A Criterion for Non-Darcy Flow in Porous Media

Transport in Porous Media (2006) 63: 57 69 Springer 2006 DOI 10.1007/s11242-005-2720-3 A Criterion for Non-Darcy Flow in Porous Media ZHENGWEN ZENG and REID GRIGG Petroleum Recovery Research Center, New Mexico Institute of Mining and Technology, Socorro, NM 87801, U.S.A. e-mails: {zeng,reid}@prrc.nmt.edu (Received: 28 October 2003; accepted: 11 February 2005) Abstract. Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid solid interactions to that by viscous resistance. It is directly related to the non-darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-darcy flow is expressed in terms of the critical non- Darcy effect. Considering a 10% non-darcy effect, the critical Forchheimer number would be 0.11. Key words: non-darcy behavior, Reynolds number, Forchheimer number, critical value. Nomenclature A sample cross-sectional area, cm 2 d characteristic length of the porous media, cm d t pore throat diameter, cm D p particle diameter, cm E non-darcy effect E c critical non-darcy effect Fo Forchheimer number Fo c critical Forchheimer number k permeability, 10 15 m 2 k 0 permeability at zero velocity, 10 15 m 2 l sample length, cm Author for correspondence

58 ZHENGWEN ZENG AND REID GRIGG M molecular weight of the fluid (gas), g/mol p pressure, atm p 1 pressure at sample inlet, atm p 2 pressure at sample outlet, atm Q flowrate, cm 3 /h Q p flowrate in the pump, cm 3 /h r radius of pore throat, cm R universal gas constant, atm-cm 3 /(g-mol-k) Re Reynolds number T temperature, K u intrinsic velocity of the fluid, cm/s v superficial velocity of the fluid, cm/s x dummy variable X direction of fluid flow y dummy variable Y direction normal to fluid flow z gas compressibility factor Z direction normal to fluid flow β non-darcy coefficient, 10 8 m 1 ϕ porosity µ fluid viscosity, Pa-s ρ fluid density, g/cm 3 ρ p fluid density in the pump, g/cm 3 1. Introduction Fluid flow in porous media is an important dimension in many areas of reservoir engineering, such as petroleum, environmental and groundwater hydrology. Accurate description of fluid flow behavior in the porous media is essential to the successful design and operation of projects in these areas. Darcy s law depicts fluid flow behavior in porous media. According to Darcy s law, the pressure gradient is linearly proportional to the fluid velocity in the porous media. The one-dimensional Darcy equation can be written as dp dx = µv k, (1) where p is the pressure, X is the direction of fluid flow, µ is the viscosity, v is the superficial velocity, and k is the permeability. The Darcy equation is an empirical relationship based on experimental observations of one-dimensional water flow through packed sands at low velocity. Efforts have been made to derive it theoretically via different approaches. Using the volumetric averaging theory, Whitaker (1969) derived the permeability tensor for the Darcy equation under low velocities. Following a continuum approach, Hassanizadeh and Gray (1980) developed a set of equations to describe the macroscopic behavior of fluid flow through porous media. Linearization of these equations yields a Darcy equation at low velocities. All these and other similar work indicate that Darcy s law

