To attain the goals of this study by means of SHM, a workflow including five subsequent steps is introduced here (Fig. 1).
Capillary pressure data classification
There are three main methods for measuring the capillary pressure curve of a core: Porous Plate, Centrifugal, and Mercury Injection Capillary Pressure (MICP) method. In comparison to the other methods, the MICP method is more common in petroleum industry because of its rapidness and low measurement costs (Dandekar 2013). Capillary pressure curves in this study are based on air-mercury MICP measurements in which the air is the wetting phase and the mercury is the nonwetting phase, respectively (Byrnes et al. 2008). According to the laboratory analysis, the corresponding cores of the MICP data in this study could be classified into four groups based on their size, sorting, and texture:
Group 1: Moderately shaly sandstones with 10–40% clay and silt (12 cores)
Group 2: Very fine sandstones (6 cores)
Group 3: Fine sandstones (11 cores)
Group 4: Medium sandstones (17 cores)
Grouping the MICP curves using this classification could increase the SHM accuracy.
Capillary pressure data correction
It is obvious that only a corrected input data can lead us to the desirable goals from SHM. Therefore, four corrections have been applied to the MICP curves subsequently.
Out of trend data elimination
There might be some out of trend curves in the capillary pressure data because of poor core condition or laboratory measurement errors. It is crucial to detect and eliminate these kinds of curves by graphical evaluation of total capillary pressure data in each group. After this correction, 5 out of 46 capillary pressure curves have been removed. Figure 2 illustrates one of the out of trend curves before the elimination.
Closure correction
In the first few steps of capillary pressure measurements, the nonwetting phase (e.g., mercury in the MICP experiment) will occupy the core surface asperities which are not the members of the actual pore system. “The pressure at which mercury commences to occupy the actual pore system of the sample being tested is called the initial pore entry pressure or closure pressure. All intrusion data recorded up to this initial entry pressure are subtracted from the MICP raw data output as the closure correction” (Shafer and Neasham 2000). Figure 3 shows the effect of closure correction on one of the curves.
Stress correction
In comparison to the reservoir conditions, stress is much lower in the laboratory measurements. This stress relief consequences in porosity and permeability increase, pore entry pressure reduction, and also capillary pressure curve changes (McPhee et al. 2015). The stress correction which is applied to the laboratory measurements is derived by Juhasz from Shell in the 1979 (Juhasz et al. 1979):
$$\begin{aligned} (P_{\mathrm{c}})_{\mathrm{stress}\text{-}{\mathrm{corr}}} = (P_{\mathrm{c}})_{\mathrm{lab}} \left( \frac{(\phi )_{\mathrm{res}}}{(\phi )_{\mathrm{lab}}} \right) ^{-0.5}. \end{aligned}$$
(1)
The effect of stress correction on one of the capillary pressure curves is illustrated in Fig. 4.
Conversion to reservoir fluid conditions
In laboratory measurements, the contact angle and also the interfacial tension will vary from reservoir condition because of different nonwetting and also wetting phase fluids. It is possible to eliminate the effect of this variation on capillary pressure curves using Eq. 2 (Purcell 1949):
$$\begin{aligned} (P_{\mathrm{c}})_{\mathrm{res}} = \frac{(\sigma \cos \theta )_{\mathrm{res}}}{(\sigma \cos \theta )_{\mathrm{lab}}}(P_{\mathrm{c}})_{\mathrm{lab}}. \end{aligned}$$
(2)
According to the input data (Byrnes et al. 2008), all laboratory measurements assume \(\sigma = 484\,\text{dyne}/\text{cm}\) and \(\theta = 140\,\text{deg}\). These data also represent the reservoir in situ gas–brine condition by \((\sigma \cos \theta )_{\mathrm{res}} = 40\,\text{dyne}/\text{cm}\).
