In this section, the properties of a sandstone core, hybrid PSO coupled with multiple linear regression to provide the required data for proposing correlations, and a discussion on averaging techniques to assign a single capillary pressure and relative permeability curve to grid blocks in the simulator is elaborated.
Sandstone core properties
The sandstone core properties are investigated regarding porosity, permeability, clay content, and wettability. Two rock types are selected, and their properties are reported in Table 1:
For RTP1, the average core porosity and permeability are reported as 0.3 and 4.5 (Darcy), respectively. For RTP2, the reported values are 0.20 and 1.50 (Darcy), respectively. RTP1 is considered a no clay formation, and RTP2 is a clayey formation. The permeability and porosity correlation for RTP1 and RTP2 is as shown in Fig. 1:
The permeability and porosity correlation for RTP1 and RTP2.
Hybrid particle swarm optimization coupled with machine learning technique for estimating core scale relative permeability and capillary pressure curves
To determine the relative permeability and capillary pressure functions for the LSW process at the core scale, the studies of similar sandstone oil reservoirs are used and categorized into different rock types. These parameters are reported in some studies, but most are not. So, there is a need to determine the matching parameters by coupling the reservoir simulator and PSO algorithm.
Studies similar to our lithology (with and without clay content) are gathered thoroughly. Their properties including formation type, injected brine salinity, wettability, and clay content are investigated and the objective of each study is elaborated. The results are summarized in Tables 2 and 3:
Moreover, the distribution of porosity and permeability in the investigated studies are compared to ours as in Fig. 2:
The distribution of porosity and permeability for RTP1 (a) and RTP2 (b).
After categorizing the conducted studies into two different rock types, the gathered data from these studies is used to determine the HM parameters of both rock types, which will be discussed in the following sections.
Coupling reservoir simulator and PSO algorithm to determine model parameters
If the relative permeability and capillary pressure parameters are not reported directly, the matching parameters must be obtained by coupling the reservoir simulator and the PSO algorithm (AHM). The Coupling procedure to obtain the optimized model parameters is described in 9 steps:
-
1.
Preparing the initial swarm; n values (n = 50 in this study) are given to each parameter of the swarm.
-
2.
Calculating relative permeability and capillary pressure values for each parameter. Corey’s model is used to calculate water and oil relative permeability:
$${{\text{k}}}_{{\text{rw}}}={{\text{k}}}_{{\text{rw}}}^{0}\times {({{\text{S}}}_{{\text{wn}}})}^{{{\text{n}}}_{{\text{w}}}}$$
(1)
$${{\text{k}}}_{{\text{ro}}}={{\text{k}}}_{{\text{ro}}}^{0}\times {(1-{{\text{S}}}_{{\text{wn}}})}^{{{\text{n}}}_{{\text{o}}}}$$
(2)
In the above equations,\({{\text{k}}}_{{\text{rw}}/{\text{o}}}^{0}\) water/oil endpoint relative permeability and \({{\text{n}}}_{{\text{o}}/{\text{w}}}\) are Corey exponent parameters, which would be obtained by AHM.
$${{\text{J}}}_{\left({{\text{S}}}_{{\text{w}}}\right)}^{{\text{m}}}=\frac{{{\text{a}}}_{1}}{1+{{\text{k}}}_{1}{{\text{S}}}_{{\text{nw}}}}-\frac{{{\text{a}}}_{2}}{1+{{\text{k}}}_{2}{(1-{\text{S}}}_{{\text{nw}}})}+{{\text{b}}}_{1}$$
(3)
$$- 1 < {\text{J}}^{{\text{m}}} { } < 1.{\text{ P}}_{{{\text{cow}}}} \left( {{\text{S}}_{{{\text{nw}}}} } \right) = {\text{P}}_{{{\text{cowmax}}}} {\text{J}}_{{{\text{S}}_{{{\text{nw}}}} }}$$
(4)
In Eq. 3,\({{\text{a}}}_{1}\), \({{\text{k}}}_{1}\),\({{\text{a}}}_{2}\), \({{\text{k}}}_{2}\), and \({{\text{b}}}_{1}\) are correlation parameters. \({{\text{P}}}_{{\text{cowmax}}}\) in Eq. 4 presents maximum capillary pressure. If the values of relative permeability and capillary pressure are not reported in the investigated studies, the constant parameters of the capillary pressure, along with Corey’s equation, would be determined by AHM.