A CRITERION FOR NON-DARCY FLOW IN POROUS MEDIA 59 is an approximation in describing the phenomenon of fluid flow in porous media, and is valid under a limited range of low velocities. Experimentally, fluid flow deviations from Darcy s law have long been observed. Various terms, such as non-darcy flow, turbulent flow, inertial flow, high velocity flow, etc., have been used to describe this behavior (Firoozabadi and Katz, 1979). Many attempts have been made to correct the Darcy equation. Forchheimer (1901) added a second order of the velocity term to represent the microscopic inertial effect, and corrected the Darcy equation into the Forchheimer equation: dp dx = µv k + βρv2, (2) where β is the non-darcy coefficient and ρ is the fluid density. Considering the macroscopic shearing effect between the fluid and the pore walls, Brinkman (1947) added the second-order derivatives of the velocity to the Darcy equation, resulting in the Brinkman equation (Civan and Tiab, 1991): p X = µv ( 2 ) k µ v Y + 2 v, (3) 2 Z 2 where X, Y and Z are mutually perpendicular directions. In most porous media, pore diameter is very small and the change of velocity across the pore throat is negligible; thus the Brinkman equation will not be further discussed in this work, and the term non-darcy is used in place of inertial effect. Non-Darcy behavior has shown significant influence on well performance. Holditch and Morse (1976) numerically investigated the non-darcy effect on effective fracture conductivity and gas well productivity. Their results show that at the near-wellbore region, non-darcy flow could reduce the effective fracture conductivity by a factor of 20 or more, and gas production by 50%. Non-Darcy effect on hydraulically fractured wells has also been confirmed by others (Guppy et al., 1982; Matins et al., 1990). Due to the importance of the non-darcy effect, efforts have been made to include it in well performance simulations (Ewing et al., 1999). However, its inclusion dramatically increases the expense of numerical simulation with a high order of approximation (Garanzha et al., 2000). A criterion for predicting non-darcy flow in porous media is needed. The earliest work on the criterion for non-darcy flow behavior in porous media was apparently published by Chilton and Colburn (1931). Due to previous belief that non-darcy flow in porous media was similar to turbulent flow in a conduit, the Reynolds number for identifying turbulent flow in conduits was adapted to describe non-darcy flow in porous media.

60 ZHENGWEN ZENG AND REID GRIGG These authors conducted fluid flow experiments on packed particles, and redefined the Reynolds number, Re, as Re= ρd pv µ, (4) where D p is the diameter of particles. Their experiments show that the critical Reynolds number for non-darcy flow to become significant is in the range of 40 80. Fancher and Lewis (1933) flowed crude oil, water, and air through unconsolidated sands, lead shot, and consolidated sandstones. Using Chilton and Colburn s definition of the Reynolds number, their experimental results show that non-darcy flow occurs at Re = 10 1000 in unconsolidated porous media and at (Re) = 0.4 3 in loosely consolidated rocks. Here the particle diameter for the loosely consolidated rocks was obtained by screen analysis from the carefully ground rock samples. Realizing the difficulty of determining the particle diameter, Green and Duwez (1951) used permeability, k, and non-darcy coefficient, β, to redefine the Reynolds number, which can be written as Re= kβρv µ. (5) They conducted N 2 flow experiments through four different porous metal samples. Results show that non-darcy behavior started at Re = 0.1 0.2. By including porosity, ϕ, and using intrinsic velocity, u (the true velocity of the fluid in the pores), Ergun (1952) modified Chilton and Colburn s definition, which can be written as Re= ρd pu 1 µ 1 ϕ. (6) From experiments with gas flow through packed particles, Ergun observed a critical value of Re= 3 10. In comments on previous work, Bear (1972) suggested a critical Reynolds number of 3 to 10; while in a review, Scheidegger (1974) noted a range of 0.1 75. Hassanizadeh and Gray (1987) believed critical value Re = 1 15, and suggested Re = 10 as a critical value for non-darcy flow; from this assumption they concluded that non-darcy flow behavior is due to the increase of the microscopic viscous force at high velocity. Since the late 1980s, numerical modeling on this topic has increased rapidly. Blick and Civan (1988) used a capillary orifice model to simulate fluid flow in porous media. Based on that model, the critical Reynolds number defined in Equation (4) for non-darcy behavior is 100, below which Darcy s law is valid.

