Feb 26, 2013 - Ðrp ¼ l kd. V. Ñ1Ю. The linearity in the Darcy's law is a direct consequence of the linearity of the Stokes (creeping) flow in porous media. At ele-.
Lattice Boltzmann Simulation of Non-Darcy Flow In Stochastically Generated 2D Porous Media Geometries M.S. Newman, SPE, Chevron Australia Pty. Ltd, and X. Yin, SPE, Colorado School of Mines
Summary It is important to consider the additional pressure drops associated with non-Darcy flows in the near-wellbore region of conventional gas reservoirs and in propped hydraulic fractures. These pressure drops are usually described by the Forchheimer equation, in which the deviation from the Darcy’s law is proportional to the inertial resistance factor (b-factor). While the b-factor is regarded as a property of porous media, detailed study on the effect of pore geometry has not been performed. This study characterized the effect of geometry on the flow transition and the b-factor using lattice Boltzmann simulations and stochastically constructed 2D porous media models. The effect of geometry was identified from a large set of data within a porosity range of 8–35%. It was observed that the contrast between pore throat and pore body triggers an early transition to non-Darcy flows. Following a quick transition where the correction to the Darcy’s law was cubic in velocity, the flows entered the Forchheimer regime. The b-factor increased with decreasing porosity or an increasing level of heterogeneity. Inspection of flow patterns revealed both steady vortices and onset of unsteady motions in the Forchheimer regime. The latter correlated well with published points-of-transition. In developing a dimensionally consistent correlation for the b-factor, we show that it is necessary to include two distinctive characteristic lengths to account for the effect of pore-scale heterogeneity. This finding reflects the fact that it is the contrast between pore bodies and throats that dictates the flow properties of many porous media. In this study, we used the square root of the permeability and the fluid-solid contact length as the two characteristic lengths. Introduction Non-Darcy flows are frequently observed in the near-wellbore region in conventional gas reservoirs and in propped hydraulic fractures. Non-Darcy flows occur at elevated fluid velocity, where the inertial effects make the pressure drop a nonlinear function of the velocity through the porous medium. Accurate determination of the pressure drops associated with non-Darcy flows is important for gas reservoir modeling and production enhancement (Buell and Crafton 1985; Miskimins et al. 2005; Ramirez et al. 2007). In Darcy flows, the relationship between the pressure and velocity is described by the linear Darcy’s law Eq. 1, where p is the pressure, l is the viscosity of the fluid, V is the fluid velocity through the porous medium, and kd is the Darcy permeability which is a property of the porous medium. rp ¼
l V: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð1Þ kd
The linearity in the Darcy’s law is a direct consequence of the linearity of the Stokes (creeping) flow in porous media. At elevated flow velocities, it has been shown both theoretically (Mei and Auriault 1991; Skjetne and Auriault 1999b; Wodie and Levy 1991) and numerically (Balhoff and Wheeler 2009; Coulaud et al. C 2013 Society of Petroleum Engineers Copyright V
This paper (SPE 146689) was accepted for presentation at the SPE Annual Technical Conference and Exhibition, Denver, 30 October–2 November 2011, and revised for publication. Original manuscript received for review 28 June 2011. Revised manuscript received for review 25 April 2012. Paper peer approved 1 May 2012.
12
1988; Hill et al. 2001; Souto and Moyne 1997) that the first correction to the Darcy’s law is cubic in the velocity. rp ¼
l V þ aV 3 : . . . . . . . . . . . . . . . . . . . . . . . . . ð2Þ kd
However, the cubic law only applies in a very narrow range of flow velocities beyond the creeping flow regime. With further increase in the flow velocity, the relation between the imposed pressure gradient and velocity becomes quadratic, yielding the well known Forchheimer’s law (Forchheimer 1901), which is perhaps the most popular correlation to describe non-Darcy flows in porous media. l rp ¼ V þ bqV 2 : . . . . . . . . . . . . . . . . . . . . . . . . ð3Þ kf In Eq. 3, kf is termed as the Forchheimer permeability and is very close but does not equal to kd (Balhoff and Wheeler 2009; Barree and Conway 2004; Barree and Conway 2005; Fourar et al. 2004) because of Eq. 2. The difference between kd and kf is shown schematically in Fig. 1. b has many names in the literature such as the non-Darcy coefficient, inertia resistance factor, or simply the beta factor. Eq. 3 does not apply at all flow velocities. In many studies (Fand et al. 1987; Huang and Ayoub 2008; Seguin et al. 1998a; Seguin et al. 1998b), departures from Eq. 3 were observed at very high flow velocities. It was proposed that Eq. 3 should be either modified with a cubic correction (Firoozabadi and Katz 1979; Forchheimer 1901) or replaced with other types of correlations (Barree and Conway 2004). In the regime where Eq. 3 applies (the Forchheimer regime), it is generally accepted that b is a property of porous medium, and it is expected that the geometry of the pores has a significant effect on its value. In this study, stochastic 2D porous media geometry models were generated to investigate the transition from Darcy to nonDarcy flows and the impact of pore geometry on the transition and the inertial resistance factor b. The flow in the pore space was resolved by a lattice Boltzmann method. This study is motivated by the fact that the effect of irregular pore geometry on the nonDarcy flow properties has not been well studied and quantified. In the literature, most laboratory studies examining non-Darcy flows used packing of regularly shaped particles (Comiti et al. 2000; Dybbs and Edwards 1984; Fand et al. 1987; Green and Duwez 1951). Such geometries are fine for studying gas flow through proppant packings. However, they bear little resemblance to consolidated reservoir rocks that usually have highly irregular geometries (Geertsma 1974). Experimental data obtained from consolidated rocks showed much larger variations in b than those obtained from unconsolidated packings (Firoozabadi and Katz 1979). It is thus of great interest to explore the effect of irregular pore geometry on the value of b. On the numerical simulation side, methods based on network models (Balhoff and Wheeler 2009; Wang et al. 1999) must rely on assumed shapes of the pore geometry. Alternative methods based on solution of the NavierStokes equation can be applied to any given pore geometry. However, so far they have only been used to study non-Darcy flows in highly simplified porous media geometries made up by rectangular posts, cylinders, or spheres (Hill et al. 2001; Koch and Ladd 1997; Lee and Yang 1997; Rojas and Koplik 1998; Fourar et al. 2004) and only a few studies (Magnico 2009; Petrasch et al. February 2013 SPE Journal
right-hand side approaches unity when V ! 0. The difference between Eq. 3 and Eq. 6 is that b is generally regarded as a material property, whereas Fo depends on both material and flow. In this work, we will use k* as defined in Eq. 6 to highlight the transition from Darcy to non-Darcy flows, but will present the pffiffiffiffiffiresults in dimensionless forms of the inertial resistance factor b kd and b/s. There are many correlations for b in the literature. Li and Engler (2001) and Lopez-Hernandez et al. (2004), among others, have presented good reviews on the existing correlations for b. A general form from the reviewed work is
–∇p/μV
1/kd
b ¼ C1 sC2 kdC3 /C4 ; . . . . . . . . . . . . . . . . . . . . . . . . . . ð7Þ 1/kf
V Fig. 1—This diagram displays qualitatively the difference between kd and kf in the paper. The circles connected by the solid line show the general trends followed by numerical and experimental data, and the dotted line indicates Forchheimer’s law.
