an actual X-ray CT im- governing forces that dominates the ﬂow. Among the MD age of a reservoir core, we applied our workﬂow to calculate studies, various issues in multi-phase processes were paid the absolute permeability of the core, the vapor–liquid H O close attention. Ten Wolde and Frenkel studied the ho- relative permeability, and capillary pressure curves. mogeneous nucleation of liquid phase from vapor (Ten Wolde and Frenkel 1998). Wang et al. studied thermody- namic properties in coexistent liquid–vapor systems with 2 Methodology liquid–vapor interfaces (Wang et al. 2001). A sharp peak and a small valley at the thin region outside the liquid– 2.1 The lattice Boltzmann model vapor interface were found to be evidence of a non-equi- librium state at the interface. In our work, we established a The Boltzmann equation describes the evolution with re- similar system to simulate forces between the ﬂuid and gard to a space–velocity distribution function from motions solid components for different ﬂuid densities. of microscopic ﬂuid particles (Atkins et al. 2006). 123 284 Pet. Sci. (2015) 12:282–292 equation implies two kinds of particle operations: stream- of þ n r f þ a r f ¼ Xðf Þ; ð1Þ x n ing and collision. The term on the left side of Eq. (2) de- ot scribes particles moving from the local site x to one of the where t is time, vector x is location, vector n is the ﬂuid neighbor sites x ? e dt within each time step. The ﬁrst term molecular velocity at time t and location x, and f is the on the right side of Eq. (2) describes the collisions con- velocity distribution function of the ﬂuid molecules, which tributing to loss or gain of the particles with a velocity of e . is equivalent to the density of the ﬂuid molecules whose After collision, the velocity distribution will relax to an velocity is n at time t on location x, a is the acceleration of eq equilibrium distribution, f . the ﬂuid molecules, and X(f) is the velocity distribution The ﬂuid density q and its velocity u at one node are function change caused by collision between ﬂuid calculated in Eqs. (3) and (4) (Qian et al. 1992). The re- molecules. lationship between s and ﬂuid kinematic viscosity m can be It is not realistic to solve the integral–differential ðÞ 2s 1 dx described as v ¼ , where dx is the lattice Boltzmann equation directly. An alternative approach is to ðÞ 6dt solve the discrete form of the Boltzmann equation. The constant and dt is the lattice time (Qian et al. 1992; Swift most widely used approach is the LBM. The key idea of et al. 1996). LBM is both the location and velocity of the particles q1 which are discretely characterized (Fig. 1). A typical LB q ¼ f ð3Þ equation can be written as (Shan and Chen 1993; Qian i¼0 et al. 1992) q1 qu ¼ f e ð4Þ i i fðÞ x þ e dt; t þ dt fðÞ x; t i i i i¼0 ð2Þ eq ¼ðÞ f ðÞ x; t fðÞ x; tþ wðÞ x; t ; eq i i We use the equilibrium distribution function f in the standard form (Yuan and Schaefer 2006) where x denotes the position vector, e (i = 0, 1,…, q - 1) is the particle velocity vector to the neighbor sites, q is the eq eq f ðx; tÞ¼ f ðq; uÞ i i number of neighbors, which depends on the lattice ge- ; ð5Þ e u ðÞ e u u i i ometry, f is the particle velocity distribution function along ¼ qx 1 þ þ eq 2 h 2h 2h the ith direction, f is the corresponding local equilibrium distribution function satisfying the Maxwell distribution, s where q and u are for q(x, t) and u(x, t), respectively, is the collision relaxation time, and w is the change in the corresponding to local ﬂuid density and macro velocity. distribution function due to the body force. The LB With this equilibrium distribution function, the ‘‘kinetic temperature’’ in standard LB models, such as ‘‘D2Q9’’ and dx ‘‘D3Q19,’’ is equal to h ¼ (Yuan and Schaefer ðÞ 3dt 2006). In this work, we use the D2Q9 model for the 2D simulations. The weighting factor and discrete velocity for this model are given below: 6 2 5 ðÞ 0; 0 ; i ¼ 0 e e e < 6 2 5 e ¼ ðÞ 1; 0ðÞ 0; 1 ; i ¼ 1; 2; 3; 4 3 ðÞ 1; 1 ; i ¼ 5; 6; 7; 8 3 0 1 < 4=9; i ¼ 0 x ¼ 1=9; i ¼ 1; 2; 3; 4 e e e 7 4 8 1=36; i ¼ 5; 6; 7; 8 7 4 8 For the implementation of the body force term, w,itis appropriate to use the exact difference method (Kupersh- tokh and Medvedev 2006; Kupershtokh 2010) eq eq wðÞ ~ x; t ¼ fðÞ q; ~ u þ D~ u f ðÞ q; ~ u ; ð6Þ i i where the change of the velocity is determined by the force F acting on the node Fig. 1 Illustration of the lattice Boltzmann model 123 Pet. Sci. (2015) 12:282–292 285 the following equations for the P–R equation to be used in Du ¼ FDt=q ð7Þ the LB model (Yuan and Schaefer 2006) F is identiﬁed in three categories: attractive (or repulsive) l r l r l r T T p p q q m m force between local ﬂuid and the ﬂuid of their neighbor ¼ ¼ ¼ ; ð14Þ l r l r l r T T p p q q lattices F , attractive force between the local lattice ﬂuid c c c c m;c m;c ff and the boundary wall F , and the macro body force, e.g., sf where the superscript l and r stand for the lattice and real the gravity, F . systems, respectively, and c means the critical state. With the body force F, the real ﬂuid velocity v should be evaluated at half the time step (Kupershtokh 2010) 2.3 Application of the molecular dynamics q1 simulation to obtain rock–ﬂuid interaction qv ¼ f e þ FDt ð8Þ i i forces F 2 sf i¼0 Thus, the most important part for LB modeling is the de- It is clear that the force acting between ﬂuid components termination of the interaction forces between neighboring and the rock boundary wall has a signiﬁcant effect on ﬂow ﬂuids F and the ﬂuid and rock wall F . ff sf modeling with LBM. In fact, this force decides the capil- lary force and relative permeability curve for multi-phase systems. 2.2 Application of the equation of state to obtain In previous studies (Huang et al. 2009, 2011; Huang and ﬂuid–ﬂuid interaction forces F ff Lu 2009; Martys and Chen 1996; Hatiboglu and Babadagli 2007, 2008), the force between the ﬂuid and boundary wall It is suggested that the ﬂuid–ﬂuid interacting force can be is simpliﬁed as obtained by solving the state equations (Shan and Chen 1993, 1994; Kupershtokh and Medvedev 2009; Yuan and F ðÞ x; t ¼uqðÞ ðÞ x; t G x dðÞ x þ e e ; ð15Þ Schaefer 2006) sf sf i i i i¼0 F ðÞ x; t ¼rUðÞ x; t ; ð9Þ ff where q is the ﬂuid component density, boolean d = 0, 1 to where U(x, t) is a function of state that can be expressed as indicate if it is the ﬂuid or solid lattice, respectively. G is sf (Kupershtokh 2010) the force strength factor between ﬂuid and solid, and it is usually determined by ﬁtting the macroscopic ﬂuid–solid UðÞ x; t ¼ pðÞ q ðÞ x; t ; TðÞ x; t q ðÞ x; t h; ð10Þ m m contact angle. where q is the speciﬁc density of the ﬂuid, which can be There are two disadvantages of this method. One is that expressed as q = q M where M is the standard molar G cannot always be obtained because of the lack of sf quality of ﬂuid component. T is temperature. p(q , T)isa contact angle data for some common ﬂuids such as certain state equation. Replacing U with the effective mass methane, nitrogen, carbon dioxide, and polymers for in- density u, we get stance. Another disadvantage is that the assumption that F sf is directly proportional to the ﬂuid effective mass density uðÞ x; t ¼jj UðÞ x; t ð11Þ yields signiﬁcant error under actual conditions. For ex- Then the interacting force between ﬂuids can be written as ample, this assumption cannot represent the adsorption force for coal gas and shale gas to the formation rock. F ðÞ x; t ¼ 2uðÞ x; truðÞ x; t ð12Þ ff To overcome these limitations, we propose to obtain the In our work, the Peng–Robinson (P–R) state equation is force between ﬂuid components and the rock through used in the expression molecular dynamic simulation. 