A CRITERION FOR NON-DARCY FLOW IN POROUS MEDIA 61 Du Plessis and Masliyah (1988) used a representative unit cell to model fluid flow in porous media. They derived a relationship between porosity and tortuosity, which further led to a correlation between Reynolds number and tortuosity. Their results show that a critical Re can be from 3 to 17. Ma and Ruth (1993) numerically simulated non-darcy behavior using a diverging converging model. They defined Reynolds number as Re= ρd tu µ (7) and a new criterion, the Forchheimer number, Fo, as Fo= k 0βρv µ, (8) where d t is the throat diameter, and k 0 is the permeability at zero velocity from Darcy s law. From that work, the authors found that the critical Reynolds number is 3 10 while the corresponding Forchheimer number is 0.005 0.02. Andrade et al. (1998) modeled fluid flow in a disordered porous media. Following the definition of Equation (5), they showed that the critical Reynolds number is 0.01 0.1. Thauvin and Mohanty (1998) used a network model to simulate the porous media. They defined the Reynolds number as Re= ρrv µ, (9) where r is the pore throat radius. Their result shows that critical Reynolds number is 0.11. In summary, there have been two types of criteria for non-darcy flow in porous media: Type-I represented by Equation (4), and Type-II by Equation (5). Critical values for non-darcy flow vary from 1 to 100 for the Type-I criterion, and from 0.005 to 0.2 for the Type-II criterion. The Type- I criterion has been applied mainly for columns of packed particles in which characteristic length, usually representative particle diameter, is available, whereas the Type-II criterion has been used mainly in numerical models, except for one in artificial porous metal samples. Due to inconsistency in definitions and thus in critical values, no widely accepted criterion for non-darcy flow in porous media is available. This paper addresses this problem. The following questions will be answered: (1) Which criterion is recommended? (2) What is the physical meaning of this criterion? (3) What is the critical value for the beginning of a significant non-darcy effect? And (4) using this criterion, what is the error if the non- Darcy effect is ignored?

62 ZHENGWEN ZENG AND REID GRIGG 2. Recommendation of the Forchheimer Number Historically the term Reynolds number was used in both the Type-I and Type-II criteria as shown in the previous section. The term Forchheimer number was later introduced to represent the Type-II criterion (Ma and Ruth, 1993). In order to avoid future confusion, the Type-I criterion is hereafter called the Reynolds number, Re, and redefined as Re= ρdv µ, (10) where d is a characteristic length of the porous media. The Type-II criterion is termed the Forchheimer number, Fo, and redefined as Fo= kβρv µ. (11) From the aforementioned review, it is seen that the Reynolds number is only applicable to columns of packed particles, unconsolidated or loosely consolidated sands. The Reynolds number has its root in the similar criterion for turbulent flow in a conduit. A logical result of this genetic connection is the inclusion of a characteristic length of the porous media in the definition, Equation (10). This characteristic length is similar to the roughness of the conduit in the definition for turbulent flow in a pipe. However, due to the complexity of the porous media structure, such a characteristic length is not easily defined and determined. In the case of packed particles, a representative diameter can be used as the characteristic length. In unconsolidated or loosely consolidated sands, this representative diameter can be determined through analysis of particle-size distribution from the crushed samples. However, the widely scattered critical values of this criterion indicate the need for a more well-defined representative diameter of the particles. Compared to its counterpart in the conduit flow, the physical meaning of the particle diameter needs to be better defined. For consolidated rocks, this criterion is not applicable without an acceptable definition of characteristic length. Therefore, the Reynolds number criterion is not recommended. In contrast, the Forchheimer number has the advantage of clear definition, sound physical meaning, and wide applicability. In Equation (11), all the involved parameters are clearly defined, and can be determined. It is obvious that this revised definition can be applied to all types of porous materials, as long as the permeability and non-darcy coefficient can be determined experimentally, or empirically when no experimental data is available (Li and Engler, 2001). In fact, several researchers have expressed their preference for using this type of criterion (Geertsma, 1974; Martins

