# Modeling and simulation of transition zones in tight carbonate reservoirs by incorporation of improved rock typing and hysteresis models

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## Abstract

Transition zones in tight heterogeneous carbonate reservoirs contain a considerable amount of original oil in place. Identifying and characterizing the petrophysical flow units of the transition zone is crucial for reserve estimation and performance prediction. This paper presents a petrophysical rock typing method based on decoding pore-size distributions from mercury injection capillary pressure (MICP) data by using Thomeer hyperboles, with proven application in a tight carbonate reservoir in the Middle East region. In this study, 150 MICP data sets were used which were type curve matched using Thomeer Hyperbolas with closure correction. Multivariate clustering method has been employed to divide the samples into a number of groups for the purpose of both representing the reservoir heterogeneity and simplifying rock typing for dynamic modeling. From the MICP curves, it has been found that mainly monomodal pore systems prevail in the targeted transition zone and five different rock types are identified. It has been observed that most of the pore throat types are meso and micro types with the negligible existence of nano-pore type. The intrinsic advantage of this rock typing method is to describe the pore system quantitatively, which is different from other rock typing methods based on the apparent poro-perm relationship. The comparison of the grouped data with the petrophysical properties showed that different pore-size distributions yield similar porosity and permeability values, reinforcing the importance of grouping rocks based on pore systems instead of their resulting properties. Additionally, a study has also been conducted to improve understanding on the relative permeability in transition zone by implementing an up-to-date model for the mix-wet condition. The work in this paper provides a guide for the further understanding of rock typing and modeling of transition zones in carbonate reservoirs.

## Keywords

Transition zones Carbonate reservoirs Rock typing Pore geometry Thomeer hyperboles Hysteresis Dynamic model## Abbreviations

- \( k_{\text{rN}}^{\text{Im}} \)
Imbibition nonwetting-phase relative permeability

- \( S_{\text{N}} \)
Nonwetting-phase saturation

- \( k_{\text{rN}}^{\text{Dr}} \)
Drainage nonwetting-phase relative permeability

- \( S_{\text{N}}^{\text{Hyst}} \)
Maximum historical nonwetting saturation

- \( S_{\text{Nr}} \)
Residual or trapped nonwetting saturation

- \( k_{\text{rN}}^{{{\text{Dr,}}\,{\text{Exp}}}} \)
Experimental drainage nonwetting-phase relative permeability

- \( k_{\text{rN}}^{{{\text{Im,}}\,{\text{Exp}}}} \)
Experimental imbibition nonwetting-phase relative permeability

- \( S_{\text{N}}^{\text{Norm}} \)
Normalized nonwetting-phase saturation

- \( S_{\text{Nr}}^{\text{Max}} \)
Maximum possible residual or trapped nonwetting saturation

- \( S_{\text{N}}^{{\text{Max}}} \)
Maximum possible nonwetting saturation

- \( k_{\text{rw}}^{\text{Im}} \)
Imbibition water or wetting-phase relative permeability

- \( k_{\text{rw}}^{{{\text{Im,}}\,{\text{Exp}}}} \)
Experimental imbibition water or wetting-phase relative permeability

- \( k_{\text{rw}}^{\text{Dr}} \)
Drainage water or wetting-phase relative permeability

- \( k_{\text{rw}}^{{{\text{Dr,}}\,{\text{Exp}}}} \)
Experimental drainage water or wetting-phase relative permeability

- \( S_{\text{wc}} \)
Critical water saturation

- \( k_{\text{rw}} \)
Water relative permeability

- \( k_{\text{ro}} \)
Oil relative permeability

- \( S_{\text{wn}} \)
Water saturation, normalized

- \( S_{\text{orw}} \)
Residual oil saturation after water injection

- \( k_{\text{row}} \)
Oil relative permeability with water injection

- \( n_{\text{wi}} \)
Water imbibition Corey exponent

- \( n_{\text{wd}} \)
Water drainage Corey exponent

- \( S_{\text{or}}^{{\text{max}}} \)
Maximum residual oil saturation

- \( S_{\text{oi}}^{{\text{max}}} \)
Maximum initial oil saturation

- \( S_{\text{wir}} \)
Irreducible water saturation

*L*(*o*,*w*),*E*(*o*,*w*),*T*(*o*,*w*)LET model equation parameters for oil and water relative permeability curves

