Numerical Simulation Modeling of Carbonate Reservoir Based on Rock Type

There are many types of carbonate reservoir rock spaces with complex shapes, and their primary pore structure changes dramatically. In order to describe the heterogeneity of K carbonate reservoir, equations of porosity, permeability and pore throat radii under different mercury injection saturations are fitted, and it shows that 30% is the best percentile. method is presented for rock typing, and six rock types are divided according to value of plugs. The porosity-permeability relationship is established for each rock type, and their relevant flow characteristics of each rock type have been studied. Logs are utilized to predict rock types of noncored wells, and a three-dimensional (3D) rock type model has been established based on the well rock type curves and the sedimentary facies constraint. Based on the relationship between J function and water saturation, the formula of water saturation, porosity, permeability, and oil column height can be obtained by multiple regressions for each rock type. Then, the water saturation is calculated for each grid, and a 3D water saturation model is established. The model can reflect the formation heterogeneity and the fluid distribution, and its accuracy is verified by the history matching.


Introduction
T oilfield is in the transitional belt between the Zagros Fold Zone and the Eastern margin of Arab Platform. This field is an elongated anticline which extends about 45 km by North-South direction and about 15 km by East-West direction. reservoir is the main developed formation in T oilfield, and its mid-depth is 4250 m, with the abnormal high pressure of 9500 psi. The storage space is dominated by secondary interstices and vugs, and there are few fractures in the reservoir. The average porosity is 12.04%, and the average permeability is 5.87 mD.
reservoir buried depth is very deep and is affected by the strong diagnosis. Its primary pore structure changes dramatically, and the reservoir physical property distribution has strong heterogeneity. Rock typing is an essential step in the process of the carbonate reservoir characterization and geological modeling, and it is the method of classifying reservoir rocks which have the same fluid flow features into groups [1][2][3][4][5][6]. Rock typing is an effective way to obtain accurate permeability, and it enables us to get better understanding of the fluid movement and to enhance oil recovery [7][8][9][10][11].
There are several well-known models for the rock type discrimination. Leverett introduced the famous function which could be used for rock typing [12,13]. Winland [14] used the mercury injection-capillary pressure curves to develop an empirical relationship among porosity, permeability, and pore throat radii to different mercury saturations and found that the 35th percentile gave the best correlation. Amaefule et al. [15,16] introduced a hydraulic flow unit concept and developed a method to predict the permeability of cored and uncored intervals; they grouped different rock samples based on their pore attributes similarity.
In this paper, we focus on the high heterogeneity of T oilfield and study the rock typing method for carbonate reservoirs. The capillary pressure curves and the relative permeability curves are assigned to each rock type, and different flow units are generated. Based on geological model, the numerical simulation model has been established, and the accuracy of the model is verified by the history production matching.

Rock Typing Method
Darcy's law is an equation that describes the flow of a fluid through porous medium. It can be expressed as below: where is volumetric flow rate, is permeability, is crosssectional area of plug, is viscosity, is length of plug, and Δ is pressure drop.
Poiseuille's equation can be used to determine the pressure drop of a constant viscosity fluid exhibiting laminar flow through a rigid pipe.
where is number of capillary tubes, is pore throat radius of th tube, and is apparent length of fluid flow path. For homogenous plug samples, porosity can be calculated by the following equation: where is effective porosity. By applying Darcy's and Poiseuille's Laws, a relationship between porosity and permeability can be derived as the following equation: where is tortuosity, = ( / ) 2 . Converting (4) into logarithmic form, we can derive the following equation: where A, B, C are constant parameters. For reservoir, porosity, permeability, and mercury injection experiments have been carried out for 206 plugs, and the throat radii with the mercury saturation ranging from 10% to 85% are calculated for each plug. Take 10 (throat radius at the mercury saturation of 10%) as an example; each plug has a 10 value; porosity, permeability, and 10 of 206 plugs consist of three vectors. By adopting the multiple regression method, the coefficients of (5) can be fitted. Using the same method, the equations of porosity, permeability, and pore throat radii under different mercury injection saturations are fitted.
It is shown in Figure 1 that the radius at mercury saturation 30% has the best correlation with porosity and permeability.
Based on the pore throat distributions of plugs, we define five typical pore throat radii; they are separately 0.

Pore Throat Distribution.
Because of the heterogeneity of carbonate reservoir, the pore throat distribution curves are complex, and they may have "bimodal" or "trimodal" characteristics. Figure 3 shows the pore throat distributions of plugs of each rock type, and the horizontal ordinate is the pore throat radius, while the vertical ordinate is the mercury saturation distribution function (MSDF). If there are measure points in the test, the mercury saturation distribution function can be calculated as follows: where MSDF is the MSDF value at the th point; Hg, +1 is the mercury saturation at ( + 1)th point; Hg, is the mercury saturation at th point; +1 is the pore throat radius at ( +1)th point; is the pore throat radius at th point. Figure 3 also shows that rock type 1 has the best rock quality, and its pore throat radius is large and the peak value is greater than 10 microns. The larger the rock type number is, the worse the rock quality is, and the smaller the pore throat radius is. Compared with other rock types, rock types 3 and 4 have more complex pore throat distribution.

