PUBLICATIONS. Water Resources Research. An adaptive lattice Boltzmann scheme for modeling two-fluid-phase flow in porous medium systems

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1 PUBLICATIONS Water Resources Research RESEARCH ARTICLE Key Points: The lattice Boltzmann method (LBM) is a useful tool in the study of multiphase flow The approach of two-phase flow to equilibrium is a slow process dominated by fluid interfaces The adaptive LBM algorithm enables the efficient and accurate simulation of flow Correspondence to: A. L. Dye, alynnd@live.unc.edu Citation: Dye, A. L., J. E. McClure, D. Adalsteinsson, and C. T. Miller (2016), An adaptive Lattice Boltzmann Scheme for modeling two-fluid-phase flow in porous medium systems, Water Resour. Res., 52, , doi:. Received 23 OCT 2015 Accepted 7 MAR 2016 Accepted article online 9 MAR 2016 Published online 3 APR 2016 VC American Geophysical Union. All Rights Reserved. An adaptive lattice Boltzmann scheme for modeling two-fluid-phase flow in porous medium systems Amanda L. Dye 1, James E. McClure 2, David Adalsteinsson 3, and Cass T. Miller 1 1 Department of Environmental Sciences and Engineering, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA, 2 Advanced Research Computing, Virginia Tech Institute and State University, Blacksburg, Virginia, USA, 3 Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA Abstract We formulate a multiple-relaxation-time (MRT) lattice-boltzmann method (LBM) to simulate two-fluid-phase flow in porous medium systems. The MRT LBM is applied to simulate the displacement of a wetting fluid by a nonwetting fluid in a system corresponding to a microfluidic cell. Analysis of the simulation shows widely varying time scales for the dynamics of fluid pressures, fluid saturations, and interfacial curvatures that are typical characteristics of such systems. Displacement phenomena include Haines jumps, which are relatively short duration isolated events of rapid fluid displacement driven by capillary instability. An adaptive algorithm is advanced using a level-set method to locate interfaces and estimate their rate of advancement. Because the displacement dynamics are confined to the interfacial regions for a majority of the relaxation time, the computational effort is focused on these regions. The proposed algorithm is shown to reduce computational effort by an order of magnitude, while yielding essentially identical solutions to a conventional fully coupled approach. The challenges posed by Haines jumps are also resolved by the adaptive algorithm. Possible extensions to the advanced method are discussed. 1. Introduction Multiphase flow in porous media is modeled on several different length scales (molecular scale, microscale, macroscale, and megascale), each of which is inherently linked to the other scales [Gray and Miller, 2014]. The microscale and the macroscale are two continuum length scales of interest in this work. At the microscale, the morphology and topology of the pore space and phase distributions are resolved. At the macroscale, the details of the system at the microscale are not resolved. Since the details of the microscale geometry are not resolved, average geometric properties are used to characterize porous medium systems, such as the fractional volume of the porous media occupied by each phase, the volume of pore space within the total volume (porosity), and interfacial area between phases per volume (specific interfacial area). In order to identify and characterize these properties, the characteristic length at the macroscale must be large enough that the volume fractions will not be effected by small changes in the length scale. Therefore, upscaling from the microscale to the macroscale is commonly done by averaging over a representative elementary volume (REV), a volume that is smaller than the domain of the system but large enough to permit a meaningful statistical average [Bear, 1972]. A primary objective of microscale study is to advance understanding of macroscopic system behavior such that the microscopic details of flow can be neglected. To accomplish this endeavor, important aspects of the microscopic physics must be represented by the macroscopic model formulation in the form of closure relations. The thermodynamically constrained averaging theory (TCAT) provides a means for obtaining macroscopic models from microscopic models via averaging schemes [Gray and Miller, 2005, 2014]. At the microscale, conservation equations are combined with thermodynamics, constitutive laws and simplifying assumptions to obtain a microscopic description of the system. To obtain a macroscopic model, the thermodynamics and conservation equations are averaged from the microscale. This ensures that all macroscopic variables are rigorously defined in terms of more familiar microscale variables, eliminating any ambiguity with respect to these variables. The macroscopic equations are then combined with simplifying assumptions and constitutive relationships, which must be determined in order to obtain a closed model. Microscale simulations provide a mechanism to study these constitutive relationships provided that simulations DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2601

