From Molecular Dynamics to Lattice Boltzmann: a New Approach for Pore-Scale Modeling of Multi-Phase ﬂow

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 Ó 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 H2O 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, Keywords Molecular dynamics Á Lattice Boltzmann Á laboratory experiments are usually expensive, not robust Multi-phase flow Á 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 [email protected] It is desirable to obtain the core properties through nu- merical modeling based on actual pore structure charac- 1 College of Engineering, Peking University, Beijing 100871, terizations. Recently, the lattice Boltzmann (LB) method, China which is based on a molecular velocity distribution func- 2 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 3 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 Edited by Yan-Hua Sun blue-colored particles to represent two types of fluids 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 H2O 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-equilibrium 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 of equation implies two kinds of particle operations: stream- þ n Ár f þ a Ár f ¼ Xðf Þ; ð1Þ ot x n ing and collision. The term on the left side of Eq. (2) describes 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 i on the right side of Eq.

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