-
3.
Creating include file.
-
4.
Run MATLAB coupled with reservoir simulator.
-
5.
Post Processing: The exclude file is an Excel file containing the results of oil recovery and pressure drop; other results would be omitted. In HM, rock properties (permeability and porosity), fluid properties (viscosity, compressibility, density, and API), and rock-fluid properties (\({{\text{k}}}_{{\text{r}}}\) and \({{\text{P}}}_{{\text{C}}}\)) are considered as inputs. If the model result does not match the experimental data, then model parameters are changed using the HM algorithm (PSO), and it is rerun.
-
6.
The final purpose is to minimize the difference between the model and experimental results, evaluating the objective function as Eq. 5:
$${\text{ERROR}}=\sqrt{{\left({(\left\{{\text{RF}}\right\}.\left\{\Delta {\text{P}}\right\})}_{{\text{Sim}}}-{(\left\{{\text{RF}}\right\}.\left\{\Delta {\text{P}}\right\})}_{{\text{Exp}}}\right)}^{2}}$$
(5)
The model can be used to predict reservoir performance when the difference between predicted \({(\left\{{\text{RF}}\right\}.\left\{\Delta {\text{P}}\right\})}_{{\text{Sim}}}\) and measured values \({(\left\{{\text{RF}}\right\}.\left\{\Delta {\text{P}}\right\})}_{{\text{Exp}}}\) is minimized4.
-
7.
Optimizing the errors, Pbest (the best solution of particle i) and Gbest (the best solution of all particles) are calculated.
-
8.
Creating a new swarm based on Pbest and Gbest.
-
9.
Repeating 2–8 steps to reach the lowest error or convergence.
The overall fellow chart of coupling the simulator and PSO to obtain optimized model parameters is shown in Fig. 3:
The overall fellow chart of coupling the simulator and PSO to obtain optimized model parameters.
By coupling the simulator and PSO algorithm, experimental results of recovery factor and pressure drop are matched with simulated results, and relative permeability and capillary pressure curves are determined as matching parameters. The summarized variation of optimized parameters for RTP1 is reported in Table 4:
The summarized variation of optimized parameters for RTP2 is reported in Table 5:
Similarly, for capillary pressure, matching parameters are also obtained in each study for both rock types. HM parameters reported in Tables 4 and 5 are computed with approximately less than 5% error and a running time of 2.5 h. High salinity relative permeability curves for studies similar to RTP1 and RTP2 were compared with those of our cases.
Proposed correlation for relative permeability parameters (machine learning technique)
Having the matching parameters in two different salinities (high and low), in other low-salinity conditions the relative permeability and capillary pressure parameters were estimated through linear interpolation, and finally by using multi-linear regression, the correlations were developed.
For RTP1 (no clay), the correlations for relative permeability and capillary pressure parameters are proposed using multiple-linear regression as Eqs. 6–10 for every desired low-salinity condition (1000–10000 ppm). The experimental conditions for developing correlations include the porosity range of 0.15–0.4, permeability range of 0.01–100 Darcy, mixed wet wettability, and clay-free formation.