A CRITERION FOR NON-DARCY FLOW IN POROUS MEDIA 63 et al., 1990; Gidley, 1991). For all these reasons, the Forchheimer number revised in Equation (11) is recommended as the criterion for non-darcy flow in porous media. 3. Theoretical Analysis of the Forchheimer Number In the Forchheimer equation, Equation (2), the left-hand-side term, (dp/dx), is the total pressure gradient. The first term in the right-handside, (µv/k), can be considered as the pressure gradient required to overcome viscous resistance. Similarly, the second term, βρv 2, is the pressure gradient needed to overcome liquid solid interactions. The ratio of the liquid-solid interaction pressure gradient to that by viscous resistance leads to (kβρv/µ), which is the Forchheimer number defined in Equation (11). Therefore, the Forchheimer number is the ratio of liquid solid interaction to viscous resistance. Defining the non-darcy effect, E, as the ratio of pressure gradient consumed in overcoming liquid solid interactions to the total pressure gradient, from Equation (2), leads to E = βρv2. dp dx (12) Using Equation (2) to eliminate (dp/dx) in Equation (12), and then combining Equations (11) and (12) gives E = Fo 1 + Fo. (13) Thus from Equation (13), it can be seen that the Forchheimer number is directly connected to the non-darcy error, i.e. the error of ignoring non- Darcy behavior. Such a connection will be useful to numerical simulation of fluid flow in porous media for practitioners to determine the trade-off on whether to include the non-darcy effect or not in their model. 4. Determination of Forchheimer Number 4.1. measurement of k and β According to Equation (11), the determination of the Forchheimer number, Fo, requires the permeability, k, the non-darcy coefficient, β, the fluid density, ρ, the superficial velocity, v and the viscosity, µ. In this section, Forchheimer numbers of three representative rocks (Dakota sandstone, Indiana limestone, and Berea sandstone) under a range of flowrates are determined using experimentally measured k and β.

64 ZHENGWEN ZENG AND REID GRIGG In order to measure k and β, gas flow experiments were conducted using a high pressure/high temperature gas flooding device (Zeng et al., 2003). Cylindrical samples of 2.54 cm (1-in.) diameter by 5.08 cm (2-in.) length were put in a triaxial coreholder. Hydrostatic pressure of 272 atm (4000 psi) was applied. After thermal equilibrium was reached in the sample at 311 K (100 F), nitrogen started to flow through the sample. The flowrate, controlled by an accurate pump, varied from 25 to 10,000 cm 3 /h under 136 atm (2,000 psi) and 300 K (80 F) in the pump. At each flowrate, the pressures, p 1 and p 2, at the inlet and outlet of the sample were measured when flow equilibrium was reached. Using these experimental measurements, k and β were calculated as follows. Under the gas flow experimental conditions, Equation (2) can be rewritten as: MA ( p1 2 ) p2 2 = 1 ( ) 2zRT µlρ p Q p k + β ρp Q p, (14) µa where M is the molecular weight of the gas (nitrogen), A is the cross-sectional area of the sample, p 1 is the pressure at the sample inlet, p 2 is the pressure at the sample outlet, z is the gas compressibility factor, R is the universal gas constant, T is the temperature in the sample, µ is the gas viscosity in the sample, l is the sample length, ρ p is the gas density in the pump, and Q p is the gas flowrate in the pump (Cornell and Katz, 1953). In Equation (14), M and R are constants; l and A are sample size; T and Q p are preset variables; p 1 and p 2 are measured under flowrate Q p ; ρ p is calculated using phase behavior simulators with preset pump pressure and temperature (CALSEP, 2002); and z and µ are calculated using phase behavior simulator with temperature and mean gas pressure, (p 1 +p 2 )/2, in the sample. Thus, only k and β are unknowns. For the sake of simplicity, denote ρ p Q p /µa as x, and MA(p1 2 p2 2 )/ 2zRT µlρ p Q p as y. Equation (14) then simplifies to y = 1 k + βx. (15) Equation (15) defines a linear equation in the x y coordinate system, with slope β and intercept on y-axis 1/k. This sets the foundation for the determination of k and β. At each flow rate, a pair of (p 1,p 2 ) are measured, and a pair of (x, y) are calculated. For different flowrates, different pairs of (x, y) are obtained, and the straight line in the x y coordinate system is defined. From this straight line, k and β are determined. Figure 1 shows the determination of k and β in Dakota sandstone. Similarly, k and β in Indiana limestone and Berea sandstone are determined. Results of k and β together with porosity of these three rocks are shown in Table I (Zeng et al., 2004).