Applying different saturation height models
“A saturation height model is an equation that represents the water saturation profile in a reservoir interval as a function of the fluid/rock properties and the distance above the Free Water Level (FWL) and is constructed from capillary pressure data” (Valentini et al. 2017). There are many Saturation Height Models with specific fitting parameters which are in use in petroleum industry. To predict the in situ water saturation in an un-cored depth of a reservoir, it is crucial to generalize these models. This generalization is possible using regression methods between fitting parameters of a model and cores’ petrophysical properties (like porosity, permeability, and also square root of permeability/porosity). In this study, we applied five important Saturation Height Models on each of the groups.
Brooks–Corey (Brooks and Corey 1964)
This model is one of the most conventional Saturation Height Models in petroleum industry. Based on the Brooks–Corey model, each capillary pressure curve could be approximated by a unique curve with three fitting parameters: \(S_{\mathrm{wirr}}\), \({P_{\mathrm{ce}}}\), and also N:
$$\begin{aligned} S_{\mathrm{w}} = S_{\mathrm{wirr}}+(1- S_{\mathrm{wirr}})\left( \frac{P_{\mathrm{ce}}}{P_{\mathrm{c}}}\right) ^{\frac{1}{N}}. \end{aligned}$$
(3)
Using the regression methods between the fitting parameters of the model and cores’ petrophysical properties, it is possible to rewrite the fitting parameters in Eq. 3 based on cores’ petrophysical properties. As an example, in part (a) of Fig. 5, each capillary pressure curve in the group 1 (in gray color) has been approximated by the Brooks–Corey model (in red color). In part (b), the relations between the fitting parameters of the model and the cores petrophysical properties (\(\phi\), K, and also \(\sqrt{\frac{K}{\phi }}\)) have been investigated using regression methods. The \(S_{\mathrm{wirr}}\) has been set to zero as a numerical artifact. Finally, and in part (c), rewriting Eq. 3 based on regression results, it is possible to calculate the capillary pressure curves using the petrophysical properties (red color curves).
Leverett-J (Leverett et al. 1941)
According to this model, if capillary pressure curves of a bundle of cores with relatively similar pore sizes re-plotted based on \(J_{(\mathrm{SW})}\) versus \(S_{\mathrm{w}}\) (in which \(J_{(\mathrm{SW})}\) is called “J-function” and has been elucidated in Eq. 4), it is possible to express each capillary pressure curve by a global function (Eq. 5):
$$\begin{aligned}J_{(\mathrm{SW})} = \frac{P_{\mathrm{c}}}{\sigma \cos \theta }\sqrt{\frac{K}{\phi }} \end{aligned}$$
(4)
$$\begin{aligned}J = a (S_{\mathrm{w}})^{b}, \end{aligned}$$
(5)
where a and b are the fitting parameters of the model. Figure 6 represents the capillary pressure curves of the group 3 (in gray color) and also their approximation after applying the Leverett-J model.
Skelt–Harrison (Skelt et al. 1995)
This model correlates the water saturation to the height above free water level (HAFWL):
$$\begin{aligned} S_{\mathrm{w}}=1-A {\mathrm{e}}^{\left( -\left( \frac{B}{h+D}\right) ^{C}\right) }, \end{aligned}$$
(6)
where A, B, C, and D are the parameters of the model, and h is HAFWL in meter.
The capillary pressure curves of the group 4 (in gray color) have been changed to the red curves after applying the Skelt–Harrison model (Fig. 7).
Lambda (Wiltgen et al. 2003)
Using Eqs. 7 and 8, this model can relate the water saturation to the HAFWL:
$$\begin{aligned}S_{\mathrm{w}}=A h^{-\lambda }+B \end{aligned}$$
(7)
$$\begin{aligned}\lambda ={\mathrm{e}}^{(A+B \ln (\phi \frac{e}{100}))}, \end{aligned}$$
(8)
where A, B, and \(\lambda\) are the parameters of the model, and h is HAFWL in meter.