Capillary Pressure Curve of Each Rock Type.
Capillary pressure curve is useful in characterizing rock type, because it is an indication of pore throat distribution within one rock type. According to the principle of rock typing, the plugs are classified into six groups. Figure 4 shows different capillary pressure curves of each rock type in reservoir. As the rock number becomes larger, the plateau of the capillary pressure curve becomes higher.
We also could find that there are some clear bandings under the 30 method; when the mercury saturation is 30 %, there is one turning point on the capillary pressure curve and the "width" of the capillary pressure curve set becomes narrow; this further verifies the reliability of 30 method. Rock type 6 is very tight with low porosity; the fluid cannot flow in this rock type. Therefore, it could be regarded as a barrier. We use function to make unique dimensionless drainage capillary pressure curves. For mixed-wet conditions, function does not work well because the value of permeability depends on the capillary pressure [17]. For reservoir, the average function value of each rock type is obtained by data regression.
where ,ave , ave , and ave are average capillary pressure, permeability, and porosity of plugs under the same water saturation; is interfacial tension; is wetting angle; wn is normalized water saturation; w is water saturation; wc is connate water saturation; or is irreducible oil saturation. An inverse transformation is performed on (8), and the capillary pressure curve of each rock type can be obtained. As shown in Figure 5, we can find that the capillary pressure curves have obvious regular rules except for rock type 1, and as the rock type number increases, the capillary pressure becomes higher under the same water saturation.

Relative Permeability Curve.
We have carried out relative permeability tests for 66 plugs; they cover from rock type 1 to rock type 5. Because of the heterogeneity of carbonate reservoir, we have not found any rules from the endpoint distributions of relative permeability curves, so we calculate the average value of each rock type, as shown in Table 1.
Corey saturation function is derived to fit the average relative permeability curves [19]. Based on the fit results and the average normalized relative permeability, the denormalized relative permeability curves are calculated and plotted for each rock type. As shown in Figure 6, the relative permeability curves show obvious regular rules except for rock type 1. As the rock type number increases, the irreducible water saturation increases, the two-phase flow region becomes narrow, and the relative permeability at isotonic point decreases. The plugs of rock type 1 have fractures, and the permeability curve and the relative permeability curve have different characteristics.

3D Rock Type Model.
Each plug has a coring depth, and we can find the logging data at corresponding depth. The rock type number and the logging data of each plug compose a sample. By using these samples, a link is set up between rock types and log responses by the KNN ( Nearest Neighbor) method. Logs are utilized to predict rock types of noncored wells according to this link, and then, the rock type curves of all wells can be obtained [18]. Using this method, a 3D rock type model is established based on well rock type curves and sedimentary facies constraint, as shown in Figure 7(a).
The porosity model is built up based on logging data, seismic attribution, and sedimentary facies control (Figure 7(b)). According to the relationship between porosity and permeability of each rock type, 3D permeability model can be established (Figure 7(c)). The capillary pressure curve and the relative permeability curve are assigned to each rock type, and different flow units are generated.

3D Water Saturation
Model. The capillary pressure can be transformed into oil column height under formation conditions; (8) can be written into the following form: where owc is the depth of water/oil contact; is the height of arbitrary point in transition zone; w and o are the density of water and oil; is the acceleration of gravity. According to (10), water saturation, porosity, permeability, and oil column height relationship can be obtained by multiple regressions. The oil column height can be calculated based on the depth difference between water/oil contact and grid depth, and the water saturation is calculated for each grid. There are different functions for each rock type; with the changes of the physical properties of formation, water saturation will show different characteristics, as shown in Figure 8.

History Performance Matching.
As the numerical simulation model has been established, we have to testify the accuracy of the model. There are 8 wells on production in reservoir, and the actual production process of the simulation area has reappeared by the history matching. We use fixed oil production rate as the inner boundary condition, and Figure 9 shows the history matching result of the field oil production rate (FOPR) and field water cut (FWCT); the fitting rate is up to 99.5%.
Take wells K2 and K8 as an example (their formation thickness is similar), we calculate the thickness proportion of different rock types of these two wells. As shown in Figure 10, rock types 4 and 5 are the main types for well K2, and for K8, types 2 and 3 are the main. The reservoir property near K8 is better than that of K2. Figure 11 shows the production performance of two wells, and the daily production rate of K8 is much higher than that of K2. This proves that the rock type classification is in coincidence with the production performance, and the model is reliable for the further design of the development plan.
Based on the rock type model, we design 15 infill wells, which mainly distribute in rock types 1∼3. The predicted production rates of these wells are good, as shown in Figure 12. The higher the oil production rate is, the shorter the stabilized production period is. We design four scenarios to optimize the field oil production rate, and when the field oil production rate is 90000 bbl/d (the average oil production rate per well is 4200 bbl/d), the stabilized production period can achieve four years, which can meet the requirement of the development plan.

Conclusions
(1) By applying Darcy's and Poiseuille's Laws, the relationships among porosity, permeability, and different pore throat radii have been fitted, and the typical pore throat radius 30 has been selected.
(2) 30 method is presented to define six petrophysical rock types with different reservoir properties. The capillary pressure curves and the relative permeability curves are assigned to each rock type, and different flow units are generated.
(3) Based on geological model, the numerical simulation model has been established, and it can reflect the heterogeneity and the fluid distribution of formation very well. This model is proved to be accurate by the production history matching and can be used to design the further development plan.

Conflicts of Interest
The authors declare that they have no conflicts of interest.     Field oil production rate (bbl/d) Figure 11: Daily oil production rates of K2 and K8.