$$\begin{aligned} {\mathbf{n}}_{{{\mathbf{w}}\left( {{\mathbf{LSW}}} \right)}} & = – {\text{2}}.{\text{32}} + \left[ {\left( {{\text{2}}.{\text{58E}} – 0{\text{5}}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] \\ &\quad+ [\left( {{\text{2}}.{\text{19}}} \right) \times ({\text{n}}_{{{\text{w}}\left( {{\text{HSW}}} \right)}} )\left] – \right[\left( {{\text{8}}.{\text{2E}} – 0{\text{6}}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)] \\ {\text{R}}^{2} & = 0 \cdot 98.~~{\text{MSE}} = 1{\text{E}} – 04 \\ \end{aligned}$$
(6)
$$\begin{aligned} {\mathbf{n}}_{{{\mathbf{o}}\left( {{\mathbf{LSW}}} \right)}} & = {1}.{14} – \left[ {({2}.{\text{9E}} – 0{5}) \times ({\text{HSW}}\,{\text{Concentration}})] + } \right[(0.{895}) \times ({\text{n}}_{{{\text{o}}\left( {{\text{HSW}}} \right)}} )] + \left[ {\left( {{2}.{\text{4E}} – 0{6}} \right) \, \times \, \left( {{\text{LSW}}\,{\text{Concentration}}} \right)} \right] \\ {\text{R}}^{2} & = 0 \cdot 98. {\text{MSE}} = 6{\text{E}} – 04 \\ \end{aligned}$$
(7)
$$\begin{aligned} {\mathbf{kr}}_{{{\mathbf{w}}\left( {{\mathbf{LSW}}} \right)}}^{{{\mathbf{end}}}} & = 0.{\text{279}} – \left[ {\left( {{\text{8}}.{\text{1E}} – 0{\text{6}}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right]\\ &\quad + [\left( {0.{\text{545}}} \right){\text{ }} \times ({\text{kr}}_{{{\text{w}}\left( {{\text{HSW}}} \right)}}^{{{\text{end}}}} )\left] + \right[\left( {{\text{6E}} – 0{\text{6}}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)] \\ {\text{R}}^{2} & = 0 \cdot 98.~~{\text{MSE}} = 7{\text{E}} – 05 \\ \end{aligned}$$
(8)
$$\begin{aligned} {\mathbf{kr}}_{{{\mathbf{o}}\left( {{\mathbf{LSW}}} \right)}}^{{{\mathbf{end}}}} & = 0.{149} + \left[ {\left( {{2}.{\text{39E}} – 0{6}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] \\ &\quad+ [\left( {0.{81}} \right) \times ({\text{kr}}_{{{\text{o}}\left( {{\text{HSW}}} \right)}}^{{{\text{end}}}} )\left] – \right[\left( {{2}.{\text{9E}} – 0{6}} \right) \times ({\text{LSW}}\,{\text{Concentration}})] \\ {\text{R}}^{2} & = 0 \cdot 97. {\text{MSE}} = 5{\text{E}} – 05 \\ \end{aligned}$$
(9)
$$\begin{aligned} {\mathbf{S}}_{{{\mathbf{or}}\left( {{\mathbf{LSW}}} \right)}} & = 0.0{97} – \left[ {({1}.{\text{3E}} – 0{6}) \times ({\text{HSW}}\,{\text{Concentration}})] + } \right[\left( {0.{46}} \right) \times ({\text{S}}_{{{\text{or}}\left( {{\text{HSW}}} \right)}} )] \\ &\quad+ [\left( {{3}.{\text{44E}} – 0{6}} \right) \times ({\text{LSW}}\,{\text{Concentration}})] \\ {\text{R}}^{2} & = 0 \cdot 95. {\text{MSE}} = 8{\text{E}} – 04 \\ \end{aligned}$$
(10)
$$\begin{aligned} {\mathbf{P}}_{{{\mathbf{Cmax}}\left( {{\mathbf{LSW}}} \right)}} & = – {13}.{44} + \left[ {\left( {{1}.{\text{2E}} – 0{4}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] \\ &\quad+ [\left( {0.{93}} \right) \times ({\text{P}}_{{{\text{Cmax}}\left( {{\text{HSW}}} \right)}} )\left] + \right[\left( {{7}.{\text{58E}} – 0{4}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)] \\ {\text{R}}^{2} & = 0 \cdot 93. {\text{MSE}} = 1 \cdot 2{\text{E}} – 02 \\ \end{aligned}$$
(11)
$$\begin{aligned} {\mathbf{a}}_{{1\left( {{\mathbf{LSW}}} \right)}} & = 0.{\text{314}} – \left[ {({\text{8}}.{\text{51E}} – 0{\text{6}}) \times ({\text{HSW}}\,{\text{Concentration}})] + } \right[({\text{1}}.{\text{35}}) \times ({\text{a}}_{{1\left( {{\text{HSW}}} \right)}} )] – \left[ {\left( {{\text{2}}.{\text{6E}} – 0{\text{5}}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)} \right] \\ {\text{R}}^{2} & = 0 \cdot 90.~~{\text{MSE}} = 4 \cdot 3{\text{E}} – 01 \\ \end{aligned}$$
(12)
$$\begin{aligned} {\mathbf{a}}_{{2\left( {{\mathbf{LSW}}} \right)}} & = 0.0{34} + \left[ {\left( {{2}.{\text{3E}} – 0{7}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] + [\left( {{1}.{22}} \right) \times ({\text{a}}_{{2\left( {{\text{HSW}}} \right)}} )\left] – \right[\left( {{1}.{\text{6E}} – 0{5}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)] \\ {\text{R}}^{2} & = 0 \cdot 99.{\text{ MSE}} = 6 \cdot 7{\text{E}} – 05 \\ \end{aligned}$$
(13)
$$\begin{aligned} {\mathbf{k}}_{{2\left( {{\mathbf{LSW}}} \right)}} & = – {68}.{44} + \left[ {({1}.{\text{3E}} – 0{3}) \times ({\text{HSW}}\,{\text{Concentration}})] + } \right[({1}.{13}) \times ({\text{k}}_{{2\left( {{\text{HSW}}} \right)}} )] + [\left( {{\text{8E}} – 0{4}} \right) \times ({\text{LSW}}\,{\text{Concentration}})] \\ {\text{R}}^{2} & = 0 \cdot 97. {\text{MSE}} = 3 \cdot 4{\text{E}} – 04 \\ \end{aligned}$$
(14)
In the above equations, (HSW Concentration) is the concentration of high-salinity water, and (LSW Concentration) is the concentration of low-salinity water. The subscript of each of the coefficients indicates high and low salinity conditions.
