Novel Method for estimation of Gas/Oil relative Permeabilities
Introduction
Phenomena such as diffusion, dispersion, viscous fingering are more dominant in porous medium especially in presence of heterogeneities. These factors are the main reason for complexity of multiphase flow in porous media [1]. The significance of multiphase flow in porous media in Soil Science, Hydrology, Chemical Engineering and Petroleum Engineering vindicate the need of well understanding of this issue [2]. Multiphase flow is strongly essential in reservoir engineering in order to recognize productivity, injectivity, and ultimate recovery. The conventional ways of describing the fluid flow in porous medium are theoretical-empirical models such as Navier–Stokes, Darcy, Brinkman and Darcy-Forchheimer. The mentioned flow models are function of relative permeability in multiphase flow systems [3]. As an example, considering the concept of effective permeability in Darcy's law shows the significance of multiphase flow in porous media [4]. One of the important concepts for description of multiphase flow in porous media is relative permeability, the ratio of effective permeability of a fluid at certain saturation to the absolute permeability. This parameter can be affected by many parameters including fluid saturation, pore structure, saturation history, wettability of surfaces, interfacial tension, viscosity of fluid, temperature, flow rate, overburden pressure and etc. [5], [6].
Relative permeability data are essential for almost all calculations of fluid flow in oil and gas reservoirs. In other words, simulation and modeling of reservoirs cannot be performed unless the relative permeability at reservoir condition is available [7]. Reservoir simulation is often employed to predict reservoir performance under different scenarios. For accurate simulation it is necessary to diminish or minimize the uncertainties of involving parameters. The main factor which causes the major uncertainty in prediction of reservoir performance is the accurate estimation of reservoir rock/fluid properties as an input data for simulator [8], [9]. Also, the relative permeability is one of the key petrophysical properties, which has a significant effect on the evaluation and forecasting of reservoir performance [10]. Undoubtedly, the relative permeabilities data are valuable required input data in oil and gas reservoir simulator and the accurate determination of their values is necessary and essential in every reservoir studies.
Generally, relative permeability curves are determined in laboratories using the analysis of multiphase fluid flow in the core [11]. Overall, laboratory methods are divided into two major groups, steady state and unsteady state [12]. For steady state method, the immiscible fluids simultaneously flow in core plugs until saturation and pressure equilibrium is attained, whereas for the unsteady state method a fluid is injected by constant rate or constant pressure to displace in-situ fluid [13]. Laboratory measurements of relative permeability curves are usually very sensitive, time consuming and costly [14]. Hence, researchers prefer to obtain these data from other methods that are quick and accurate. Empirical correlations and analytical mathematical models [15], [16] are widely used for forecasting relative permeability data. Purcell in 1949 introduced the first analytical mathematical models to estimate the relative permeability of water–oil and gas-oil systems using capillary pressure data [17]. In 1954, Corey developed an empirical correlation to estimate relative permeability of gas-oil systems based on relative permeability measurements on a large number of cores from several formations [18]. In 1982, Honarpour et al. developed empirical correlations for water–oil and gas-oil systems in which the impacts of wettability and rock type had been considered [15]. In 2005, Lomeland et al. proposed a new smooth and flexible three parameter analytical correlations for relative permeability of water–oil, gas-oil and gas-water systems. Their model has flexibility for entire saturation range and also shows good agreement with experimental data in high or low fluid saturation [19]. The impact of pressure, fluid viscosity and flow rate were studied by Mosavat et al. [5]. Their proposed correlation estimates relative permeability of water–oil systems based on Corey's model. Xu et al. in 2015 proposed an empirical model to consider the impact of displacement pressure gradient for relative permeability of water phase in water–oil system [13]. Overall, presented correlations are divided into two categories. The first category includes such correlations which predict relative permeability as function of capillary pressure, absolute permeability, porosity, fluid saturation, and interfacial tension. The drawback of these correlations is the difficulties in obtaining some required data. The second category refers to the parametric models. The required parameters are obtained by implementing measured experimental data. Indeed, these models cannot be used in the absence of laboratory measured data.
