# Co-estimation of saturation functions (*k*_{r} and *P*_{c}) from unsteady-state core-flood experiment in tight carbonate rocks

## Abstract

The production from oil and gas reservoirs is greatly affected by rock and fluid properties of the porous rock. Capillary pressure (*P*_{c}) and relative permeability (*k*_{r}) are two important properties employed in the mathematical simulation reservoirs for predicting oil recovery from underground hydrocarbon resources. In this study, various core-flood experiments were performed using different tight carbonate rock samples for oil–water and oil–gas systems. The objective of this research is to investigate the multi-phase flow functions (*k*_{r} and *P*_{c}) in tight formations. The *k*_{r} curves of each sample were obtained by two different mathematical methods: the history-matching (ant colony optimization) technique and analytical method (JBN). The comparison between the relative permeability of the history-matching technique with that of the JBN method revealed a significant discrepancy between them. The modeling of an experiment using *k*_{r} of JBN revealed a significant difference between experimental and simulation oil production, whereas the relative permeability of history matching accurately reproduced the experimental oil recovery. This observation highlights the inadequacy of the JBN technique for determination of relative permeability in particular in the tight rock where capillary forces are important. In addition to the relative permeability, the capillary pressure values as a function of saturation were estimated from core-flood tests using a history-matching technique. The comparison between oil–water capillary pressures obtained from centrifuge tests with those of core-flood experiments depicted good agreement, whereas the capillary pressure of oil–gas system measured from core-flood tests was considerably different from centrifuge experiment results. This outcome demonstrated that the capillary pressure obtained from centrifuge experiments in some cases may not be representative of dynamic capillary pressure governing the multi-phase flow in porous media.

## Keywords

Relative permeability Capillary pressure Porous media Oil Gas Water## Introduction

Due to the great importance of relative permeability (*k*_{r}) and capillary pressure (*P*_{c}) curves for the mathematical modeling of the multi-phase flow in porous media (e.g., petroleum reservoir, carbon dioxide geological storage, and underground water resources), many types of research have been directed toward the prediction of these parameters. Over the past years, various numerical and experimental methods have been developed for the direct measurement and estimation of both *k*_{r} and *P*_{c} (O’Meara Jr and Lease 1983; Watson et al. 1988; Nordtvedt et al. 1993; Shahverdi et al. 2011; Shahverdi 2012; Pini and Benson 2013). Shahverdi (2012) developed a new method for simultaneous estimation of three-phase *k*_{r} and *P*_{c} values based on history matching of unsteady-state experimental data using the genetic algorithm. He verified his method by making use of two sets of three-phase experimental data published in the literatures and then applied his methodology to calculate three-phase *k*_{r} values of different WAG cycles. Pini and Benson (2013) used three different fluid pairs and measured *k*_{r} and *P*_{c} drainage curves simultaneously on a single Berea Sandstone core. They proved reliability of the measured *k*_{r} curves by comparing their measured data on Berea Sandstone cores and with various gas/liquid pairs and those obtained from the literature obtained.

The most common experimental methods for the measurement of relative permeability are steady-state and unsteady-state experiments. In the steady-state method, two immiscible fluids are injected co-currently at a specific ratio through the core until the same production ratios are achieved from the outlet of the core. In this method, the relative permeability values can be calculated directly using Darcy’s law. Although a steady-state test facilitates the direct determination of *k*_{r} curves, it is a very time-consuming and costly method with some limitations such as neglecting capillary effects, one-dimensional flow in the core, and isothermal and incompressible fluid (Cinar et al. 2007). Moreover, the ratio of the injected fluids is not technically allowed to be less than a certain value, hence the relative permeability at low saturations cannot be measured. An alternative technique for determining the relative permeability values is the unsteady-state method which is much quicker than the steady-state test. In the unsteady-state test, one fluid (which is immiscible with the fluid in the core) is injected through the core to displace the resident fluid inside the core. Then the fluid production and pressure across the core are measured against time. Since the flow in the porous media is occurring under the unsteady-state condition, the relative permeability data cannot be obtained directly from Darcy’s equation. Hence, two general methods have been devised for the measurement of relative permeability from unsteady-state tests: the analytical (explicit) method and the optimization (implicit) method. In the analytical method, the Buckley–Leverett theory (Buckley and Leverett 1942) is used to derive the relative permeability from the production and pressure profile (Johnson et al. 1959). However, in the optimization technique, the relative permeability curve is tuned in a simulation–optimization procedure to reach an acceptable match between the laboratory results of fluid production and pressure with the corresponding values calculated by the model (Shahverdi et al. 2011). These methods are described in detail later in this manuscript.

