Lattice Boltzmann Simulation of Immiscible Two-Phase Displacement in Two-Dimensional Berea Sandstone
Lattice Boltzmann Simulation of Immiscible Two-Phase Displacement in Two-Dimensional Berea Sandstone
Gu, Qingqing;Liu, Haihu;Zhang, Yonghao
Article Lattice Boltzmann Simulation of Immiscible Two-Phase Displacement in Two-Dimensional Berea Sandstone 1 2 1, ID Qingqing Gu , Haihu Liu and Yonghao Zhang * James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK; firstname.lastname@example.org School of Energy and Power Engineering, Xi’an Jiaotong University, 28 West Xianning Road, Xi’an 710049, China; email@example.com * Correspondence: firstname.lastname@example.org Received: 16 July 2018; Accepted: 26 August 2018; Published: 31 August 2018 Abstract: Understanding the dynamic displacement of immiscible ﬂuids in porous media is important for carbon dioxide injection and storage, enhanced oil recovery, and non-aqueous phase liquid contamination of groundwater. However, the process is not well understood at the pore scale. This work therefore focuses on the effects of interfacial tension, wettability, and the viscosity ratio on displacement of one ﬂuid by another immiscible ﬂuid in a two-dimensional (2D) Berea sandstone using the colour gradient lattice Boltzmann model with a modiﬁed implementation of the wetting boundary condition. Through invasion of the wetting phase into the porous matrix, it is observed that the viscosity ratio plays an important role in the non-wetting phase recovery. At the viscosity ratio (l) of unity, the saturation of the wetting ﬂuid is highest, and it linearly increases with time. The displacing ﬂuid saturation reduces drastically when l increases to 20; however, when l is beyond 20, the reduction becomes less signiﬁcant for both imbibition and drainage. The front of the bottom ﬁngers is ﬁnally halted at a position near the inlet as the viscosity ratio increases to 10. Increasing the interfacial tension generally results in higher saturation of the wetting ﬂuid. Finally, the contact angle is found to have a limited effect on the efﬁciency of displacement in the 2D Berea sandstone. Keywords: lattice Boltzmann method; immiscible displacement; porous media; interfacial tension; contact angle; viscosity ratio 1. Introduction The simultaneous ﬂow of two immiscible ﬂuids in porous media is often found in nature and industrial processes such as non-aqueous phase liquid (NAPL) contamination of groundwater, carbon dioxide injection and storage, and enhanced oil recovery (EOR) operations. For the spontaneous imbibition of oil, carbon dioxide or water, which is immiscible to the oil phase, is usually imbibed into permeable rock formations which are characterized by low porosity and low in situ permeability. The displacement process is affected by matrix properties such as the heterogeneity, wettability, and ﬂuid properties of each phase . Experimental efforts have been made to understand interactions between fluids and rock during fluid displacement [2,3]. Theoretical works usually use simplified pore geometries which allow theoretical solutions to the microscopic flow patterns . As a complementary method to the experimental and theoretical approaches, accurate and reliable numerical tools are needed to understand the underlying mechanisms of multiphase displacement. Conventional macro-scale simulations have achieved great success in solving the continuity, momentum, and species balance equations which rely on the empirical estimation of relative permeabilities. This approach, however, may not be able to capture effects related Appl. Sci. 2018, 8, 1497; doi:10.3390/app8091497 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 1497 2 of 14 to micro-scale structure in a multiphase system. On the contrary, pore-scale simulation can provide local detailed information about the flow field. The previous works on pore-scale simulation of immiscible two-phase flows in porous media [5–10] have been shown to be capable of capturing the effects of various factors, e.g., capillary number, viscosity ratio, surface wettability, and media heterogeneity. However, due to the inherent complexity related to interface evolution, heterogeneous geometry, dominant capillarity, and moving contact line, the computational simulation of multiphase displacement in realistic porous media still remains a research challenge. As a mesoscopic method, the lattice Boltzmann method (LBM) has been widely accepted as a useful alternative for simulating multiphase ﬂows, in