For RTP2 (clayey formation), the proposed correlations are as Eqs. 11–15 for every desired low-salinity case. The experimental conditions for developing correlations include the porosity range of 0.1–0.35, permeability range of 0.00001–100 Darcy, weakly water wet wettability, and clayey formation. Moreover, these correlations are used for the salinity range between 1000 and 10,000 ppm:
$$\begin{aligned} {\mathbf{n}}_{{{\mathbf{w}}\left( {{\mathbf{LSW}}} \right)}} & = {2}.0{63} – \left[ {\left( {{\text{1E}} – 0{5}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] + [\left( {0.{6}0{1}} \right) \times ({\text{n}}_{{{\text{w}}\left( {{\text{HSW}}} \right)}} )\left] – \right[\left( {{\text{2E}} – 0{5}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)] \\ {\text{R}}^{2} & = 0 \cdot 94. {\text{MSE}} = 9 \cdot 09{\text{E}} – 05 \\ \end{aligned}$$
(15)
$$\begin{aligned} {\mathbf{n}}_{{{\mathbf{o}}\left( {{\mathbf{LSW}}} \right)}} & = – 0.{611} – \left[ {\left( {{\text{8E}} – 0{6}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] \\ &\quad+ [\left( {{1}.{17}} \right) \times ({\text{n}}_{{{\text{o}}\left( {{\text{HSW}}} \right)}} )\left] + \right[\left( {{1}.{\text{62E}} – 0{5}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)] \\ {\text{R}}^{2} & = 0 \cdot 98. {\text{MSE}} = 1 \cdot 3{\text{E}} – 05 \\ \end{aligned}$$
(16)
$$\begin{aligned} {\mathbf{kr}}_{{{\mathbf{w}}\left( {{\mathbf{LSW}}} \right)}}^{{{\mathbf{end}}}} & = 0.0{\text{25}} – \left[ {\left( {{\text{2}}.{\text{2E}} – 0{\text{6}}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] \\ &\quad+ [\left( {0.{\text{778}}} \right) \times ({\text{kr}}_{{{\text{w}}\left( {{\text{HSW}}} \right)}}^{{{\text{end}}}} )\left] + \right[\left( {{\text{7}}.{\text{73E}} – 0{\text{6}}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)] \\ {\text{R}}^{2} & = 0 \cdot 97.~~{\text{MSE}} = 3 \cdot 37{\text{E}} – 06 \\ \end{aligned}$$
(17)
$$\begin{aligned} {\mathbf{kr}}_{{{\mathbf{o}}\left( {{\mathbf{LSW}}} \right)}}^{{{\mathbf{end}}}} & = – 0.{89} + \left[ {\left( {{1}.{\text{99E}} – 0{5}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] \\ &\quad+ [\left( {{1}.{57}} \right) \, \times \, ({\text{kr}}_{{{\text{o}}\left( {{\text{HSW}}} \right)}}^{{{\text{end}}}} )\left] – \right[\left( {{3}.{\text{9E}} – 0{6}} \right) \, \times \, ({\text{LSW}}\,{\text{Concentration}})] \\ {\text{R}}^{2} & = 0 \cdot 94. {\text{MSE}} = 1 \cdot 1{\text{E}} – 05 \\ \end{aligned}$$
(18)
$$\begin{aligned} {\mathbf{S}}_{{{\mathbf{or}}\left( {{\mathbf{LSW}}} \right)}} & = 0.0{97} – \left[ {({9}.{\text{6E}} – 0{6}) \times ({\text{HSW}}\,{\text{Concentration}})] + } \right[({1}.{45}) \times ({\text{S}}_{{{\text{or}}\left( {{\text{HSW}}} \right)}} )] \\ &\quad+ [\left( {{3}.{\text{17E}} – 0{6}} \right) \, \times \, ({\text{LSW}}\,{\text{Concentration}})] \\ {\text{R}}^{2} & = 0 \cdot 99. {\text{MSE}} = 3 \cdot 1{\text{E}} – 05 \\ \end{aligned}$$
