A general correlation for the non-Darcy flow coefficient with respect to permeability, porosity, and tortuosity using a wide variety of dam from consolidated and unconsolidated porous formations is presented. This correlation implicitly includes the effect of the overburden stress through permeability, porosity. and tortuosity and the effect of the existence of other fluid phases through the effective permeability. Introduction Because of its simplicity, the Forchheimer equation has been popular to represent the non-Darcy flow behaviour in petroleum reservoir. The Forchheimer equation for horizontal and one dimensional single phase fluid flow through a homogenuous porous media given by Forchheimert (1). Equation (1) (Available in full paper) and for multiphase flow by modifying the equation given by Schulenberg and Muller (2). In Equation (1) and (2), a is the distance, p, u, µ and ρ are the pressure, volumetric flux, viscosity and density of the fluid, k, and β are the the permeability and non-Darcy flow coefficient of the porous media, l, denotes the l db phase: k1 andβ 1 are the effective permeability and the effective non-Darcy flow coefficient.. Development of correlations for the β coefficient has been of continuing interest (Janiek and Katz (3), Geertsma (4), Norman and Archer (5), Evans et al. (6), Evans and Evans (7) and Kutasov (8)). By-and-large, the previous studies dealing with single-phase flow have overlooked the importance of tortuosity, t, and correlated the β coefficient with respect to permeability k and porosity Ø alone. Some of these studies (6,7) included the overburden stress, although its effect is implicitly included through k and Ø and the tortuosity, t. A few studies (4,5) involving multiphase flow have incorporated the effect of the existence of other fluids in terms of the saturation correction on the fractional pore volume occupied by the phase of interest, whereas this effect is already accounted for by means of the effective permeability k1. FIGURE 1: Correlating non-Darcy flow coefficient with permeability, porosity and tortuosity for a small number of rocks. (Available in full paper) Formulation The data such as porosity, permeability and tortuosity can provide a direct measure of the apparent characteristics of the pore structure and can be readily determined by means of well logging and/or well testing at in situ reservoir conditions. The tortuous paths in porous media have been recognized as a major cause of non-Darcy flow (1–9). Tortuosity provides useful information about the representative length of tortuous flowpaths in pore structures. Therefore, it is reasonable to take into account tortuosity, t, in the correlation of the non-Darcy flow coefficient, β, in addition to the porosity, Ø, and the permeability, k. To demonstrate the importance of including the tortousity, t, the data by Cornell and Katz (10) who measured porosity, permeability, tortousity and non-Darcy flow coefficient for a number of rocks including sandstones, carbonates and dolomites have been used. Their laboratory data we plotted in Figure 1 to correlate the non-Darcy flow coefficient first with permeability, porosity and tortuosity.

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The effects of immobile and mobile liquid saturations on the non-Darcy flow coefficient in propped fractures have been investigated. It was observed that an immobile liquid saturation of up to 20% PV can triple the non-Darcy flow coefficient and a small mobile liquid saturation will increase the non-Darcy flow coefficient by nearly an order of magnitude (over that of the dry case). The linear relationship between the logarithm of the non-Darcy flow coefficient and the logarithm of the proppant permeability for propped fractures, as proposed by Cooke, is shown to be invalid in the presence of an immobile or mobile liquid saturation. Constants are presented that can be used to obtain a more accurate approximation than has previously been available on the non-Darcy flow coefficient for single-phase flow in fractures propped with 10/20-and 20/40-mesh Ottawa sand. The experimental data obtained from this research are compared with those of others reported in the literature. Fig. 1-Slmulated hydraulic fracture. Nitrogen Sourc.

Co 5(% /wobsummary.The results of an experimental research program to investigate the effects of immobite liquid saturations on the non-Darcy flow coefticieitt are presented. Sandstone cores of absolute permeabilities ranging from 50 to 800 md were used to investigate non-Darcy flow phenomena under mtdtiphase conditions. Immobile liquid saturations were varied from 8 to 30% PV. The multiphase experiments were conducted with N* gas as the flowing phase and glycerin as the imrnObile liquid P~se. It was found that the non-Darcy flow coefficient for the multiphase case may be estimated with a dry-core non-Darcy-flow-coefficient/ permeability relationship developed for the rock in question and the effective gas permeabiliV at a given saturation.For the immobile multiphase cases, the non-Darcy flow coefficient consistently increised with increased ?.aNrarion. The experimental data obtained from this research were compared with the limited mukiphase data in the Ikeratme. Where comparisons could be made, the daq rcpmted in this paper agreed favorably with the existing published &ta.An analysis of the experimental data revealed that a unique relationship existed between the non-Darcy flow coefficient and the effective gas permeability, porosity, liquid saturation, and effective overburden pressure at a given ti,mperantre.(@relations were developed from this analysis to prtilct the non-Darcy flow coefficient as a function of rock and fluid properties. Where possible, the correlations were used to predict the nomDamy coeftkient measmed by other researchers and were compared with the dry-core correlations developed "by liudcek and Katz and with a theoretical equation developed by Geemma.

An analytical theory is presented that permits the formulation of a mathematical model to describe the variation of relative permeability with temperature in a water/oil system. The theory develops analytical equations for temperature-dependent relative permeability in terms of water saturation, irreducible water saturation, and differential change in irreducible water saturation with temperature. These equations predict and agree reasonably well with experimental results reported by other researchers.The implications of temperature-dependent relative-permeability data on reservoir performance of a thermal process are also presented. These data together with the application of the Buckley-Leverett frontal-advance theory and the fractional-flow equation are used to predict oil recovery.