2 There are a variety of minerals in the reservoir rock. But it is q RT aaðÞ T q m m p ¼ ð13Þ mainly composed of minerals including quartz (SiO ), calcite 2 2 1 bq m 1 þ 2bq ðÞ bq m m (CaCO ), dolomite (CaMg(CO ) ), feldspar (KAlSi O , 3 3 2 3 8 2 2 where a = 0.458R T /p , b = 0.0778RT /p , a(T) = NaAlSi O ,CaAl Si O mixture), and clay (Al Si O (OH) ) c c c c 3 8 2 2 8 3 2 5 4 0.5 2 [1 ? f(x)(1-(T/T ) )] where T and p are the critical with a mixing ratio. In our preliminary investigation, we chose c c c temperature and pressure of ﬂuid components, x is the monocrystalline silicon to represent the rock solid, although acentric factor which depends on the ﬂuid composition itself. elemental silicon does not occur in reservoir rocks. It is According to the correspondence principle, the molar however convenient for model validation against theoretical density q , viscosity l, etc. are the same when the two and experimental results. In our on-going research, we now ﬂuids have the same values of T/T and p/p . Following this use the more realistic assumption that the rock grains are c c principle, the lattice ﬂuid and the real ﬂuid should satisfy composed of SiO and other components. 123 286 Pet. Sci. (2015) 12:282–292 The SPC/E model (Jorgensen et al. 1983) was used to y directions. The length along these two directions is both model water molecules. This model speciﬁes a 3-site rigid 43.45 Angstrom, and the length along the z direction is 60.14 water molecule with charges and Lennard-Jones pa- Angstroms. The solid walls were represented by layers of rameters assigned to each of the 3 atoms. The bond length DM (face-centered cubic) silicon atoms (1152 9 2). The of O–H is 1.0 Angstrom, and the bond angle of H–O–H is speed and the force applied on the Si atoms were both set as 109.47. zero to make them represent boundary walls and to keep a The well-known LJ potential and the standard constant volume of the system. Coulombic interaction potential were applied to calculate For different cases, at the beginning of simulation, water the intermolecular forces between all the molecules. The molecules of different numbers were sandwiched by the parameters are listed in Table 1. The cross parameters of two solid walls. LJ potential were obtained by Lorentz Berthelot combining Firstly, using the constant NVT (certain number, vol- rules. As originally proposed, this model was run with a 9 ume, and temperature–canonical ensemble) time integra- Angstrom cut-off for both LJ and Coulombic terms. tion via the Nose/Hoover method, the whole system was set LAMMPS code was used to construct the simulation at a uniform temperature of 373.15 K (110 C). system. Then an annealing schedule of 340,000 steps was ap- As shown in Fig. 2, the simulation domain was a box with plied to allow the system to reach equilibrium (there were periodic boundary conditions applied in the x and four cycles altogether in the annealing schedule. In one single cycle, the temperature is raised from 373.15 to 423.15 K in 20,000 steps, to 473.15 K in 20,000 steps, and Table 1 Parameters for LJ potential and Coulombic potential then reduced to 423.15 K in 20,000 steps, to 373.15 K in Parameters r, Angstroms e, kcal/mole q,e 20,000 steps). After that, the equilibrium state of the sys- tem was reached at 373.15 K (110 C). Then the NVT Silicon 3.826 0.4030 0 ensemble was changed into NVE (certain number, volume, Oxygen 3.166 0.1553 -0.8476 and energy– micro-canonical ensemble) and run for 20,000 Hydrogen 0.0 0.0 0.4238 steps. In these two processes, the equivalent length scale for the simulations was 0.5 feets to ensure energy conservation. With the equilibrium system, the force between Si and H O can be calculated. As Fig. 2 shows, since the distance between domain 1 and the solid wall is larger than the cut- off radius (9 Angstroms), the water molecules in domain 1 can be considered as free molecules. So the mass densities of free water in different cases were calculated based on the molecules in this domain. For the water molecules in do- main 2, the force F applied by the silicon solid wall was directly calculated according to the potential ﬁeld in the simulation. Then we can get the stress r = F/A, where A is the area of the wall. 2.4 Integrated workﬂow from MD to LBM simulation For the ﬁrst step, we build a MD model as presented in Sect. 2.3 to calculate the rock–ﬂuid interaction forces F sf between any solid component of the wall