(19)
$$\begin{aligned} {\mathbf{P}}_{{{\mathbf{Cmax}}\left( {{\mathbf{LSW}}} \right)}} & = – {24}.{83} + \left[ {\left( {{6}.{\text{95E}} – 0{4}} \right) \times \left( {{\text{HSW}}\,{\text{Concentration}}} \right)} \right] \\ &\quad+ [\left( {0.{55}} \right) \times ({\text{P}}_{{{\text{Cmax}}\left( {{\text{HSW}}} \right)}} )\left] + \right[\left( {{8}.{\text{98E}} – 0{4}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)] \\ {\text{R}}^{2} & = 0 \cdot 92. {\text{MSE}} = 2 \cdot 3{\text{E}} – 02 \\ \end{aligned}$$
(20)
$$\begin{aligned} {\mathbf{a}}_{{1\left( {{\mathbf{LSW}}} \right)}} & = {1}.{4} – \left[ {({3}.{\text{6E}} – 0{5}) \times ({\text{HSW}}\,{\text{Concentration}})] + } \right[({1}.{17}) \times ({\text{a}}_{{1\left( {{\text{HSW}}} \right)}} )] – \left[ {\left( {{2}.{\text{5E}} – 0{5}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)} \right] \\ {\text{R}}^{2} & = 0 \cdot 97. {\text{MSE}} = 6 \cdot 5{\text{E}} – 04 \\ \end{aligned}$$
(21)
$$\begin{aligned} {\mathbf{a}}_{{2\left( {{\mathbf{LSW}}} \right)}} & = – 0.{37} + \left[ {({4}.{\text{5E}} – 0{6}) \times ({\text{HSW}}\,{\text{Concentration}})] + } \right[({1}.{7}) \times ({\text{a}}_{{2\left( {{\text{HSW}}} \right)}} )] – \left[ {\left( {{\text{3E}} – 0{5}} \right) \times \left( {{\text{LSW}}\,{\text{Concentration}}} \right)} \right] \\ {\text{R}}^{2} & = 0 \cdot 90. {\text{MSE}} = 7 \cdot 2{\text{E}} – 02 \\ \end{aligned}$$
(22)
$$\begin{aligned} {\mathbf{k}}_{{2\left( {{\mathbf{LSW}}} \right)}} & = – {6}.{2} + \left[ {({2}.{\text{8E}} – 0{7}) \times ({\text{HSW}}\,{\text{Concentration}})] + } \right[({1}.0{1}) \times ({\text{k}}_{{2\left( {{\text{HSW}}} \right)}} )] + [\left( {{4}.{\text{19E}} – 0{4}} \right) \times ({\text{LSW}}\,{\text{Concentration}})] \\ {\text{R}}^{2} & = 0 \cdot 99. {\text{MSE}} = 7 \cdot 5{\text{E}} – 05 \\ \end{aligned}$$
(23)
Since few similar studies were available, the results of Rivett for RTP142, and Shojaei for RTP250 were used to validate the developed correlations and is presented in Table 6:
Although the data from these two studies were not used to develop oil and water relative permeability correlations, a suitable match was observed (R2 = 0.95, MSE = 1.2E-04 for RTP1 and R2 = 0.97, MSE = 8E-05 for RTP2). Therefore, it is possible to use these proposed correlations to determine the relative permeability and capillary pressure functions at every desired low-salinity condition, within the scope of the mentioned conditions with high accuracy.
