Formation resistivity depends on the Resistivity factor and Resistivity index and is useful in interpretation of well logging, evaluation of porosity and water saturation. Archie equation for the relationship between formation resistivity factor and porosity, also resistivity index and saturation are presented by use of parameters Cementation factor and Saturation exponent. The Archie equation is valid only when the rock is strongly wate rwet and clay free, and may not be applicable to carbonate rocks. Investigating the effects of confining pressure, temperature, wettability and capillary pressure can lead to correct use of Archie equation. In this paper, series of experiments are performed in order to find influence of imbibitions and/or drainage on Archie parameters by changing wettability, for carbonate rocks of one of Iranian petroleum reservoir. The results show that the saturation exponent value is decreased by wettability alteration of rock samples from oil wet to water wet.

The determination of fluid saturation is vital for the estimation of hydrocarbon in place which is the base for economic decision making about reservoir production. Archie’s equations are the most prevalent equations used for saturation calculation. One of the important parameters in these equations is saturation exponent.Saturation exponent has fundamental difference comparing with other parameters which are cementation exponent and tortuosity. Cementation exponent and tortuosity mostly depend on rock type but saturation exponent depends on the distribution of fluids and therefore the hysteresis of saturation during imbibitions and drainage and also wettability. The purpose of this research is an investigation into the effect of saturation hysteresis and wettability on saturation exponent. The saturation exponent was measured from primary drainage to secondary drainage with a four-electrode system at reservoir temperature with reservoir fluids on some cylindrical core samples. Dynamic porous plate technique was used, for which full capillary pressure curves and wettability with combined Amott-USBM were determined simultaneously. It was concluded that the saturation hysteresis affects saturation exponent in different samples during imbibition and drainage and cause it to range from 0.1 to 0.66; the variations were higher in oil-wet samples than water-wet samples.

Evaluation of hydrocarbon saturation is very important in oil reservoirs calculations. Archie equations are the basic relations for evaluating rock saturation. The coefficients of these equations are determined by laboratory experiments. Archie presented these coefficients, which are constants for sandstone. However, since carbonate rocks show drastic variations in texture and pore type, Archie coefficients, which are more sensitive to pore type, should be determined for different types of carbonate rocks. Uncertainty in these coefficients causes many errors in saturation evaluation especially in the determination of the volume of oil in place. Cementation exponent is the main factor, which causes error in determining saturation. In this study Archie parameters (a and m) are determined in the laboratory for various petrofacies based on petrographic studies and CT-scan images. Due to high dependence of seismic wave velocity on pore shape in carbonate rocks, petrofacieses were also determined by using wave velocity deviation logs. Subsequently, Archie parameters were determined for each petrofacies. The results showed that cementation exponent increases with increasing velocity deviation values. It is concluded that correlation coefficient in different petrofacies obtained from velocity deviation logs is comparatively higher than the same for petrographic study. Considering the fact that petrophysical data are more accessible than the core-based petrography data, the method presented in this study seems to be a more useful approach in determining Archie coefficients.

Evaluating the characteristics of carbonate reservoirs, given their significant heterogeneity, is always accompanied by challenges and high uncertainties. Water saturation is a crucial parameter in assessing these reservoirs, and the Archie equation is commonly used for water saturation estimation. Moreover, the effect of water saturation on the mechanical behavior of rock is recognized as an important phenomenon in geotechnical engineering. The accuracy of water saturation calculated through the Archie equation depends on the precision of its parameters, including cementation exponent, saturation exponent, and tortuosity exponent. The heterogeneity of carbonate reservoirs significantly affects the Archie equation coefficients and, consequently, water saturation calculations. In this study, various methods, including the electrical efficiency, current zone indicator, and Winland method, were employed to manage reservoir heterogeneity. Subsequently, Archie parameters were calculated for each category, and water saturation was determined and compared with Dean-Stark water saturation. Furthermore, the influential parameters on the accuracy of water saturation were discussed and examined. To achieve the study objectives, 157 Dean-Stark water saturation data, 57 core plug samples for formation resistivity factor (FRF) determination, 20 core plug samples for measuring formation resistivity index (FRI), 1114 porosity and permeability measurements from core plug samples and 1368 thin sections were utilized from an exploration well in the western Gulf of Persian. Our findings highlight the significance of exploring electrical behavior characteristics and pore throat radii as crucial elements influencing the precision of water saturation calculations. As per the results, employing constant Archie parameters leads to an overestimation of water saturation and, consequently, an underestimation of hydrocarbon reserves. Our analysis illustrates that effectively managing reservoir heterogeneity through the electrical efficiency method significantly improves the accuracy of predicted water saturation compared to other approaches. Conversely, the Winland method exhibits the highest uncertainty in predicting water saturation.