particular, with the advantage of dealing with complex geometries such as porous media . A number of multiphase LBM models have been proposed, e.g., the colour gradient model [12,13], the pseudo-potential model , the free energy model  and the mean-ﬁeld model . The colour gradient model is known to be the ﬁrst multiphase model. Later, a new multiphase LBM model was proposed by Shan and Chen , where the interparticle forces (whether attractive or repulsive) mimicking the molecular potentials are introduced by a modiﬁed equilibrium velocity in the equilibrium distribution functions. The free energy model is thermodynamically consistent, and it considers a generalised equilibrium distribution function that includes a nonideal pressure tensor term. The original free energy model suffers from a lack of Galilean invariance, but later works have mitigated this problem [17,18]. The mean-ﬁeld model describes interfacial dynamics based on the mean-ﬁeld theory, and uses a pressure distribution function instead of its density counterpart to reduce discretization errors in the calculation of density gradient and to improve the numerical stability for variable density. Extensive reviews of these LBM multiphase models can be found in previous research [8,19]. The viscosity contrast is large while the density difference is small for the water or carbon dioxide involved oil recovery process. However, most LBM models can only access a limited range of viscosity ratios for dynamic ﬂuid displacement. Yang et al.  performed a systematic comparison of three popular multiphase models, i.e., the pseudo-potential model, the free energy model, and the colour gradient model, for ﬂow simulations in porous media. They concluded that the pseudo-potential model is a promising tool for liquid-gas systems, but may not be the optimal solution for the simulation of immiscible ﬂows. Also, it produces high spurious velocities and thick interfaces for immiscible ﬂows . Jonathan et al.  used the pseudo-potential model to simulate two-phase ﬂows in a two-dimensional channel with different viscosity ratios. A viscous-ﬁngering-like phenomenon was observed for viscosity ratios of up to 10, and the simulations become unstable for larger viscosity ratios. Hatiboglu et al.  studied the spontaneous imbibition of water into a water-wet micro-model which was initially saturated with oil. It was veriﬁed that the maximum allowable viscosity ratio is four. The free energy model and colour gradient model show similar capabilities in modelling the ﬂow of binary ﬂuids with high viscosity contrast, but the latter produces a relatively thinner interface than the former one. In addition, the colour gradient model is able to control the interfacial diffusion and adjust the interfacial tension and viscosity independently to facilitate the numerical investigation. The colour gradient LBM was ﬁrst proposed by Rothman and Keller , and has been widely used for the simulation of immiscible ﬂuids [9,12,23–26]. To obtain an accurate and detailed understanding of the pore-scale mechanisms within porous media, Xu et al.  proposed a new algorithm for imposing the contact angle on the solid surface in the colour gradient LBM. The simulation results were validated against the experimental observations of Wu et al. . However, these simulations only dealt with the simpliﬁed porous media, where the solid grains are regular circles and rectangles. In realistic porous media, the pore structures are extremely complex and irregular, which allow us to further assess the capability of numerical model. We apply the colour gradient LBM for dynamic displacement of immiscible ﬂuids in a complex porous ﬂow structure extracted from a realistic Berea sandstone sample, and study the effects of the viscosity ratio, interfacial tension, and contact angle on displacement process. Appl. Sci. 2018, 8, 1497 3 of 14 2. Colour Gradient Lattice Boltzmann Model for Two-Phase Flow The colour gradient LBM was used in this work. In this model, the “Red” and “Blue” distribution R B functions ( f and f ) are used to represent two different ﬂuids. The total distribution function is i i R B deﬁned as f = f + f . Each colored distribution function undergoes the collision and streaming i i steps expressed by the following equation: k k k k f (x + c Dt, t + Dt) = f (x, t) + W ( f (x, t)), (1) i i i i where x and t are the position and time, k = red or blue, i represents the discrete velocity directions for the two-dimensional 9-velocity (D2Q9) model, Dt is the time step, and c denotes the lattice velocity vectors in the i-th discrete velocity direction. For the two-dimensional 9-velocity (D2Q9) model used in this work, the lattice velocity vector is given by (0, 0), i = 0 c = (2) (cos[p(i 1)/2], sin[p(i 1)/2]), i = 1, 2, 3, 4 2(cos[p(i 5)/2 + p/4], sin[p(i 5)/2 + p/4]), i = 5, 6, 7, 8. The collision step includes self- and cross-interactions with the other type of particles which can be written as (3) (1) (2) k k k k W = (W ) (W ) + (W ) , (3) i i i i (1) (2) (3) k k k where (W ) is the single-phase collision operator, (W ) is the perturbation operator, and (W ) i i i is the recolouring operator. 