Averaging relative permeability and capillary pressure curves
Among the rock properties, capillary pressure and relative permeability are more challenging, as are functions of fluids’ saturation. TEM function is calculated by having relative permeability, permeability, porosity, and viscosity. Equation 24 is similar to the J function and is extensively used to scale the capillary pressure curve27. The average relative permeability should be determined for each rock type as a function of the normalized saturation. The weighted relative permeability approach (Eq. 25) determines the average relative permeability curve for RTP1 and RTP2. The obtained pseudo relative permeability curve in this method varies between zero and one27,36.
In Eq. 17, \({{{\text{kr}}}_{{\text{a}}}}_{({\text{Ave}})}\) is the average phase relative permeability for each rock type, \({{\text{T}}}_{{\text{x}}}\) is the phase transmissibility, \({\upmu }_{{\text{a}}}\) is the phase viscosity, \(\mathrm{\varphi }\) is the rock’s porosity, and \({\text{k}}\) is the rock’s absolute permeability.
$${{\text{TEM}}}_{{\text{a}}}=\frac{{{\text{kk}}}_{{\text{ra}}}}{\mathrm{\varphi }{\upmu }_{{\text{a}}}}$$
(24)
$${{{\text{kr}}}_{{\text{a}}}}_{({\text{Ave}})}=\frac{\sum_{{\text{i}}=1}^{{\text{n}}}{[{\text{T}}}_{{\text{x}}}{{\text{k}}}_{{\text{ra}}}{]}_{{\text{i}}}}{\sum_{{\text{i}}=1}^{{\text{n}}}{[{\text{T}}}_{{\text{X}}}{]}_{{\text{i}}}}=\frac{\sum_{{\text{i}}=1}^{{\text{n}}}[{{\text{TEM}}}_{{\text{a}}}{]}_{{\text{i}}}}{\sum_{{\text{i}}=1}^{{\text{n}}}[\frac{{\text{k}}}{\mathrm{\varphi }{\upmu }_{{\text{a}}}}{]}_{{\text{i}}}}=\frac{\sum_{{\text{i}}=1}^{{\text{n}}}[\frac{{{\text{kk}}}_{{\text{ra}}}}{\mathrm{\varphi }{\upmu }_{{\text{a}}}}{]}_{{\text{i}}}}{\sum_{{\text{i}}=1}^{{\text{n}}}[\frac{{\text{k}}}{\mathrm{\varphi }{\upmu }_{{\text{a}}}}{]}_{{\text{i}}}}$$
(25)
“The Leverett J” is a dimensionless number relating capillary pressure to rock and fluid properties such as porosity, interfacial tension, and mean pore radius. It is the most common method used for averaging capillary pressure curves and can be calculated using Eq. 266:
$${{\text{J}}}_{({\text{Sw}})}=0\cdot 2166\times {\text{Pc}}\times \frac{\sqrt{\frac{{\text{k}}}{\mathrm{\varphi }}}}{\upsigma \times \mathrm{cos\theta }}$$
(26)
Pc is capillary pressure (psi), σ is the interfacial tension (0.27 dynes/cm), θ is the fluid’s contact angle (θ = 0), and J is a dimensionless value. To estimate the average capillary pressure using J-function, the following steps are required:
-
1.
Equation 26 converts the capillary pressure values to the J-function.
-
2.
Calculation of the average J-function for a block by using the pore volume averaging is as Eq. 27:
$${{\text{J}}}_{{\text{ave}}}=\frac{\sum_{{\text{i}}=1}^{{\text{n}}}{{\text{J}}}_{{\text{i}}}{{\text{V}}}_{{\text{i}}}{\mathrm{\varphi }}_{{\text{i}}}}{\sum_{{\text{i}}=1}^{{\text{n}}}{{\text{V}}}_{{\text{i}}}{\mathrm{\varphi }}_{{\text{i}}}}$$
(27)
\({{\text{V}}}_{{\text{i}}}\) and \({\mathrm{\varphi }}_{{\text{i}}}\) stand for each rock sample’s bulk volume and porosity, respectively.
-
3.
The average capillary pressure is finally calculated (Eq. 28) by having the average J-function values (Eq. 27).
$${{\text{P}}}_{\mathrm{c }({\text{ave}})}=\frac{{{\text{J}}}_{{\text{ave}}}\times\updelta \times }{0\cdot 2166}\sqrt{\frac{{{\text{K}}}_{{\text{ave}}}}{{\mathrm{\varphi }}_{{\text{ave}}}}}$$
(28)