and the ﬂuid of different density. Input parameters for this step include molecular species of the ﬂuid and boundary solid and the potential parameters. The result of this step is the rela- tionship between the solid–ﬂuid interface and the ﬂuid Fig. 2 The molecular dynamics model of the Si–H O system. density. L = L = 43.45 Angstrom, L = 60.14 Angstrom. Green atom is x y z Then we calculate u(q) of different ﬂuid density q from for Si, pink for O, and white for H. The density of water phase is P–R EOS calculations as presented in Sect. 2.2. The ﬂuid– determined by H O molecules in domain 1. Force between water and ﬂuid interaction force F is then determined from Eq. (12). the boundary is determined by H O molecules in domain 2 2 sf 123 Pet. Sci. (2015) 12:282–292 287 Input parameters for this step include ﬂuid density, critical (a) (b) pressure and temperature, and acentric factor of the ﬂuid 1000 component. With an actual core X-ray scan image for lattice grid construction and the interaction forces being determined from MD and EOS calculation, we use the LB method to simulate the ﬂuid ﬂow in porous media and to obtain the absolute permeability, relative permeability, and capillary (c) (d) pressure curves. The integrated workﬂow is illustrated in Fig. 3. 3 Results 3.1 LBM simulation on liquid–vapor phase transition process of water Fig. 4 Mass density (kg/m ) distribution of water vapor and liquid at different time steps t. a t = 250. b t = 500. c t = 1000. d t = 2000 To validate the EOS model, the single-component ﬂuid phase transition process was simulated using the model low (about 0.76 kg/m ), which indicates it is saturated with described in Sect. 2.2. water vapor. It is clearly shown in this ﬁgure that the small Water is the ﬂuid we choose in our simulation. Its cri- liquid droplets aggregate to form bigger ones as time in- tical temperature is 373.99 C, critical pressure is creases. Eventually, all these small droplets coalesce to 22.06 MPa, critical density is 322.0 kg/m , and its acentric form bigger droplets and the rest of the space of the factor is 0.344. computational domain is occupied by vapor only. All these A 2D 200 9 200 square lattice was used in this simulation results for the single-component phase transi- simulation, and the periodic boundary conditions were tion process are consistent with those reported by Zeng applied on the four boundaries. The average mass density et al. (2009) and Qin (2006). of the computational domain was set to be 500 kg/m .At We then repeated the simulation at different tem- the beginning of simulation, the mass density was ho- peratures for model validation. Figure 5 shows the density mogenously initialized with a small (0.1 %) random perturbation. Figure 4 shows the mass density contour in the com- putation domain at different time steps (t = 250, 500, 1000 and 2000) when the temperature is 110 C. The density in the pink area is high (793 kg/m ), which indicates it is saturated with water liquid. The density in the green area is Single component multi-phase flow in porous media Interaction between Interaction between fluid component fluid and rock media Liquid density, Exp. P–R EOS MD simulation Vapor density, Exp. Liquid density, EOS Lattice Boltzmann model Pore network structure Vapor density, EOS Liquid density, LBM Vapor density, LBM Absolute Relative Capillary permeability permeability pressure 0 200 400 600 800 1000 ρ, kg/m Fig. 3 Schematic illustrations of the workﬂow. First, the interaction forces between ﬂuids are determined from P–R EOS calculation, and Fig. 5 Saturated water (in blue) and vapor (in red) densities versus the rock–ﬂuid interaction force is determined from MD simulation. temperature. ‘‘o’’ is the result of LB simulation. The solid line is the Then we run the lattice Boltzmann simulation based on the grid solution of the P–R equation using the Maxwell equal-area construc- constructed from the core X-ray scan image to obtain the absolute tion. asterisk Denotes the experimental data from Handbook of permeability, relative