## Introduction

The interval from the water–oil contact (OWC) to the height where water saturation attains its irreducible level is called transition zone of a reservoir (Bera and Belhaj 2016; Masalmeh et al. 2007; Spearing et al. 2014). In tight carbonate reservoirs, the transition zone contains a considerable amount of original oil in place due to high capillary pressure phenomena. The flow units are mainly characterized by different size and geometry of pore systems. Pore systems provide primary control on hydrocarbon distribution in the reservoir. They also control the interaction between rock and fluid in terms of capillary pressure, relative permeability, and microscopic displacement efficiency (Clerke 2007; Clerke et al. 2008). Therefore, understanding the petrophysical flow units within the transition zone and correlations of capillary pressure and relative permeability study is important for performance prediction, transition zone reservoir modeling and simulation (Shi et al. 2017). Selection of a suitable mathematical model for the correlation study also signifies the importance of capillary pressure and relative permeability hysteresis incorporation in the reservoir model for transition zone behavior prediction. Precise rock typing is another important issue in transition zone reservoir modeling and simulation. The purpose of rock typing is to classify the reservoir rocks into different units with respect to pore system, texture, and lithotypes.

It is also important to mention that in some cases, the amount of oil contained in transition zones of the reservoirs is relatively low compared to the total oil in place. Therefore, development of such reservoirs is not beneficial for economic purpose. But if a reservoir with transition zone contains a significant amount of oil then it is very important to develop such reservoirs for oil production to perforate the transition zones. For this purpose, it is necessary to understand the reservoir fluid flow behavior in transition zones. As it is well known that both oil and water are movable in the transition zone, it is essential to understand the complexity of the transition zone before modeling and simulation to predict the production performance. However, if the transition zone in the reservoir is large enough, then the uncertainty exists in the variation of relative permeability, capillary pressure, and residual oil saturation with the initial water saturation (Spearing et al. 2014).

*R*

_{35}method, and Leverett J-function method. These methods have been extensively used in the literature before (Amaefule et al. 1993; Francesconi et al. 2009; Rincones et al. 2000; Skalinski et al. 2005; Venkitadri et al. 2005). The cutoff-based clustering method is based on porosity or permeability ranges or even a combination of both terms. Amaefule et al. (1993) introduced FZI and RQI methods for rock typing and thereafter these methods have been used widely in the literature (Gunter et al. 1997; Porras and Campos 2001; Jennings Jr and Lucia 2003; Rincones et al. 2000; Soto et al. 2001). The following equations are used to calculate the FZI and RQI methods:

*Φ*

_{e}is the effective porosity and

*k*is permeability.

*R*

_{35}method was developed by Winland and latterly improved by Pittman (1992) based on sandstone field study. It is an empirical law equation, defining

*R*

_{35}, the pore throat radius at

*S*

_{Hg}= 35% from MICP data, as the largest connected pore throat in a rock with intergranular porosity. The rock types are then generated by

*R*

_{35}values.

*P*

_{c}is the capillary pressure,

*σ*is the interfacial tension and

*θ*represents the contact angle.

The above-mentioned rock typing methods, however, are based on petrophysical properties or empirical knowledge. To overcome these limitations, rock typing based on Thomeer hyperboles has been implemented to capture the geometry of different pore systems and can be tied with different geological information of the reservoir.

In the present work, rock typing was done by Thomeer hyperbole method with decoding pore system for 150 MICP curves of a Middle East carbonate reservoir. Previously this method has been used in our work for 64 MICP curves (Pinto et al. 2016). In addition, MICP data clustering was also conducted to distinguish different rock types. Different hysteresis models were tested to find out the best-fitted one for dynamic modeling purpose. Incorporation of hysteresis in the dynamic model showed different results when compared to the model built without hysteresis.