2 are able to accommodate the complex solid morphology and large domain sizes needed to produce results that can be scaled to a macroscopically significant length scale (REV) and to adequately resolve the relevant physical mechanisms. Due to the simplicity by which fluid and solid interfaces are treated, the lattice Boltzmann method (LBM) has become a primary tool for simulation of multiphase flow in porous media at the microscale [Bao and Schaefer, 2013; Aidun and Clausen, 2010; Ahrenholz et al., 2008; T olke et al., 2006; Zheng et al., 2006]. The significance of this role is evidenced by widespread LBM investigations of multiphase flow behavior due to well-documented deficiencies in existing macroscopic model formulations [Miller and Gray, 2002; Jackson et al., 2009]. Traditional closure relations state a functional dependence between capillary pressure and fluid saturation [van Genuchten, 1980; Brooks and Corey, 1966] that depend upon the system history due to nonwetting phase entrapment and other pore-scale effects. It has been posited that the specification of specific interfacial area and other state variables may reduce or eliminate the hysteresis observed in traditional closure relations [Li et al., 2005; Bradford et al., 2000; Celia et al., 1993]. LBM simulations of drainage and imbibition have been widely used to investigate the functional dependence of capillary pressure in a two-fluid-phase porous medium system [Dye et al., 2015; Porter et al., 2009; Sukop et al., 2008; Schaap et al., 2007; Pan et al., 2004]. Alternatively, the microscale physics of twofluid-phase flow have also been studied experimentally using microtomography imaging techniques [Armstrong and Berg, 2013; Wildenschild and Sheppard, 2013; Brusseau et al., 2006]. Simulations of drainage and imbibition processes are performed typically by varying the external fluid pressures and allowing the system to equilibrate before the next step change in fluid pressures. Equilibrium is usually assumed when the change in capillary pressure and the change in fluid saturations over time becomes negligible. However, recent data have shown that the interfacial curvature of the system continues to change long after the boundary pressures and saturations approach their equilibrium state, suggesting previous LBM simulations as well as experiments were not carried out to true equilibrium [Gray et al., 2015; Armstrong et al., 2012]. To accurately study the dependency of capillary pressure on state variables in a two-fluid-phase porous medium system using an LBM, simulations of drainage and imbibition have to be performed until the interfacial curvature reaches an equilibrium state. To obtain data needed to assess the equilibrium relationship, many drainage and imbibition equilibrium states must be simulated. Each equilibrium state can take on the order of hours to days to compute based on a variety of factors, including the size of the domain, discretization of the system, the size of the pressure step taken, and the phase distributions of the initial system. The inherent expense involved with extant LBM simulation approaches for modeling two-fluid-phase flow motivates a need for more efficient methods. The goal of this work is to increase the overall efficiency of simulating both dynamic and equilibrium states in two-fluid-phase porous medium systems. The specific objectives of this work are 1. to evaluate the characteristics of the flow phenomena being modeled; 2. to design an algorithm to reduce the computational effort of two-phase flow simulations compared to existing approaches; 3. to validate that the proposed method simulates the physical systems of focus accurately; and 4. to assess the computational performance of the method compared to a standard approach. 2. LBM Model Formulation We implemented a two-dimensional, nine velocity vector (D2Q9) formulation to model two-fluid flows in complex geometries. Fluid-phase pressures, saturations, and interfacial curvatures are computed from the resulting microscale state of the system. The formulation is based on a color method, which have been applied successfully to model two-fluid flows in three-dimensional porous medium systems [McClure et al., 2014; Ahrenholz et al., 2008]. In our approach, lattice Boltzmann equations (LBEs) are constructed to recover conservation laws in the limit of low Mach number. To model two-fluid flows, LBEs are constructed to track the mass transport for each fluid component, and a third LBE is used to model the momentum transport. In this section, we describe the formulation, beginning with the momentum transport LBE and then describing the LBEs used to model the mass transport. DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2602

3 The LBE used to model the momentum transport uses a MRT scheme to recover the Navier-Stokes equations with additional terms that are needed to model the effect of interfacial forces. The method advances a set of velocity distribution functions ff i ji50; 1;...8g fromonediscretetimeleveltothe next on a regular two-dimensional lattice d s Z 2, corresponding to nine fixed velocity vectors per lattice site fe i ji50; 1;...8g, using both a collision operator and an advective streaming equation. The fluid pressure and velocity depend linearly on the distribution functions. The nine discrete velocities of a D2Q9 lattice are defined by formulating a quadrature scheme for the molecular distributions [He and Luo, 1997] ( ) e i 5 : (1) The discrete velocities e i characterize the connection of each lattice site x k 2 d s Z 2 to eight neighboring lattice sites fx ki ji50; 1;...8g. The distribution functions f associated with the discrete velocities e i at each lattice site x k evolve according to an MRT form of the lattice Boltzmann equation f i ðx k 1e i dt; t n 1dtÞ2f i ðx k ; t n Þ5C i ; (2) where dt is the time step, t n is discrete time, and C i is a collision operator. The collision operator for the MRT-LBM model can be formulated as C i 5M 21^S m eq ðx k ; t n Þ2mðx k ; t n Þ ; (3) where M is an orthogonal transformation matrix which linearly maps the distribution functions f to the velocity moments m, written as m5m f: (4) The values of the transformation matrix M are chosen based on a Gram-Schmidt orthogonalization constructed using polynomials of the discrete velocities e i [Lallemand and Luo, 2000]. The nine velocity moments fm i ji50; 1;...8g include the density variation dq and momentum j, which are conserved moments in the system. The pressure p, which is proportional to the density q, is determined by summing over all distributions: p5c 2 s q5 1 3 p where c s 51= ffiffiffi 3 is the speed of sound for the model. The momentum is defined as j5q 0 u5 X8 i50 X 8 i50 f i ; (5) f i e i ; (6) where u is the velocity and q 0 is a reference density. The remaining moments are nonconserved quantities that relate to energy and the strain-rate tensor. The MRT formulation allows each moment m to relax toward its equilibrium value m eq at a rate specified by the diagonal matrix ^S. The relaxation rates k i for the conserved moments are set to zero since they are not affected by collisions, which is written as For the nonconserved moments, the relaxation rates are set as and where the parameter k l is related to the dynamic viscosity by k 0 5k 3 5k 5 50: (7) k 1 5k 2 5k 7 5k 8 5k l (8) k 4 5k 6 58 ð22k lþ ð82k l Þ ; (9) DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2603

4 l5 1 3k l : (10) The equilibrium moments m eq are constructed to recover the Navier-Stokes equations, as outlined in Lallemand and Luo [2000], with additional terms that account for interfacial stresses: m 0eq 5q; (11) m 1eq 522q1 3 ðjx 2 q 1j2 y Þ2rjCj; (12) 0 m 2eq 5q23ðjx 2 1j2 y Þ; (13) m 3eq 5j x ; (14) m 4eq 52j x ; (15) m 5eq 5j y ; (16) m 6eq 52j y ; (17) m 7eq 5 1 ðjx 2 3q 2j2 y Þ rjcjðn2 x 1n2 yþ; and (18) m 8eq ðj xj y Þ1 1 2 rjcjðn xn y Þ; (19) where the parameter r is related linearly to the interfacial tension c wn. The color gradient of the phase density field / is computed as C5 3 X 8 cs 2d w i e i /ðt; x k 1e i dtþ: (20) t i50 The dependence of the momentum transport LBE on / couples the mass and momentum transport. The nondimensional phase density field is defined as /5 q w2q n q w 1q n ; (21) where q w and q n are the densities of the wetting and nonwetting phases, respectively. The value of / is constant in the bulk of each fluid phase and varies within the interfacial region between the two fluid phases. The value of / is set to a constant value within the solid phase, written as /ðx k Þ5/ s for x k 2 X s : (22) The desired contact angle between the fluid-fluid interface and the solid is obtained by setting the value of / s. A mass transport solution is obtained using a separate set of distributions to track q w and q n. A recoloring scheme is applied to prevent mass fluxes across the interface for either component [Leclaire et al., 2011; Latva-Kokko and Rothman, 2005]. The mass transport distributions are defined as g iw 5w i q w 11 3 cs 2 e i u 1f q wq n n e i and (23) q w 1q n g in 5w i q n 11 3 cs 2 e i u 2f q wq n q w 1q n n e i ; (24) where the parameter f determines the interfacial thickness, and the unit normal vector is defined as The quadrature weights are given as n5 C jcj : (25) DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2604

