# Analysis and Application of Fractured Carbonate Dual-Media Composite Reservoir Model

Authors: Heng Song; Yuan Hu; Erhui Luo; Congge He; Xing Zeng; Shaoqi Zhang
Journal: Geofluids
 (2022)

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## Abstract

Fractured carbonate reservoirs are different from conventional reservoirs and generally develop fractures. Conventional reservoir well test performance analysis can only obtain the formation parameters and discharge radius around a single well. For vertical wells in fractured carbonate reservoirs, we can establish a dual-porosity composite reservoir model for well testing interpretation and analysis based on the actual conditions of the reservoir and combined with geological knowledge. The model can be applied to formations with different reservoir properties in the near-well and far-well areas around a single well to analyze the fracture parameters (elastic reserve ratio and channeling coefficient) and the effective flow capacity of formation fluids in the two areas. On this basis, we use the interpretation of actual production wells to verify the accuracy of the model and analyze the influence of reservoir parameters and fracture parameters on the theoretical curve.


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## Body

## 1. Introduction

Compared with conventional oil and gas reservoirs, fractured carbonate reservoirs generally have natural fractures, so one well has large controlled reserves and high production [1–7]. Carbonate reservoirs generally have two systems: matrix system and fracture system [8–16]. The matrix system is the storage space of oil and gas, and the fracture system is the migration channel of oil and gas in the formation [17–22]. It is of great significance to study the development degree of reservoir fractures and the channeling ability of matrix to fractures for formulating oil and gas field development plans and adjusting development performance.In 1960, the former Soviet scholar Barenblatt et al. [23] established the dual-media seepage model and gave the general solution on the infinite plane, so as to establish the semilogarithmic curve chart of pressure drop well test and pressure recovery well test curve. In 1963, Warren and Root believed that the fractured carbonate reservoir can be divided into two systems [24]. The fracture system is simplified into vertical and orthogonal plane fractures, which cut the whole reservoir into small pieces of matrix rocks. The fluid flows through the matrix to the fractures and then into the wellbore. Two important parameters are used to characterize the condition of fractures and matrix in the reservoir, namely, elastic storage capacity ratio (ω) and channeling coefficient (λ). The elastic storage capacity ratio represents the space occupied by the fracture, and the channeling coefficient represents the conductivity of the fluid in the matrix to the fracture. At present, this model has been widely used in the study of oil and gas seepage in oil and gas reservoirs with natural fractures. In 1988, Houze et al. [25] studied the variation characteristics of bottom hole pressure of vertically fractured wells in dual-media reservoir. The research shows that parameters such as elastic storage capacity ratio, channeling coefficient, fracture half length, and fracture conductivity have varying degrees of influence on the variation of bottom hole pressure. In 2001, Verga [26] studied the bottom hole pressure response characteristics of a horizontal well in dual-media reservoir and drew the chart of the model by using Laplace transform and Stehfest numerical inversion method. In 2012, Yang et al. [27] researched that the dual-media well test curve represents the insignificant flow of fractures in the near-wellbore zone to the wellbore. In 2014, Kuchuk and Biryukov [28] considered that Warren and Root model [29–31] is the result of idealizing the fracture length, width, thickness, distribution, and angle of natural fractures. This model has limitations in practical application, but it has not formed a practical and effective new model for the characterization of dual-media strata. In 2019, Zhang et al. [32] established a MFV model that considered the interaction of flow regimes in the matrix, discrete natural fractures, large fractures, and pores to simulate fluid flow in fractured-cavity reservoirs. In 2020, M. Wang et al. [33] combined the discrete fracture model (DFM) to simulate the local effects of discrete fractures and established a generalized multiscale multicontinuum model for fractured-cavity carbonate reservoirs. In 2021, Smeraglia et al. [34] developed a 3D discrete fracture network (DFN) model and characterized the fracture distribution in the low displacement (similar to 50 m) damage zone. In 2021, Li et al. [2] proposed a theoretical well test model that takes into account gravity effects and natural fractures connecting the caves and validated this type of reservoir.Based on the oil-gas seepage theory, this study establishes a well test model suitable for different reservoir physical properties of single well near-well zone and far-well zone in fractured carbonate reservoir, makes a well test interpretation analysis for well A-21 in the fractured carbonate reservoir, and studies the influence of different parameters on the shape of double logarithm curve. This study provides important theoretical support for the inversion of fracture development, matrix channeling ability to fractures, and reservoir physical properties.