2008) have applied them to complex geometries such as ceramics and metallic foams. In this study, the simulation data obtained were analyzed using dimensionless variables. Typically, a Reynolds number is used to quantify the transition from Darcy to Forchheimer flow regime. The Reynolds number measures the relative importance between fluid inertia and the viscous effects and is defined as Re ¼
qVl ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð4Þ l
where l is a characteristic length of the porous medium. Many definitions of the Reynolds number exist in the literature, for example qVd ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð5aÞ l pffiffiffiffiffi qV kd Rek ¼ ; . . . . . . . . . . . . . . . . . . . . . . . . . . . ð5bÞ l
Red ¼
Res ¼
qV : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð5cÞ ls
Red is typically used when the internal length scale of the porous medium is known. For example, d can be the width of the conductive channels in the geometry of the porous medium or the size of particles that make up the packing. Rek is defined based on the square root of the Darcy permeability (also referred to as the Brinkman length); Res (for 2D porous media) is based on s, the perimeter of fluid/solid contact per unit area of the porous medium. The analog of s in 3D porous media is the surface area per unit volume of porous medium that is open to fluid flow. Of the three definitions, calculation of Red requires a characteristic pore size or grain size from a microscale study. In contrast, kd and s can be obtained relatively easily from bulk measurements. In view of this relative convenience, in this study we will present our data in terms of Rek and Res. In an alternative formulation (Geertsma 1974; Lopez-Hernandez et al. 2004; Ruth and Ma 1992; Zeng and Grigg 2006), it was proposed that the departure from Darcy’s law be described by
where s is the tortuosity and C1 through C4 are constants that are formation-specific. In Eq. 7, C2 must be 0.5 to make Eq. 7 dimensionally correct. However, only the correlations proposed by Ergun (1952), Geertsma (1974), McFarland and Dranchuk (1976), Macdonald et al. (1979), and Comiti et al. (2000) satisfied this requirement. Changes in a flow regime are usually connected to changes in the flow patterns in porous media. The transition from Darcy to Forchheimer flow regimes in particular is associated with the inertial effects manifested when fluid particles take tortuous paths to move through porous media (Hlushkou and Tallarek 2006). Flows in the Forchheimer regime feature streamlining (Stanley and Andrade 2001) that is also described as inertial cores (Dybbs and Edwards 1984), localized kinetic energy dissipation (Andrade et al. 1999; Skjetne and Auriault 1999a), and onset of unsteady motions (Hill and Koch 2002; Seguin et al. 1998b). Using the advantages of a lattice Boltzmann method, we were able to present data in this paper showing the various interstitial flow patterns in complex porous media geometries that cannot be easily observed in laboratory experiments. This paper is organized as follows: we first introduce the lattice Boltzmann method. Second, the grain geometry creation method and the properties of the created geometries are presented. In the Results section, we first discuss the outcome of the numerical convergence study; we then follow with results on the point of transition between the Darcy and the Forchheimer flow regimes and on the inertial resistance factor b. The effect of porous media geometry is discussed and quantified. Finally, the fluid-flow patterns in the Forchheimer flow regime are presented. Methodology Lattice Boltzmann Method. In this investigation, a lattice Boltzmann method was used to simulate fluid flow. It does not directly solve the nonlinear Navier-Stokes equations, but rather solves a simplified kinetic model for the fluid molecular velocity distribution, the moments of which recover the Navier-Stokes equations. As a new technique for fluid-flow simulation, it has attracted much interest because of its ability to simulate fluid-dynamics problems with complex boundaries. These fluid-dynamics problems include suspensions, emulsions, single- and multiphase flows in porous media, and viscoelastic flows (Lallemand and Luo 2000). Many reviews of the method and its applications are available in the literature [Chen and Doolen (1998) and, more recently, Aidun and Clausen (2010)]. The lattice Boltzmann method consists of two processes. The first process describes “particle” advection; the second describes “particle” collision on a regular lattice. The term “particle” is defined as a collection of molecules that travel with velocity ca at node rj. The fluid nodes (rj) are defined on a space filling square (2D) or cubic (3D) lattice. The amount of particles that travel with velocity ca at node rj is described by the particle velocity distribution fa ðrj ; tÞ: The lattice Boltzmann advection and collision can be written collectively as
1 kd kd rp ¼ 1 þ Fo: . . . . . . . . . . . . . . . . . . ð6Þ ¼ ¼ k kapp lV
fa ðrj þ ca Dt; t þ DtÞ ¼ fa ðrj ; tÞ þ Xa ð fa Þ: . . . . . . . . . . . ð8Þ
In Equation 6, k* is the dimensionless apparent permeability and Fo is the Forchheimer number. The benefit of Eq. 6 is that the
Xa ðÞ is an operator that yields the effect of the particle collision. The post collision population fa ðrj ; tÞ þ Xa ð f Þ is then advected to
February 2013 SPE Journal
13
6
2
7 α Index
Lattice Pathway 4
1
5 Lattice Node
9
3 Advection: Particles redistribute along the lattice after collision
Collision: Redistributed particles collide
8
Advection: Each node receives redistributed particles
Fig. 2—Repeated advection and collision processes in the lattice Boltzmann method as shown at one lattice location (D2Q9 representation).
location rj þ ca Dt in the next timestep. Through the advection process, particles are redistributed on the lattice; through the collision all particles on a lattice location rj exchange momentum in a given fashion that conserves net momentum and mass. The method iterates between the two processes, as shown in Fig. 2. On the basis of the kinetic theory, for isothermal systems, in the lattice Boltzmann method the microscopic density (q) and momentum ( j) at each node can be calculated as follows: X X fa . . . . . . . . . . . . . . . . . . . . . . . . . ð9Þ q¼ faeq ¼ X X j¼ ca faeq ¼ ca fa : . . . . . . . . . . . . . . . . . . . . . ð10Þ In this investigation, the D2Q9 (D2: two dimensions; Q9: nine flow pathways) advection model was employed for 2D simulations. Several models are available for calculating collisions. Essentially all satisfy the conservation of mass and momentum, but with varying limitations and assumptions. The most common and simplest is the BGK scheme. The equilibrium distribution faeq is found using a single equation (Eq. 11). The collision is then calculated using a relaxation parameter sr (Eq. 12) where u is the hydrodynamic velocity at an individual lattice point. 9 3 2 2 eq . . . . . . . ð11Þ fa ¼ wa q 1 þ 3ðca uÞ þ ðca uÞ u 2 2 Xa ðfa Þ ¼
1 ½ fa ðrj ; tÞ faeq : . . . . . . . . . . . . . . . . . . . ð12Þ sR
In Eq. 11, wa is the weighting parameter and equals 4/9, 1/9, and 1/36 for a ¼ 1, 2 through 5, and 6 through 9, respectively. In this investigation, the multiple relaxation time (MRT) collision method was used. This method introduces MRTs to control the rates of relaxation of the various moments of the velocity distribution, and has advantages over the widely used BGK collision scheme for its stability and accuracy (Pan et al. 2006). The drawback, however, is the approximately 30% increase in computational time. In the MRT method, the MRTs are implemented by introducing the following matrix equation: 14
fa ðrj þ ca Dt; t þ DtÞ ¼ fa ðrj ; tÞ þ M1 S M fa ðrj ; tÞ: ð13Þ In this equation, M is the linear operator that calculates the kinetic moments of fa ðrj ; tÞ, and S is a diagonal matrix that contains the relaxation rates for the various moments. While mass and momentum are conserved before and after collision, energy conservation is not considered because the system is assumed to be isothermal. Other moments such as the stress tensors are not conserved through the collision process. The relaxation rates of these other moments are related to the shear and bulk viscosities of the fluid. The reader is referred to d’Humie`res (2002) and Lallemand and Luo (2000) for an in-depth description of the method. This lattice Boltzmann method is second-order accurate, and the compressibility error is proportional to the square of the Mach number Ma ¼ u=cs : Here, u is the hydrodynamic velocity in the unit of Dx=Dt where Dx is the lattice grid pffiffiffi spacing, and cs is the lattice speed of sound which equals 1= 3Dx=Dt: In this study, Ma was kept below 0.1 in all simulations. Therefore, the fluid is essentially incompressible. In the advection step, when the particle velocity distribution is directed toward a solid boundary, a link-bounceback scheme (Frisch et al. 1987) was used to recover the no-slip boundary condition at locations halfway between the solid and the fluid nodes. This boundary condition is first-order accurate in the lattice spacing. The numerical-convergence/grid-resolution tests presented later in the Results section show that the scheme is sufficient and the data are accurate. Compared with second-order accurate schemes [e.g., Bouzidi et al. (2001) and Ginzburg and d’Humie´res (2003)], the linkbounceback scheme is much simpler to code, is numerically stable, and is directly applicable to a random porous medium, the geometry of which is usually in the form of a discrete set of fluid and solid nodes (characterized by computed tomography). The potential benefit of higher-order schemes is to offer a similar accuracy at a lower grid resolution. However, many higher-order schemes also require additional information such as the fractional location where a fluid-solid nodal connection is intersected by a February 2013 SPE Journal
800
800
600
600
180
160 400
400
200
200
140
0
0
200
400
600
800
0
0
200
400
600
800
120 520
540
560
580
Fig. 3—Examples of porous-media geometries studied. On the left, no grains were removed; in the middle, a few randomly selected grains were removed to create enlarged pore bodies; a squared section of the graph in the middle is enlarged to show a typical grid made up by lattice points used to conduct lattice Boltzmann simulations. In this example, the channel is resolved by 16 lattice points across. The x and y axes are in the unit of lattice (grid) spacing Dx.
