## Highlights

- Pore size dependent capillary pressure is coupled with the phase equilibrium equations.
- Different PVT properties associated with oil and gas-condensates have been calculated.
- The developed thermodynamic model can simulate the PVT properties of oil and gas-condensates in shale and tight reservoirs.

#### Authors

Arash Kamari^{a}, Lei Li^{a}, James J. Sheng^{a,b}

^{a}Bob L. Herd Department of Petroleum Engineering, Texas Tech University, P.O. Box 43111, Lubbock, TX 79409, United States. ^{b}Southwest Petroleum University, Chengdu, China

Received 28 November 2016 Accepted 16 June 2017

## Abstract

Shales make up a large proportion of the rocks with extremely low permeability representing many challenges which can be complex in many cases. A careful study of rock and fluid properties (i.e. PVT of shales) of such resources is needed for long-term success, determining reservoirs quality, and increased recovery factor in unique unconventional plays. In this communication, a comprehensive thermodynamic modelling is undertaken in which capillary pressure is coupled with the phase equilibrium equations.To this end, the data associated with both shale oil and gas-condensates of Eagle Ford shale reservoir located in South Texas, U.S., is used. Different properties, including bubble and dew point pressures, capillary pressure, interfacial tension, liquid and gas densities, and liquid and gas viscosities, are predicted observing the effects of rock pore sizes by the thermodynamic modelling performed in this study.

The results demonstrate that the thermodynamic model developed in this study is capable of simulating the PVT properties of oil and gas-condensates in shale and tight reservoirs. For a binary mixture 25:75 C_{1}/C_{6}, the bubble point pressure at different reservoir temperature is increased by increasing the pore sizes from 10 to 50 nm. Furthermore, an increase in pore sizes from 10 to 50 nm can increase the dew point pressure for a studied binary mixture 75:25 C_{1}/C_{6}.

## Introduction

By increasing global demands for the energy of oil and gas during last decades, recovery of unconventional hydrocarbon resources can be counted as one of the most attractive tasks in petroleum engineering disciplines [1]. As a consequence, shale and tight reservoirs have gained significant attention because of their high potential for hydrocarbon production. In unconventional shale plays, the shale plays the role of source, reservoir, and seal. However, recovery of such hydrocarbon resources is much difficult and challenging due to the reservoir properties with extremely low permeability and low porosity.

In other words, shale and tight reservoirs are considered as complicated reservoirs with many challenges to identify, map, determine the appropriate locations of wells, as well as for enhanced oil recovery, drilling, considering environmental factors, etc. Therefore, appropriate selection of technologies as well study of rock and fluid properties (i.e. PVT of shales) of such resources is needed for long-term success, determining reservoirs quality, and increased recovery factor in unique unconventional plays.

As mentioned above, shale and tight reservoirs include a large proportion of the rocks with extremely low permeability and low porosity. Therefore, the pore size of shale and tight rocks and its effects on the reservoir fluid properties have key roles in determining the reservoirs quality and recovery factor. Over the years, there are numerous studies focus on the effects of pore sizes on the PVT properties of shale oil [2], [3], [4], [5], [6], [7], but not considerable researches studying the pore size effects on gas-condensates.

Sigmund et al. [8] analysed the experimental and numerical impact of porous media on phase behaviour of hydrocarbon binary mixtures. In another study, the impact of capillary pressures on the phase behaviour of a multicomponent system was investigated thermodynamically by Brusllovsky [9]. Qi et al. [10] developed a new model to calculate gas-oil phase equilibrium in deep gas-condensate reservoirs i.e. fluid and reservoir properties near the wellbore. The results indicated that interfacial tension can increase the dew point pressure.

Wang et al. [11] integrated the capillary pressure effect into the Peng-Robinson equation of state to calculate the phase behaviour of oil. Zhang et al. [12] found that the small pores can decrease the bubble-point pressures, increase the upper part dew pressure and decrease lower part pressure. Nojabaei et al. [13] developed a thermodynamic model to observe the effects of capillary pressures on the phase behaviour of Bakken shale oil. However, they did not present their followed approach clearly and in detail in order to check and examine its reliability and accuracy. Furthermore, capability of their model was not examined for real gas-condensate samples.

The current study aims to observe the effects of pore sizes on the entire pressure-temperature (PT) phase envelope for various hydrocarbon mixtures. In this paper, a thermodynamic modelling is conducted in which capillary pressure is coupled with the phase equilibrium equations. To this end, the data associated with both shale oil and gas-condensates of Eagle Ford shale reservoir located in South Texas, US, is used. Different properties, including bubble and dew point pressures, capillary pressure, interfacial tension, liquid and gas densities, and liquid and gas viscosities, are predicted observing the effects of rock pore sizes by the thermodynamic modelling performed in this study.