## 2. Method

### 2.1. Physical Model and Its Assumptions

Dual media is a porous medium with two types of pores: fracture and pore. Carbonate reservoir has a dual pore system with obvious fractures and pores. In the formation, due to the invasion of drilling fluid, acid fracturing, and gas injection, the properties of the reservoir near the wellbore change, making the reservoir properties in the near-wellbore zone and the far-well zone inconsistent. Therefore, a dual-media composite reservoir model is established to describe the reservoir conditions (see Figures1 and 2).Figure 1
Schematic diagram of fluid flow in a dual-media composite reservoir model (cite from KAPPA).Figure 2
Dual-media composite reservoir model (cite from Feng’s work [35]).The basic assumptions of the dual-media composite reservoir model are as follows:(1)
Fractured carbonate reservoirs are composed of matrix and fractures. Fluids in the fracture system flow directly into the wellbore. Fluids in the matrix system flow first into the fracture and then into the wellbore. The channeling flow between the fracture and the matrix system is pseudo-steady-state channeling(2)
The reservoir stratum is horizontal and equal thickness, the two layers are radially compounded, and the radius of the inner layer isR(3)
The fluid in the formation is an incompressible oil phase, and the influence of gravity and capillary force on the flow of crude oil in the formation is not considered(4)
The fluid in the fracture and matrix system conforms to Darcy’s law of seepage and isothermal seepage(5)
The fluid flow in the wellbore considers the wellbore storage effect and the skin factor, where the wellbore storage factor isC and the skin factor is S(6)
The original formation pressure at the initial time of matrix system and fracture system in the formation ispi(7)
There is no additional pressure drop at the transition interface between the inner and outer areas of the reservoir


### 2.2. Establishment and Solution of Mathematical Model

According to the assumptions, factors such as wellbore storage effect and skin effect are considered when establishing a dual-media composite reservoir model.In order to simplify the model and facilitate solution, dimensionless variables are first introduced:(1)p1fD=K1fhpi−p1f1.842×10−3qBμ,where p1fD is the dimensionless fracture pressure in the inner zone, k1f is the fracture permeability in the inner zone, h is the effective stratum thickness, pi is the initial formation pressure, p1f is the fracture pressure in the inner zone, q is the ground yield, B is the crude oil volume factor, and μ is the crude oil viscosity.
(2)p1mD=K1fhpi−p1m1.842×10−3qBμ,where p1mD is the dimensionless matrix pressure in the inner zone andp1m is the matrix pressure in the inner zone.
(3)p2fD=K1fhpi−p2f1.842×10−3qBμ,where p2fD is the dimensionless fracture pressure in the outer zone, k2f is the fracture permeability in the outer zone, and p2f is the fracture pressure in the outer zone.
(4)p2mD=K1fhpi−p2m1.842×10−3qBμ,where p2mD is the dimensionless matrix pressure in the outer zone andp2m is the matrix pressure in the outer zone.
(5)pwfD=K1fhpi−pwf1.842×10−3qBμ,where pwfD is the dimensionless wellbore flow pressure and pwf is the wellbore flow pressure.
(6)tD=3.6K1ftϕVCt1μrw2,where tD is the dimensionless time, t is time, rw is the wellbore radius, and ϕVCt1 is the product of porosity, reservoir volume, and comprehensive elastic compressibility of the inner zone.
(7)CD=0.156C2πϕVCt1hrw2,where CD is the dimensionless wellbore storage factor and C is the wellbore storage factor.
(8)rD=rrw,where rD is the dimensionless radius and r is the radius.The seepage differential equation is as follows:(9)1rD∂∂rDrD∂p1fD∂rD=ω1∂p1fD∂tfmD+1−ω1∂p1mD∂tfmD1≤rD≤rfD,where ω1 is the elastic storage capacity ratio of the inner zone and RD is the dimensionless radius of the inner zone.
(10)1−ω1∂p1mD∂tfmD=λ1p1fD−p1mD,where λ1 is the channeling coefficient of the inner zone.
(11)1rD∂∂rDrD∂p2fD∂rD=η12ω2∂p2fD∂tfmD+1−ω2∂p2mD∂tfmDrfD≤rD≤∞,where ω2 is the elastic storage capacity ratio of the outer zone and η12 is the ratio of the conduction pressure coefficient of the inner zone to the outer zone.
(12)1−ω2η12∂p2mD∂tfmD=λ2p2fD−p2mD.The initial conditions are as follows:(13)p1fDrD,0=p1mDrD,0=0,p2fDrD,0=p2mDrD,0=0.The internal boundary conditions are as follows:(14)CDdpwfDdtfmD−rD∂p1fD∂rDrD=1=1,pwfD=p1fD−S∂p1fD∂rDrD=1.The connected surface conditions are as follows:(15)p1mDrD=RD=p2mDrD=RD,p1fDrD=RD=p2fDrD=RD.The outer boundary condition is as follows:(16)limrD⟶∞p2mD=limrD⟶∞p2fD=0.Laplace transformation of dimensionless timetfmD is performed on the above mathematical model to obtain the dimensionless pressure solution of the model in pull space (cite from Zhang’s work [32]):
(17)pwfD=A1I0gf1g−Sgf1gI1gf1g+A2K0gf1g+Sgf1gK1gf1g.In Equation (17),
(18)f1g=ω11−ω1g+λ11−ω1g+λ1.In Equation (17), I0x and K0x are the zero-order Bessel functions of the first kind and the second kind; I1x and K1x are the Bessel functions of the first order and the second kind; and A1 and A2 are the coefficients determined by internal and external boundary conditions.


