Variation of gas/oil IFT during gas injection processes from immiscible to miscible conditions has been observed in lab and field experiences. The possible effects of variation of IFT on kr have been investigated by several researchers. Although there is some conflict with the findings of different researchers, in general, all agree that any reduction in IFT causes an increase in kr. It has been observed that as IFT decreases toward zero, the kr increases, its curvature reduces, and less hysteresis effect is observed. Also, most of the research studies have introduced a critical IFT value above which there are no significant effects of IFT variations on relative permeabilities while below the critical value IFT significant changes are observed in relative permeabilities. A summary of the literature review is presented in the following paragraphs.
Bardon and Longeron (1980) performed a series of coreflood (gas injection) experiments on Fontainebleau sandstone core using a binary mixture of two pure hydrocarbons (C1-nC7/C1-nC10). The IFT was controlled by varying the equilibrium pressure of the mixture. For the range of IFT from 0.001 to 12.6 mNm-1, they introduced a critical IFT value of 0.04 mNm-1. As shown in Figure 5-1, for the IFT values greater than the critical IFT, they obtained a single curve for gas relative permeability (krg) but a family of curves for oil relative permeability (kro) with kro increasing with decreasing IFT. However, for the IFT values less than the critical IFT, there is a great variation in the shapes of both kro and krg curves. The variation is mainly in the kro and krg curvature towards a straight-line as IFT decreases. i.e., both kro and krg increase, as IFT decreases.
Harbert (1983) performed coreflood experiments on outcrop and reservoir rock samples using an alcohol, brine, and oil fluid system to investigate the effect of low IFT on oil and water relative permeabilities. They found that IFT reduction had more pronounced effect on the non-wetting phase kr than on the wetting phase kr.
Fulcher (1985) conducted a series of steady-state oil/water kr measurements on fired Beria sandstone, to determine whether the capillary number or its constituents cause any changes in the two-phase relative permeabilities. They introduced a critical oil/water IFT value of 2 mNm-1, below which value both oil and water relative permeabilities increased with decreasing IFT, and the curves straightened out at very low IFT values. Moreover, the increase in oil (non-wetting) kr was observed to be more significant than the increase in water (wetting) kr as IFT reduced.
Asar and Handy (1988) carried out a study similar to Bardon and Longeron (1980) but for a narrower IFT range of 0.03 to 0.82 mNm-1. They used a methane/propane mixture to represent a gas-condensate system and performed steady-state core flood experiment to measure oil and gas kr curves. They also concluded that the oil and gas kr curves tend to straighten and residual oil saturation decreases to zero as IFT approaches zero. In their experiments, the shape of kro and krg curves deviated from high IFT curvature only at conditions close to the fluid critical point. Moreover, kro increases more rapidly than krg as IFT decreases.
McDougall et al. (1997) developed an unsteady-sate pore-scale simulator to investigate the effect of gas/oil IFT on kr and interpret the results of a series of core flood experiments which had been previously performed for a gas/oil IFT range of 0.019 to 9.76 mNm-1. The pore-scale model was suitably anchored to the experimental rock samples. The relative permeabilities calculated from the results of simulations were found to exhibit the same IFT sensitivity as the relative permeabilities calculated from experimental results. They concluded that as IFT decreased, the oil (wetting phase) kr curve remained unaffected while the gas (non-wetting phase) kr increased significantly.
Henderson et al. (1997) (1998) investigated the effect of flow rate and IFT on kr in a gas condensate system using long sandstone cores at high pressure. The IFT range in their study was from 0.05 to 0.4 mNm-1. They observed that gas (non-wetting) kr was more sensitive to IFT changes than the condensate (wetting) kr.
Chen et al. (1999) investigated the effects of IFT and flow rate on gas/oil kr. They performed coreflood experiments under reservoir conditions using rock and fluid samples from two North Sea gas condensate reservoirs. They observed a greater change in kro than in krg when IFT decreased.
Blom et al. (2000) measured two-phase kr curves for a binary fluid system of methanol/n-hexane at near-critical conditions. The fluid system exhibited a critical point at ambient conditions and could be representative of a near-critical gas/condensate or gas/volatile oil system. The measurements were done at IFT range from 0.10 to 0.51 mNm-1 and for different superficial velocities. They concluded that there was a strong dependency of kr on IFT and superficial velocity. Their results showed that the non-wetting phase kr was more affected by IFT reduction and the kr to the wetting phase remained unaffected until IFT was reduced to below 0.06 mNm-1.