## Eagle Ford reservoir

The Eagle Ford shale has become one of the most promptly developing shale gas plays in the U.S. after successful development of the Barnett Shale in Fort Worth Basin, Haynesville Shale in Louisiana and Texas, and the Marcellus Shale in Pennsylvania [14]. As a report in 2009, the estimated natural gas and oil recoverable from Eagle Ford Shale were 21 Tcf and 3 billion barrels, respectively [15]. Eagle Ford shale is a calcareous shale play located in South Texas, US, which lies beneath the Austin chalk and extends laterally all the way across Texas from the southwest to the northeast part of the state [16], [17], [18].

**Fig. 1.** Lateral extent of Eagle Ford shale reservoir [18].

The range of depth in Eagle Ford shale reservoir is 2500 to 14,000 ft; the range of thickness is between 50 to more than 300 ft, the pressure gradients range from 0.4 to 0.8 psi/ft, and the range of TOC is from 2 to 9% [17]. As a core analysis, the gas saturation ranges from 83 to 85%, and permeability varies between 1 and 800 nD [16]. Fig. 1[18] illustrates the general areas where the reservoir produces oil (top), high liquids or condensate (middle), and predominately dry gas (bottom).

Table 1, Table 2, Table 3, Table 4[19] summarize the results of PVT analysis associated with for Eagle Ford shale oil and gas-condensates. Table 1, Table 2[19] list the compositional data for Eagle Ford shale oil and gas-condensate, respectively. Furthermore, Table 3, Table 4[19] summarize binary coefficients utilized for flash calculation of Eagle Ford shale oil and gas-condensate, respectively.

**Table 1.** Compositional data for Eagle Ford oil [19].

**Table 2.** Compositional data for Eagle Ford gas condensate [19].

**Table 3.**Peng-Robinson interaction coefficients for Eagle Ford oil [19].

**Table 4.** Peng-Robinson interaction coefficients for Eagle Ford gas-condensate [19].

## Thermodynamic modelling

In this study, several steps have been pursued for thermodynamic modelling of the different PVT properties of shale and tight gas reservoirs i.e. bubble and dew point pressures, viscosity, density, interfacial tension, capillary pressure considering effect of rock pore sizes.

Step 1:

The first step is to determine the parameters required in the thermodynamic modelling which are the number of components (NC), overall compositions (z), acentric factors (ω), critical temperature (Tc), critical pressure (Pc), critical volume (Vc), molecular weight (MW), interaction coefficients (IC), Parachor constant, reservoir temperature (T), size of pore (r).

Step 2:

In this study, the generalized form of cubic equation of state is applied as follows [20], [21]:

where u = 2, ω=1, a, and b are calculated according to Peng-Robinson as follows [20]:

T_{r} is the reduced temperature in equation (2)

In this step, a trial value of bubble point pressure (P_{b}) should be assumed. T_{o} assume an accurate and reasonable value, applying Wilson’s equation for calculating K_{i} can be a good start as follows [22]:

Solving for the bubble-point pressure gives:

Step 4:

Using the assumed bubble point pressure, the equilibrium ratio (Ki) for each component is calculated at the system temperature. This Ki can be computed by Wilson’s Equation [23], [24]:

Step 5:

In this step n_{v}, n_{L}, xi, yi should be calculated. As a result, the mixture is then flashed in order to determine the vapor and liquid compositions. For two phases, a mass balance on 1 mol of mixture yields the following:

where z_{i} is the overall composition of a component in the system and n_{L} is the mole fraction of the mixture which is presented in the liquid phase. Following expressions for the liquid and gaseous molar fractions of each components are given by using Ki=y_{i}/x_{i} into the equation above:

Regarding the fact that the sum of all mole fractions in each phase must be one, we can combine equations above to yield [25]:

Eq. (11) can be iteratively solved to obtain n_{v} and n_{L}. To calculate n_{v}, the Newton-Raphson iteration technique is applied for the number of moles of the vapor phase. To this end, the procedure below should be followed.

Any arbitrary value of n_{v} between 0 and 1 should be assumed e.g. such as n_{v} = 0.5. A good assumed value may be calculated from the following relationship [22]:

Thus, the function f(nv) in Eq. (11) is estimated using the assumed value of n_{v}. As a result, if the absolute value of the function f(nv) is smaller than a pre-set tolerance e.g. 10^{−6}, then the assumed value of nv is the desired solution [24]. On the other hand, if the absolute value of f(nv) is greater than the pre-set tolerance, then a new value of nv is calculated by the following expression [24]:

where f ‘(nv) is given as follows:

where (n_{v})_{new} is the new value of nv which is utilized for the next iteration. This procedure is repeated with the new value of n_{v} until convergence is achieved [24].

Step 6:

In this step, the parameters (a_{L} and b_{L}) of liquid phase associated with the equation of state applied in this study should be calculated as follows [26]:

Next, the compressibility factor (Z) is calculated as follows:

Here it should be noted that the equation above has 3 roots. The smallest and positive root is considered as compressibility factor for liquid phase (Z_{L}) [22]. Afterward, the molecular weight for both gas and liquid phased should be calculated.