### 2.3. Establishment and Characteristic Analysis of Bottom Hole Pressure Characteristic Curve Plate

Through Stehfest numerical inversion, the double logarithm characteristic curve of bottom hole pressure difference curve and pressure difference derivative curve of dual-media composite reservoir model can be obtained, as shown in Figure3.Figure 3
Theoretical double logarithm characteristic curve of dual-media composite reservoir model.The basic parameters for calculating the characteristic curve are shown in Table1.Table 1
basic parameter table.
Formation parametersResultk1 (mD)4ω10.05λ12×10−5k2 (mD)3.6ω20.05λ25.6×10−6C (m3/MPa)3S-5.7R (m)30pi (MPa)50h (m)380q (m3/d)200As shown in Figure3, according to the characteristics of double logarithm curve, the bottom hole pressure response characteristics of fractured carbonate dual-media composite reservoir can be divided into six flow stages:The first stage is wellbore storage stage. The pressure difference curve and pressure difference derivative curve coincide at this stage, and the slope is 1.The second stage is skin effect stage. The pressure difference derivative curve shows the skin effect stage with the characteristics of “hump.”The third stage is the pseudo-steady-state channeling stage from the matrix system to the fracture system in the inner zone. The “concave” feature appears on the pressure difference derivative curve.The fourth stage is flow transitional stage from the outer zone to the inner zone. Due to the interaction between different flow stages of the dual-media in the inner and outer zone, the shape of the pressure difference derivative curve is complex, and the position of the high point is not fixed. Due to the earlier transition phase, the radial flow in the inner zone is covered on the curve.The fifth stage is the pseudo-steady-state channeling stage from the matrix system to the fracture system in the outer zone. The “concave” feature appears on the pressure difference derivative curve.The sixth stage is plane radial flow stage in the outer zone. The horizontal section appears in the pressure difference derivative curve, which is a typical feature of radial flow in the outer system of the formation.


## 2.1. Physical Model and Its Assumptions

Dual media is a porous medium with two types of pores: fracture and pore. Carbonate reservoir has a dual pore system with obvious fractures and pores. In the formation, due to the invasion of drilling fluid, acid fracturing, and gas injection, the properties of the reservoir near the wellbore change, making the reservoir properties in the near-wellbore zone and the far-well zone inconsistent. Therefore, a dual-media composite reservoir model is established to describe the reservoir conditions (see Figures1 and 2).Figure 1
Schematic diagram of fluid flow in a dual-media composite reservoir model (cite from KAPPA).Figure 2
Dual-media composite reservoir model (cite from Feng’s work [35]).The basic assumptions of the dual-media composite reservoir model are as follows:(1)
Fractured carbonate reservoirs are composed of matrix and fractures. Fluids in the fracture system flow directly into the wellbore. Fluids in the matrix system flow first into the fracture and then into the wellbore. The channeling flow between the fracture and the matrix system is pseudo-steady-state channeling(2)
The reservoir stratum is horizontal and equal thickness, the two layers are radially compounded, and the radius of the inner layer isR(3)
The fluid in the formation is an incompressible oil phase, and the influence of gravity and capillary force on the flow of crude oil in the formation is not considered(4)
The fluid in the fracture and matrix system conforms to Darcy’s law of seepage and isothermal seepage(5)
The fluid flow in the wellbore considers the wellbore storage effect and the skin factor, where the wellbore storage factor isC and the skin factor is S(6)
The original formation pressure at the initial time of matrix system and fracture system in the formation ispi(7)
There is no additional pressure drop at the transition interface between the inner and outer areas of the reservoir