5 8 4 9 ; for q 2 f0g >< 1 w i 5 ; for q 2 1; 2; 3; 4 9 f g >: 1 ; for q 2 5; 6; 7; 8 36 f g: The density values for the subsequent time step can then be computed as (26) q a ðx k ; t n 1dtÞ5 X8 i50 g ia ðx k 2e i dt; t n Þ for a5w; n : (27) Continuum expressions for conservation of mass for each fluid phase are provided by combining equation (27) with the mass transport distributions defined in equations (23) and (24) Geometric Analysis Average phase pressures and saturations, and interfacial curvatures can be computed by integration of microscale LBM simulations. The regular lattice used for the LBM simulations provides a straightforward way to evaluate the averaged quantities. The phase density field / computed using equation (21) is used to identify the regions of the pore space occupied by the wetting (w) phase, the nonwetting (n) phase, and the wetting-nonwetting (wn) interface. Because the position of the solid (s) phase is specified, all phases and interfaces can be identified explicitly. Based on the phase density field, the domain X of each entity of interest is defined according to X w 5fx : /ðxþ < 0g ; (28) X n 5fx : /ðxþ > 0g ; and (29) X wn 5fx : /ðxþ50g : (30) Once the entity associated with each lattice site is identified according to equations (28) and (30), the average fluid saturations and pressures, and the fluid-fluid interfacial curvature can be computed. The phase saturations are computed according to X 1 sa X x k 2X a 1 ; (31) x k 2X w[x n where s a is the saturation of fluid phase a. Similarly, the phase pressures are calculated by X p p a X x k 2X a a 1 ; (32) x k 2X a where p a is the average pressure of phase a and p a is the pressure observed within a lattice site occupied by fluid phase a. Based on the interface definitions in equation (30), the normal vector to the w phase can be computed as the gradient of the phase density field / evaluated on the wn interface (/50), given as n w 5 r/ jr/j : (33) /50 The normal vector n w is calculated at the lattice sites using a second-order central finite difference scheme. The mean curvature of the wn interface is then evaluated on the lattice by taking the divergence of the normal vector n w, defined by J w 5rn w : (34) Using finite difference approximations for the spatial derivative, the right hand side of equation (34) is determined at each lattice site. The interfacial curvature calculation was validated by J. E. McClure et al., (Connecting pore-scale simulation to macroscale theory for two-fluid-phase flow in porous media, DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2605

6 submitted to Journal of Fluid Mechanics, 2016). To approximate the mean curvature for the entire wn interface, the mean curvatures are averaged according to X Jw wn J X x k 2X w wn 1 ; (35) x k2x wn where J wn w is the average mean curvature of the wn interface Parameter Estimation In the LBM three parameters must be established: f, which controls the interfacial width, r, which determines the interfacial tension, and / s, which controls the contact angle. A f of 0.9 was used based on previous analysis that determined the parameter is independent of interfacial tension and there is no advantage to having a large interfacial thickness [McClure et al., 2014]. The r parameter was set to 0.01 and a twodimensional bubble test in the absence of a solid phase was used to measure the corresponding interfacial tension. Based on the chosen value for r, a constrained bubble test in a capillary tube was run to establish an appropriate value of / s. The bubble test is used in multiphase LBMs to measure the interfacial tension [Chen et al., 2014; Philippi et al., 2012], and it is a validation that shows that the measured difference in the phase pressures is equivalent to the product of the interfacial tension c wn and the radius of curvature R as defined by Laplace s law p n 2p w 5 c wn R ; (36) where p n is the nonwetting phase pressure and p w is the wetting pressure. In the bubble test, a circular shaped bubble of n phase with a defined radius R was immersed in the wetting phase. Periodic boundary conditions were imposed in both the x and y directions and the system was allowed to reach steady state. When equilibrium was established, the radius of curvature R was determined based on the area of the immersed bubble and the values of the p n and p w were extracted by using the phase indicator field / to identify the maximum pressure value within each phase. Equilibrium was defined when the change in p n p w between time steps was less than Simulations for various bubble sizes were carried out for r Plotting p n p w as a function of 1/R for the various bubble sizes resulted in a slope of , which corresponded to the c wn in lattice units used in this work. The contact angle of the LBM was defined using a constrained bubble test in a capillary tube as suggested by Huang et al. [2007]. In this case, a two-dimensional capillary tube with radius R was used to provide a constraint on the equilibration of the bubble. Initially, a circular shaped bubble of nonwetting phase with a radius R was placed in the center of the capillary tube and immersed in the wetting phase. Periodic boundary conditions were imposed in the x direction, with the y boundary being closed to flow. The system was allowed to progress to a steady state. When equilibrium was established, the values of the p n and p w were measured. The difference in fluid pressures was related to the contact angle u ws;wn between the wetting phase-solid and wetting-nonwetting phase interfaces through the two-dimensional Young-Laplace equation given by p n 2p w 5 c wncos u ws;wn R : (37) Simulations in a capillary tube of radius R 5 20 were performed for various values of / s. Given the known values for c wn and R, the contact angle u ws;wn for each / s value was determined by computing p n p w. / s was chosen to be 0.95, which corresponded to a contact angle of Validation In the LBM, the interfacial tension c wn is set by choosing the parameter r and the phase pressure difference and interfacial curvature are computed directly from the microscale. Therefore, Laplace s law at equilibrium can be used to verify the LBM. At equilibrium, the phase pressures, interfacial tension and interfacial curvature are constant in the example considered, which means that p n 2p w 52c wn J wn w : (38) DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2606