2.1. Single-Phase Collision Operator To achieve high accuracy and good stability, we implemented the collision step in accordance with the two-relaxation-time (TRT) model [28–30]—a special multiple relaxation time (MRT) model with only two relaxation rates . With the TRT model, the single-phase collision operator is expressed as (1) k,eq k 1 k (W ) = (M SM) ( f f ), (4) ij i j where M is the transformation matrix and is given by 0 1 1 1 1 1 1 1 1 1 1 B C 4 1 1 1 1 2 2 2 2 B C B C 4 2 2 2 2 1 1 1 1 B C B C B C 0 1 0 1 0 1 1 1 1 B C B C M = 0 2 0 2 0 1 1 1 1 . (5) B C B C 0 0 1 0 1 1 1 1 1 B C B C 0 0 2 0 2 1 1 1 1 B C B C @ A 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 The diagonal relaxation matrix S is deﬁned as S = (1, s , s , 1, s , 1, s , 0, s , s , s ). (6) e # q q q n n For the conserved moments of density and momentum, the relaxation rates were set to 1. s and s e n are related to the bulk and shear viscosities, and s and s are free parameters. Following the # q recommendations of Ginzburg et al. , these relaxation rates were taken as Appl. Sci. 2018, 8, 1497 4 of 14 1 (2 s ) s = s = s = , s = 8 , (7) e # n q t (8 s ) where t is the dimensionless relaxation time. The mass conservation for each ﬂuid distribution function and the total momentum conservation require k,eq k,eq k k r = f = f , ru = f c = f c , (8) k å å åå i åå i i i i i i i i k i k where r is the total density, and r is the density of ﬂuid k, and u is the local ﬂuid velocity. k,eq The equilibrium distribution function ( f (x, t)) is a truncated Taylor series expansion up to the second order in Mach number of the Maxwell–Boltzmann equilibrium distribution function, which was chosen to satisfy Equation (8): " # c u (c u) u k,eq i i f (r , u) = r W 1 + + , (9) k k i 2 4 2 c 2c 2c s s s in which, c = 1/ 3 is the lattice sound speed, and the weight coefﬁcients are W = 4/9, W = 1/9 0 1 4 and W = 1/36 for the D2Q9 lattice model. As we focused on the oil-water two-phase systems, 5 8 the density ratio of both ﬂuids was set to be unity. To ensure a constant viscosity across the interface when both ﬂuids were of different viscosities, the following harmonic mean  was employed to determine the viscosity of the ﬂuid mixture: N N 1 1 + r 1 r = + , (10) h 2h 2h R B where h (k = R or B) is the dynamic viscosity of ﬂuid k, and h is the dynamic viscosity of the ﬂuid 2 N mixture which is related to the dimensionless relaxation time t by h = c r(t 0.5)dt; r is the indicator function used to distinguish different ﬂuids, and it is deﬁned as r (x, t) r (x, t) R B r (x, t) = . (11) r (x, t) + r (x, t) R B 2.2. Perturbation Operator k (2) (W ) takes effect in the mixed interfacial region to generate an interfacial tension. Using the continuum surface force (CSF)  model, a body force term was added to the macroscopic momentum equation, and it reads as F = sKrr , (12) where s is the interfacial tension coefﬁcient, and K is the local interface curvature calculated by  K = r n, (13) where r = (I nn) r is the surface gradient operator, I is the second-order identity tensor, N N and n = rr / rr is a unit vector normal to the interface. Such a body force was then incorporated into LBM through the forcing scheme proposed by Guo et al.  which can signiﬁcantly improve computation accuracy and reduce spurious velocities effectively . The perturbation operator k (2) (W ) is expressed as k (2) 1 k (W ) = M (I S)F , (14) 2 Appl. Sci. 2018, 8, 1497 5 of 14 with F (x, t) = a [0, 6(u F + u F ), k x sx y sy 6(u F + u F ), F , F , F , F (15) x sx y sy sx sx sy sy 2(u F u F ), u F + u F ] , x sx y sy x sy y sx where a is the fraction of interfacial tension contributed by ﬂuid k and satisﬁes a = 1. k k k The ﬂuid velocity was corrected to recover the Navier–Stokes (N-S) equations in the interfacial region where two ﬂuids coexist ru = f c + F . (16) i s åå i i k 2.3. Recolouring Operator The