Shen et al. (2006) performed a series of steady-state two-phase flow experiments to measure oil/water kr at IFT range from 0.01 to 34 mNm-1. They showed that there was a critical oil/water IFT value (of 3 mNm-1) above which, IFT had little effect on kr while below this critical value, kr to both oil and water increased with a decrease in IFT. The results of their experiments showed that IFT variations have considerable impact on the water (wetting phase) kr in comparison with the oil (non-wetting) kr (Figure 5-2).
Calisgan et al. (2006) conducted unsteady state displacement experiments on a carbonate core using a binary gas condensate fluid sample at near-critical conditions. The experimental results showed a strong dependence of krg on IFT and superficial velocity. This dependency was more pronounced in the presence of immobile water.
Al-Wahaibi et al. (2006) investigated the behaviour of two-phase kr at near-miscible conditions. They performed unsteady-state displacement in a linear two dimensional bead-pack. The results of experiments showed that kr increased as IFT decreased from 24.2 to 0.03 mNm-1. Furthermore, non-wetting phase kr showed more rapid increase than the wetting phase kr and had less hysteresis effect as IFT reduced.
In contrast to the above literature, there are some research studies such as Delclaud et al. (1987) that did not report any significant change in two-phase kr with IFT changes.
From the aformentioned literature (except the Delclaud’s), it can be concluded that the IFT reduction increases the kr of both phases but not necessarily equally. It has been observed that as the system moves from immiscible toward miscible conditions, the kr increases, its curvature reduces, and less hysteresis effect is observed
IFT Scaling Methods
Coats (1980) proposed an empirical treatment on kr for IFT change which was not based on any theory or experimental results directly. However, it was based on the general idea that reduction in IFT must increase kro and krg towards straight lines that at the time was assumed to be the case for completely miscible fluids, such as alcohol and water in a pipe (rather than in a porous medium). This treatment was devised to show the Chapter 5– Effect of Gas/Oil IFT on Two- and Three-Phase Relative Permeability and Residual Oil Saturation expected behaviour of kr curves when IFT decreased. i.e., moving from immiscible conditions toward miscible conditions (reduction in IFT), the curvature of kr decreases and residual oil saturation approaches zero. The Coats method was, in fact, an interpolation between immiscible and miscible relative permeabilities, using a weighting function with only one parameter (Equations 5-1 and 5-2). Other research work on the behaviour of kr at near-miscible conditions for gas injection processes, and at near-critical conditions, for gas condensate reservoirs, showed that this flow function must depend on the ratio of viscous to capillary forces on a pore scale, known as capillary number (Nc). Therefore, several authors tried to modify the Coats equations by including the capillary number in the weighting function equation, residual saturation, and kr at miscible conditions. As a consequence, the number of required parameters increased.
Fulcher et al. (1985) performed kr measurements to determine the dependency of two-phase kr on the capillary number and its constituents. They developed a kr model based on the results of core experiments. The model is basically a Corey model with capillary number dependent coefficients. It requires seven parameters to predict residual saturation and Corey coefficient. Other authors tried to include the capillary number in the prediction of Corey coefficients differently with fewer parameters.
Blom and Hagoort (1998) reviewed and analysed fifteen different methods proposed for including the capillary number in the gas condensate kr functions and categorized them into two groups: Corey functions with Nc-dependent coefficients and Interpolation functions between immiscible and miscible kr with a Nc-dependent weighting factor. They mentioned that Corey functions were highly non-linear and could not accurately represent the convex-concave kr shapes, and these are the main disadvantages of this category in comparison with Coats type of interpolation functions. They concluded that weighting factors introduced by Coats (1980) were one of the most appropriate factors.
The Coats (1980) method has been implemented in some commercial reservoir simulators, e.g. Eclipse and CMG for the purpose of IFT scaling for compositional simulations. It seems that since this method requires fewer parameters, in comparison with other methods, it has been selected for use in such more operational simulators.