## Methodology

### Theoretical background of Thomeer hyperbola rock typing

*G*,

*P*

_{d}, and \( \left( {V_{\text{b}} } \right)_{{P_{\infty } }} \) as shown in Eq. (5):

*G*is the shape factor of the hyperbola curve, determined by the pore geometry and related to the range of pore throats (typically 0.1 <

*G*< 2). P

_{d}is the displacement pressure required for mercury to intrude the largest pore-throat in the rock (in psi) and

*P*

_{c}is the capillary pressure. \( \left( {V_{\text{b}} } \right)_{{P_{\text{c}} }} \) is the bulk volume occupied at capillary pressure and \( \left( {V_{\text{b}} } \right)_{{P_{\infty } }} \) represents the percentage of bulk volume invaded by mercury at infinite pressure (\( P_{\infty } ) \) (Fig. 1). For a rock with unimodal pore system, one hyperbola is used to fit with the experimental data while for a rock with bimodal or trimodal pore system, two or three hyperbolas can be combined together to match the data, with each pore system’s geometry (

*G*factor) and volume (\( \left( {V_{\text{b}} } \right)_{{P_{\infty } }} \)) represented.

### MICP data clustering

*K*-means clustering technique.

*K*-means clustering aims to partition the input observations into different clusters in which each observation belongs to the cluster with the nearest mean, and the center of each cluster is taken as the average capillary pressure curve. Their error bounds can be obtained as follows (Buiting and Clerke 2013; Buiting 2011):

### Rock typing and decoding pore system

### Modeling and simulation

Numerical modeling of mixed-wet systems with a thick transition zone typically requires the full hysteresis option to be invoked in the reservoir simulator, and this requires preparing all the bounding curves for each rock type. It is crucial to express the relationship between capillary pressure, relative permeability, and saturation to facilitate generating scanning curves for dynamic modeling.

Previously, a number of models have been proposed to correlate capillary pressure and relative permeability with water saturation (Killough 1976; Carlson 1981; Skjaeveland et al. 2000; Masalmeh et al. 2007; Nono et al. 2014). However, Killough and Carlson models are developed for a water-wet condition which does not represent the real situation in the transition zone. The models also make use of Land’s type *S*_{oi} and *S*_{or} relationship (Land 1968), which is not fit for our mixed-wet transition zone case. Skjaeveland et al. (2000) introduced the oil-wet branch referring from the Brooks and Corey method of water-wet branch equation and extended it to the mixed-wet reservoir for capillary pressure correlation, Skjaeveland’s relative permeability hysteresis model shows better predictions but requires a lot of inputs that are not always available at laboratory scale. Masalmeh et al. (2007) introduced a modified Darcy method which is fit for the intermediate-wet system. Nono et al. (2014) presented their hysteresis experiment on two unimodal and bimodal pore system of carbonate rocks and present a new k_{r} hysteresis model, using the bounding k_{r} (relative permeability) curves and incorporating wettability change, which is quite promising for dynamic modeling of the transition zone. Hysteresis models for different wetting conditions are discussed in the following sections.

#### The Killough (1976) hysteresis model

*k*

_{rw}) for a given trapped nonwetting-phase saturation (

*S*

_{N}) can be calculated by using the following equation:

#### The Masalmeh et al. (2007) model (modified Corey model)

*S*

_{or}(

*S*

_{oi}), \( k_{\text{ro}}^{\text{Dr}} \left( {S_{\text{oi}} } \right) \), and \( n_{\text{o,Im}} \left( {S_{\text{oi}} } \right) \) are the residual oil saturation, the endpoint oil relative permeability in drainage, and the Corey exponent, respectively, and

*c*(

*S*

_{oi}) is a fitting parameter to adjust the oil relative permeability to match the experimental values. The constraint for

*c*(

*S*

_{oi}) is that its maximum value is 0.005.