recolouring step was then applied to promote phase segregation and maintain a sharp interface  which is given as r r r (eq) R R B R (3) (W ) = f + b f (r, u = 0) cos(j ), (17) i i i r r r r r (eq) B (3) B R B (W ) = f b f (r, u = 0) cos(j ), (18) i i r r where f = f denotes the post-perturbation, pre-segregation value of the total distribution i i eq k,eq function along the i-th discrete velocity direction, and f = å f is the total equilibrium distribution i i function. b is the segregation parameter related to the interface thickness, and its value must be between 0 and 1 to ensure positive particle distributions—it was chosen to be 0.7 here. j is the angle between the indicator function gradient rr and the lattice vector c , which is deﬁned by c rr cos(j ) = . (19) jc jjrr j To minimize the discretization error, a fourth-order isotropic ﬁnite difference was used to calculate the partial derivatives. For example, for a variable y, the partial derivatives can be calculated by ry(x, t) = W y(x + c d , t)c . (20) å i i t i 2.4. Wetting Boundary Condition For the colour gradient LBM, the most widely-used wetting boundary condition is to set a virtual density at solid surfaces . However, it has been found in recent works [38,39] that numerical errors resulting from nonphysical mass transfer and spurious velocities may be accumulated during simulations which potentially render meaningless results. To overcome this, the algorithm proposed by Xu et al.  was implemented to impose the contact angle on solid surfaces. This algorithm is able to precisely control the contact angle for both static and dynamic problems which is essential for accurate simulations of two-phase displacement in porous media. 3. Relative Permeability Validation: Concurrent Flow in a 2D Channel We ﬁrst used the above model to simulate a pressure driven ﬂow through a porous media. In the single-phase case, the intrinsic permeability K can be determined according to the Darcy’s law  K mu u = rP, K = , (21) m rP Appl. Sci. 2018, 8, 1497 6 of 14 where u is the Darcy velocity in the porous medium. Note that the intrinsic permeability does not depend on the nature of the ﬂow ﬁeld but only on the geometry of the porous medium. In case of two-phase ﬂows, the relative permeability K can be given as  r,i KK m u r,i i i u = rP , K = , i = w, nw. (22) i i r,i m KrP i i Here, i = w and nw refer to the wetting and non-wetting ﬂuids respectively, m is the dynamic viscosity of ﬂuid i, and rP is the pressure gradient of ﬂuid i. First, we tested whether the LBM model can accurately predict relative permeability for a simple immiscible two-phase cocurrent ﬂow. Speciﬁcally, we calculated the relative permeabilities of the layered two-phase ﬂow in a channel under different saturation values. In the simulation, a constant pressure difference was applied on the inlet and outlet boundaries following Zou and He , and a non-slip (bounce-back) boundary condition was applied on the upper and lower walls using the halfway bounce-back scheme. The wetting phase ﬂowed along the walls (a < y b) and non-wetting j j phase ﬂowed in the central region (jyj < a). One can obtain the analytical solutions for the relative permeability, a function of wetting phase saturation S and viscosity ratio l , 3 3 1 2 2 K = (1 S ) l + (1 S ) (1 l) , K = S (3 S ), (23) r,nw w w r,w w 2 2 2 where S is the wetting phase saturation, which is deﬁned as S = 1 a/b, and l is the viscosity ratio w w of the non-wetting to wetting ﬂuids, i.e., l = m /m . nw w For two typical viscosity ratios of l = 1 and l = 0.1, Figure 1 shows excellent agreement between the computed relative permeabilities and the analytical solutions. 1 1 0.8 0.8 nw 0.6 0.6 nw 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 S S w w (a) (b) Figure 1. Relative permeabilities as a function of wetting phase saturation for two-phase concurrent ﬂow in a 2D channel for (a) l = 1 and (b) l = 0.1. The open circles represent the simulation results, and the solid lines are the analytical solutions from Equation (23). 4. Results and Discussion This section describes the conduction of a systematic investigation on a pressure-driven immiscible two-phase ﬂow in a 2D micromodel extracted from Boek et al. , which was engineered at Schlumberger Cambridge Research based on a thin section of a 3D Berea sandstone rock sample. We deﬁned the lattice units for length, mass and time as lu, ts, and mu, respectively. To match the parameters in the lattice unit towards their physical values, three basic physical quantities were chosen as the reference values: the length scale (l ), the mass scale (m ) and the time scale (t ). 