As an example, Fig. 8 illustrates the capillary pressure curves of the group 4 before and after applying the Lambda model.
Thomeer (Thomeer et al. 1960)
According to this model, the pore geometric factor (G) could affect the shape and location of capillary pressure curves dramatically. The original equation is based on bulk mercury saturation which is the product of porosity and mercury saturation. It could be rewritten in the form of Eq. 9:
$$\begin{aligned} S_{\mathrm{w}} = S_{\mathrm{wirr}}+(1- S_{\mathrm{wirr}})\left( 1-{\mathrm{e}}^{\frac{G}{\ln (\frac{p_{\mathrm{ce}}}{p_{\mathrm{c}}})}}\right) . \end{aligned}$$
(9)
Figure 9 illustrates the group 3 curves (in gray color) and the subsequent red curves after applying the Thomeer model.
By plotting each of the constants of a specific SHM equation versus porosity, permeability, and square root of permeability/porosity and also using regression methods, it is possible to rewrite the equation as a function of the mentioned petrophysical properties.
Acquiring a global in situ water saturation function
Using porosity and permeability logs of each well as the inputs of the generalized models and also using Eq. 10 to change the HAFWL into capillary pressure, the water saturation values in all depths of each well have been calculated:
$$\begin{aligned} \text{HAFWL}=\frac{ (P_{\mathrm{c}})_{\mathrm{res}}}{(\rho _{\mathrm{wat}}-\rho _{\mathrm{gas}})_{\mathrm{reservoir}} (g)}, \end{aligned}$$
(10)
where \(\rho _{\mathrm{wat}}\) and \(\rho _{\mathrm{gas}}\) are the water and the gas density, respectively, and g is the gravitational constant.
Comparing the calculated water saturations from each model with the water saturation curve obtained from the well logs as the base one, it is possible to determine the most accurate model. Figure 10 illustrates the calculated water saturation curves from the Lambda model of the group 4 and also water saturation curves from well logs in medium sandstone intervals of all three wells in Piceance basin simultaneously. By eye balling, it can be concluded that the accuracy of this model is low, medium, and high from the left to the right, respectively.
Although curve illustrations can give us a bird’s eye view in selecting the best saturation height model, but it is necessary to use statistical analysis like standard error of estimate (SEE) or regression methods (Sohrabi et al. 2007). To select the best SHM method, we used the SEE analysis for all of the wells in this study (Eq. 11):
$$\begin{aligned} \text{SEE}=\sqrt{\frac{\sum _{1}^{n}(\text{SW}_{\mathrm{model}}-\text{SW}_{\mathrm{log}})^{2}}{n-1}}. \end{aligned}$$
(11)
As an example, Table 1 represents the SEE values for one of the wells in the Washakie basin considering medium sandstone intervals (all values have been calculated from the group 4 models). According to the total analysis of the SEE values in all of the wells, the Brooks–Corey model is the most accurate method, while the Leverett-J model is the less accurate one (this could be a consequence of oversimplified assumptions of this method which are based on a bundle of straight and perfect capillary tubes).
Table 1 Water saturation SEE values of medium sandstone intervals of a well in the Washakie basin (based on the group 4 models) Calculating accurate permeability values using SHM
Reducing uncertainty in permeability values and increasing their accuracy are extremely important in reservoir characterization and also 3D properties modeling. As water saturation log, HAFWL value, and porosity log are available for different intervals of wells, the permeability profiles have been calculated based on the most reliable model (Brooks–Corey) saturation height function in each group (Fig. 11).
High regression coefficient of determination \((R^{2})\) between the calculated and core permeability values approves the result (Fig. 12). Here, the \(R^{2}\) is equal to 0.652 and the regression formula is:
$$\begin{aligned} \log (K_{\mathrm{core}})=1.745528 ({\mathrm{e}}^{\log (K_{\mathrm{model}})})-1.708569. \end{aligned}$$
(12)