solid surface, which is only available accurately for geometries with analytical boundaries. 2D Geometry Generation. The geometry of a porous medium is an important factor for its flow and transport properties. Because one of the objectives of this study is to expose the effect of pore geometry on flow transition and the b-factor, we simulated fluid flow in stochastically constructed 2D porous-media geometry models where one can have full control over the geometry parameters. The geometries were generated using an algorithm based on Voronoi diagrams. Starting from a random set of points, the Voronoi diagram divides a 2D domain into many nonoverlapping cells (Voronoi 1907). In our algorithm, the edges of the Voronoi diagrams were converted into flow channels. The parameters of the algorithm can be adjusted to produce fully percolated geometries with both low and high effective porosities (10 to 50%). In addition, as a part of the geometry algorithm, the user can remove randomly selected grains with a prescribed probability. As Fig. 3 shows, the geometry with all grains retained simulates a porous medium texture that is homogeneous on the pore level; it features a single characteristic length scale: the channel width. The geometry with certain grains removed, on the other hand, simulates a porous medium texture with isolated large pore bodies. The latter type of geometry will be henceforth mentioned as microscopically heterogeneous or “vuggy” in this paper. Geometries of the latter type have more than one characteristic length (pore body or vug vs. pore throat). Grain textures with removed grains can be used to model porous media with heterogeneous geometry that contains many large pore bodies. The porosity occupied by large pore bodies created from removing grains is termed /p ; which can be calculated from: /p ¼ / /orig ; . . . . . . . . . . . . . . . . . . . . . . . . . . . ð14Þ where / is the porosity of the system and /orig is the porosity of the system if no grains were removed. The increased proportion of pore space occupied by the “vugs” created from grain removal is characterized by n; which is defined as follows: n¼
/p 1 ¼ 1: . . . . . . . . . . . . . . . . . . . . . . . . ð15Þ 1/ c
Here, c is the input parameter representing the proportion of grains retained. If c ¼ 100%; n ¼ 0; if c ¼ 95%; n ¼ 0:0526; and if c ¼ 90%; n ¼ 0:111: It should be noted that the input percentage into the grain geometry algorithm does not precisely remove 5 or 10% of grains. The parameter c is simply a probability threshold for the algorithm to decide if a grain is kept or removed. Setup of Simulation and Data Interpretation. To investigate the impact of grain geometry on non-Darcy flows, 11 sets (types) February 2013 SPE Journal
of geometries were created (Table 1). Each set contains five geometries generated with the same input parameters that have slightly different porosities because of statistical fluctuations in the creation of these synthetic media. These geometries covered a range of porosities from 8.8 to 30.3%, and contain both homogeneous grain textures (A, B, D, E, I, K) and “vuggy” grain textures (C, F, H, J, L). Because the geometries generated were periodic (periodic groups of Poisson points were used to generate the Voronoi geometries), periodic flow boundary conditions were applied to the sides of the computational domain. The pore fluid was initially stationary; driven by a body force that is equivalent to a pressure gradient, the fluid accelerates until the friction from the solid walls and the driving force reaches equilibrium. The superficial velocity of the fluid is then calculated. For each geometry, 30 or more simulations were run at varying pressure gradients and the velocities were measured. When applied pressure gradients are small, the Darcy permeability of the geometry kd can be determined. With increasing pressure gradients, from the pressure gradient, measured fluid velocity, and the previously obtained Darcy permeability, the dimensionless apparent permeability k* can be obtained and analyzed using dimensionless plots as exemplified in Fig. 4, where 1/k* is plotted against Rek. As demonstrated by Fig. 4, the point of transition away from Darcy flow can be clearly seen as a deviation away from the horizontal line. This deviation in the non-Darcy flow regime is then used to compute the inertial resistance factor (b) in the Forchheimer equation, which is generally accepted to be a function of porous-medium geometry. pffiffiffiffiffi The dimensionless inertial resistance p factor ffiffiffiffiffi b2 kd was then found using a plot of ðrp þ lV=kd Þ kd =qV against Rek. According to Eq. 3, in the Forchheimer regime, b ¼ ðrp þ lV=kf Þ=qV 2 : Because the initial cubic correction to Darcy’s law is only evident in a narrow range of Rek, kf is very close to kd, and we can effectively replace kf in the expression for b with kd. As exemplified in Fig. pffiffiffiffiffi 5, starting from zero (the Darcy’s limit), ðrp þ lV=kd Þ kd =qV 2 increases with increasing Rek and levels off at a constant value, indicating that the flow has entered pffiffiffiffiffi the Forchheimer regime. The constant value is then taken as b kd : Results Numerical Convergence. In all numerical simulations, adequate resolution is essential. Adequate lattice (grid) resolution is particularly important for non-Darcy flows. One must increase the lattice resolution progressively with increasing Reynolds number to resolve the momentum boundary layer, the thickness of which is inversely proportional to the square root of the Reynolds number (Hill and Koch 2002; Hill et al. 2001a; Hill et al. 2001b; Tenneti 15
TABLE 1—2D GRAIN TEXTURES CREATED FOR LATTICE BOLTZMANN FLOW SIMULATION Geometry Type
16
No. of Texture Grains
System Size (Dx)
Channel Grains Porosity Darcy Permeability Size (Dx) Retained (/) kd (Dx2)
Geometry A
1 2 3 4 5
101 97 99 102 103
1,600800
20
100%
31.1% 29.9% 29.6% 30.1% 30.9%
5.53 5.31 5.12 5.51 5.59
Geometry B
1 2 3 4 5 6
108 106 104 107 107 -
1,600800
16
100%
23.6% 23.4% 23.3% 24.1% 24.2% 24.2%
2.58 2.56 2.53 2.77 2.65 2.88
Geometry C
1 2 3 4
104 93 96 90
1,600800
16
90%
27.4% 33.9% 29.6% 31.5%
2.90 5.13 3.42 3.81
Geometry D
1 2 3 4 5
59 57 57 58 58
1,600800
16
100%
17.3% 17.1% 17.3% 17.6% 17.3%
1.73 1.73 1.76 1.84 1.87
Geometry E
1 2 3 4 5
58 58 57 58 58
1,600800
20
100%
21.3% 21.5% 21.6% 20.8% 21.1%
3.45 3.68 3.59 3.61 3.70
Geometry F
1 2 3 4 5
55 52 51 54 52
1,600800
16
90%
21.0% 21.0% 24.8% 23.1% 26.6%
2.44 2.19 2.47 2.10 2.43
Geometry H
1 2 3 4 5
49 52 48 58 50
1,600800
20
90%
29.9% 25.1% 33.3% 23.3% 27.1%
4.95 4.06 5.90 3.79 4.09
Geometry I
1 2 3 4 5
35 35 35 33 32
1,600800
16
100%
12.7% 12.6% 12.0% 12.0% 12.6%
1.26 1.28 1.21 1.26 1.27
Geometry J
1 2 3 4
33 29 31 31
1,600800
16
95%
15.7% 20.9% 19.8% 22.2%
1.37 1.57 1.65 1.76
Geometry K
1 2 3 4 5
20 19 20 20 20
3,2001,600
16
100%
8.9% 8.5% 8.8% 9.0% 8.8%
0.79 0.78 0.82 0.90 0.87
Geometry L
1 2 3 4 5
42 46 45 44 46
3,2001,600
16
95%
13.1% 16.1% 11.4% 13.3% 13.4%
0.83 0.92 0.85 0.93 0.87 February 2013 SPE Journal
1.3
0.2 Non-Darcy Flow
1.25
√k
d
0.16
∇p + μV/kd pV 2
1/k*
1.2 1.15 1.1
0.12 0.08 0.04
Darcy Flow
1.05
0 0
1 0.01
0.1
1
0.5
1
Fig. 4—Example of an inertia plot. In this system, porosity is 16.9%. The data obtained in this figure corresponds to Geometry Type J, Texture 1.
et al. 2011). To find the optimum lattice resolution for the simulation of non-Darcy flows in porous media, a study on numerical convergence was carried out. First, a geometry that contains 101 grains and is 800400 ½ðDxÞ2 in size with a prescribed channel width of 8Dx was created where Dx is the lattice spacing. The geometry was then successively enlarged to 1,600800 ½ðDxÞ2 and 2,4001,200 ½ðDxÞ2 and the channel widths became 16Dx and 24Dx; respectively. Applying different force densities across the grain geometry allowed us to measure the change in the apparent permeability with increasing fluid velocity. To compare the different resolutions, a dimensionless plot was used where the dimensionless apparent permeability k* was calculated and plotted as a function of Rek. Fig. 6 shows the effect of channel resolution to the dimensionless apparent permeability. It appears that at low Reynolds numbers where deviation from Darcy flow begins, all three channel widths provided similar results. However, when Rek is approximately 1, the results from the geometry with a channel width of
2
Rek
10
Rek
1.5
pffiffiffiffiffiffi Fig. kd is obtained at higher Rek, where 2(p1lV/kd) pffiffiffiffiffiffi 5—b kd /qV 2 becomes nearly independent of Rek. Geometry J, Texture 1.