A CRITERION FOR NON-DARCY FLOW IN POROUS MEDIA 65 y, 10 12 m -2 1000 800 600 400 200 y = 157.88x + 287.36 R 2 = 0.99 k = (287.36*10 12 m -2 ) -1 = 3.48*10-15 m 2 β = 157.88*10 8 m -1 0 0 1 2 3 4 x, 10 4 m -1 5 Figure 1. Determination of permeability and non-darcy coefficient from gas flow experiment in Dakota sandstone at 311 K (100 F), 34 atm (500 psi) pore pressure and 272 atm (4000 psi) hydrostatic confining pressure. Table I. Measured permeability and non-darcy coefficient Parameter Dakota sandstone Indiana limestone Berea sandstone k, 10 15 m 2 3.48 21.6 196 β, 10 8 m 1 157.88 36.00 2.88 φ 0.14 0.15 0.18 4.2. calculation of superficial velocity The superficial velocity, v, in the sample is calculated for each flowrate as follows: Assuming there is no loss of nitrogen in the flow system due to leakage, reaction or any other reasons, the mass flowing out from the pump equals that flowing through the sample. Therefore, the mass flowrate in the pump is the same as that in the core sample at flow equilibrium, though the volumetric flowrates can be different due to the change of pressures and temperatures. On the other hand, the mass flowrate equals the product of the density and the volumetric flowrate, therefore, ρq= ρ p Q p, (16) where ρ is the nitrogen density and Q is the volumetric flowrate, both under the sample conditions. Similar to the calculation of z and µ, the density of nitrogen in the sample is calculated using phase behavior simulators (CALSEP, 2002). Q can be defined as Q = va, (17) therefore, the superficial velocity in the core sample at each flowrate is v = ρ pq p ρa. (18)

66 ZHENGWEN ZENG AND REID GRIGG 4.3. calculation of Fo So far, all the variables in Equation (11) have been defined. Therefore the Forchheimer number can be calculated. Table II shows the superficial velocity and the Forchheimer number of the three rocks under each flowrate. Figure 2 shows that the Forchheimer number increases and is nonlinear, with superficial velocity. This implies that superficial velocity alone does not serve as a criterion for the non-darcy behavior. Referring to the permeability of these three rocks shown in Table I, it is observed that the higher the permeability, the higher the flowrate required to deviate from the linear relationship. This is consistent with the fact that non-darcy behavior is more severe in low permeability porous media. 5. Critical Forchheimer Number One reason that a new criterion is needed is the inconsistency of a critical value for the beginning of non-darcy flow behavior from existing criteria. If a critical value can be given, it would be very helpful for practitioners to decide when to include, or not to include, non-darcy behavior in calculating fluid flow in porous media. Previous efforts have focused on finding the starting point of the departure of the linear Darcy prediction from the observed, non-linear performance in Table II. Forchheimer number and superficial velocity in the three rocks at 311 K (100 F), 34 atm (500 psi) pore pressure and 272 atm (4000 psi) hydrostatic confining pressure Pump flowrate Dakota sandstone Indiana limestone Berea sandstone Q p,cm 3 /h v, cm/s Fo v, cm/s Fo v, cm/s Fo 25 0.006 0.006 50 0.011 0.012 100 0.022 0.024 200 0.044 0.048 0.041 0.069 300 0.066 0.071 0.062 0.104 400 0.087 0.095 0.082 0.139 600 0.127 0.143 0.123 0.208 0.107 0.125 800 0.166 0.190 0.163 0.277 0.170 0.200 1000 0.203 0.237 0.203 0.347 0.192 0.225 1500 0.286 0.355 0.301 0.520 0.320 0.376 2000 0.363 0.471 0.396 0.693 0.426 0.502 3000 0.489 0.701 0.576 1.037 0.637 0.753 4000 0.592 0.928 0.742 1.380 0.846 1.004 6000 0.755 1.373 1.035 2.060 1.256 1.505 8000 0.876 1.804 1.280 2.732 1.658 2.008 10,000 0.975 2.227 1.489 3.395 2.046 2.509