7 Figure 1. The difference in phase pressures p n p w as a function of the product of the interfacial tension c wn and interfacial curvature Jw wn as determined using a series of two-dimensional bubble tests in the absence of a solid phase. The slope of the line is The macroscopic nonwetting phase pressure, p n, and the macroscopic wetting phase pressure, p w, were calculated by averaging the microscale phase pressure values over the domain using equation (32). The interfacial curvature of the wettingnonwetting interface was computed directly at the microscale and averaged over the fluid-fluid interface to obtain the macroscopic curvature Jw wn (equation (35)). Simulations for various bubble sizes were carried out for r The results for each bubble test performed are plotted in Figure 1. The slope of the line is It is shown from the LBM simulations that the difference in phase pressures p n p w agrees with the product of the interfacial tension and interfacial curvature at equilibrium, validating Laplace s law. 3. Computational Challenges Figure 2. Two-dimensional porous media in which the solid is represented by gray and the pore space is represented by white. The D2Q9, MRT LBM described in section 2 was used to simulate a drainage sequence in a two-dimensional porous media containing a distribution of circular grains with a porosity of 0.52 depicted in Figure 2. The twodimensional case simulated a microfluidic cell, which is an experimental approach used routinely to advance the fundamental understanding of porous medium systems [Dye et al., 2015; Karadimitriou et al., 2013; Zhang et al., 2011; Kim et al., 2012; Sohrabi et al., 2008]. In the LBM simulation, the grid resolution of the porous media was lattice sites, resulting in approximately 68 lattice sites per mean grain diameter. The initial condition was a fully wetting-phase-saturated system. Constant pressure boundary conditions were set on one inlet face for one fluid and on the other inlet face for the second fluid, with no-flow boundaries on all other boundaries. The pressure boundary conditions were adjusted in discrete time steps and the simulator was run until an equilibrium state was achieved, observing both the final equilibrium state and the dynamics of the approach to the equilibrium state. The LBM results were used to calculate the average phase pressures p n and p w, wetting phase saturation s w, and average interfacial curvature Jw wn at each time step. The simulation of each pressure step was run until the system approached an equilibrium state as determined by the microscale interfacial curvature. After each time step, the L 2 norm of the curvature at every point on the fluid-fluid boundary, denoted J w, was computed and compared to the value of the L 2 norm of J w at the previous time step, yielding the difference in the L 2 norm of the curvature over a time step denoted as DJ w. The approach to equilibrium was terminated when DJ w The simulation results were used to analyze the timescales over which the changes in phase pressures, wetting phase saturation, and interfacial curvature approach an equilibrium state in light of the experimental observations DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2607

8 Figure 3. The phase pressure difference, wetting phase saturation, and capillary pressure values as they evolved in time from one equilibrium state to the next. The LBM data were normalized to bring all values into the range of [0,1], where 0 represents the initial equilibrium state of the system and 1 represents the final equilibrium state. made by Gray et al. [2015] using a microfluidic imaging technique. Based on the LBM simulation, the phase pressures, wetting phase saturation, and interfacial curvature change rapidly during the first 5,000 time steps followed by a period over which they relax at more gradual rates toward equilibrium, as shown in Figure 3. The phase pressures relax at a slow rate until approaching an equilibrium value at about 1,30,000 time steps. The saturation changes at a much slower rate than the phase pressures until eventually approaching an equilibrium value at roughly the same time the phase pressures reach an equilibrium state. Similar to the data reported by Gray et al. [2015], the interface continues to relax to an equilibrium state long after the phase pressures and saturation come to equilibrium. The interface approaches an equilibrium state at approximately 220,000 time steps. The LBM simulation confirmed that the approach of two-fluid-phase flow to an equilibrium state involves a relatively slow process in which the fluid interfaces relax to their equilibrium state. 4. Adaptive LBM Algorithm Computational performance for the LBM is determined by the number of arithmetic operations that must be performed, and the efficiency with which they can be done given memory limitations. In an effort to improve the efficiency of LBM simulations for two-fluid-phase flow, a temporally adaptive domain decomposition LBM algorithm was developed based on the physical phenomena detailed in section 3. Both microfluidic experiments and LBM simulations show that the bulk fluid relaxation time is considerably less than the interfacial relaxation time. This observation is the motivation for an adaptive LBM that aims to increase the computational efficiency by decreasing the size of the computational domain in accordance with the rate of relaxation of the system toward an equilibrium state. The interfacial speed g is used to track the relaxation of the system. To evaluate the interfacial speed g, a level-set approach is used. The level-set method was introduced by Osher and Sethian [1988] and has become a widely used method for simulating multiphase flows [Prodanović and Bryant, 2006; Olsson and Kreiss, 2005; Tryggvason et al., 2001; Adalsteinsson and Sethian, 1999]. The level-set equation is defined 1gjr/j50 ; where / is the phase density field defined by equation (21) and g is the speed of the interface in the normal direction [Sethian, 1999]. Rearranging equation (39) g52 jr/j ; (40) an expression for the interfacial speed that can be approximated numerically. The vector r/ is computed at each time step of the LBM as discussed in section 2. The time derivative of / is approximate using a second-order finite difference approximation /ðt1dtþ2/ðt2dtþ : (41) 2dt DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2608