8Dx started to deviate from those with higher resolutions. These results also show that there is limited benefit to increasing the channel width to 24Dx because little difference exists between data obtained with resolution of 16Dx and that with 24Dx throughout the range of Reynolds numbers tested. Therefore, a channel width of 16Dx was used in our 2D simulations to investigate nonDarcy flows. Darcy Permeability. The Darcy permeabilities of the test geometries in Table 1 were summarized in Fig. 7. The Darcy permeability kd is nondimensionalized using s, the perimeter of fluid/solid contact per unit area of porous medium. It may be observed that permeabilities from homogeneous geometries have much less scatter than those from heterogeneous geometries, and the normalized permeabilities can clearly be correlated as a simple function of porosity in the Kozeny form consistent with nondimensional analysis (Bird et al. 2002). kd s2 ¼ 0:0902/3 : . . . . . . . . . . . . . . . . . . . . . . . . . . ð16Þ
1
k*
0.9
0.8
0.7 0.001
0.01
0.1
1
10
Rek 800x400 System Size, Channel Size of 8Δx
1600x800 System Size, Channel Size of 16Δx
2400x1200 System Size, Channel Size of 24Δx Fig. 6—Numerical convergence study. The effect of lattice (grid) resolution on the dimensionless apparent permeability is shown. February 2013 SPE Journal
17
0.01 0.003 0.002 0.001 0.001
0.000 0.1
0.2
0.3
kd S 2
0
0.0001
0.00001 0%
5%
10%
15%
20%
25%
30%
35%
40%
Porosity ξ=0
ξ = 0.0526
ξ = 0.111
Fig. 7—Permeability-porosity cross plot. The insert in the upper left corner shows the Kozeny equation fit (Eq. 16) for data with n 5 0 with an R2 value of 0.99. The horizontal axis of the insert is porosity, and the vertical axis is kds2.
The permeabilities in heterogeneous geometries are lower than those from homogeneous geometries of the same porosity, because the permeabilities are controlled by the pore throats, and the porosities in isolated large pore bodies do not control the permeability.
Transition From Darcy to Non-Darcy Flow. The transition from Darcy to non-Darcy flow is analyzed using dimensionless plots k* vs. Rek (Fig. 8). Clearly, grain geometry impacts the point of transition. In Fig. 8a, homogeneous grain textures that are completely channelized make transition consistently when Rek is between 0.3 and 0.4. For heterogeneous “vuggy” geometries (Fig. 8b and 8c) with isolated pore bodies, the point of transition occurs when Rek is between 0.1 and 0.2, and with a greater degree of scatter. There is no clear difference between heterogeneous textures with n ¼ 0:0526 (Fig. 8c) and those with n ¼ 0:111 (Fig. 8b). For homogeneous geometries, a trend was observed that the point of transition increased with increasing porosity. However, no systematic dependence of the point of transition on the porosity was observed for the heterogeneous geometries. The higher fluctuations observed in the heterogeneous geometries are likely to be related to the modest number of grains removed (on average four to 10). The number of grains removed is limited by the size of computational domain that can be practically simulated. If larger computational domains can be used, increasing the number of grains removed is likely to improve the statistics and may lead to better demonstration of the effect of porosity and distinction between Fig. 8b and 8c. Our explanation for the difference in the point of transition between homogeneous and heterogeneous geometries is as follows: The contrast between pore body and pore throat triggers a rapid change in fluid velocity. As fluid velocity increases, this contrast causes an early transition between viscous flows in the Darcy regime to inertia-dominated flows in the non-Darcy regime. The scaling of the initial departure from the Darcy’s law is examined in Fig. 9. Starting from Eq. 2, one can derive kd rp al 1 ¼ ln 2 þ 2lnðRek Þ: . . . . . . . . . . ð17Þ ln lV q kd rp In Fig. 9, we plotted ln 1 vs. lnðRek Þ: It is clear lV that the initial departure from the Darcy’s law follows the slope of 18
two, indicating that the cubic law is valid. However, Eq. 2 only holds when 0.1 < Rek < 0.3. When Rek becomes 13, the data began to follow the slope of one, indicating that the flow entered the Forchheimer regime. This result is consistent with the 2D simulations conducted by Koch and Ladd (1997) over regular and random arrays of cylinders as well as the 2D simulations conducted by Fourar et al. (2004) over a face-centered squre cell with a single cylinder. Heterogeneous geometries have larger intercepts at lnðRek Þ ¼ 0; indicating that they have higher a values. Again, the observation confirms that flows in heterogeneous geometries would become non-Darcy at lower Reynolds numbers (early transition). Although the initial departure from the Darcy’s law is clearly cubic, it is also evident from the small inserts in Fig. 8a through 8c that the cubic regime is very short and the difference between kd and kf is very small. Therefore, the quadratic Forchheimer equation with kf approximated as kd should be acceptable for all practical purposes. We did not observe any systematic deviation from the Forchheimer equation on the high-Re end (Rek < 10). The Inertial Resistance Factor. pffiffiffiffiffi Fig. 10 shows the dimensionless inertial factors b/s and b kd as functions of porosity. It can be observed that the b-factor decreases with increasing porosity, which agrees with many experimental results. In Fig. 10a, b/s obtained from microscopically homogeneous geometries have a very low scatter, indicating that they can be well correlated as a function of porosity. This observation agrees with the outcome of a dimensional analysis: In a microscopically homogeneous porous medium where there is only one characteristic length (here, 1/s is the length pffiffiffiffiffi scale), b/s must be a function of the porosity. In Fig. 10b, b kd from homogeneous geometries have a larger scatter than in Fig. 10a. Still, they can be well correlated as a function of porosity. A more important observation from Fig. 10 is that the inertial resistance factors obtained from microscopically heterogeneous geometries are significantly higher than those obtained from homogeneous geometries. In Fig. 10a, b/s obtained from heterogeneous are approximately 3–5 times larger; in Fig. 10b, pffiffiffiffigeometries ffi b kd obtained from heterogeneous geometries are approximately 2–3 times larger. Clearly, if one were to design a “universal” correlation for the inertial resistance factor that can be applied to both homogeneous and heterogeneous porous media geometries, one must bring in an additional parameter to reflect the presence February 2013 SPE Journal
1.05 (a)
1.00 0.95
1.6
0.80
1.4
0.75
1/k*
1.8
0.85
k*
0.90
1.2 1.0
0.70
0.8 0
0.65
2
4
6
8
10
Rek 0.60 0.001
0.01
0.1
1
10
Rek Geometry A
Geometry B
Geometry D
Geometry E
Geometry I
Geometry K
1.05 (b) 1.00 0.95 0.90 1.6 0.85
1/k*
k*
1.4 0.80
1.2
0.75
1.0 0.70 0.8 0.65
0.0
0.2
0.4
0.6
Rek
0.60 0.001
0.01
0.1
1
10
Rek Geometry C
Geometry F
Geometry H
1.05 (c) 1.00 0.95
0.85
1.2
0.80
1/k*
1.3
k*
0.90
1.1
0.75
1.0
0.70
0.9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Rek
0.65 0.60 0.001
0.01
0.1 Rek Geometry J
1
10
Geometry L
Fig. 8—Collection of dimensionless normalized permeability k* plotted against Rek from all simulations showing the transition from Darcy to non-Darcy flow. f 5 0 in A, f 5 0.0526 in B, and f 5 0.111 in C. The small inserts show the 1/k* plotted against Rek.