A CRITERION FOR NON-DARCY FLOW IN POROUS MEDIA 67 Superficial velocity, cm/s 2.5 2.0 1.5 1.0 0.5 0.0 Dakota SS Indiana LS Berea SS 0 1 2 3 4 Forchheimer number Figure 2. Change of superficial velocity with Forchheimer number in Dakota sandstone, Indiana limestone and Berea sandstone. different forms. The most commonly mentioned ones include friction factor versus Reynolds number curve, and the pressure drop versus flowrate curve. Because of the restriction of resolution, the visually identified critical point for the starting of non-darcy behavior is usually not accurate, and thus less dependable. On the other hand, the Forchheimer number is directly related to the non-darcy effect, as shown in Equation (13). Denoting E c as the critical value for non-darcy effect, from Equation (13) the critical Forchheimer number would be Fo c = E c. (19) 1 E c Equation (19) offers the user the choice of selecting the critical Forchheimer number based on the features of the problem. For example, if 10% is the limit of the non-darcy effect, Equation (19) would give a critical Forchheimer number of 0.11. Although this value is much higher than that from numerical simulation (Ma and Ruth, 1993), it is quite close to the critical value observed in experiments of gas flow through porous metal samples (Green and Duwez, 1951). Values can easily be selected that are within ranges that can be experimentally determined in reservoir core samples and thus can be considered as a good reference for the critical Forchheimer number. 6. Conclusions (1) The two types of non-darcy criteria, the Reynolds number and the Forchheimer number, for fluid flow in porous media have been reviewed. A revised Forchheimer number defined in Equation (11) is recommended due to the clear meaning of variables involved.

68 ZHENGWEN ZENG AND REID GRIGG (2) The Forchheimer number represents the ratio of pressure drop consumed by liquid solid interactions to that by viscous resistance. It has a direct relation to non-darcy effect, as shown in Equation (13). (3) Forchheimer numbers for nitrogen flow at varied flowrates in Dakota sandstone, Indiana limestone and Berea sandstone are determined using experimentally measured permeability and non-darcy coefficient. The results add considerably to the array of Forchheimer numbers for characterizing fluid flow in consolidated rocks. (4) An expression for the critical Forchheimer number is given in terms of non-darcy effect limit. This non-darcy effect-based expression allows the user to define the critical Forchheimer number according to the features of the problem, and should be scientifically more reasonable in comparing to a fixed critical value. A good reference for the critical Forchheimer number is 0.11, which corresponds to a 10% non-darcy effect. Acknowledgements The experiments were conducted in the gas flooding laboratory at the New Mexico Petroleum Recovery Research Center (PRRC) with the assistance of S. Ganda, D. B. Gupta, and Bob Svec. Elizabeth Bustamante located some obscure literature for this research. The research is financially supported by the US Department of Energy and the state of New Mexico, USA. References Andrade, J. A. Jr., Costa, U. M. S., Almeida, M. P., Makse, H. A. and Stanley, H. E.: 1998, Inertial effects on fluid flow through disordered porous media, Phys. Rev. Lett. 82(26), 5249 5252. Bear, J.: 1972, Dynamics of Fluids in Porous Media, American Elsevier, New York, pp. 125 129. Blick, E. F. and Civan, F.: 1988, Porous media momentum equation for highly accelerated flow, SPE Reserv. Engng. 1048 1052. Brinkman, H. C.: 1947, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. A 1, 27 34. CALSEP: 2002, Help on PVTsim 12, Calsep Inc., Denmark, pp. 55 60. Chilton, T. H. and Colburn, A. P.: 1931, Pressure drop in packed tubes, Ind Engng. Chem. 23(8), 913 919. Civan, F. and Tiab, D.: 1991, Effects of external boundary conditions on steady and semisteady radial flow equations based on Darcy, Forchheimer, Brinkman, and capillaryorifice models, SPE 22923, in: Proceedings of the SPE Annual Technical Conference, Dallas, Texas, USA, Oct. 6 9. Cornell, D. and Katz, D. L.: 1953, Flow of gases through consolidated porous media, Indu. Engng. Chem. 45(10), 2145 2152.