9 In order to evaluate equation (41), a brief history of the phase density field /, including /ðx; t2dtþ; /ðx; tþ, and /ðx; t1dtþ, must be retained. Once the time derivative of / has been computed at the lattice sites along the wn interface, the value of g at the interfacial lattice sites can be approximated using equations (40) and (41). The interfacial velocity at each lattice site on the wn interface is then computed as w wn 5gn w : (42) This method for calculating interfacial velocity was validated by McClure et al. (submitted manuscript, 2016). Initially, the LBM is run on all fluid phase lattice sites. At each time step X wn is identified according to equation (30) and the interfacial speed g is calculated at each lattice site along the interface using equation (40). When the interfacial speed is decreasing monotonically between time steps and falls below j w, the computational domain is reduced to a subdomain of the total system domain, denoted as X H. For this work, j w was set to , which is the maximum speed of the interface in lattice units per time step. The subdomain consists of an interfacial region X I and a halo surrounding the interfacial region to account for movement of the wn interface as the system relaxes. The interfacial region represents the portion of the computational domain that is the slowest to approach an equilibrium state. Within the interfacial region the phase properties such as the pressure and density deviate from the bulk of the fluid phase. The phase density field / obtained from the LBM has a value of 61 in the bulk phase and varies continuously within the interfacial region. Therefore, the phase density field information can be used to distinguish between the interfacial region and the bulk phase region according to X P 5fx : jr/j < eg ; and (43) X I 5fx : jr/j eg ; (44) where e is chosen so that surface contributions to the continuum variables are negligible within the bulk phase region X P. For this work, e was set to The width of the halo surrounding the interfacial region is dependent on the interfacial speed g as well as the number of LBM time steps b used to advance the state of the subdomain X H. The halo surrounding the interfacial region X I has a width of h lattice units that varies along the boundary of X I denoted C I. The width of the halo at a given boundary node k CI is set based on the number of LBM time steps b that will be carried out on the subdomain X H and the interfacial speed g. The number of time steps that will be carried out on the subdomain is set according to h max b5 max Xwn ðgðxþþ ; (45) where h max is the maximum allowable width of the halo. h max 53 was used in this work. Due to the form of equation (41), the interfacial speed at time t cannot be computed until t 1 dt. Therefore to estimate the distance the wn interface will move between time t and t 1 bdt, g at time t dt was used. The width of the halo at each lattice site on the leading edge of the interfacial region was set according to h l ðk CI Þ5 2ðbdtgðxÞÞ dx : (46) A safety factor of 2 was included when computing h l ðk CI Þ because the interfacial domain X I cannot reach the boundary of the subdomain X H, denoted C H, for the simulation to be valid. If the interfacial region were to reach C H, the evolution of the wn interface would not be accurately captured because only the region within the subdomain is being updated. Microscale physics suggests that the trailing edge of the interfacial region will progress forward as the leading edge of the interfacial region advances. But to accommodate for any small fluctuations along the trailing edge of the interfacial region, the number of lattice units in the halo at each lattice site on the trailing edge of the interfacial region is set according to h t ðk CI Þ5 bdtgðxþ dx : (47) DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2609

10 Algorithm 1 Adaptive LBM for two-fluid-phase flow. while L 1 ½gðx; tþš > e w do while L 1 ½gðx; tþš > j w or L 1 ½gðx; tþš > L 1 ½gðx; t21þš do Full LBM solve Compute gðx; tþ end while Compute b i51 while i < b and X I ~C H do Adapted LBM Solve State Identify X I i ¼ i11 end while Full LBM solve Compute gðx; tþ end while h l ðk CI Þ and h t ðk CI Þ are reduced to an integer number of lattice units by rounding up to the nearest integer value. After the subdomain X H is established, a maximum of b time steps of the LBM are run on the subdomain. To ensure that X I X H during the adapted LBM solves, X wn is computed after every solve. When either b time steps of the LBM are run on X H or X I reaches within one lattice site of C H, which is denoted ~C H, one LBM time step is run on all fluid phase lattice sites and the interfacial speed is computed. If the termination condition e w for the interfacial speed is met, the system has approached an equilibrium state and the simulation is complete. For this work, e w 51x10 8. The adaptive LBM algorithm is summarized in Algorithm 1. A condition is included on the second line of the algorithm to ensure a monotonically decreasing maximum velocity of the front as a condition for an adaptive solve. This is necessary to capture Haines jumps. 5. Results 5.1. Evaluation of the Adaptive LBM Algorithm Two-dimensional LBM simulations of a drainage and imbibition sequence were performed for the porous media depicted in Figure 2 using both the nonadaptive LBM approach and the adaptive LBM algorithm. The computational results generated by each algorithm are shown in Figure 4. A grid independence study was performed to ensure that the results were independent of the grid size. In both sets of LBM simulations, the grid resolution was lattice sites with constant pressure boundary conditions being set on one inlet face for one fluid and on the other inlet face for the second fluid, and no-flow boundaries on all other boundaries. Once the pressure boundary conditions were set, the system was allowed to equilibrate before the next step change in fluid pressures. The final equilibrium state at a given pressure step was based on the interfacial curvature Jw wn in the nonadaptive LBM simulations and on the interfacial velocity w wn in the adaptive LBM simulations. The dimensionless pressures from the LBM simulations were scaled to a physical system using the Young- Laplace equation, given by ðp n 2p w Þ 5 p nlb2p wlb ; (48) c wn Dc wnlb DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2610