of multiple length scales—pore bodies (vugs) and pore throats (channels)—in microscopically heterogeneous porous media. The fact that kds2 are different between homogeneous and heterogeneous geometries in Fig. 7 indicates that kd and s are not February 2013 SPE Journal
fully correlated [i.e., kd 6¼ f ðs; /Þ]. Therefore, they can be used simultaneously in a correlation to provide two distinctive length scales. A dimensionally consistent correlation for b that includes both kd and s can be of the following form 19
0.00
–1.00
–2.00
In(–kd∇p/μv–1)
–3.00
–4.00
–5.00
–6.00
–7.00 Texture A1 –8.00
–9.00 –4.00
–3.00
–2.00
Texture A5
Texture A2 Texture C1
Texture A3 Texture C2
Texture A4 Texture C3
Texture C4 Texture E4
Texture E1 Texture E5
Texture E2
Texture E3
–1.00
0.00
1.00
2.00
3.00
In(Rek) dp Fig. 9—Plot of ln klV 1 vs. lnðRek Þ: Data were from simulations in Geometry A, Geometry C, and Geometry E (Table 1). The dashed lines have slopes of 2 and 1 to assist chart reading.
b ¼ D1 kdD2 s2D2 þ1 f ð/Þ . . . . . . . . . . . . . . . . . . . . . . . ð18Þ In this equation, D1 and D2 are fitting parameters and f ð/Þ is an arbitrary function of the porosity. After some trial and error, we find that the following equation: b
pffiffiffiffiffi /ðkd s2 Þ1=2 . . . . . . . . . . . . . . . . . . ð19Þ kd ¼ 0:00993 pffiffiffiffiffiffiffiffiffiffiffiffi 1/
provides a good fit to our numerical data. The quality of fitting can be viewed in Fig. 11. Eq. 19 fits our numerical data with an average deviation of 16%, and the maximum deviation is 47%. Data from heterogeneous geometries show larger scatters, indicating that the locations of the voids are important and have not been totally eliminated from averaging. To suppress the scatter, larger representative elemental areas (simulation domains) may be needed. Even so, the ability of Eq. 19 to bring together b-factors that are far apart (Fig. 10) suggests that, when developing predictive models for porous media flow properties, it is important to account for the multiple length scales that are inherently present in many porous-media geometries. Flow Patterns in Non-Darcy Flow Regimes. It has been noted that flow pattern in porous media changes with increasing Reynolds number (Dybbs and Edwards 1984). Specifically, localized high-speed flow regions with intense energy dissipation will appear (Andrade et al. 1999; Stanley and Andrade 2001), and these regions are referred to as “inertial cores” or “streamlines” in the literature. In addition, in numerical simulations of flow through packed beds of spheres, it was found that when the Reynolds number (based on particle size) is increased to approximately 30, flow becomes unsteady, which is considered as the onset of turbulence (Hill et al. 2001b). In experiments performed in packed bed of spheres, unsteady motions were observed when 20
Reynolds number based on particle size reached 180 (Seguin et al. 1998a). A high-porosity 2D grain texture was created to investigate the flow patterns in Darcy and non-Darcy regimes. This geometry has a porosity of 61.4%. The use of a high-porosity medium amplifies the contrast between pore throat and pore bodies such that its effect on inertial flow patterns becomes more obvious for observation. As found previously, the contrast between pore throat and pore body has a major influence over flow-regime transition. Although porosity in this case is in excess of what is typically found in sand packs and reservoir rocks, it shows the contrast between pore body and pore throats over a small volume; diagenetic processes that remove whole grains could create local high porosity and a pore body to pore throat ratio that is even more significant than what is shown here. As such, this geometry is not totally removed from reality. This geometry is 400400 [(Dx)2] in size, and the channels have a width of 20Dx. In the following subsections, the flow patterns in the observed interstitial pore space in this geometry are discussed using visual examples from the data circled in Fig. 12. In Fig. 13, we focus on a small section of the entire porous medium. In order to display the data effectively, a velocity arrow is shown for only every fourth fluid node. The velocity plots provide qualitative detail, where red colors indicate higher velocity and blue colors indicate lower velocity. In a Darcy regime, viscous flow predominates. As the fluid moves from a pore throat to a pore body, the fluid spreads evenly to fill the pore space. Fig. 13a and 13c show the flow pattern observed at the start of the Darcy flow regime and in the transition to the Forchheimer flow regime. At these two Reynolds numbers, little difference in flow pattern was observed. The sweeping motion of the fluid is very efficient, and this regime exhibits the highest dimensionless apparent permeability. In the non-Darcy flow regimes, the dimensionless apparent permeability decreases with increasing Reynolds number. Fig. 13d and 13e show the flow velocities observed at the beginning of February 2013 SPE Journal
40 (a)
β/s
30
20
10
0 0.0
0.1
0.2
0.3
0.4
φ 0.30 (b) 0.25
β(kd)1/2
0.20 0.15 0.10 0.05 0.00 0.0
0.1
0.2
0.3
0.4
φ pffiffiffiffiffiffi Fig. 10—The dimensionless inertial resistance factors (a) b/s and (b) b kd plotted as functions of porosity. The blue diamonds were obtained from microscopically homogeneous geometries (A, B, D, E, I, K); the red squares were obtained from heterogeneous geometries where n 5 0.0526 (J and L); the green triangles were obtained from heterogeneous geometries where n 5 0.111 (C, F, H).
(Rek ¼ 0.879) and midway through (Rek ¼ 6.45) the Forchheimer regime. The flow patterns changed significantly between these two points. In Fig. 13d, before the onset of the Forchheimer regime, the flow pattern still has many of the characteristics
observed in the Darcy regime. But with increased Reynolds numbers, “streamlines,” which refer to streams of high-speed flows that bypassed much of the pore space and reduced the sweeping efficiency, began to develop through the porous medium, which is
0.25
12
0.20
1
0.15
0.8
0.10 k*
β (Δx –1) (Correlation)
0.30
0.6
0.05 0.4 0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.2
β (Δx –1) (LBM) Fig. 11—Parity plot showing the quality of fitting of Eq. 19 to the numerically obtained inertial resistance factors. The blue diamonds were obtained from microscopically homogeneous geometries (A, B, D, E, I, K); the red squares were obtained from heterogeneous geometries where n 5 0.0526 (J and L); the green triangles were obtained from heterogeneous geometries where n 5 0.111 (C, F, H). February 2013 SPE Journal
0 0.001
0.01 0.1 1 Reynolds Number, Rek
10
Fig. 12—Semilog plot of the dimensionless apparent permeability k* vs. Rek. The flow patterns at the circled locations are presented in further detail. 21
-5
x 10
340
-9
x 10
340
3 320
320
2.2
300
300
2
2.5
280
1.8
280 2
260
1.6
260
1.4 240
240 1.2
1.5
220
220 1
200
200
0.8
1 180
180
160
160
0.5
140
140
120
120
100
150
(a)
200
250
0.6 0.4 0.2 100
150
(b)
200
250
-3
x 10
340
7
320
-3
340
D
320 6
300 280
5
260
x 10 5 4.5
300
4
280 3.5 260
4
240 220
3 240 2.5
220 3
200
2
200 2
180 160
180
1.5
160
1
140
0.5
1 140 120
100
150
(c)
200
120
250
100
150
(d)
200
250
-3
340
0.035
D
320
C
0.03
300 280
0.025
260 240 220 200
x 10
340 320
16
300
14
280 12 260
0.02
B 0.015
10
240
8
220 200
A
0.01
180
6
180 4
160
160
0.005 140 120
2
140
100
150
200
250
(e)
120
100
150
200
250
(f)