*S*

_{or}(

*S*

_{oi}) which is determined by a linear correlation, shown as the following equation:

*S*

_{wx}is obtained by the below equation:

#### The Nono et al. (2014) hysteresis model

*K*

_{rw}at

*S*

_{o}

^{Hyst}, \( K_{\text{rw}}^{\text{Im,Exp}} \) corresponds to the bounding imbibition

*K*

_{rw}, a combination of Killough’s and Skjaeveland’s approach of construction.

*S*

_{oi}as follows:

*S*

_{oi, max}which is taken as intermediate-wet condition curve. \( S_{\text{orw}}^{{\text{max}}} \) represents the residual oil saturation achieved for bounding imbibition curve.

## Results and discussion

### Analysis of pore throat mode

*P*

_{d}), the results of 150 MICP curves matching have been shown in Fig. 4. It is clear from Fig. 4 that most of the capillary entry pressures lie in the range of 100–400 psi and pore throat diameter ranges from 0.8 to 2.0 µm for the majority of the core plugs. It should be mentioned that some of the core plugs show very ‘tight’ property as the maximum pore throat diameter is smaller than 0.8 µm as shown in Fig. 4 (right). Results depicted in Fig. 5 show modality of the maximum pore throat diameter. The peak of maximum pore throat is about 1 µm which carries most of the porosity in pore system. The maximum pore throat diameter, i.e., 10 µm has a low-frequency distribution in these measured core plugs indicating the ‘tight’ property of this carbonate reservoir.

### Rock-type identification

*S*

_{w}) log for all the 150 MICP curves is shown in Fig. 6. The

*S*

_{w}is derived from the buoyancy model, which assumes that at a certain depth above free water level (FWL), the buoyancy pressure (

*P*

_{b}) between water and oil is in an equilibrium state with the capillary pressure

*P*

_{cwo}. The buoyancy pressure and capillary pressure of oil and water in the reservoir have been expressed by the following equations:

*g*is the gravity of acceleration, and

*H*represents the height above free water level (HAFWL). In Eq. 19,

*σ*signifies the interfacial tension (IFT), and

*θ*indicates the contact angle obtained from laboratory measurements. Using standard values of these parameters, the equation is validated and converted to the following:

Figure 6 shows a good correlation between the MICP-derived water saturation and the log-based water saturation values. Meanwhile, the core plug samples capture some *S*_{w} characteristics that are not observed in log *S*_{w}.

*S*

_{w}derived from RTs and

*S*

_{w}log data versus rock types has been plotted in Fig. 7. From Fig. 7, it has been observed that as the number of RTs increases the variance decreases. However, the variance shows very little change for five RT’s and above. Hence, with the purpose of describing the heterogeneity and minimizing the number of RTs in the dynamic model, the optimum number of rock types has been selected as five. The resulting five RTs are then used to derive the water saturation and compared with log

*S*

_{w}data. It has been found from Fig. 8 that 5 RTs still follow the log

*S*

_{w}trend although some detailed information is missing due to clustering. After the grouping of samples into rock types, it was decided to overlay the rock types on a porosity–permeability scatter plot as shown in Fig. 9 to observe whether any correlation exists between the rock types and the joint distribution of these petrophysical properties.

It has been seen that different pore systems of rock types can produce similar porosities and permeabilities. But the present study is still able to distinguish the rock types with separation in the porosity–permeability domain. This leads to a conclusive remark that the petrophysical rock typing techniques based on porosity and permeability alone may have low resolution in classifying cores with different pore systems whereas grouping rocks based on their pore system features can help avoid this misleading issue.

### Dynamic modeling

#### Generating scanning curves for dynamic modeling

Oil–water relative permeability correlations are critically important for transition zone modeling. Several hysteresis models are available in the industry, but mostly failed to describe the mixed-wet property and complex *S*_{oi} versus *S*_{or} relationship in the transition zone. In this study, two up-to-date hysteresis models by Masalmeh et al. (2007) and Nono et al. (2014), suitable for the mix-wet condition, are applied to perform dynamic modeling of the transition zone.