11 Figure 4. Pressure-saturation curves obtained from LBM simulations performed using the nonadaptive LBM and the adaptive LBM algorithm. where p n p w is the pressure difference between the two first-kind boundary conditions for the fluids in the physical system, c wn is the interfacial tension of the physical system, D is the lattice resolution, and LB denotes a value in the LBM model. D is introduced to scale the dimensionless LBM variables. For this work, D 5 0.5lm per lattice unit and c wn was set to 24 dynes/cm, the interfacial tension between nitrogen and decane. Nitrogen and decane were the two fluids used in Gray et al. [2015]. The mean absolute error of the magnitude of saturations for comparable pressures of the two methods was less than 4x10 5 for all cases. The fluid distributions for the adapted and nonadapted simulations are virtually identical Computational Performance of the Adaptive LBM Algorithm Performance was analyzed for the nonadaptive LBM approach and the adaptive LBM algorithm, and compared. For the nonadaptive LBM model considered in this work, there are 490 floating point operations (FLOPs) per lattice site in the w phase and n phase. An additional 84 FLOPs per lattice site have to be performed on the lattice sites located on the wn interface to calculate the interfacial curvature Jw wn. The adaptive LBM algorithm performs 490 FLOPs per lattice site in the w phase and n phase that are located within the simulated domain and an additional 116 FLOPs per lattice site on the wn interface to compute the interfacial speed g. The runtime of both the nonadaptive LBM approach and the adaptive LBM algorithm was calculated for six of the pressure steps shown in Figure 4, three on the primary drainage curve and three on the main imbibition curve. Each algorithm was run in serial on an Intel Xeon E v3 (2.60GHz). The runtime for each algorithm to go from an initial equilibrium state to the next equilibrium state simulated is shown in Table Discussion Table 1. Runtime on an Intel Xeon E v3 (2.60GHz) for a Range of Pressure Steps Using Implementations of the Nonadaptive LBM Approach and the Adaptive LBM Algorithm Initial p n p w (kpa) Final p n p w (kpa) Nonadaptive Runtime (s) Adaptive Runtime (s) , , , , , , Figure 4 shows the computational results of the nonadaptive LBM approach compared to those from the developed adaptive LBM algorithm for a sequence of pressure steps. The data from the adaptive LBM algorithm matched the data collected using the nonadaptive LBM approach for both primary drainage and main imbibition. The runtime required for each algorithm to go from one equilibrium state to the next for a range of pressure steps is shown in Table 1. The comparison between the nonadaptive LBM approach and adaptive LBM algorithm revealed that the adaptive algorithm took approximately a tenth of the runtime of the nonadaptive approach to obtain essentially the same result. The adaptive LBM algorithm increased the computational efficiency of an equilibrium state simulation by adapting the active domain size in accordance with the physical phenomenon observed as the system relaxed toward equilibrium. Figure 5 depicts the time evolution of the wn interface position as the two-fluidphase system relaxes toward equilibrium in an equilibrium state simulation performed with the adaptive LBM algorithm. As time progresses in the simulation, the DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2611

12 Figure 5. The time evolution of the wn interface position as the two-fluid-phase system relaxes toward equilibrium. Black represents the initial distribution of the n phase (t 5 0), red represents the change in the n phase distribution observed after the first 25,000 time steps (t 5 25,000), yellow represents the change in the n phase distribution observed between the 25,000 and 50,000 time steps (t 5 50,000), green represents the change in the n phase distribution observed between 50,000 and 75,000 time steps (t 5 75,000), and blue represents changes in n phase distribution observed between 75,000 and 1,00,000 time steps (t 5 1,00,000). w phase is displaced by the n phase and a distinct fingering pattern emerges. Other than a main finger that continues to take form in Figure 5, the majority of the wn interface remains immobile after the first 25,000 time steps of the simulation. The time evolution of the active fraction of the domain for the equilibrium state simulation is shown in Figure 6. By the time the simulation has reached 5,000 time steps, the active domain has converged onto an interfacial subdomain X H. The size of the subdomain continues to decrease rapidly before leveling off to a more gradual rate of change at approximately 30,000 time steps. The size of the active domain as a function of time will not be a monotonically decreasing function if a Haines jump occurs as the system is relaxing toward equilibrium. The active fraction of the domain as a function of time for an equilibrium state simulation where a Haines jump occurs is shown in Figure 7. Similarly to Figure 6, the active domain converges onto an interfacial subdomain X H by 5,000 time steps into the simulation. The size of the subdomain continues to decrease rapidly until approximately 30,000 time steps into the simulation when the domain size suddenly increases. By 35,000 time steps the active domain constitutes all the lattice sites in the w phase and n phase. The active domain size was increased to capture a potential Haines jump. A Haines jump does occur in the system between 35,000 and 40,000 time steps. The time evolution of the wn interface position between 35,000 and 40,000 time steps is depicted in Figure 8. After the Haines jump occurs, the active domain converges back to an interfacial subdomain by 40,000 time steps. The size of the subdomain continues to gradually decrease until the system reaches an equilibrium state. Therefore, there will always be a reduction in computational time by using the adaptive LBM algorithm unless a Haines jump occurs in the system at a frequency such that continuous large scale movement of interfaces occurs at all times, in which case the computational time of the adaptive LBM algorithm would be equal to the nonadaptive LBM algorithm. Active fraction of domain Time Steps Figure 6. The time evolution of the active fraction of the domain for the equilibrium state simulation depicted in Figure 5. During the LBM simulation of primary drainage, a wn interface only existed between the bulk of the w phase and the bulk of the n phase. However during imbibition, disconnected, entrapped residual n phase features formed in the two-fluid-phase system. Examples of the simulated fluid displacement patterns for both primary drainage and main imbibition are depicted in Figures 9 and 10, respectively. All the disconnected n phase features formed during the imbibition process are represented by the set of disconnected n phase regions that remain after imbibition is complete. At equilibrium, each residual n phase DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2612