Fig. 13—Fluid velocity map: distribution of viscous dissipation and pressure in the test porous medium geometry at different Reynolds numbers. (a) The velocity field at Rek 5 0.0054; (b) the viscous dissipation r:r/2 in the pore space (r is the viscous stress tensor) at Rek 5 0.0054; (c) the velocity field at Rek 5 0.313; (d) the velocity field at Rek 5 0.879; (e) the velocity field at Rek 5 6.45; (f) the viscous dissipation in the pore space at Rek 5 6.45; (g) the pressure distribution for (d); (h) the pressure distribution for (e). Features highlighted by arrows: (A) streamlines; (B) vortices in the large pore bodies; (C) vortices in the channels; (D) flow reversal perpendicular to the imposed pressure gradient. The horizontal and vertical axes were labeled in terms of lattice (grid) spacing. The kinematic viscosity of the fluid used in this simulation was m 5 1/300 (Dx2/Dt) except in (d), where m 5 1/75 (Dx2/Dt). The velocity scale bars are in the unit of (Dx/Dt)
marked by A in Fig. 13e. These streamlines became further defined as the Reynolds number increased. This characteristic is caused by fluid inertia and is sometimes referred to as the inertial core (Dybbs and Edwards 1984). Also observed in this region is the onset of steady-state vortices downstream to the predominating flow direction. Steady-state 22
vortices developed first in larger pore spaces (B in Fig. 13e) and then, with subsequent increase in the Reynolds number, spread into smaller pore spaces (C in Fig. 13e). Another observation made in regard to Fig. 13d and 13e is that the flow in a pore channel perpendicular to the imposed pressure gradient can reverse direction with increased Reynolds number (D in Fig. 13d and February 2013 SPE Journal
340
320
320
300
300
280
280
260
260
240
240
Δy
Δy
340
220
220
200
200
180
180
160
160
140
140
120
100
150
200
120
250
100
150
200
Δx
250
Δx
(g)
(h)
Fig. 13 (continued)—Fluid velocity map: distribution of viscous dissipation and pressure in the test porous medium geometry at different Reynolds numbers. (a) The velocity field at Rek 5 0.0054; (b) the viscous dissipation r:r/2 in the pore space (r is the viscous stress tensor) at Rek 5 0.0054; (c) the velocity field at Rek 5 0.313; (d) the velocity field at Rek 5 0.879; (e) the velocity field at Rek 5 6.45; (f) the viscous dissipation in the pore space at Rek 5 6.45; (g) the pressure distribution for (d); (h) the pressure distribution for (e). Features highlighted by arrows: (A) streamlines; (B) vortices in the large pore bodies; (C) vortices in the channels; (D) flow reversal perpendicular to the imposed pressure gradient. The horizontal and vertical axes were labeled in terms of lattice (grid) spacing. The kinematic viscosity of the fluid used in this simulation was m 5 1/300 (Dx2/Dt) except in (d), where m 5 1/75 (Dx2/ Dt). The velocity scale bars are in the unit of (Dx/Dt)
In this numerical study, we did not observe a change in the bfactor even at the highest Reynolds number attempted. Therefore, our simulations did not reach the trans-Forchheimer regime, where a change in the b-factor is expected. Our simulations indicate that streamlines, vortices, and flow instability all exist in the Forchheimer regime. Therefore, streamlining cannot be the sole cause for the transition from the Forchheimer to the trans-Forchheimer flow regime, as a few previous studies have speculated (Balhoff and Wheeler 2009; Barree and Conway 2009; Hlushkou and Tallarek 2006). It is likely that the deviation away from the Forchheimer regime at very high Reynolds numbers is caused by intensification of fluctuations and full development of turbulence in the porous media. Conclusions In this study, we simulated the transition from Darcy to nonDarcy flows in synthetically generated 2D porous media models using a lattice Boltzmann method. It is confirmed that the initial departure from Darcy’s law is cubic in velocity. However, the
320
320
300
300
280
280
260
260
240
240
Δy
Δy
13e). This flow reversal is caused by the Bernoulli effect that reduces the pressure in the high-speed inertial core, as evidenced by the plots of the pressure fields (Fig. 13g and 13h). The viscous dissipation in the Darcy flow regime, as shown in Fig. 13b, is localized in the pore throats and near the walls. In contrast, the viscous dissipation in the Forchheimer flow regime, as shown in Fig. 13f, is not distributed evenly in pore throats but focused along the streamlines. With further increase in the Reynolds number, the onset of fluctuations in the velocity and pressure field was observed. Hill et al. (2001) investigated flow in ordered and random arrays of packed spheres and noticed that flow instability occurred at Red of approximately 30 to 70. Results from this investigation correlate well with those published in Hill et al. (2001). Fluctuations in the average flow velocity indicating weak flow instability were first observed at Rek ¼ 4.4, which corresponds with Red ¼ 21 and became more pronounced with increasing Re. Fig. 14 shows qualitatively the pressure distribution within the porous medium for a flow with Rek ¼ 21.4 at two different times. The vortices also change their sizes and strengths over time.
220
220
200
200
180
180
160
160
140
140
120
100
150
Δx
200
250
120
100
150
Δx
200
250
Fig. 14—Pressure distribution at two different times with Rek 5 21.4. Hot colors are higher in pressure; cooler colors are lower in pressure. The circles are used to highlight the change in the flow field with time. February 2013 SPE Journal
23
cubic law only applies until Rek becomes O(0.1). When Rek reaches approximately O(1), the flow enters the Forchheimer regime, where the apparent permeability becomes inversely proportional to the flow velocity. For the 2D geometries studied, the difference between kd and kf (Eqs. 2 and 3) is very small, and the quadratic law (Forchheimer equation) is sufficiently accurate up to the highest Reynolds number tested, Rek ¼10. The effect of pore geometry on the transition point is examined. It is found that geometries that are microscopically heterogeneous (vuggy) experience early transitions to the non-Darcy flow regimes. The early transition in heterogeneous geometries is caused by the contrast in the size of pore bodies (vugs) and the size of pore throats (channels) that amplifies the effect of fluid inertia. The inertial resistance factors obtained in heterogeneous geometries were also higher than those obtained in homogeneous geometries. We show that after using both the Darcy permeability and the fluid/solid contact length per unit area of porous medium as fitting parameters, the effect of multiple length scales in microscopically heterogeneous geometries can be adequately accounted for. The correlation that provided a good fit for our numerical data is of the following form: b
pffiffiffiffiffi /ðkd s2 Þ1=2 kd ¼ 0:00993 pffiffiffiffiffiffiffiffiffiffiffiffi : 1/
This correlation is dimensionally consistent and has an average error of 17%. Because this correlation is based on 2D simulations, we do not expect it to hold for 3D porous media. However, we believe that a finding of the paper that will equally apply to 3D geometries is that at least two length scales should be incorporated in the b factor if the objective is to develop a “universal” correlation. For 3D non-Darcy flows, numerical simulations conducted over periodic cells of cylinders (2D) and spheres (3D) (Fourar et al. 2004), over random arrays of cylinders (Koch and Ladd 1997) and spheres (Hill et al. 2001b), and over sphere packings (Maier et al. 1998) all show that 3D flows have faster transition to the Forchheimer (quadratic) flow regime (Eq. 3) than 2D flows. These observations suggest that the difference between kf and kd is going to be smaller for 3D flows than for 2D flows, and Eq. 3 should work even better. When simulating 3D non-Darcy flows, attention should be paid to the grid resolution. Specifically, the grid resolution must be progressively increased with increasing Reynolds number so that the momentum boundary layer, the thickness of which is inversely proportional to the square root