Recently, the application of intelligent systems such as artificial neural network (ANN), adoptive neuro-fuzzy interface system (ANFIS) and least square support vector machine (LSSVM) for prediction of petro-physical properties and modeling of engineering problems have been reported extensively in published papers [20]. The work done by Mohamadi-Baghmolaei et al. for prediction of Z-factor of using different intelligent models is an appropriate illustration [21]. Other physical properties such as dew pint or bubble point prediction also have been the aim of researchers for prediction. Take Baghban et al. attempt for prediction of dew point using ANFIS and LSSVM model [22]. It is worth mentioning that, relative permeability which is a key parameter in oil and gas engineering has been the subject of several researches. Ahmadi et al. used an intelligent model which was capable to predict the oil/water relative permeabilities. They used ANN model with different learning algorithms [23]. The gas/oil relative permeabilities also, have been modeled by Ahmadi who utilized LSSVM model successfully [24].
Despite this growing rate of predicting petro-physical properties using intelligent systems, it should be mentioned that, dealing with these systems cause some severity. The intelligent systems are highly structural dependent and always available source of data is necessary for development of models. Moreover, these systems do not provide an applicable clear mathematical function that can be employed in modeling and simulation as well. It is worth mentioning that, among intelligent models, those which provide mathematical relationship between input data and output are most favorable. In this study, the Multi-gene genetic programming (MGGP) approach is utilized to develop a new consolidated robust equation which predicts relative permeability of gas/oil systems. This approach is talent to learn key information from input data considering multidimensional information domain.
Section snippets
Methodology
The evolutionary algorithms are promising and encouraging methods for optimization problems particularly in engineering purposes [21], [25]. Different kinds of evolutionary algorithms are Genetic Algorithms (GA), Evolution Strategies (ES), Evolutionary Programming (EP) and Genetic Programming (GP) [26]. The general scheme of evolutionary algorithms are shown Fig. 1.
The Darwinian evolution theory is the origin of evolutionary algorithms [27]. In fact, these algorithms are inspired by process of
Data Set
The development of any predictive model strictly depends on accuracy and reliability of input data. Also, quantity and number of independent variables directly affects the robustness of developed correlations. Traditionally, the proposed correlations of gas/oil system relative permeability are function of normalized oil saturations. This normalized parameter is commonly function of oil phase saturations (So), residual oil saturation (Sor), Connate water saturation (Swc), Critical gas phase
Results and Discussion
In order to develop new correlations using MGGP approach, some important configuration should be set including population size, maximum number of generations to run for number of genes, tournament size and maximum depth. The development process 75% of data related to relative permeability of oil and gas were allocated for training purposes and the rest for test. The model has been developed based on genetic programming toolbox for the identification of physical systems (GPTIPS) [49]. The
Conclusion
The smart evolutionary technique, Multi-gene genetic programming, was applied to developed two general mathematical equations for prediction of gas/oil relative permeabilities. The main aim of this study was to introducing new equations which are able to predict gas/oil relative permeability of different types of rock and gas. The results strongly admit the implemented technique (MGGP) and accuracy of developed mathematical equations. The performed error analysis proved the priority of new
Nomenclature
- API
American Petroleum Institute gravity
- kabs
Absolute permeability
- krg
Gas phase relative permeability
- krg , Max
Maximum point in gas phase relative permeability curve
- kro
Oil phase relative permeability
- kro , Max
Maximum point in oil phase relative permeability curve
- MWg
Gas molecular weight
- Sg
Gas phase saturation
- Sgc
Critical gas saturation
- Sg , Max
Maximum gas phase saturation
- So
Oil phase saturation
- Sor
Residual oil saturation
- So , Max
Maximum oil phase saturation
- Swc
Connate water saturation
- φ
Porosity
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