*S*

_{oi}value at the given HAFWL:

- (a)
The residual oil saturation as a function of the initial oil saturation,

*f*(*S*_{oi}).

This relationship can be determined experimentally. The *S*_{oi}/*S*_{or} relationship model includes Land’s type, Suzanne type (plateau type), and Linear correlation type. It needs to be determined by special core analysis (SCAL) data.

*c*) is given as follows:

*S*_{oi}*c* is the maximum the initial oil saturation above which there is no dependency of *S*_{oi}/*S*_{or}.

*S*

_{oi}and

*S*

_{or}exists in the experimental data.

- (b)
The bounding drainage and imbibition oil/water relative permeability curves.

*K*

_{r}versus

*S*

_{w}relationship. The LET equation is recently developed and can describe

*K*

_{r}versus

*S*

_{w}relationship more precisely with three tuning parameters for oil and water as depicted in Table 1. From Fig. 11, it is observed that the LET equation matched with the drainage relative permeability curve of core No. 22 more accurately compared with Corey model.

Matching parameters for the well in steady-state drainage/imbibition experiment data

Lo | Eo | To | Lw | Ew | Tw | |
---|---|---|---|---|---|---|

Core 22 | ||||||

Drainage | 1.472 | 3.476 | 0.603 | 3.562 | 0.974 | 0.15 |

Imbibition | 2.268 | 9.238 | 1.247 | 4.009 | 1.275 | 0.252 |

Core 72 | ||||||

Drainage | 2.458 | 1.054 | 0.327 | 0.892 | 6.271 | 0.854 |

Imbibition | 0.819 | 28.874 | 2 | 2.448 | 1.032 | 0.329 |

Core 113 | ||||||

Drainage | 1.857 | 1.33 | 0 | 5.512 | 0.773 | 0 |

Imbibition | 3.036 | 0.515 | 0 | 1.769 | 2.394 | 0.961 |

Core 114 | ||||||

Drainage | 1.359 | 4.049 | 1.109 | 3.308 | 1.706 | 0.636 |

Imbibition | 3.094 | 1.597 | 0.159 | 0.937 | 5.687 | 1.074 |

Core 9 | ||||||

Drainage | 2.146 | 1.06 | 0.379 | 3.797 | 1.531 | 0.607 |

Imbibition | 2.147 | 6.985 | 1.245 | 3.169 | 1.107 | 0.158 |

Core 15 | ||||||

Drainage | 1.163 | 8.054 | 1.407 | 4.457 | 0.285 | 0 |

Imbibition | 1.472 | 1.654 | 0 | 3.615 | 6.477 | 0.784 |

Core 4 | ||||||

Drainage | 1.151 | 4.925 | 1.462 | 3.818 | 1.135 | 0.27 |

Imbibition | 2.592 | 2.109 | 0.322 | 3.599 | 2.092 | 0.371 |

Core 6 | ||||||

Drainage | 1.635 | 1.896 | 0.64 | 3.562 | 1.379 | 0.48 |

Imbibition | 1.056 | 15.806 | 1.856 | 3.067 | 1.597 | 0.309 |

Core 138 | ||||||

Drainage | 0.138 | 0.101 | 0.052 | 1.52 | 11.586 | 1.469 |

Imbibition | 3.707 | 0.483 | 0 | 4.063 | 1.487 | 0.411 |

Core 139 | ||||||

Drainage | 2.126 | 1.354 | 0.482 | 1.688 | 7.525 | 1.308 |

Imbibition | 3.131 | 0.55 | 0 | 3.118 | 2.863 | 0.803 |

*S*

_{oi}) versus residual oil saturation (

*S*

_{or}) relationship from the well for steady-state imbibition experiments, it is seen that

*S*

_{or}is about 0.1 when

*S*

_{oi}is larger than 0.25; Plug 113, 114,138 and 139 show higher

*S*

_{or}values. This may be due to the effect of pore geometry. The experimental results show a plateau-type relationship between

*S*

_{oi}and

*S*

_{or}; hence, a plateau type model is used to generate scanning curve in the program. Figure 12 shows the interface used to generate the relative permeability scanning curves for each of the five rock types.