permeability, and capillary pressure curves Chemical Engineering (Liu et al. 2001) T, ºC 288 Pet. Sci. (2015) 12:282–292 distribution curves of both vapor and liquid when they vapor density is smaller than 0.787 kg/m or the liquid reach phase equilibrium at different temperatures. This density is greater than 793 kg/m . The relationship be- ﬁgure indicates that the density curve calculated with LBM tween density and stress is plotted in Fig. 7. It clearly is nearly identical to that obtained from P–R EOS using the shows that the stress varies with the ﬂuid density nonlin- Maxwell equal-area construction (Yuan and Schaefer early, thus demonstrates the assumption that ‘‘the force 2006). It suggests that the LBM, combined with EOS, between ﬂuid and rock is supposed to be proportional to the provides an accurate method to model single-component ﬂuid density’’ used by former studies (Martys and Chen gas–ﬂuid two-phase ﬂow. 1996; Hatiboglu and Babadagli 2007, 2008) is not It is noted that there are some differences between the appropriate. theoretical solution using EOS and the experimental results for liquid density calculations, and this is originated from 3.3 Calculation of monocrystalline silicon–water the inadequacy of P–R EOS and is in agreement with contact angle (Atkins and William 2006). Contact angle reﬂects the interaction forces at the phase 3.2 MD simulation in determination of force interfaces and the wettability. Thus, it is a signiﬁcant pa- between ﬂuid components and the boundary rameter to describe the interaction between oil, gas, water, wall and rock in petroleum reservoirs (Wolf et al. 2009). We simulated the contact angle of the monocrystalline In the MD simulation, 42 sets of systems with different Si–water liquid–water vapor system by applying our inte- H O densities were generated. We select 4 cases for il- grated MD–LBM approach and then compared the value lustration in Fig. 6. with literature for validation. As shown in Fig. 5, at the simulation temperature Figure 8 demonstrates the equilibrium state of the sys- (110 C), the force calculation is only meaningful when the tem after LBM simulation. In the simulation, the compu- tational domain is 100 9 50, where the upper and lower boundaries are solid walls, and the east and west boundary is periodic. For the solid node, before the streaming step, a bounce–back algorithm was implemented to mimic the non-slip wall boundary condition. In Fig. 8, the gray grid on the top and bottom is for Si, the red is for water liquid, and the blue is for water vapor. The interaction force between water ﬂuids was calculated by EOS, and the interaction force between water and Si was obtained by MD. This ﬁgure shows that the macro contact angel is approximate 101, which is consistent with the result given by (Williams and Goodman 1974) in which it states that the contact angle is near 90. Therefore, the method of MD used in ﬂuid–solid interaction force calcu- lation incorporated into LBM is reasonable in porous me- dia ﬂow simulation. 3.4 Rock permeability determination based on an actual X-ray CT image of a reservoir core Similar to the methodology in previous literature (Guo and Zheng 2009; Huang et al. 2009, 2011; Huang and Lu 2009; Hatiboglu and Babadagli 2007, 2008; Jorgensen et al. 1983), we calculated the relative permeability curves by simulating a two-phase ﬂow in the porous media. Figure 9 is a post-processed 2D X-ray image of a reservoir core in Tarim Basin conglomerate with a length L = 2.7 mm and width A = 1.35 mm. The domain is gridded into Fig. 6 Si–H O system in equilibrium with 4 different water densities. 