Another experimental method for estimating relative permeability that was first introduced by Purcell (1949) involves using the capillary pressure data. In this method, the porous medium is considered as a bundle of capillary tubes with different diameters that are not connected to each other. Purcell (1949) derived an integral formula representing the relative permeabilities as a function of capillary pressure data. This method was later modified by other researchers (Fatt and Dykstra 1951; Burdine 1953).

In addition to the experimental methods, modeling approaches such as pore-network modeling and empirical correlations have been developed for predicting relative permeability (Rajaram et al. 1997; Laroche and Vizika 2005; Sinha and Wang 2007). In the pore-network modeling approach, the void space of a rock is described as a network of pores connected by the throats. The pores and throats are assigned some idealized geometry and rules are developed to determine the multi-phase fluid configurations and transport in these elements. The rules are combined in the network to compute effective transport properties on a macroscopic scale some tens of pores across (Blunt et al. 2002). The triumph of pore-scale network models depends strongly upon the quality of representing real pore spaces based on their topographical and geometrical characteristics (Xiong et al. 2016).

Various empirical correlations were developed for predicting two-phase and three-phase relative permeability as a function of fluid saturation (Corey 1954; Hirasaki 1973; Sigmund and McCaffery 1979; Honarpour et al. 1982; Chierici 1984; Lomeland et al. 2005; Shahverdi and Sohrabi 2013; Shahverdi and Sohrabi 2014). These correlations contain some parameters depending on the rock and fluid condition that should be adjusted using experimental data.

The geological specification of the rock is one of the key factors that significantly affect multi-phase flow in porous rocks. Many studies have been directed toward investigations of flow functions (*k*_{r} and *P*_{c}) in sandstone rocks (Oak 1991; Dana and Skoczylas 1999; Haugen et al. 2014; Krause and Benson 2015; Moghadasi et al. 2015; Kianinejad et al. 2016), while much fewer works have been carried out to investigate *k*_{r} in carbonate rocks (Spearing et al.; Schneider and Owens 1970; Sola et al. 2007). In general, the carbonate rocks have complex textures and pore networks with high degrees of heterogeneity, unlike sandstone rocks which are more homogeneous in terms of pore structure. In addition to the above features, the capillary forces play a substantial role in governing the fluid flow in carbonate rocks because they typically have low permeability. Hence capillary pressure should be taken into account in simulation (Roehl and Choquette 2012).

In this study, various core-flood experiments are performed on carbonate rocks (with the order of 1 mD permeability) using the oil–water and oil–gas system in order to investigate the performance of the relative permeability in the very low permeability formations. Different mathematical methods are used to estimate the relative permeability and capillary pressure curves from the core-flood experiments. Then the accuracy of each method is discussed. The capillary pressure curves measured from core-flood experiments are also compared to those obtained from centrifuge tests based on which some conclusions are drawn.

