Determination of Resistivity Index, Capillary Pressure, and Relative Permeability

It is known that the three important parameters: resistivity, capillary pressure, and relative permeability, are all a function of fluid saturation in a porous medium. This implies that there may be a correlation among the three parameters. There have been many papers on the approach to inferring relative permeability from capillary pressure data. However the literature on the interrelationship between resistivity index, capillary pressure and relative permeability has been few. The models representing such relationships have been proposed in this study. It has been shown that the other two parameters could be determined using these models if one of the three parameters (capillary pressure, relative permeability, and resistivity) is known. Using this approach, it would be possible to quickly obtain a distribution of capillary pressure and relative permeability characteristics as a function of depth and location across an entire reservoir. INTRODUCTION Capillary pressure and relative permeability are the key parameters that govern fluid flow in porous media and have been adopted in petroleum reservoir engineering, geothermal reservoir engineering, soil science, and many other industries. Determination of capillary pressure and relative permeability are traditionally conducted in the laboratory. However it is expensive, difficult, and time-consuming to measure capillary pressure and relative permeability in many cases, especially in cases in which phase transformation and mass transfer exist between the two phases as pressure changes. These include steamwater flow (Li and Horne, 2001 and 2004), gascondensate flow (App and Burger, 2009; Kumar, et al., 2006; Li and Firoozabadi, 2000), CO2-oil flow (Dria, et al., 1993), CO2-water flow (Bennion and Bachu, 2008), etc. It is also difficult to maintain exact reservoir conditions in taking a core sample out from reservoirs and bringing it to the surface, and it is almost impossible either to obtain capillary pressure and relative permeability in real time. On the other hand, it is easier to measure resistivity in the reservoir; a large number of resistivity measurements are available from well logging, even in real time. Literature on the relationship between capillary pressure and resistivity index has been scarce. A brief discussion on this is presented as follows. Szabo (1974) proposed a linear model to correlate capillary pressure with resistivity by assuming the exponent of the relationship between capillary pressure and water saturation is equal to that of the relationship between resistivity and water saturation. This assumption may not be reasonable in many cases. The linear model proposed by Szabo (1974) can be expressed as follows: c t bP a I R R + = = 0 (1) where Ro is the resistivity of rock at a water saturation of 100%, Rt is the resistivity at a specific water saturation of Sw, I is the resistivity index, Pc is the capillary pressure, a and b are two constants. The results from Szabo (1974) demonstrated that a single straight line, as predicted by the model (Equation 1), could not be obtained for the relationship between capillary pressure and resistivity index. Longeron et al. (1989) measured the resistivity index and capillary pressure under reservoir conditions simultaneously. Longeron et al. (1989) didn’t attempt to correlate the two parameters. Li and Williams (2006) developed a correlation between resistivity and capillary pressure theoretically. The model was derived according to the fractal modeling of porous media. As mentioned previously, it is difficulty to measure both capillary pressure and relative permeability. But it is relatively easier to measure capillary pressure, especially when mercury-intrusion approach is applied. It may be because of this that several mathematical models have been proposed to infer relative permeability from capillary pressure data. In 1949, Purcell (1949) developed a method to calculate the permeability using capillary pressure curves measured by mercury-injection. Later, Burdine (1953) introduced a tortuosity factor in the model. Corey (1954) and Brooks and Corey (1966) summarized the previous work and modified the method by representing capillary pressure curve as a power law function of the wetting-phase saturation. The modified model was known as the Brooks and Corey relative permeability model. Li and Horne (2005, 2006) reported that steam-water relative permeability could be calculated from capillary pressure data. It would be helpful to establish the relationship between relative permeability and resistivity index. However, literature on the relationship between relative permeability and resistivity index has been scarce as well (Pirson et al., 1964; Li, 2007). Routine well testing can only provide the effective permeability of the rock at one specific value of water saturation (usually at irreducible water saturation). Most of the existing approaches to evaluating absolute permeability from resistivity well logging are based on empirical relationships between porosity and permeability. Note that absolute permeability is a concept in single phase fluid flow but resistivity well loggings are usually conducted in rock in which multi-phase flow exists near wellbores. It is not surprised if the absolute permeability data estimated from resistivity well logging (conducted in multi-phase flow) are not consistent with those determined in single-phase flow using other techniques. In this study, analytical mathematical models correlating resistivity index, capillary pressure, and relative permeability were proposed. It is shown that capillary pressure and relative permeability can be inferred from resistivity data. Actually the other two could be inferred using these models if one of the three parameters (capillary pressure, relative permeability, and resistivity) is known. Experimental data of capillary pressure, relative permeability, and resistivity index were used to test these models. MATHEMATICS Resistivity, capillary pressure, and relative permeability have similar features. For example, all are a function of fluid saturation in a porous medium. This implies that there should be a correlation among the three parameters. The models representing such relationships are discussed in this section. Relationship between Wetting-phase Relative Permeability and Resistivity Index Li (2007) derived the relationship between relative permeability and resistivity index: I S k w rw 1 * = (2) krw is the relative permeability of the wetting phase. * w S is the normalized saturation of the wetting-phase and is expressed as follows: wr wr w w S S S S − − = 1 * (3) where Swr is the residual saturation of the wetting phase. The resistivity index, as a function of the wettingphase saturation, can be represented using the Archie’s equation (1942): n w t S R R I − = = 0 (4) where n is the Archie’s saturation exponent. Relative permeability of the wetting-phase can be calculated using Eq. 2 from resistivity index data once the residual saturation of the wetting-phase is available. Note that the residual saturation of the wetting-phase can be obtained from the experimental measurement of resistivity in the porous medium. Calculation of Nonwetting-phase Relative Permeability The wetting-phase relative permeability can be inferred from the resistivity data based on Eq. 2. However the relationship between nonwetting-phase relative permeability and resistivity has not been established. The computation of nonwetting-phase relative permeability will be described as follows. According to Li and Horne (2006), the wetting-phase relative permeability can be calculated using the Purcell approach (1949): λ λ + = 2 * ) ( w rw S k (5) where λ is the pore size distribution index and can be calculated from capillary pressure data. After the relative permeability curve of the wetting-phase is obtained using Eq.2, the value of λ can be inferred using Eq. 5. According to the Brooks-Corey model (1966) and the study by Li and Horne (2006), the relative permeability of the nonwetting-phase can be calculated once the value of λ is available. The equation is expressed as follows: ] ) ( 1 [ ) 1 ( 2 * 2 * λ λ + − − = w w rnw S S k (6) One can see that the entire relative permeability set (both wetting and nonwetting phases) can be inferred from resistivity index data using Eqs. 2 and 6. Relationship between Capillary Pressure and Resistivity Index There are two approaches to determining capillary pressure once resistivity index data are available. The first approach is to calculate capillary pressure using the Brooks and Corey capillary pressure model (Brooks and Corey, 1966): λ / 1 * ) ( − = w cD S P (7) where PcD is the dimensionless capillary pressure (Pc/pe); pe is the entry capillary pressure and λ is the pore size distribution index. As pointed out previously, the value of λ could be inferred once resistivity index data are available. Therefore the dimensionless capillary pressure can be determined using Eq. 7 with the value of λ. The second approach to determining capillary pressure is the application of the model developed by Li and Williams (2006):