3 3 540 9 270 = 145,800 cells with each one being a In vapor phase: a q = 0.43 kg/m and b q = 0.66 kg/m . In liquid 3 3 5 9 5 lm square. In this image, the part in black phase: c q = 842 kg/m and d q = 911 kg/m 123 Pet. Sci. (2015) 12:282–292 289 0.8 5.0 (b) (a) 0.6 4.5 0.4 4.0 0.2 3.5 0 0.2 0.4 0.6 0.8 700 800 900 1000 1100 ρ, kg/m ρ, kg/m Fig. 7 The stress–density relationship. a in the vapor region, linear ﬁtted by the least squares method and b in the liquid region, quadratic ﬁtted by the least squares method represents pores and white for the rock grains. Porosity / the image in Fig. 9. The system is at temperature = 43.5 %. T = 110 C. At the initial state, each cell was saturated We applied a single-vapor phase ﬂow of water to cal- with H O steam with a density of 0.754 kg/m . In order to culate the absolute permeability of the core represented by use the MD simulation results on interaction forces be- tween H O and Si consistently, we assumed that the rock grain is made of Si (in our on-going research, we use the more realistic assumption that the rock grains are com- posed of quartz (SiO ) and other minerals). As in Fig. 10, we applied a virtual body force g and periodic boundary conditions in calculating permeability based on LBM solution of the water ﬂow problem. The body force is equivalent of adding a pressure drop Dp ¼ qgL (q ¼ q =n is the average density of water in the i¼1 Body force g Fig. 8 Equilibrium state simulated from LBM for a water liquid drop in contact with the Si surface. The gray grid on the top and bottom is for Si, the red for water liquid, and the blue for water vapor L = 2.7 mm Periodic boundary condition Fig. 10 The ﬂow problem in calculating absolute permeability. Closed boundaries on the top and bottom. A virtual body force g towards right is applied to the simulation domain, and periodic Fig. 9 A 2D CT image of a reservoir rock in Tarim Basin boundary condition is set on left and right sides of the simulation conglomerate domain σ, MPa A = 1.35 mm σ, MPa 290 Pet. Sci. (2015) 12:282–292 simulation domain, n is the total number of the grids ﬁlled with water) between the left and right side of the core. By applying different values of g, we can calculate ﬂow ve- locity under different pressure drops. For a given pressure drop Dp, the ﬂow velocity of H O steam reaches equilibrium after some steps of simulation. The volumetric velocity of H O steam ﬂow is calculated as v ¼ / v =n (Cihan et al. 2009). Figure 11 depicts the g i i¼0 calculated relationship between v and Dp. The linearity between v and Dp shown in Fig. 11 is in Fig. 12 Distribution of vapor (in pink) and liquid (in blue) phases of H O in rock pores agreement with Darcy’s law v = KDp/(lL), where l is the viscosity of water vapor, L is the length of the core, and K is permeability of the core. -6 represent different saturations (Fig. 12). At the simulation Plug L = 2.7 mm and l = 12.4 9 10 Pa s into temperature, 110 C, saturated water vapor density is Darcy’s law and use least square linear ﬁtting of calculated 0.787 kg/m , and saturated water liquid density is 793 kg/ points in Fig. 11, we calculate the permeability of the rock m . Then the average density of water in simulation do- as 636 mD. main, q, was calculated for different saturations. Given that Dp ¼ qgL, the pressure drop for any values of g can be 3.5 Vapor–liquid two-phase relative permeability obtained. determination based on an actual X-ray CT At the simulation temperature, the viscosities of image of a reservoir core -6 saturated water vapor and liquid are l = 12.4 9 10 Pa -6 s and l = 252 9 10 Pa s. Relative permeability, as a function of phase saturation, We calculated the permeability of the vapor and liquid could be modeled by multi-phase ﬂow simulation using the phases using Darcy’s law for different phases, K = v l L/ proposed workﬂow. Similar to the conditions set for ab- g g g Dp, K = v l L/Dp. K and K at different saturations solute permeability simulation, virtual body force g and w w w g w were obtained from repeated simulations and calculations periodic boundary condition were applied to the simulation of volumetric velocity of each phase. Finally, the relative domain, and temperature was set at 110 C. permeability K = K /K and K = K /K is obtained as At initial state, the core was randomly ﬁlled with vapor– rg g rw w shown in Fig. 13. liquid two-phase water at different mixing ratios to -3 1.0 x 10 rw 0.9 rg 0.8 0.7 0.6 0.5 0.4 0.3 2 0.2 