A CRITERION FOR NON-DARCY FLOW IN POROUS MEDIA 69 Du Plessis, J. P. and Masliyah, J. H.: 1988, Mathematical modeling of flow through consolidated isotropic porous media, Transport Porous Media 3, 145 161. Ergun, S.: 1952, Fluid flow through packed columns, Chem. Engng. Prog. 48(2), 89 94. Ewing, R. E., Lazarov, R. D., Lyons, S. L., Papavassiliou, D. V., Pasciak, J. and Qin, G.: 1999, Numerical well model for non-darcy flow through isotropic porous media, Computat. Geosci. 3, 185 204. Fancher, G. H. and Lewis, J. A.: 1933, Flow of simple fluids through porous materials, Ind. Engng. Chem. 25(10), 1139 1147. Firoozabadi, A. and Katz, D. L.: 1979, An analysis of high-velocity gas flow through porous media, J. Petrol. Technol. 211 216. Forchheimer, P.: 1901, Wasserbewegung durch boden, Zeit. Ver. Deutsch. Ing. 45, 1781 1788. Garanzha, V. A., Konshin, V. N., Lyons, S. L., Papavassiliou, D. V. and Qin, G.: 2000, Validation of non-darcy well models using direct numerical simulation, in: Chen, Ewing and Shi (eds), Numerical Treatment of Multiphase Flows in Porous Media, Lecture Notes in Physics, 552, Springer-Verlag, Berlin, pp. 156 169, Geertsma, J.: 1974, Estimating the coefficient of inertial resistance in fluid flow through porous media, Soc. Petrol. Eng. J., 445 450. Gidley, J. L.: 1991, A method for correcting dimensionless fracture conductivity for non- Darcy flow effect, SPE Prod. Engng. 391 394. Green, L. Jr. and Duwez, P.: 1951, Fluid flow through porous metals, J. Appl. Mech. 39 45. Guppy, K. H., Cinco-Ley, H. and Ramey, H. J.: 1982, Pressure buildup analysis of fractured wells producing at high flowrates, J. Petrol. Technol. 2656 2666. Hassanizadeh, M. and Gray, W. G: 1980, General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow, Adv. Water Resour. 3, 25 40. Hassanizadeh, S. M. and Gray, W. G: 1987, High velocity flow in porous media, Transport Porous Media 2, 521 531. Holditch, S. A. and Morse, R. A.: 1976, The effects of non-darcy flow on the behavior of hydraulically fractured gas wells, J. Petrol. Technol. 1196 1179. Li, D. and Engler, T. W.: 2001, Literature review on correlations of the non-darcy coefficient, SPE 70015, in: Proceedings of the SPE Permian Basin Oil and Gas Recovery Conference, Midland, Texas, USA, May 15 16. Ma, H. and Ruth, D. W.: 1993, The microscopic analysis of high Forchheimer number flow in porous media, Transport Porous Media 13, 139 160. Martins, J. P., Milton-Taylor, D. and Leung, H. K.: 1990, The effects of non-darcy flow in propped hydraulic fractures, SPE 20790, in: Proceedings of the SPE Annual Technical Conference, New Orleans, Louisiana, USA, Sept. 23 26. Scheidegger, A. E.: 1974, The Physics of Flow through Porous Media (3rd edn), University of Toronto Press, 152 170. Thauvin, F. and Mohanty, K. K.: 1998, Network modeling of non-darcy flow through porous media, Transport Porous Media 31, 19 37. Whitaker, S.: 1969, Advances in theory of fluid motion in porous media, Ind. Eng. Chem. 61(12), 14 28. Zeng, Z., Grigg, R. and Ganda, S.: 2003, Experimental study of overburden and stress on non-darcy gas flow in Dakota sandstone, SPE 84069, in: Proceedings of the SPE Annual Technical Conference, Denver, Colorado, USA, Oct. 5 8. Zeng, Z., Grigg, R. B, and Gupta, D. B.: 2004, Laboratory investigation of stress-sensitivity of non-darcy gas flow parameters, SPE 89431, in: Proceedings of the SPE/DOE Fourteenth Symposium on Improved Oil Recovery Conference, Tulsa, Oklahoma, USA, April 17 21.