High-Velocity Flow in Porous Media (includes associated papers 31033 and 31169 )
- Abbas Firoozabadi (Reservoir Engineering Research Inst.) | L.K. Thomas (Phillips Petroleum Co.) | Bert Todd (Phillips Petroleum Co.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- May 1995
- Document Type
- Journal Paper
- 149 - 152
- 1995. Society of Petroleum Engineers
- 1 in the last 30 days
- 334 since 2007
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High-velocity gas flow causes a pressure drop greater than that predicted by Darcy's law. This subject has been addressed from a theoretical and an experimental standpoint. The purpose of this paper is to summarize previous literature on this phenomenon and to clarify recent misconceptions that have arisen. Additional flow experiments have been made and analyzed to support the conclusions of this work.
The literature of a number of disciplines, including civil,1 chemical,2,3 and petroleum4,5 engineering and hydrology,6contains a large body of experimental data and theoretical analyses relating to the inadequacy of Darcy's law at high velocities. In the petroleum literature, this subject has received even greater attention than other disciplines7-9 owing to the significance of gas flow in the wellbore region and the need to interpret gas well testing properly.
As early as 1901, Forchheimer10 recognized that increased velocity in porous media results in a pressure drop greater than proportional to velocity increase and suggested that a nonlinear term should be added to account for the additional pressure drop. Green and Duwez11 derived the differential form of the equation proposed by Forchheimer from dimensional analysis by
where ?=fluid density and ß=a property of porous media independent of the dimension. The first term on the right side of Eq. 1 represents the contribution from the Darcian flow and the second term is the contribution of high-velocity flow. Forchheimer and later investigators12 have pointed out that at very high rates, higher-order terms in velocity, ?, may be required to represent the flow.
The high-gas-velocity phenomenon has been the subject of many experimental and theoretical studies. A wide range of opinions exists in the literature on the mechanisms of high-velocity flow. Early authors believed high-velocity flow resembles turbulence.3,13 This was conclusively rejected.1,11 Later, inertial effects were described to account for the added pressure drop primarily because of the form of the pressure-drop equation (Eq. 1).14 Firoozabadi and Katz15 summarize various discussions concerning the causes for the extra pressure drop beyond that obtained from Darcy's equation and suggest the use of high-velocity rather than non-Darcy flow. Hassanizadeh and Gray16 rejected the notion of inertial effects for high-velocity flow in porous media on the basis of fundamental laws of continuum mechanics. They showed that, at high velocities, the inertial forces are as much as three orders of magnitude smaller than interfacial drag forces. Hassanizadeh and Gray attributed the rise of nonlinear terms to the effects of increased microscopic drag forces on the pore walls. Their theory supports the point of view expressed in general terms in Ref. 15. These authors also concur with the suggestion that the term "high-velocity flow"is appropriate when nonlinear effects become significant. (Ref. 16 gives the derivation of a generalized form of the Forchheimer equation.)
In departure from the above literature, Temeng and Horne17 suggested that Darcy's law is adequate to describe the high-velocity flow when nonlinearities in the flow equations are accounted for. On the basis of their work, they concluded that "...the likelihood that some flows that have been described as non-Darcy might not be so at all, but may result from changes in fluid properties over the length of the system." They also concluded that the velocity coefficient, ß, which previously was believed to be a function of rock properties (i.e., k and f) is also a function of length and stated that the continued use of this parameter, based on standard empirical correlations that do not include length, is incorrect. Selective testing of data in the literature apparently support the above conclusions.17,18
The purpose of this paper is to examine the validity of Temeng and Horne's conclusions. We will present new gas flow data, compare the data with the models of Temeng and Horne17 and Katz et al.,19 and draw conclusions on the basis of that comparison. This work will show that the high-velocity flow effects cannot be explained by changes in fluid properties over the core length and that ß is not a function of core length.
Experimental Data and Analysis
A cylindrical Berea sandstone core of 2.51 cm in diameter and 23.4 cm long (f=22%) was used. Nitrogen at room temperature was used in the flow experiments, and flow tests were conducted at backpressures of 0, 689.5, and 3447.5 kPa. High backpressures of 689.5 and 3447.5 kPa were used to minimize the Klinkenberg effect.20 Pressure and flow-rate measurements were plotted with the equation
Note that on the basis of Eq. 2, a plot of [MA(p12- p22)]/2µzRTLw vs. w/Aµ should give a straight line. Fig. 1 shows all the data for the 9.2-in.-long core at backpressures of 0, 700, and 3500 kPa. All the data points are represented by the straight line y=0.0028x+1.57. The permeability of the Berea is 645 md, and the high-velocity coefficient is 2.8×105 cm-1. Next, the core was cut into two and flow tests were performed on each half. Table 1 shows the dimensions and properties of the two halves.
Fig. 2 shows the plot of the measured data for the first half core. Except for the first two data points the rest of the measured data follow a straight-line relationship. The straight-line equation, y=0.0032x+1.79 represents all the data well. This line gives a permeability of 565 md and a high-velocity coefficient of 3.2×105 cm-1. Both values are close to the results from the complete core. Flow data for the other half of the core (results not shown) give y=0.0028x+1.59, which represents all the data. A permeability of 640 md and a high-velocity coefficient of 2.8×105 cm-1 are given by e straight line. It is interesting that both permeability and high-velocity flow coefficients are nearly identical for the second half core and complete core. These results show that length does not effect ß. Table 1 provides k and ß values for the complete core and the two half cores. Tables 2 through 4 give the flow data.
In this section, we will use Temeng and Horne's17 method (TH model) to calculate flow rate from
where Tsc=288.9 K and psc=101.3 kPa. Eq. 3 is the same as Eq. 32 of Ref. 17. It is based on Darcy's law and accounts for the squared-gradient nonlinearity of the flow equation.
In Eq. 3, c is the coefficient of the squared-gradient term averaged over the core length and is given by
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