0 0 0 6 13 7 In the present study, l = 8 10 m, m = 5.12 10 kg, and t = 7.2 10 s. Therefore, 0 0 0 r Appl. Sci. 2018, 8, 1497 7 of 14 m 5.12 10 phy 0 3 3 the density can be obtained by r = r = = 10 kgm , the dynamic viscosity by 3 6 3 l (8 10 ) phy 5.12 10 m = m = 0.167 = 0.0148 Pas, and the interfacial tension displaced displaced 6 7 l t 8 10 7.2 10 0 0 phy 2 by s = sm /t . For the interfacial tension values used in the present work, the corresponding 0 0 2 2 2 1 1 physical values for 0.03 muts , 0.015 muts , and 0.0005 muts were 0.0296 Nm , 0.0148 Nm , and 0.0004938 Nm , respectively. 4.1. Problem Statement The 2D micromodel is shown in Figure 2, and its porosity is 0.33. The entire micromodel is divided into L H = 2644 2117 lu with a resolution of 0.67 m per lu. The narrowest throat has a width of 12 lu which is ﬁne enough to produce grid-independent results in two-phase displacement simulations .Twenty-two extra layers of lattices are added to both the inlet and outlet to facilitate the implementation of boundary conditions, so the actual domain size is 2688 2117 lu . Here, the ﬂow velocity for each ﬂuid is so small that the inertial effect becomes insigniﬁcant. The constant pressure boundary condition developed by Zou and He  was imposed on the inlet and outlet, which are located at the left and right sides of the domain, respectively. The halfway bounce-back scheme and wetting boundary condition were applied at the walls that contain the surfaces of solid grains as well as the upper and lower sides of the domain. The densities of both wetting and non-wetting ﬂuids were set to be unity as the density difference between oil and water is 1 1 insigniﬁcant. The dynamic viscosity of the displaced ﬂuid was kept as 0.167 mulu ts , and the dynamic viscosity of the displacing ﬂuid was varied to achieve different viscosity ratio values (l = m /m ). Initially, for x 22 lu, the domain was ﬁlled with the displaced ﬂuid (represented displaced invading 1 2 in red) at rest, and a constant pressure p = 1/3 mulu ts was imposed at the outlet, while for out x < 22 lu, the region was occupied by the displacing ﬂuid (represented in blue) with a higher pressure at the inlet. The simulations stopped when the displacing ﬂuid broke through at the outlet to avoid the capillary end effect . Figure 2. The 2D pore network used in the present lattice Boltzmann method (LBM) simulations. The solid grains are represented in gray, while the displacing and displaced ﬂuids are represented in blue and red, respectively. The domain size is 1774 m by 1418 m. By studying the dependency of porosity on the size of the rock sample, we calculated the porosity using different sizes of Berea sandstone to ﬁnd its representative elementary volume (REV). The results are shown in Figure 3, which demonstrates that the chosen size of 2688 2117 lu could be a good REV of the Berea sandstone. This guarantees that the simulated geometry included a sufﬁcient number of pores for the meaningful statistical average, which is generally required by the continuum concept . Appl. Sci. 2018, 8, 1497 8 of 14 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.4 0.6 0.8 1 x/L, y/H Figure 3. Porosity as a function of the domain size. L is the length of the entire micromodel, while H is its width; x refers to the distance to the left boundary, and y is the distance to the bottom boundary. For each point shown above, the same scaling was applied to x and y, i.e., x/L = y/H. 4.2. The Viscosity Ratio Effect The viscosity ratio is one of the most important dimensionless numbers in two-phase displacement through a porous media, and its effect was studied for a constant contact angle of 120 , corresponding to a weak wetting property of the displacing ﬂuid. The inlet pressure is also the same for different 1 2 viscosity ratios, i.e., p = (p + 0.0125) mulu ts . Figure 4 shows the ﬂuid distributions inside out in the pore network when breakthrough of the displacing ﬂuid occurs for viscosity ratios of 1, 2, 5, and 10. It can be seen that for all the viscosity ratios considered, two or three preferential paths are generally formed. As the viscosity ratio increases, it becomes harder for the ﬁnger to ﬂow into the large pore as highlighted by the black elliptical circles. The front of the bottom ﬁnger is ﬁnally halted at the position near the inlet for the viscosity ratio of 10. Also, the breakthrough ﬁnger