## Experiment

Rock properties and core dimensions used in the experiments

Property | Core #1 | Core #2 | Core #3 | Core #4 |
---|---|---|---|---|

Porosity (%) | 16.51 | 24.16 | 13.90 | 11.05 |

Absolute permeability (mD) | 0.676 | 1.238 | 0.459 | 0.3 |

Connate water saturation (%) | 20.48 | 20.94 | 16.04 | 27.59 |

Core diameter (cm) | 3.81 | 3.81 | 3.81 | 3.81 |

Core length (cm) | 5.5 | 5.5 | 5.5 | 5.5 |

Fluid properties used in the core-flood tests (water-flooding and gas injection) at a temperature of 25 °C

Brine viscosity (cp) | Brine density (g/cc) | Brine salinity (g/l) | Oil viscosity (cp) | Oil density (g/cc) | Gas viscosity (cp) |
---|---|---|---|---|---|

1.4 | 1.19 | 200 | 4.92 | 0.81 | 0.0183 |

- 1.
The core plugs were cleaned with toluene and methanol through a Soxhlet extraction apparatus at 60–70 °C to remove residual hydrocarbons, formation brine, salts and other contaminants. Then they were dried in an oven at 70 °C for 8 h. A helium porosimeter and liquid permeameter were used to measure the porosity and permeability of the core.

- 2.
Each core plug was placed in the core holder and then flooded continuously with brine using a constant pressure pump until the core was fully saturated with the brine. As fluid injection proceeds, brine injection rate is measured at different time steps; if it reaches a stable constant value, the only produced fluid is brine which means the core is fully saturated with brine.

- 3.
Crude oil was injected into the core sample to displace the brine and establish the connate water saturation (

*S*_{wi}) in the core. The oil was injected at the representative fluid velocity in the reservoir and the flooding continued until no further brine was produced, i.e., any remaining water in the core is immobile. - 4.
The core was aged in the crude oil at the temperature of 65 °C for 15 days to restore reservoir wettability condition.

- 5.
Water or gas was then injected to displace the oil in the core. The cumulative oil production and fluid injection rates were recorded versus time. The fluid injection continued until no further oil was produced from the core, i.e., residual oil saturation (

*S*_{orw}for water injection test and*S*_{org}for gas injection test) was attained in the core. It should be noted that during the whole process the pressure drop across the core was held constant.

Initial and boundary conditions used in the core-flood tests (water-flooding and gas injection)

Core number | Initial saturation (%) | Pressure drop (psi) | |||
---|---|---|---|---|---|

Water | Oil | Gas (nitrogen) | Water injection | Gas injection | |

1 | 20.48 | 79.52 | 0 | 165 | 110 |

2 | 20.94 | 79.06 | 0 | 240 | 100 |

3 | 16.04 | 83.96 | 0 | 605 | 465 |

4 | 27.59 | 72.41 | 0 | 1725 | 1000 |

## Mathematical method

*k*

_{r}values from the unsteady-state test is an analytical method that implements the Buckley–Leverett theory (Buckley and Leverett 1942) to derive the relative permeability curve from the results of core-flood test (e.g., production and pressure data). This method was originally devised by Johnson–Bossler–Naumann (Johnson et al. 1959). Later in 1998, (Grader and O’Meara Jr 1988) modified the JBN method to present a more straightforward formulation and procedure for estimating relative permeability curve from the experimental results. In this method, first the saturation of the phase

*j*at the outlet of the core is calculated by the following equation:

*j*,

*PV*is the amount of the injected fluid as a fraction of pore volume, \( R_{j} \) is the fluid production of phase

*j*as a fraction of pore volume, and

*S*

_{ j }is the saturation of phase

*j*at the outlet of the core.

*S*

_{ j }, the corresponding relative permeability at this saturation can be computed using:

*k*

_{rj}is the relative permeability of phase

*j*,

*q*

_{t}is the total production flow rate (cc/h), \( f_{j} \) is the fractional flow of phase

*j,*which is equal to the ratio of \( q_{j} \) to

*q*

_{t}, \( k \) is absolute permeability (mD), \( A \) is the cross section of the core (cm

^{2}), \( \mu_{j} \) is the viscosity of phase

*j*(cp),

*L*is the core length (cm), \( \Delta P \) is the pressure drop across the core (atm), PV is pore volume of the core (cm

^{3}).

Despite the simplicity of the JBN method, the formative assumptions of this method (e.g., numerical differentiation error, neglecting the effect of capillary forces and flow in one direction) are very limiting and lead to inaccurate results in many cases. The numerical differentiation of the experimental data (in Eqs. 1–3) can lead to erroneous results, in particular, when the experimental data points are affected by fluctuations. Also ignoring the effect of capillary forces can result in a significant inaccuracy of the relative permeability data especially for the tight rocks.