of the Reynolds number (based on pore size), can be adequately resolved. Although very large grids are needed to accurately simulate non-Darcy flow in a 3D porous medium of a meaningful REV size, the computational resource needed is not totally out of touch with parallel computing. The flow patterns in Darcy and non-Darcy flow regimes were examined. Our results validate that Darcy flows and Forchheimer flows are dominated by viscous and inertial properties of the fluid, respectively. Streamlines and vortices appear in the Forchheimer regime and grow with increasing Reynolds number. When Rek was increased to approximately 4.4, fluctuations generated by flow instability were observed. The point of onset of the fluctuations correlates well with published numerical and experimental results. Nomenclature ca ¼ lattice Boltzmann velocity vector cs ¼ lattice speed of sound C ¼ constants d ¼ width of a conductive channel (e.g., pore diameter) D ¼ fitting parameters fa ¼ lattice Boltzmann particle velocity distribution faeq ¼ lattice Boltzmann equilibrium velocity distribution Fo ¼ Forchheimer number j ¼ momentum k* ¼ dimensionless apparent permeability, k* ¼ kapp/kd 24
kapp kd kf l M Ma rj Re s
¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼
S u V wa rp a b c Dt Dx l n q r s sR / /orig
¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼
/p ¼ Xa ¼
apparent permeability darcy permeability Forchheimer permeability length scale transformation matrix Mach number lattice fluid node location matrix Reynolds number wetted perimeter of the grains per unit area in a 2D porous medium diagonal matrix with relaxation rates velocity at an individual lattice point superficial velocity lattice Boltzmann weighting parameter pressure gradient cubic velocity coefficient inertial resistance factor proportion of grains retained lattice Boltzmann timestep one lattice (grid) spacing kinematic viscosity increased proportion of microvug pore space density viscous stress tensor tortuosity relaxation parameter porosity porosity from a synthetic porous media before grain removal porosity from the microvugs collision operator
Subscripts j ¼ Lattice index x, y, z ¼ x, y, or z direction in Cartesian coordinates a ¼ Order of the distribution Acknowledgments We acknowledge the work that Feng Xiao, Shaoyan Ji, and Mitra Azizian (Colorado School of Mines students) have done in testing the 2D simulation program. The computational part of the work was supported by the Golden Energy Computing Organization at the Colorado School of Mines, using resources acquired with financial assistance from the National Science Foundation and the National Renewable Energy Laboratory. References Aidun, C.K. and Clausen, J.R. 2010. Lattice-Boltzmann Method for Complex Flows. Annual Review of Fluid Mechanics 42 (1): 439–472. http://dx.doi.org/doi:10.1146/annurev-fluid-121108-145519. Andrade, J.S. Jr., Costa, U.M.S., Almeida, M.P. et al. 1999. Inertial Effects on Fluid Flow through Disordered Porous Media. Phys. Rev. Lett. 82 (26): 5249–5252. http://dx.doi.org/10.1103/PhysRevLett. 82.5249. Balhoff, M.T. and Wheeler, M.F. 2009. A Predictive Pore-Scale Model for Non-Darcy Flow in Porous Media. SPE J. 14 (4): 579–587. SPE110838-PA. http://dx.doi.org/10.2118/110838-PA. Barree, R.D. and Conway, M. 2009. Multiphase Non-Darcy Flow in Proppant Packs. SPE Prod & Oper 24 (2): 257–268. SPE-109561-PA. http://dx.doi.org/10.2118/109561-PA. Barree, R.D. and Conway, M.W. 2004. Beyond Beta Factors: A Complete Model for Darcy, Forchheimer, and Trans-Forchheimer Flow in Porous Media. Paper SPE 89325 presented at the SPE Annual Technical Conference and Exhibition, Houston, 26–29 September. http:// dx.doi.org/10.2118/89325-MS. Barree, R.D. and Conway, M.W. 2005. Reply to Discussion of SPE 89325, “Beyond Beta Factors: A Complete Model for Darcy, Forchheimer, and Trans-Forchheimer Flow in Porous Media.” J Pet Technol 57 (8): 73–74. February 2013 SPE Journal
Bird, R.B., Stewart, W.E., and Lightfoot, E.N. 2002. Transport Phenomena, second edition. New York: John Wiley & Sons Publishing. Bouzidi, M.H., Firdaouss, M., and Lallemand, P. 2001. Momentum transfer of a Boltzmann-lattice fluid with boundaries. Phys. Fluids 13 (11): 3452–3459. http://dx.doi.org/10.1063/1.1399290. Buell, R.S. and Crafton, J.W. 1985. The Application of the Cubic and Quadratic Forchheimer Equations in the Design of High-Rate, GravelPacked Gas Well Completions. Paper SPE 14448 presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, USA, 22–26 September. http://dx.doi.org/10.2118/14448-MS. Chen, S. and Doolen, G.D. 1998. Lattice Boltzmann Method for Fluid Flows. Annual Review of Fluid Mechanics 30: 329–364. http:// dx.doi.org/10.1146/annurev.fluid.30.1.329. Comiti, J., Sabiri, N.E., and Montillet, A. 2000. Experimental characterization of flow regimes in various porous media — III: limit of Darcy’s or creeping flow regime for Newtonian and purely viscous non-Newtonian fluids. Chem. Eng. Sci. 55 (15): 3057–3061. http://dx.doi.org/ 10.1016/s0009-2509(99)00556-4. Coulaud, O., Morel, P., and Caltagirone, J.P. 1988. Numerical modelling of nonlinear effects in laminar flow through a porous medium. J. Fluid Mech. 190: 393–407. http://dx.doi.org/doi:10.1017/S0022112088001375. d’Humie`res, D. 2002. Multiple–relaxation–time lattice Boltzmann models in three dimensions. Phil. Trans. R. Soc. Lond. A 360 (1792): 437–451. http://dx.doi.org/10.1098/rsta.2001.0955. Dybbs, A. and Edwards, R.V. 1984. A new look at porous media fluid mechanics. In Fundamentals of Transport Phenomena in Porous Media, ed. J. Bear and M.Y. Carapcioglu, No. 82, 199–265. New York: NATO ASI Series—Applied Sciences, Martinus Nijhoff. Ergun, S. 1952. Fluid Flow Through Packed Columns. Chem. Eng. Prog. 48 (2): 89–94. Fand, R.M., Kim, B.Y.K., Lam, A.C.C. et al. 1987. Resistance to the Flow of Fluids Through Simple and Complex Porous Media Whose Matrices Are Composed of Randomly Packed Spheres. J. Fluids Eng. 109: 268–274. Firoozabadi, A. and Katz, D.L. 1979. An Analysis of High-Velocity Gas Flow Through Porous Media. J Pet Technol 31 (2): 211–216. SPE6827-PA. http://dx.doi.org/10.2118/6827-PA. Forchheimer, P.F. 1901. Wasserbewegung durch Boden. Zeitschrift des Vereines deutscher Ingenieure 45 (5): 1781–1788. Fourar, M., Radilla, G., Lenormand, R. et al. 2004. On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Adv. Water Resour. 27 (6): 669–677. http:// dx.doi.org/10.1016/j.advwatres.2004.02.021. Frisch, U., d’Humie`res, D., Hasslacher, B. et al. 1987. Lattice gas hydrodynamics in two and three dimensions. Complex Systems 1: 649–707. Geertsma, J. 1974. Estimating the Coefficient of Inertial Resistance in Fluid Flow Through Porous Media. SPE J. 14 (5): 445–450. SPE4706-PA. http://dx.doi.org/10.2118/4706-PA. Ginzburg, I. and d’Humie`res, D. 2003. Multireflection boundary conditions for lattice Boltzmann models. Physical Review E 68 (6): 066614. http://dx.doi.org/10.1103/PhysRevE.68.066614. Green, L. and Duwez, P. 1951. Fluid flow through porous metals. J. Appl. Mech. 18: 39–45. Hill, R.J. and Koch, D.L. 2002. The transition from steady to weakly turbulent flow in a close-packed ordered array of spheres. J. Fluid Mech. 465: 59–97. http://dx.doi.org/doi:10.1017/S0022112002008947. Hill, R.J., Koch, D.L., and Ladd, A.J.C. 2001. The first effects on fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448: 213–241. http://dx.doi.org/10.1017/S0022112001005948. Hlushkou, D. and Tallarek, U. 2006. Transition from creeping via viscousinertial to turbulent flow in fixed beds. J. Chromatogr. A 1126 (1–2): 70–85. http://dx.doi.org/10.1016/j.chroma.2006.06.011. Huang, H. and Ayoub, J. 2008. Applicability of the Forchheimer Equation for Non-Darcy Flow in Porous Media. SPE J. 13 (1): 112–122. SPE102715-PA. http://dx.doi.org/10.2118/102715-PA. Koch, D.L. and Ladd, A.J.C. 1997. Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J. Fluid Mech. 349: 31–66. http://dx.doi.org/doi:10.1017/S002211209700671X. Lallemand, P. and Luo, L.-S. 2000. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Physical Review E 61 (6): 6546–6562. http://dx.doi.org/ 10.1103/PhysRevE.61.6546. February 2013 SPE Journal