switches to a lower position at the outlet when the viscosity ratio increases to 10. Figure 4. Fluid distributions at breakthrough for a wetting ﬂuid invading a porous medium initially saturated with a non-wetting ﬂuid. The wetting ﬂuid (indicated in blue) is injected from the left side, and the viscosity ratio l equals 1, 2, 5, and 10, respectively. Porosity Appl. Sci. 2018, 8, 1497 9 of 14 Figure 5 shows the time evolution of the displacing (wetting) ﬂuid saturation for a viscosity ratio changing from 1 to 30. It can be clearly seen that a higher viscosity ratio results in a lower ﬂuid saturation which is less favourable for the recovery of displaced ﬂuid. This is consistent with previous experimental  and numerical results . At the viscosity ratio of unity, the highest displacing ﬂuid saturation (around 0.37) is observed, and the displacing ﬂuid saturation keeps increasing linearly with time. Although the displacing ﬂuid saturation reduces with l, the reduction becomes less signiﬁcant at larger l values. 0.4 =1 2 5 10 20 30 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.2 0.4 0.6 0.8 1 t/t Figure 5. Evolution of the displacing ﬂuid saturation for the viscosity ratio l increased from 1 to 30 at s = 0.015 muts and q = 120 (imbibition). Note that the time is normalized by t , deﬁned as the time at which the breakthrough occurs. We also tested whether the above observations are valid for varying wettability and interfacial tension. Four pairs of contact angle and interfacial tension were considered (see Table 1), and the simulation results are shown in Figure 6. It is clear that the displacing ﬂuid saturation drops rapidly when the viscosity ratio increases from 1 to 20 and then stays almost unchanged for all the cases considered. For each viscosity ratio, increasing contact angle or interfacial tension always leads to a higher displacing ﬂuid saturation or displacement efﬁciency. In addition, the limitation of the current model was tested by decreasing the viscosity of the displacing ﬂuid, and it was found that the maximum viscosity ratio that it can successfully access is as high as 50. Table 1. Parameters used in four typical cases of immiscible displacement. Case DP s cos q I 0.0125 0.015 0.5 II 0.0125 0.0005 0.5 III 0.0125 0.0005 0.5 IV 0.0125 0.015 0.5 Saturation of displacing fluid Appl. Sci. 2018, 8, 1497 10 of 14 σ=0.015; θ=120° σ=0.0005; θ=120° σ=0.0005; θ=60° 0.8 σ=0.015; θ=60° 0.6 0.4 0.2 10 20 30 40 50 Figure 6. Displacing ﬂuid saturation as a function of viscosity ratio for the four typical cases listed in Table 1. 4.3. The Interfacial Tension Effect The interfacial tension between the displacing and displaced ﬂuid is known to play an important role at the pore scale for two-phase displacement through a porous media. The same viscosity ratio (l = 1) was used and the contact angle was ﬁxed at 120 . Figure 7 shows the ﬂuid distributions at 2 2 2 breakthrough for the interfacial tensions of 0.03 muts , 0.015 muts , and 0.0005 muts . From this ﬁgure, it is seen that when the interfacial tension is as low as 0.0005 muts , the viscous force prevails, the main ﬁngers get thinner and even break up, and many ﬁngers only occupy partial pores or throats that they pass by. Also a number of small blobs of displaced ﬂuid are left behind at low levels of interfacial tension. When the interfacial tension increases to 0.015 muts , larger blobs of the displaced ﬂuid are trapped by the displacing ﬂuid, and the displacing ﬂuid ﬂows into the large pore more easily due to higher capillary pressure, as highlighted by the black elliptical circles in Figure 7. As the interfacial tension is further increased to 0.03 muts , we surprisingly found that the front of the displacing ﬂuid can be ﬂat inside some pores, e.g., the one surrounded by the black rectangle box in Figure 7, though most interfaces remain curved due to large capillary pressure. Meanwhile, much less of the displaced phase is left behind as the invading phase continues to move forward. Figure 7. Fluid distributions at breakthrough for a wetting ﬂuid invading a porous medium initially saturated with a non-wetting ﬂuid. The wetting ﬂuid (indicated in blue) is injected from the left side 2 2 2 and the values of the interfacial tension are 0.03 muts , 0.015 muts , and 0.0005 muts from the left to right. Figure 8 shows the evolution of the displacing ﬂuid saturation for an interfacial tension of 2 2 2 0.0005 muts , 0.015 muts and 0.03 muts . It is seen that generally higher interfacial tension results in a higher saturation. The linear relationship between saturation and the evolving time is again observed. Saturation of displacing fluid Appl. Sci. 2018, 8, 1497 11 of 14 0.4 =0.0005 =0.015 =0.03 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.2 0.4 0.6 0.8 1 t/t Figure 8. Evolution of the displacing ﬂuid saturation for the interfacial tension of 0.0005 muts , 2 2 0.015 muts and 0.03 muts . 