13 Active fraction of domain Time Steps Figure 7. The active fraction of the domain as a function of time for an equilibrium state simulation in which a Haines jump occurs. feature has a curvature corresponding to the capillary pressure at which the disconnected feature was formed. Due to the formation of disconnected n phase regions during main imbibition, the domain X H in the adaptive LBM algorithm may be composed of several separate subdomains that each constitute an independent interface. The interfaces of X H include the wn interfaces of the residual n phase features that have approached an equilibrium state and the wn interface between the connected w phase and the connected n phase, which dynamically change. In the LBM simulations performed, the wn interface of a residual n phase feature remained immobile once the disconnected feature had reached equilibrium at the capillary pressure at which it was formed. Therefore the adaptive LBM algorithm inherently excludes the residual n phase features at equilibrium by establishing the width of the halo region h from the interfacial speed g. Figure 11 shows the initial state of an equilibrium simulation during main imbibition where the red interface identifies the wn interface that is mobile during the simulation. The disconnected n phase features in Figure 11 were formed under previous capillary pressures. The wn interface of the disconnected features remain immobile for the duration of the simulation. In addition to equilibrium simulations, we have examined a second case where two fluids are flowing simultaneously in the porous media depicted in Figure 2 and relax dynamically to a flowing steady state. Such simulations would be of use in deducing a macroscale resistance tensor, also needed to close evolving theoretical models [Dye et al., 2015; Gray and Miller, 2014]. The initial condition was a randomized distribution of a wetting phase and nonwetting phase. The fluid-fluid interfacial tension and contact angle were set to match the values used for the equilibrium simulations discussed in section 5. Full periodic boundaries were set on the inlet face and outlet face, and no-flow boundaries on all other sides of the domain. Flow was driven by specifying a body force, which was oriented in the y direction. The simulation was run until the microscale field velocity reached a steady state. Each algorithm was run in serial on an Intel Xeon E v3 (2.60GHz). The runtime of both the nonadaptive LBM approach and the adaptive LBM algorithm was calculated. The results for this example show a factor of eight speedupcomparedtoanonadaptive t = 35,000 simulation. Thus we have conclusively t = 40,000 demonstrated that the algorithm is applicable to additional cases of importance and that the performance is impressive for the additional case investigated as well. Figure 8. The time evolution of the n phase distribution during a Haines jump. Black represents the state of the n phase distribution before the Haines jump (t 5 35,000). Red represents the n phase distribution observed after the Haines jump (t 5 40,000). In order to simulate larger domain sizes and accelerate the solution time for the developed adaptive LBM algorithm, parallel implementation is necessary. Scaling the LBM to run on a large number of processors requires a domain decomposition strategy that DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2613

14 Figure 9. Fluid distributions for a set of simulated equilibrium states along the primary drainage curve shown in Figure 4. Gray represents the solid phase, black represents the nonwetting fluid phase, and white represents the wetting fluid phase. evenly distributes the computational load among processors, while minimizing the amount of communication that must be performed. The computational load scales with the volume of lattice sites not in the solid phase within a subdomain, while the communication scales with the surface area of the subdomain. Due to the nonuniform evolution of the wn interface in time and space and the need for both global and interfacial domain simulations, efficient domain decomposition becomes a challenge. Achieving high efficiency is critical because many multiphase LBM simulations of porous medium systems of concern are at, or beyond, current computational limits for even the most advanced computers. To increase the scope of the adaptive LBM, work is needed to develop a parallel, three-dimensional extension of this work. However, due to the morphology and topology of two-fluid-phase displacement processes, the fraction of the domain that is expected to be active in three-dimensional systems as interfaces relax would be much smaller than for the two-dimensional systems considered in this work. We have done a preliminary analysis of a three-dimensional, two-fluid-phase porous medium system relaxing to equilibrium. A three-dimensional, 19-velocity-vector (D3Q19), MRT LBM [McClure et al., 2014] was used to analyze the active fraction of the computational domain as a function of time for an equilibrium state simulation. A three-dimensional, synthetic porous medium system was generated using a sphere packing algorithm [Dye et al., 2013]. The isotropic sphere pack consisted of 1,500 uniform size spheres arranged in a nonoverlapping fashion with a porosity of 0.42 and periodic boundaries. The lattice size necessary to achieve a grid-independent solution was The fluid-fluid interfacial tension and contact angle within the LBM simulation was set to match the values used for the two-dimensional simulations described above. As done with the two-dimensional system, constant pressure boundary conditions were set on one inlet face for one fluid and on the other inlet face for the second fluid, and no-flow boundaries on all other boundaries. Once the pressure boundary conditions were set, the system was allowed to dynamically relax toward an equilibrium state. Note that the simulated system mimics classical pressure-saturation DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2614

15 Figure 10. Fluid distributions for a set of simulated equilibrium states along the main imbibition curve shown in Figure 4. Gray represents the solid phase, black represents the nonwetting fluid phase, and white represents the wetting fluid phase. experiments that are performed routinely in porous medium physics [Porter et al., 2010; Culligan et al., 2006; Cheng et al., 2004]. The final equilibrium state corresponded to a specific interfacial curvature. The active fraction of the domain was recorded every 5000 time steps. Based on these large-scale simulations, the adaptive LBM algorithm could potentially decrease the computational time of a three-dimensional system by about a factor of 20. This suggests that an efficient three-dimensional implementation could save even more than the order of magnitude savings observed in this work. Figure 11. The initial state of an equilibrium simulation during main imbibition where the red interface identifies the wn interface that is mobile during the simulation. 7. Conclusions Two-dimensional, LBM simulations revealed that the approach of a two-fluidphase flow porous medium system to an equilibrium state involved a relatively slow process in which the fluid interfaces relax to their equilibrium state. The time DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2615