Step 7:

The Step 7 is organised to calculate the capillary pressure (P_{cap}) as follows:

γ_{i} is the Parachor constant for each component. Here it should be mentioned that we need the density of gas phase in the equations above, but it is a function of pressure of gas phase and cannot be calculated now. Therefore, an initial guess for density of gas phase is considered, ρ^{V}=0. Having guessed the density of gas, the pressure of gas phase is obtained as follows:

Step 8:

Similar to Step 6, the parameters (a_{v} and b_{v}) of gas phase associated with the equation of state applied should be calculated as follows:

Step 9:

In this step, the fugacity and fugacity coefficient for every component in each phase are calculated as follows [24]:

The system is in equilibrium when the following is true for all components:

If the equation above is not true, then a new value of K_{i} should be considered on the basis of the following equation and also the previous steps should be repeated as long as the system reaches the equilibrium [24]:

Step 11:

To calculate the bubble point pressure of the system at pore size r, the following equation should be satisfied:

If such condition is not happened, another pressure should be guessed and steps 4 to step 11 should be repeated until the above equation is true [22]. The next guess for pressure can be obtained by following equation [27]:

Step 12:

In this step, the density of gas phase is calculated as follows:

Next, a new P_{cap} is calculated again with the new density of gas phase by Eqs. (22), (24). Check P_{cap} equality. If the new P_{cap} is approximately equal to the old P_{cap}, the calculations are finished and P = bubble point pressure. If the new Pcap is not equal to the old P_{cap}, repeat step 3 to step 11 with new P_{cap} and ρ^{V}.

Step 13:

The approach for calculating dew point pressure is similar to bubble point, but for the dew point pressure calculation approach, we have [22]:

Hence, a starting assumption for pressure, step 3, can be obtained by applying Wilson’s equation [22], [23]:

Solving this expression can give an initial estimate of dew point as:

Step 14:

Final step is organised to calculate natural gas and oil/condensate viscosities. To this end, the correlations developed by Lee, Gonzalez and Eakin [28] and Lohrenz, Bray and Clark [29] have been employed in the thermodynamic modelling performed in this study for prediction of viscosities of gas and liquid phases, respectively.

## Results and discussion

In order to validate this model, the published data of a numerical model with capillary pressure by Nojabaei et al. [7] was used and recalculated. The two-phase envelope of the binary mixtures with different model fractions (C_{1}:C_{6} = 7:3, 5:5, and 3:7) were examined at the pore radius of 20 nm, as shown in Fig. 2. The consistency of the results from our model and Najabaei’s model illustrates that our modified model is accurate and applicable when calculating the effect of nanopore size on capillary.

**Fig. 2.** Phase behaviour for various C_{1}/C_{6} mixtures at pore radius of 20 nm (the grey curves are the original curves in Nojabaei et al.’s paper [7]).

An unconventional shale reservoir called Eagle Ford was considered to observe the effects of nanopores on the PVT properties i.e. bubble and dew point pressures, capillary pressures, interfacial tension, fluids densities and viscosities. Different reservoir temperatures have been examined to see the increasing and decreasing trends of those parameters versus pore sizes. Although the model developed in this study is able to examine the effects of pores in any sizes, a range of 10–50 nm has been considered to perform a regular trend analysis for all properties investigated.

Two fluid samples including oil and gas-condensate belong to the Eagle Ford shale reservoir are considered in this study. Furthermore, two binary mixtures 25:75 C_{1}/C_{6} and 75:25 C_{1}/C_{6} for oil and gas-condensate shale samples are respectively used to illustrate the effects of pore sizes on the PVT properties of Eagle Ford reservoir graphically. Here it is worth noting that the model developed in this study is capable to simulate those PVT properties versus different pore sizes for any number and percentage of components. Therefore, the components and their percentages mentioned above are chosen just to show the performance of the model developed graphically as oil and gas-condensate reservoirs include heavier and lighter components, respectively.

Fig. 3 indicates the trend analysis of bubble point pressures versus pore sizes at different reservoir temperatures. As it is clear from the figure, an increase in the pore size from 10 to 50 nm increases the bubble point pressures of Eagle Ford shale oil particularly in lower reservoirs temperatures. In other words, the difference between bubble point pressures curves versus pore sizes at lower reservoir temperatures is higher than the curves at higher ones.

**Fig. 3.** Trend analysis of effects of pore sizes on the bubble point pressures and capillary pressures of Eagle Ford oil at different reservoir temperatures for a 25:75 C_{1}/C_{6} mixture. (BPP: bubble point pressure; PC: critical pressure).

This means that the slope becomes zero when temperature approaches the critical temperature. Another panel sketched in Fig. 3 is the prediction of capillary pressures at different pore sizes and reservoir temperatures.

The capillary pressures are decreased by increasing pore sizes at reservoir temperatures from 100 to 200 °F. Table 5 summarizes the PVT properties predicted by the model developed in this study for pore sizes from 10 to 50 nm. As shown in the table, the bubble point pressure has been increased from 522.7 psi at pore size of 10 nm (bubble point pressure be compressed by 52.27%) to 721.44 psi at pore size of 50 nm (by 27.6%). This means that the effect of pore size on the bubble point pressures at lower radiuses is greater than higher ones.

**Table 5.** The PVT properties predicted by the model developed in this study at different pore radiuses.