*k*

_{r}and

*P*

_{c}).

*S*cc), \( k \) is absolute permeability (mD), \( k_{\text{rwo}} \) is two-phase water relative permeability, \( k_{\text{row}} \) is two-phase oil relative permeability, \( \nabla P_{\text{w}} \) is pressure gradient of water (atm/cm), \( q_{{S{\text{w}}}} \) is source/sink term related to water at the standard condition (cc/h), \( \emptyset \) is porosity, \( S_{\text{w}} \) is water saturation,

*t*is time (h), \( \rho_{\text{wo}} \) is oil density at standard condition (g/cc), \( \mu_{\text{o}} \) is oil viscosity (cp), \( B_{\text{o}} \) is oil formation volume factor (cc/Scc), \( S_{\text{o}} \) is oil saturation and \( q_{\text{So}} \) is source/sink term related to water at standard condition.

*P*

_{o},

*P*

_{w},

*S*

_{w}, and

*S*

_{o}) and fluid production are obtained. It should be noted that similar equations are employed for the oil–gas system.

In this study, a modified version of the ant colony optimization (ACO) technique is used as an optimization tool in conjunction with a core-flood simulator (as described above) to obtain the relative permeability and capillary pressure simultaneously from the unsteady-state tests (Yaralidarani and Shahverdi 2016).

In the above equation, \( \sigma_{\text{o}} \) and \( \sigma_{\text{I}} \) are standard deviations of experimental values of the oil production and fluid injection rates, respectively. These quantities represent the errors associated with the core-flood experiments, which can be obtained by repetition of the experiment. The terms \( W_{\text{o}} \) and \( W_{\text{I}} \) are weighted coefficients compensating for the magnitude difference between the two quantities (oil production and fluid injection rate). Finally, *Q*_{o} represents the oil production and *q*_{f} is the fluid injection rate. The proper adjustment of these factors allows us to match fluid production and pressure to the same extent. (Yaralidarani and Shahverdi 2016).

*S*

_{w}is water saturation,

*S*

_{o}is oil saturation, \( S_{w}^{*} \) is normalized water saturation, \( S_{wi} \) is irreducible water saturation (connate water), \( S_{\text{orw}} \) is residual oil saturation, \( k_{\text{rwo}} \) and \( k_{\text{row}} \) are water and oil relative permeability in oil–water system, respectively and \( P_{cow} \) is oil–water capillary pressure. In Eqs. (9) and (10), parameter

*T*describes the upper part of the curve,

*L*parameter describes the lower part, and parameter

*E*controls the curvature or elevation of the

*k*

_{r}curves. The parameters, \( k_{\text{rwo}}^{\text{o}} \) and \( k_{\text{row}}^{\text{o}} \) have physical meaning and represent the maximum two-phase relative permeability of water and oil, respectively. In Eq. (12), \( c_{\text{w}} \) and \( a_{\text{w}} \) account for the positive part of capillary pressure curve while \( c_{\text{o}} \) and \( a_{\text{o}} \) control the negative part.

*S*

_{g}is gas saturation, and

*S*

_{org}represents residual gas saturation.

*c*

_{w}’ and ‘

*a*

_{w}’ are set equal to zero manually, because they are representative of positive part (spontaneous imbibition) of

*P*

_{c}curve which does not apply to our oil–water experiments. Also note that ending criterion has been defined as 5% discrepancy between the experimental and history-match data.