16 scale at which a two-fluid-phase system relaxes to equilibrium makes resolving equilibrium states using a nonadaptive approach computationally expensive. Based on the physical processes by which a two-fluid-phase system relaxes to equilibrium, a temporally adaptive domain decomposition LBM algorithm was developed to resolve the equilibrium and dynamic states of a two-fluid-phase porous medium system. The adaptive LBM algorithm increased the overall efficiency of an equilibrium simulation state by decreasing the size of the computational domain in accordance with the relaxation of the system toward an equilibrium state. When compared to the nonadaptive LBM approach, the adaptive LBM algorithm achieved essentially the same result in about one tenth of the runtime. The developed adaptive LBM algorithm is an accurate and efficient method for simulating two-fluid-phase flow in porous medium systems that exhibit multiple timescale behavior. Acknowledgments This work was supported by Army Research Office grant W911NF , National Science Foundation grant , and by Department of Energy grant DE-SC The computations were supported in part by an allocation of resources under the Department of Energy INCITE program. Data may be obtained by contacting Amanda L. Dye, alynnd@live.unc.edu. References Adalsteinsson, D., and J. A. Sethian (1999), The fast construction of extension velocities in level set methods, J. Comput. Phys., 148, Ahrenholz, B., J. T olke, P. Lehmann, A. Peters, A. Kaestner, M. Krafczyk, and W. Durner (2008), Prediction of capillary hysteresis in a porous material using lattice-boltzmann methods and comparison to experimental data and a morphological pore network model, Adv. Water Resour., 31(9), Aidun, C. K., and J. R. Clausen (2010), Lattice-Boltzmann method for complex flows, Annu. Rev. Fluid Mech., 42, Armstrong, R. T., and S. Berg (2013), Interfacial velocities and capillary pressure gradients during Haines jumps, Phys. Rev. E, 88(4), , doi: /physreve Armstrong, R. T., M. L. Porter, and D. Wildenschild (2012), Linking pore-scale interfacial curvature to column-scale capillary pressure, Adv. Water Resour., 46, Bao, J., and L. Schaefer (2013), Lattice Boltzmann equation model for multi-component multi-phase flow with high density ratios, Appl. Math. Modell., 37(4), Bear, J. (1972), Dynamics of Fluids in Porous Media, Elsevier, N. Y. Bradford, S. A., T. J. Phelan, and L. M. Abriola (2000), Dissolution of residual tetrachloroethylence in fractional wettability porous media: Correlation development and application, J. Contam. Hydrol., 45, Brooks, R. H., and A. T. Corey (1966), Properties of porous media affecting fluid flow, J. Irrig. Drain. Div., 92(2), Brusseau, M., S. Peng, G. Schnarr, and M. Costanza-Robinson (2006), Relationships among air-water interfacial area, capillary pressure, and water saturation for a sandy porous medium, Water Resour. Res., 42, W03501, doi: /2005wr Celia, M. A., H. Rajaram, and L. A. Ferrand (1993), A multi-scale computational model for multiphase flow in porous media, Adv. Water Resour., 16(1), Chen, L., Q. Kang, B. Carey, and W.-Q. Tao (2014), Pore-scale study of diffusion-reaction processes involving dissolution and precipitation using the lattice Boltzmann method, Int. J. Heat Mass Transfer, 75, Cheng, J.-T., L. J. Pyrak-Nolte, D. D. Nolte, and N. J. Giordano (2004), Linking pressure and saturation through interfacial areas in porous media, Geophys. Res. Lett., 31, L08502, doi: /2003gl Culligan, K. A., D. Wildenschild, B. S. B. Christensen, W. G. Gray, and M. L. Rivers (2006), Pore-scale characteristics of multiphase flow in porous media: A comparison of air-water and oil-water experiments, Adv. Water Resour., 29(2), Dye, A. L., J. E. McClure, C. T. Miller, and W. G. Gray (2013), Description of non-darcy flows in porous medium systems, Phys. Rev. E, 87(3), , doi: /physreve Dye, A. L., J. E. McClure, W. G. Gray, and C. T. Miller (2015), Multiscale modeling of porous medium systems, in Handbook of Porous Media, 3rd ed., edited by K. Vafai, pp. 3 45, Taylor and Francis, London, U. K. Gray, W. G., and C. T. Miller (2005), Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview, Adv. Water Resour., 28(2), Gray, W. G., and C. T. Miller (2014), Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems, Springer, Switzerland, doi: / Gray, W. G., A. L. Dye, J. E. McClure, L. J. Pyrak-Nolte, and C. T. Miller (2015), On the dynamics and kinematics of two-fluid-phase flow in porous media, Water Resour. Res., 51, , doi: /2015wr He, X., and L. S. Luo (1997), A priori derivation of the lattice Boltzmann equation, Phys. Rev. E, 55, R6333 R6336. Huang, H., D. T. J. Thorne, M. G. Schaap, and M. C. Sukop (2007), Proposed approximation for contact angles in Shan-and-Chen-type multicomponent multiphase lattice Boltzmann models, Phys. Rev. E, 76(6, Part 2). Jackson, A. S., C. T. Miller, and W. G. Gray (2009), Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 6. Two-fluid-phase flow, Adv. Water Resour., 32(6), Karadimitriou, N. K., M. Musterd, P. J. Kleingeld, M. T. Kreutzer, S. M. Hassanizadeh, and V. Joekar-Niasar (2013), On the fabrication of PDMS micro-models by rapid prototyping, and their use in two-phase flow studies, Water Resour. Res., 49, , doi: /wrcr Kim, Y., J. Wan, T. J. Kneafsey, and T. K. Tokunaga (2012), Dewetting of silica surfaces upon reactions with supercritical co2 and brine: Pore-scale studies in micromodels, Environ. Sci. Technol., 46(7), Lallemand, P., and L. S. Luo (2000), Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61(6), Latva-Kokko, M., and D. Rothman (2005), Static contact angle in lattice Boltzmann models of immiscible fluids, Phys. Rev. E, 72(4, Part 2), Leclaire, S., M. Reggio, and J.-Y. Trepanier (2011), Isotropic color gradient for simulating very high-density ratios with a two-phase flow lattice Boltzmann model, Comput. Fluids, 48(1), Li, H., C. Pan, and C. T. Miller (2005), Pore-scale investigation of viscous coupling effects for two-phase flow in porous media, Phys. Rev. E, 72(2), , McClure, J. E., J. F. Prins, and C. T. Miller (2014), A novel heterogeneous algorithm to simulate multiphase flow in porous media on multicore CPU-GPU systems, Comput. Phys. Commun., 185(7), , doi: /j.cpc Miller, C. T., and W. G. Gray (2002), Hydrogeological research: Just getting started, Ground Water, 40(3), Olsson, E., and G. Kreiss (2005), A conservative level set method for two phase flow, J. Comput. Phys., 210(1), DYE ET AL. ADAPTIVE LATTICE BOLTZMANN SCHEME 2616