The capillary pressure J-function is a dimensionless measure of the capillary pressure of a fluid in a porous medium. The function was derived based on a capillary bundle model. However, the dependence of the J-function on the saturation Sw is not well understood. A prediction model for it is presented based on capillary pressure model, and the J-function prediction model is a power function instead of an exponential or polynomial function. Relative permeability is calculated with the J-function prediction model, resulting in an easier calculation and results that are more representative.
Citation: Xu WS, Luo PY, Sun L, Lin N (2016) A Prediction Model of the Capillary Pressure J-Function. PLoS ONE11(9): e0162123. https://doi.org/10.1371/journal.pone.0162123
Editor: Timothy W. Secomb, University of Arizona, UNITED STATES
Received: March 16, 2016; Accepted: August 17, 2016; Published: September 7, 2016
Copyright: © 2016 Xu et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are contained within the paper and its Supporting Information files.
Funding: The funding organization (Tarim Oilfield Company) did not play a role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript and only provided financial support in the form of authors' salaries. I have reviewed the statements relating to the author contributions. The funder provided support in the form of salaries for authors [Employee ID No. 00699778], but did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. The specific roles of these authors are articulated in the ‘author contributions’ section.
Competing interests: The authors have declared that no competing interests exist and Tarim Oilfield Company along with any other relevant declarations relating to employment, consultancy, patents, products in development, or marketed products, etc do not alter our adherence to PLOS ONE policies on sharing data and materials.
A capillary pressure curve is determined based on the measurements of a small rock sample, and therefore, it represents only one part of a reservoir. As a method to compare capillary pressure data for a range of practical applications, Leverett  proposed the semi-empirical J-function (1) where J(Sw) is the J-function, Sw is the wetting-phase saturation, pc(Sw) is the capillary pressure, σ is the interfacial tension (IFT), θ is the contact angle, k is permeability, and φ is porosity.
The J-function, which synthesizes the fluid IFT, wettability, permeability, and porosity, is used to represent the characteristics of the reservoir capillary pressure curve. It is an effective method for analyzing data on capillary pressure. Leverett postulated that the J-function of similar lithology has universal significance, and in most cases, all capillary pressures of a reservoir processed with the J-function are simplified to a single monotonic curve. Brown  studied the J-function in more detail and concluded that the J-function of the same strata and lithology provides a better comparison, which supports the view of Rose and Bruce . Therefore, the J-function can be used to represent the capillary pressure curve of a reservoir.
However, the initial J-function expression was simply proposed but not derived specifically by Leverett . Though some researchers [4–7] attempted to modify the J-function, such as with the Gao fractal model , their models have no fundamental improvement because the origin of the initial J-function is unknown.
Considering the research background and technical level of the field at the time that the J-function was proposed, we conclude that the J-function formula was derived based on the capillary bundle model. Because only capillary pressure varies in the J-function expression, the J-function prediction formula starts with a mathematical model. The fitting trend for J-function is a power function rather than an exponential or polynomial one through comprehensive investigations of the capillary pressure quantitative relation. Finally, relative permeability is calculated with the J-function prediction formula. The calculation is easier than with earlier formulas, and its results are more representative.
2. Realization of J-Function Derivation
Research shows that rocks should be described using the direct-mapping pore network model [8–12] However, owing to the research background and technical level of the Leverett era, rocks were described using the bundle-of-tubes model or the capillary bundle model. The capillary bundle model was widely applied as the classical model [13–15] such as in the derived Kozeny–Carman model . Thus, the capillary bundle model was used to derive the J-function.
When the capillary bundle model (Fig 1) is used to describe a rock, the flow of one capillary tube according to the Poiseuille equation (n capillary tubes) is given as (2) where q is the flow, r is the capillary tube radius, μ is the viscosity, l1 is the length of the capillary tube, and Δp is the pressure difference.
Substituting Eq 3 into Eq 5, and re-arranging the expression, we get (6) where l1 is not equal to l2, and the ratio between them is referred to as tortuosity. Tortuosity is constant for a certain core; hence, (7)
3. Prediction Model about the Dependence of J-Function on Sw
In recent years, rocks have been described using an interconnected network model, which is a significant improvement on the capillary bundle model because it can accurately predict the capillary pressure curve and simulate drainage and the imbibition process [9–10,12]. However, the network model has multiple and complex pore structure parameters; therefore, it is difficult to find analytical expressions for the J curve based on network models. Considering that only capillary pressure is variable in the J-function expression, the prediction about the dependence of J-function on Sw should start with the dependence of capillary pressure on Sw, that’s called the capillary pressure mathematical model.
The main mathematical models for capillary pressure are the Thomeer , Brooks–Corey , van , and the three-constant hyperbolic models [20–21]. The Brooks–Corey model is widely used owing to its accuracy [22–25], and it is expressed as (9) Or (10) where pd is the threshold pressure, Se is the effective saturation or the normalized saturation, and λ is the pore size distribution index; Se is defined as (11) where Sw is the wetting phase saturation and Sr is the minimum wetting phase saturation.
Therefore, the fitting trend for the average capillary pressure J-function is a power function rather than an exponential or a polynomial one.
Because the J-function of the same strata and lithology provides a better comparison [2–3,16,24], 10 representative rocks from the WJ Well, shan 2–2 strata of the ZY oil field in China were selected for the practical calculation. The capillary pressure raw data (S1 Data) are shown in Fig 2, and they have a sandstone lithology (Fig 3). The practical calculation about the dependence of J-function on the wetting phase saturation Sw for the ten rocks is shown in Fig 4.
Fig 4 shows that the coefficient of determination (R2) of the power form is greater than that of the exponential form; hence, the fitting precision of the power form is more accurate. This verifies the correctness of the J-function prediction model.
4. Application of the J-Function Prediction Model
The main application of the J-function is determining the average capillary pressure, which is the representative capillary pressure of a reservoir layer. The relative permeability calculated with the average capillary pressure obtained using the J-function average is more representative. The calculation can be further simplified and made more convenient by substituting the J-function expression into the empirical formula of relative permeability. The theoretical expression of the relative permeability calculated with the J-function prediction model is given below.
Because the Purcell  relative permeability model was derived based on the capillary bundle model, which is far from the true rock pore structure, Burdine  introduced tortuosity into the Purcell model to improve precision. The Burdine model is widely applied as a classical model [16,27–29]. The wetting phase relative permeability model of Burdine [16,27,29] is (14)
The calculation result of Eq 19 is (20)
Therefore, a set of new and more representative saturation functions, which are the basic input parameters for the Eclipse SCAL Section, can be composed of the analytic expressions discussed in this section. This new set of saturation functions is summarized as follows: (22)
- The Leverett J-function is derived based on the capillary bundle model.
- The prediction J-function model is a power function that points the way for its application.
- The empirical formula of the relative permeability calculation is easier to use and more convenient with the J-function prediction formula than with the earlier formula; the calculated results are more representative.
- Conceptualization: PYL.
- Data curation: NL.
- Formal analysis: NL.
- Investigation: NL.
- Methodology: WSX.
- Project administration: LS.
- Resources: LS.
- Software: NL.
- Supervision: PYL.
- Validation: PYL WSX.
- Visualization: PYL.
- Writing – original draft: WSX.
- Writing – review & editing: WSX.
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