Range of initial guesses for estimating unknown variable in the ACO algorithm for oil–water and gas–oil tests

Parameter | Minimum | Maximum | Parameter | Minimum | Maximum |
---|---|---|---|---|---|

\( k_{\text{rwo}}^{\text{o}} \) | 0.15 | 0.45 | \( k_{\text{rog}}^{\text{o}} \) | 0.25 | 0.90 |

\( L_{\text{w}} \) | 1.0 | 3.0 | \( L_{\text{o}} \) | 3.0 | 8.0 |

\( E_{\text{w}} \) | 1.0 | 2.5 | \( E_{\text{o}} \) | 2.0 | 6.0 |

\( T_{\text{w}} \) | 0.5 | 2.0 | \( T_{\text{o}} \) | 0.5 | 1.5 |

\( S_{\text{orw}} \) | 0.15 | 0.35 | \( S_{\text{org}} \) | 0.1 | 0.2 |

\( k_{\text{row}}^{\text{o}} \) | 0.10 | 0.50 | \( k_{\text{rgo}}^{\text{o}} \) | 0.10 | 0.60 |

\( L_{\text{o}} \) | 0.50 | 3.0 | \( L_{\text{g}} \) | 5.0 | 9.0 |

\( E_{\text{o}} \) | 1.5 | 4.5 | \( E_{\text{g}} \) | 4.0 | 9.0 |

\( T_{\text{o}} \) | 0.8 | 1.0 | \( T_{\text{g}} \) | 0.6 | 0.8 |

\( c_{\text{w}} \) | 0.0 | 0.0 | \( c_{\text{w}} \) | – | – |

\( a_{\text{w}} \) | 0.0 | 0.0 | \( a_{\text{w}} \) | – | – |

\( c_{\text{o}} \) | 1.0 | 125 | \( c_{\text{g}} \) | 4.5 | 6.0 |

\( a_{o} \) | 0.5 | 1.5 | \( a_{g} \) | 1.4 | 1.6 |

## Results and discussion

- (a)
*Water injection test*

*k*

_{r}obtained from the history matching is much more realistic than that of the analytical method. The history-matching technique takes into account different kinds of forces (i.e., capillary, viscous and gravity) in the flow calculation, whereas the JBN method ignores the capillary and gravitational forces. The results shown in Fig. 3 depict that the

*k*

_{r}obtained from the JBN method cannot adequately predict the

*k*

_{r}of history matching. This significant disagreement is mainly attributed to the impact of capillary pressure which is pronounced for the tight rocks. This observation is very important because it suggests that the routine approach (JBN method) to deriving the relative permeability curve from the unsteady-state experiments for use in reservoir engineering calculations can produce erroneous results in particular for the tight rocks (carbonate). In addition, JBN method assumes 1D flow and may give incorrect results in case of multi-dimensional flows. Besides,

*k*

_{r}values are calculated by taking numerical derivative of production and pressure data which may cause erroneous results.

The *k*_{r} obtained from three-dimensional HM and JBN techniques was employed in the core simulation to reproduce the oil recovery and injection rate of the experiment.

*k*

_{r}from the JBN method substantially overestimates or underestimates the oil production and injection rates of all experiments. These results confirm the previous conclusion that the JBN method cannot accurately estimate relative permeability from core-flood tests.

*P*

_{c}curves.

Parameters of the relative permeability (LET) and capillary pressure (Skjæveland) models estimated by history-matching (ACO) technique in water injection test

Parameter | Core #1 | Core #2 | Core #3 | Core #4 |
---|---|---|---|---|

| ||||

\( K_{\text{rwo}}^{\text{o}} \) | 0.3680 | 0.1962 | 0.3400 | 0.2750 |

\( L_{\text{w}} \) | 2.2672 | 2.1654 | 1.5512 | 1.9097 |

\( E_{\text{w}} \) | 1.3396 | 1.9139 | 0.9091 | 1.1186 |

\( T_{\text{w}} \) | 0.6793 | 0.6276 | 0.7453 | 1.4497 |

| 0.1970 | 0.2200 | 0.3200 | 0.2900 |

\( K_{\text{row}}^{\text{o}} \) | 0.4173 | 0.2376 | 0.2177 | 0.1686 |

\( L_{\text{o}} \) | 1.0884 | 1.3758 | 2.8709 | 1.8009 |

\( E_{\text{o}} \) | 3.0220 | 2.1939 | 3.8244 | 1.6085 |

\( T_{\text{o}} \) | 0.8402 | 0.9101 | 0.9703 | 0.8947 |

| ||||

\( c_{\text{w}} \) | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

\( a_{\text{w}} \) | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

\( c_{\text{o}} \) | 10.0444 | 6.0000 | 95.0000 | 112.8569 |

\( a_{\text{o}} \) | 0.8402 | 1.4043 | 0.7500 | 0.7500 |

- (b)
*Gas injection test*

*k*

_{r}from the HM and the JBN methods. Figure 9 shows the cumulative gas production versus time obtained from experiment and simulations using