0.1 1 Simulated result Least-squares method fitted result 0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 200 250 300 Δ p, Pa Fig. 13 Calculated vapor–liquid two-phase water relative perme- Fig. 11 Calculated water ﬂow velocity versus pressure drop ability curves v , m/s r Pet. Sci. (2015) 12:282–292 291 0.20 Different from the simulation of absolute permeability, the multi-phase ﬂow modeling results using a statistical method such as LBM could be unstable, thus causing the volumetric velocity obtained at different times be slightly 0.15 different. This means that the relative permeability of each phase at a given saturation could be a distribution as shown in Fig. 13. The relative permeability curves were con- structed by taking the average values of K at each r 0.10 saturation. 3.6 Capillary pressure curve determination based 0.05 on an actual X-ray CT image of a reservoir core We also applied our methodology to calculate the capillary pressure curve. As shown in Fig. 14, constant pressure 0.00 0 0.2 0.4 0.6 0.8 1.0 boundary conditions (p on the left and p on the right) and 1 2 closed boundary conditions on the top and bottom were applied to the simulation domain. Fig. 16 Calculated vapor–liquid H O capillary pressure curve In the initial state, the domain was ﬁlled with saturated water vapor. We injected saturated water liquid from the 4 Conclusions left side under constant pressure p . Since p [ p , water 1 1 2 vapor will ﬂow out of the domain from the right side. By In this work, we proposed a new systematic workﬂow to applying different pressure drops Dp = p -p , the injected 1 2 integrate MD simulation with the LB method to model liquid ﬂowed into the pores to yield different phase multi-phase ﬂow in porous media. As an improvement, this saturations. At an equilibrium state as shown in Fig. 15, the new approach avoids parameter ﬁtting or incorrectly as- capillary pressure is equal to the pressure drop, i.e., suming a linear relationship between the rock–ﬂuid inter- p (S ) = Dp. The calculated capillary pressure curve is c g action force and ﬂuid density. We have validated this shown in Fig. 16. approach by simulating a two-phase separation process and a gas–liquid–solid three-phase contact angle. The success of MD–LBM results in agreement with published EOS solution, and experimental results demonstrated a break- Liquid Vapor through in pore-scale, multi-phase ﬂow modeling. Based inflow outflow on an actual X-ray CT image of a reservoir core, we ap- plied our workﬂow to calculate absolute permeability of p p 1 2 the core, vapor–liquid water relative permeability, and capillary pressure curves. With the application of this workﬂow to a more realistic model considering actual Fig. 14 The ﬂow problem in calculating capillary pressure curve. A reservoir rock and ﬂuid parameters, the ultimate goal is to constant pressure boundary is set on both sides of the simulation develop an accurate method for prediction of permeability domain tensor, relative permeability, and capillary curves based on 3D CT image of the rock, actual ﬂuid, and rock components. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, dis- tribution, and reproduction in any medium, provided the original author(s) and the source are credited. References Ahlrichs P, Du¨nweg B. 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Petroleum Science – Springer Journals

Published: Mar 28, 2015

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From molecular dynamics to lattice Boltzmann: a new approach for pore-scale modeling of multi-phase flow

From molecular dynamics to lattice Boltzmann: a new approach for pore-scale modeling of multi-phase flow

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

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From molecular dynamics to lattice Boltzmann: a new approach for pore-scale modeling of multi-phase flow

From molecular dynamics to lattice Boltzmann: a new approach for pore-scale modeling of multi-phase flow

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