There are two approaches for measuring hydrocarbon saturation; well log interpretation and usual developed formulas. Archie’ s equation is one of the most fundamental equations for water saturation calculation. Archie’ s equation includes three factors Cementation factor, tortuosity and saturation exponent. Archie determines those factors based on lab results in sandstone and provides fixed value for them. Carbonate reservoirs have variety of texture, shape and distribution of pores, therefore mentioned factors, especially cementation are not considered constant. Several methods are presented for calculation of cementation factor continuously. In none of these methods, the Change in the lithology type and conversion of limestone to dolomite is not considered. In this study the relation between cementation factor and density log was examined. Because cementation factor can be defined as a parameter that have a close relation with density. By calculating matrix density and accordance factor between matrix density and cementation factor from core’ s analysis, the log will be generated that estimate the variation of cementation factor around the borehole. The regression of calculated values with the core data is 84. 09 percent, that shows this method is useable for calculation the cementation factor in carbonate rock.

Calculation of water saturation plays a major role in formation evaluation. Archie formula is one of the fundamental methods for calculating the water saturation of formation which contains laboratory measured parameters named Archie coefficients. When there is no access to laboratory measurement results, the Archie constants are used. Due to the heterogeneity and structural variety in different carbonate rocks, Archie constants lead to so many calculation errors. In this study, Archie coefficients have been determined for Asmari formation carbonate rocks through experiments; then some important empirical equations have been used in order to calculate Archie coefficients and compare them with the measured ones; eventually a relationship has been established between porosity and cementation factors.Results indicate that Borai formula has the best accordance with experimental data; but in order to use its formula, we must swap the coefficients with observed experimental data. Results also suggest that the experimental data have coefficient of correlation equaling 76 percent and we will witness better results provided that there are isolate samples with porosity lower than 5 percent.

Water saturation (Sw) is one of the most important petro-physical parameter for evaluating the clastic horizons of hydrocarbon reservoir, which can be calculated using Archie’s equation. The Archie’s parameters (m, a) are the major source of uncertainty in the calculation of Sw. In order to obtain Archie’s parameters, a total number of 117 sandstone samples, having resistivity measurements from Asmari Formation were studied. Due to scattered data points on the Log-Log plot of F versus Æ and to obtain reliable values for m and a parameters, the data were classified based on current zone indicator (CZI) into 6 classes of electrical flow unit (EFU). The values of parameters m and a obtained from F-Æ cross plots with excellent correlation coefficient. To avoid from data diversity and to make data applicable, the average values of m and a were obtained with considering the number of samples in each class. To assess the validation of the calculated F based on proposed values of m and a, the measured values of F versus calculated one’s using Archie, Tixier and Humble formulas and proposed values were compared. The plot shows that the calculated F using Archie Tixier and Humble formulas are lower than calculated F using proposed and measured values. Applying the determined values seems to reasonably minimize the error in calculating F and therefore Sw%.

In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation (S.D.E.). So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.

This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation‎. Additionally, ‎ an efficient method is proposed to solve these equations by defining four solution categories: Controllable‎, ‎Tolerable Solution Set (TSS)‎, Controllable ‎Solution Set (CSS)‎, ‎and Algebraic Solution Set (ASS)‎. ‎ Furthermore, ‎ the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions‎. ‎ The practicality of the proposed method is demonstrated through numerical examples.