Lee, S.L. and Yang, J.H. 1997. Modeling of Darcy-Forchheimer drag for fluid flow across a bank of circular cylinders. Int. J. Heat Mass Transfer 40 (13): 3149–3155. http://dx.doi.org/10.1016/s0017-9310(96)00347-x. Li, D. and Engler, T.W. 2001. Literature Review on Correlations of the Non-Darcy Coefficient. Paper SPE 70015 presented at the SPE Permian Basin Oil and Gas Recovery Conference, Midland, Texas, USA, 15–17 May. http://dx.doi.org/10.2118/70015-MS. Lopez-Hernandez, H.D., Valko, P.P., and Pham, T.T. 2004. Optimum Fracture Treatment Design Minimizes the Impact of Non-Darcy Flow Effects. Paper SPE 90195 presented at the SPE Annual Technical Conference and Exhibition, Houston, 26–29 September. http://dx.doi.org/ 10.2118/90195-MS. Macdonald, I.F., El-Sayed, M.S., Mow, K. et al. 1979. Flow through Porous Media-the Ergun Equation Revisited. Industrial & Engineering Chemistry Fundamentals 18 (3): 199–208. http://dx.doi.org/10.1021/ i160071a001. Magnico, P. 2009. Analysis of permeability and effective viscosity by CFD on isotropic and anisotropic metallic foams. Chem. Eng. Sci. 64 (16): 3564–3575. http://dx.doi.org/10.1016/j.ces.2009.04.036. Maier, R.S., Kroll, D.M., Kutsovsky, Y.E. et al. 1998. Simulation of flow through bead packs using the lattice Boltzmann method. Phys. Fluids 10 (1): 60–74. http://dx.doi.org/10.1063/1.869550. McFarland, J.D. and Dranchuk, P.M. 1976. Visualization of the Transition To Turbulent Flow In Porous Media. J Can Pet Technol 15 (2): 71–78. PETSOC-76-02-10. http://dx.doi.org/10.2118/76-02-10. Mei, J.C. and Auriault, J.-L. 1991. The effect of the weak inertia on flow through a porous medium. J. Fluid Mech. 222: 647–663. http:// dx.doi.org/10.1017/S0022112091001258. Miskimins, J.L., Lopez-Hernandez, H.D., and Barree, R.D. 2005. NonDarcy Flow in Hydraulic Fractures: Does it Really Matter? Paper SPE 96389 presented at the SPE Annual Technical Conference and Exhibition, Dallas, 9–12 October. http://dx.doi.org/10.2118/96389-MS. Pan, C., Luo, L.-S., and Miller, C.T. 2006. An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Computers & Fluids 35 (8–9): 898–909. http://dx.doi.org/10.1016/j.compfluid.2005.03.008. Petrasch, J., Meier, F., Friess, H. et al. 2008. Tomography based determination of permeability, Dupuit–Forchheimer coefficient, and interfacial heat transfer coefficient in reticulate porous ceramics. Int. J. Heat Fluid Flow 29 (1): 315–326. http://dx.doi.org/10.1016/j.ijheatfluidflow. 2007.09.001. Ramirez, B.A., Kazemi, H., Ozkan, E. et al. 2007. Non-Darcy Flow Effects in Dual-Porosity, Dual-Permeability, Naturally Fractured Gas Condensate Reservoirs. Paper SPE 109295 presented at the SPE Annual Technical Conference and Exhibition, Anaheim, California, USA, 11–14 November. http://dx.doi.org/10.2118/109295-MS. Rojas, S. and Koplik, J. 1998. Nonlinear flow in porous media. Physical Review E 58 (4): 4776–4782. http://dx.doi.org/10.1103/PhysRevE. 58.4776. Ruth, D. and Ma, H. 1992. On the derivation of the Forchheimer equation by means of the averaging theorem. Transport Porous Media 7 (3): 255–264. http://dx.doi.org/10.1007/BF01063962. Seguin, D., Montillet, A., and Comiti, J. 1998a. Experimental characterisation of flow regimes in various porous media—I: Limit of laminar flow regime. Chem. Eng. Sci. 53 (21): 3751–3761. http://dx.doi.org/ 10.1016/s0009-2509(98)00175-4. Seguin, D., Montillet, A., Comiti, J. et al. 1998b. Experimental characterization of flow regimes in various porous media—II: Transition to turbulent regime. Chem. Eng. Sci. 53 (22): 3897–3909. http://dx.doi.org/ 10.1016/s0009-2509(98)80003-1. Skjetne, E. and Auriault, J.-L. 1999a. Homogenization of Wall-Slip Gas Flow Through Porous Media. Transport Porous Media 36 (3): 293–306. http://dx.doi.org/10.1023/a:1006572324102. Skjetne, E. and Auriault, J.-L. 1999b. New insights on steady, non-linear flow in porous media. Eur. J. Mech. B. Fluids 18 (1): 131–145. http:// dx.doi.org/10.1016/S0997-7546(99)80010-7. Souto, H.P.A. and Moyne, C. 1997. Dispersion in two-dimensional periodic porous media. Part I. Hydrodynamics. Phys. Fluids 9 (8): 2243–2252. http://dx.doi.org/10.1063/1.869365. Stanley, H.E. and Andrade, J.S. Jr. 2001. Physics of the cigarette filter: fluid flow through structures of randomly-packed obstacles. Physica A 295 (1–2): 17–30. http://dx.doi.org/10.1016/S0378-4371(01)00140-6. 25
Tenneti, S., Garg, R., and Subramaniam, S. 2011. Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres. Int. J. Multiphase Flow 37 (9): 1072–1092. http://dx.doi.org/10.1016/j.ijmultiphaseflow. 2011.05.010. Voronoi, G. 1907. Nouvelles applications des parame`tres continus a` la the´orie des formes quadratiques. Journal fu¨r die Reine und Angewandte Mathematik 133: 97–178. Wang, X., Thauvin, F., and Mohanty, K.K. 1999. Non-Darcy flow through anisotropic porous media. Chem. Eng. Sci. 54 (12): 1859–1869. http:// dx.doi.org/10.1016/S0009-2509(99)00018-4. Wodie, J.-C. and Levy, T. 1991. Correction non line´arie de la loi de Darcy (Nonlinear recification of Darcy’s law). C.R. Acad. Sci., Ser. IIa: Sci. Terre Planets 312: 157–161. Zeng, Z. and Grigg, R. 2006. A Criterion for Non-Darcy Flow in Porous Media. Transport Porous Media 63 (1): 57–69. http://dx.doi.org/ 10.1007/s11242-005-2720-3.
Appendix In this paper, while we have used dimensionless numbers wherever possible, it should be pointed out that the raw unit for domain size, channel width, Darcy permeability kd, and the b factor is the fundamental length Dx(¼1 always), the grid (lattice) spacing, in the lattice Boltzmann method. Dx can be cast to arbitrary length units. The permeability kd with a unit of Dx2 and the b factor with a unit of Dx1 can be converted to real units using nondimensional analysis: kd kd ¼ Dx2 LBM Dx2 REAL
26
and ½bDxLBM ¼ ½bDxREAL : For example, geometry A1 has a Darcy permeability of 5.53 Dx2 : If Dxis cast to 1.3 lm, the channel size of A1 will be 30 lm, and the Darcy permeability of A1 will be 9.47 darcy, close to the permeability of a packing made up by 100-mesh sand used in many hydraulic fracturing operations. The size of A1 (representative volume) is therefore 2.08 mm1.04 mm. The b factor of A1, which is 0.0342Dx, reflects a real b factor of 2:63 104 m1 or 8:02 103 ft1 : The results presented in this paper, therefore, are not tied to any particular porous medium; they can be cast to cases with different real permeabilities, provided that the flow is governed by the Navier-Stokes equation. Michael Newman currently works for Chevron Australia Pty. Ltd. and has held roles in production and reservoir engineering. His research interests include reservoir modeling and porous-media fluid flow. Newman holds an undergraduate degree from the University of Adelaide and an MS degree in petroleum engineering from the Colorado School of Mines (CSM). Xiaolong Yin is an assistant professor in the Petroleum Engineering Department at CSM. Before joining the faculty at CSM, he was a postdoctoral researcher in chemical engineering at Princeton University. Yin’s research focuses on computational fluid dynamics, suspension, porous media, and phase behavior. He holds a BSc degree in theoretical and applied mechanics from Peking University, an MSc degree in mechanical engineering from Lehigh University, and a PhD degree in chemical engineering from Cornell University. He is an Associate Editor of the SPE Journal.
February 2013 SPE Journal