*k*

_{r}from the HM and the JBN methods. As shown in these figures (Figs. 8 and 9), similar to the oil–water system, the simulation results using

*k*

_{r}from the JBN technique are largely different from the experimental results, demonstrating the inaccuracy of the relative permeability data estimated using the JBN technique.

These results indicate the significant impact of oil–gas capillary on the relative permeability. In other words, neglecting capillary pressure in the JBN technique leads to the erroneous oil–gas relative permeability curve.

Parameters of the LET model and the Skjæveland model estimated using the ACO model in a gas injection process considering capillary effects

Parameter | Core #1 | Core #2 | Core #3 | Core #4 |
---|---|---|---|---|

| ||||

\( K_{\text{rog}}^{\text{o}} \) | 0.8338 | 0.7525 | 0.3725 | 0.3495 |

\( L_{\text{o}} \) | 4.8174 | 5.6939 | 4.1173 | 7.2834 |

\( E_{\text{o}} \) | 3.6763 | 5.1313 | 2.3899 | 4.5778 |

\( T_{\text{o}} \) | 0.9836 | 0.6915 | 1.2738 | 1.2260 |

\( S_{\text{org}} \) | 0.1244 | 0.1549 | 0.1662 | 0.1332 |

\( K_{\text{rgo}}^{\text{o}} \) | 0.1766 | 0.2327 | 0.5573 | 0.4130 |

\( L_{\text{g}} \) | 7.5538 | 8.3270 | 5.3311 | 8.3836 |

\( E_{\text{g}} \) | 6.6417 | 4.2471 | 4.9065 | 8.0087 |

\( T_{\text{g}} \) | 0.6596 | 0.7271 | 0.7768 | 0.6771 |

| ||||

\( c_{\text{g}} \) | 4.8524 | 5.3375 | 5.5456 | 4.7250 |

\( a_{\text{g}} \) | 1.4958 | 1.5389 | 1.5468 | 1.5513 |

## Conclusion

The saturation functions (*k*_{r} and *P*_{c}) were successfully co-estimated from unsteady-state core-flood experiments performed in tight carbonate rocks, using ant colony optimization (ACO) technique. In order to accurately incorporate the physic of the flow, the simulation model was built in three dimensions (*X*, *Y*, and *Z*) with fine grids considering the porosity and permeability distribution (to account for rock heterogeneity). Moreover, the good agreement between simulation and experimental results was obtained at reasonably low iteration number in optimization technique.

The results depicted that neglecting capillary pressure in core-flood simulation can lead to erroneous relative permeability data. Although adding the capillary pressure function in history-matching process can increase the uncertainty of estimated functions, having an approximate value for *P*_{c} (from other tests such centrifuge) as initial guess can modify the estimated *k*_{r} and *P*_{c}.

The comparison between the results obtained from the analytical method (JBN) and those calculated from the optimization technique revealed significant difference between the two methods. The difference is primarily because the JBN technique ignores the effect of capillary and gravitational forces. In addition to this, the analytical method assumes one-dimensional flow with the homogeneous rock properties and the constant fluid properties through the core.

The oil–water capillary pressure derived from the core-flood experiment reasonably agreed with that obtained from the centrifuge test, while the capillary pressure of the oil–gas system for the core-flood test was significantly different from the centrifuge test. This has serious implications for reservoir simulation studies that use centrifuge data as simulation inputs.

## Notes

### Acknowledgement

We are deeply grateful to the National Iranian South Oil Company (NISOC) for supporting this research.

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