4.4. The Contact Angle Effect The contact angle was measured from the displaced ﬂuid. All the contact angles were larger than 90 in this part to simulate the imbibition process. Figure 9 shows the ﬂuid distributions when the displacing ﬂuid breaks through at the outlet for the contact angles of 162 , 150 , 135 , and 120 . It is observed that the invading paths of the wetting ﬂuid do not change signiﬁcantly except for a few branches. An obvious difference can be seen at the contact angle of 135 where the bottom ﬁnger advances furthermost. Figure 10 shows the evolution of the displacing ﬂuid for different contact angles. The case of q = 120 has the lowest wetting ﬂuid saturation but its difference from the others is very small. It can also be seen from Figure 6 that the contact angle does not have a big impact on the saturation. Therefore, the contact angle has a limited effect on the efﬁciency of displacement in the micro-model of Berea sanstone. Figure 9. Fluid distributions at breakthrough for a wetting ﬂuid invading a porous medium initially saturated with a non-wetting ﬂuid. The wetting ﬂuid (indicated in blue) is injected from the left side and the values of the contact angle are 162 , 150 , 135 , and 120 , respectively. Saturation of displacing fluid Appl. Sci. 2018, 8, 1497 12 of 14 0.4 =162 =150 =135 =120 0.35 0.3 0.25 0.2 0.374 0.15 0.372 0.37 0.1 0.368 0.05 0.97 0.98 0.99 0 0.2 0.4 0.6 0.8 1 t/t Figure 10. Evolution of the displacing fluid saturation for contact angles of 162 , 150 , 135 , and 120 . 5. Conclusions The colour gradient two-phase LBM was used to study dynamic displacement of immiscible ﬂuids in a 2D micromodel of Berea sandstone. The effects of the viscosity ratio, interfacial tension, and contact angle on the ﬂuid distributions at breakthrough and the evolution of displacing ﬂuid saturation were systematically investigated. When the viscosity ratio is no more than 20, a higher viscosity ratio results in a lower displacing ﬂuid saturation which is less favourable for the recovery of the displaced ﬂuid. However, when the viscosity ratio is larger than 20, the saturation is almost constant for both imbibition and drainage, regardless of the interfacial tension. At the viscosity ratio of unity, the displacing ﬂuid saturation has the highest value, and it linearly increases with time. The interfacial tension affects the ﬂow pattern. When the interfacial tension is as low as 0.0005 muts , thin viscous ﬁngers appear. A number of small drops of displaced ﬂuid are left 2 2 behind. As the interfacial tension increases from 0.0005 muts to 0.015 muts , the size of the trapped blobs increases, and the number decreases. The displacing ﬂuid ﬂows into large pores more easily due to the increased capillary pressure. When the interfacial tension continues to increase, smaller amount of the displaced phase is left behind as the invading phase continues to move forward. In addition, increasing interfacial tension results in a higher saturation of wetting ﬂuid. The contact angle generally does not change preferential ﬂow paths except for a few branches in the imbibition, suggesting that the contact angle has limited effect on the efﬁciency of displacement within 2D Berea sandstone. Author Contributions: Q.G. performed the simulations, analysed the results and wrote the paper, H.L., Y.Z. helped to analyse the results and write the paper. Funding: UK Engineering and Physical Sciences Research Council (EPSRC) under grant number EP/L00030X/1 and EP/K000586/17. Acknowledgments: The authors would like to thank the “UK Consortium on Mesoscale Engineering Sciences (UKCOMES)” (www.ukcomes.org) funded by the UK Engineering and Physical Sciences Research Council (EPSRC) under grant number EP/L00030X/1 to provide computing time on ARCHER. The simulations were also performed on ARCHIE-WeSt funded by EPSRC (EP/K000586/17). Conﬂicts of Interest: The authors declare no conﬂict of interest. Saturation of displacing fluid Appl. 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Lattice Boltzmann Simulation of Immiscible Two-Phase Displacement in Two-Dimensional Berea Sandstone