*S*

_{w}. Finally, 59 scanning curves were generated and input into the simulation model.

Drainage/imbibition relative permeability curves used in this study

Core plug no. | Depth (ft) | Thomeer rock type | SCAL |
---|---|---|---|

113 | 9819.83 | 2 | Drainage/imbibition |

114 | 9820.83 | 2 | Drainage/imbibition |

9 | 9707.13 | 2 | Drainage/imbibition |

15 | 9712.13 | 2 | Drainage/imbibition |

22 | 9718.17 | 2 | Drainage/imbibition |

25 | 9720.79 | 2 | Imbibition |

72 | 9759.83 | 2 | Drainage/imbibition |

4 | 9703.67 | 3 | Drainage/imbibition |

6 | 9704.88 | 3 | Drainage/imbibition |

138 | 9842.5 | 4 | Drainage/imbibition |

139 | 9843.13 | 4 | Drainage/imbibition |

#### Single well model

The simulation was run with and without hysteresis for a mixed-wet reservoir to see the effect and importance of incorporation of hysteresis model for transition zone reservoir modeling and simulation with this single well model showed overestimated oil production with lower water cut when running without hysteresis. To overcome the problem, it is important to use hysteresis in the dynamic model for transition zone simulation.

#### Production performance in transition zone

Many experiments performed on core plugs from carbonate reservoir transition zone demonstrate that a mixed-wet condition widely exists in the transition zone. The mixed-wet formation will lead to different recovery mechanisms compared to that in a normal water-wet formation. However, in the industry, the mixed-wet condition and relative permeability hysteresis phenomenon are often ignored when performing transition zone simulation runs. In this work, an attempt has been taken to incorporate relative permeability hysteresis under mixed-wet condition into simulation runs for better understanding the production performance of a transition zone. The most widely used commercial simulation software does not have any built-in hysteresis model designed for the mixed-wet condition. Hence, alternatively, the scanning curves are generated by the mixed-wet Masalameh relative permeability correlation in an independent software (Masalmeh et al. 2007), and input into the simulation runs manually.

However, the simulation running with mixed-wet condition shows a higher water cut compared with the simulation with no hysteresis model. As the formation changes from water-wet to mixed-wet, the water relative permeability increases. This is important because ignoring the mixed-wet condition when running a simulation for economic evaluation purposes may lead to higher oil estimation, and delayed water breakthrough time, which is an opposite condition to reality.

## Conclusion

- 1.
The Thomeer hyperboles method is successfully applied to decode the pore system of carbonate reservoir rock samples in the Middle East region, based on 150 MICP curves.

- 2.
An execution program is developed to generate scanning curves for transition zone simulation. This program includes the most up-to-date relative permeability correlation. Different

*S*_{oi}and*S*_{or}relationship models are also included in the program. It is helpful for simulation runs of different wettability cases. - 3.
The

*S*_{oi}/*S*_{or}correlation strongly depends on wettability, pore structure, and pore-size distributions. Land’s correlation may not be valid for oil-wet carbonates because it was derived on the basis of a water-wet assumption. Extensive experimental measurements using representative rock/fluid systems are required to establish this correlation. From the steady-state imbibition experiments of core plugs of the well, a linear-type relationship between*S*_{oi}and*S*_{or}is observed. - 4.
In capillary transition zones, it is very important to use the correct hysteresis model to do performance prediction. The conventional no hysteresis-included simulation may lead to inaccurate higher oil production estimation.

## Notes

### Acknowledgements

The authors would like to acknowledge the Petroleum Institute, Abu Dhabi, A Part of Khalifa University of Science and Technology, Sas Al Nakhl Campus for hosting this research work. The authors also thank the other individuals who are directly or indirectly associated with this project from different levels of officials.

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