3.2 Automatic History Matching
The applied workflow for the automated history matching process is shown in Figure 1. The mathematical representation of the flow functions, the coreflood simulation, and the optimization algorithm have been demonstrated as the main components of the process. The process starts with an initial guess for the parameters of the kr functions. The kr values will be calculated according to the initial parameters and be provided to the numerical simulation. The difference between experimental and simulation results, referred to as objective function or misfit, is minimized iteratively by adjusting the parameters of kr functions until a certain error tolerance is achieved.
An objective function is formulated as a sum of squared differences between the measured data and the corresponding values calculated from the mathematical model of the experiment. The measured data can be divided into two types of internal and external measurement. The in-situ saturation and pressure profile during the dynamic flow are examples of internal measurements. These data are rarely available as they need more sophisticated facilities to measure them. Moreover, most of the saturation measurement methods are indirect, and the measured data should be analysed and converted to the saturation values. The externally measured data includes cumulative productions and pressure drop across the core. These data are the most available data from a coreflood experiment. The definition of a misfit (objective) function can be altered according to a number of phases available in a coreflood experiment and also the type of available measured data. Equation 3-1 can be used as a misfit function for a three-phase flow system when the cumulative production of all phases, the pressure drop across the core and the water saturation at different cross sections and also at various times are available as the measured data.
where Q is the cumulative production of each phase, ΔP is the pressure drop across the core sample, S is the saturation, and W is the weighting factor. The superscripts Exp and Sim represent the experimental and simulated quantities, respectively; the subscript i represents the value at time point i; and the subscript j represents the saturation value at the spatial location j. The subscripts o, w and g represent the oil, water and gas phases respectively. Nt and Nts are the number of sampling time points for external data and internal data respectively. Nls is the number of cross sectional slices.
The weighting factor for each data type, Wx, is defined as the combination of user defined weighting factor (w) and the maximum-likelihood weighting factor (1/ σ2x) Presented in Eq. 3-2. σ2x is the variance of the experimental measurement errors for data type x. In this study, the mean squared value of each data type has been used instead of the variance and w is equal one.
3.2.2 Functional Form of Relative Permeability (kr)
There are two functional forms for representing the relative permeability; parametric and non-parametric representations. The parametric presentations are usually analytical correlations which define the relationship between kr and saturation. The Corey (1954) model is a simple power law function with only one empirical parameter and is the most widely used model. It defines kr curves by endpoints and exponential factors. The Corey model has limitations to exhibit the flexibility to capture the different observed shape of kr curves. Chierici (1984) proposed an exponential formulation with two parameters for each kr curve. This correlation is more flexible than the Corey model. However, as the results of studies by Lomeland et al. (2005) and Moghadasi et al. (2015) showed, the Corey and Chierici models are not flexible enough to capture the entire saturation range of kr curves. Lomeland et al. (2005) proposed a three-parameter model which shows enough flexibility to capture different shapes of kr over a wide saturation range. This model is known as LET model.
Unlike parametric models, the nonparametric models are more flexible. One of the most commonly used nonparametric models is B-spline. Although B-spline curves are very flexible and have more degree of freedom, they cannot always guarantee the monotonicity of kr curves. Also, the estimated kr curves may have one or several breaks (Eydinov et al. (2007)) and sometimes irregular shapes. It is worth mentioning that B-spline models may introduce more non-uniqueness to the estimation due to their numerous unknowns. The higher the number of unknowns in B-spline model, the more uncertainty will be associated to the solution.
In this study, a power-law model (Corey) which has the least number of parameters for history matching and also the flexible and versatile model of LET are used to represent the kr curves as two separate options.
Corey (1954) introduced a simple power law function with only one empirical parameter. Using the Corey correlation for oil and water system, the oil and water relative permeabilities are as follow
For two-phase gas/oil system at connate water saturation (irreducible water saturation) with gas injection, the kr curves can be obtained using Corey model as follow:
Having these parameters, the LET model behaves flexibly and produces smooth and physically meaningful curves of kr. The model can reconcile most of the measure kr data and capture the variable behaviour of kr curve across the entire saturation range (Moghadasi (2015), Ebeltoft et al. (2014), Lomeland et al., (2005)).
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