## 2.2. Establishment and Solution of Mathematical Model

According to the assumptions, factors such as wellbore storage effect and skin effect are considered when establishing a dual-media composite reservoir model.In order to simplify the model and facilitate solution, dimensionless variables are first introduced:(1)p1fD=K1fhpi−p1f1.842×10−3qBμ,where p1fD is the dimensionless fracture pressure in the inner zone, k1f is the fracture permeability in the inner zone, h is the effective stratum thickness, pi is the initial formation pressure, p1f is the fracture pressure in the inner zone, q is the ground yield, B is the crude oil volume factor, and μ is the crude oil viscosity.
(2)p1mD=K1fhpi−p1m1.842×10−3qBμ,where p1mD is the dimensionless matrix pressure in the inner zone andp1m is the matrix pressure in the inner zone.
(3)p2fD=K1fhpi−p2f1.842×10−3qBμ,where p2fD is the dimensionless fracture pressure in the outer zone, k2f is the fracture permeability in the outer zone, and p2f is the fracture pressure in the outer zone.
(4)p2mD=K1fhpi−p2m1.842×10−3qBμ,where p2mD is the dimensionless matrix pressure in the outer zone andp2m is the matrix pressure in the outer zone.
(5)pwfD=K1fhpi−pwf1.842×10−3qBμ,where pwfD is the dimensionless wellbore flow pressure and pwf is the wellbore flow pressure.
(6)tD=3.6K1ftϕVCt1μrw2,where tD is the dimensionless time, t is time, rw is the wellbore radius, and ϕVCt1 is the product of porosity, reservoir volume, and comprehensive elastic compressibility of the inner zone.
(7)CD=0.156C2πϕVCt1hrw2,where CD is the dimensionless wellbore storage factor and C is the wellbore storage factor.
(8)rD=rrw,where rD is the dimensionless radius and r is the radius.The seepage differential equation is as follows:(9)1rD∂∂rDrD∂p1fD∂rD=ω1∂p1fD∂tfmD+1−ω1∂p1mD∂tfmD1≤rD≤rfD,where ω1 is the elastic storage capacity ratio of the inner zone and RD is the dimensionless radius of the inner zone.
(10)1−ω1∂p1mD∂tfmD=λ1p1fD−p1mD,where λ1 is the channeling coefficient of the inner zone.
(11)1rD∂∂rDrD∂p2fD∂rD=η12ω2∂p2fD∂tfmD+1−ω2∂p2mD∂tfmDrfD≤rD≤∞,where ω2 is the elastic storage capacity ratio of the outer zone and η12 is the ratio of the conduction pressure coefficient of the inner zone to the outer zone.
(12)1−ω2η12∂p2mD∂tfmD=λ2p2fD−p2mD.The initial conditions are as follows:(13)p1fDrD,0=p1mDrD,0=0,p2fDrD,0=p2mDrD,0=0.The internal boundary conditions are as follows:(14)CDdpwfDdtfmD−rD∂p1fD∂rDrD=1=1,pwfD=p1fD−S∂p1fD∂rDrD=1.The connected surface conditions are as follows:(15)p1mDrD=RD=p2mDrD=RD,p1fDrD=RD=p2fDrD=RD.The outer boundary condition is as follows:(16)limrD⟶∞p2mD=limrD⟶∞p2fD=0.Laplace transformation of dimensionless timetfmD is performed on the above mathematical model to obtain the dimensionless pressure solution of the model in pull space (cite from Zhang’s work [32]):
(17)pwfD=A1I0gf1g−Sgf1gI1gf1g+A2K0gf1g+Sgf1gK1gf1g.In Equation (17),
(18)f1g=ω11−ω1g+λ11−ω1g+λ1.In Equation (17), I0x and K0x are the zero-order Bessel functions of the first kind and the second kind; I1x and K1x are the Bessel functions of the first order and the second kind; and A1 and A2 are the coefficients determined by internal and external boundary conditions.


## 2.3. Establishment and Characteristic Analysis of Bottom Hole Pressure Characteristic Curve Plate

Through Stehfest numerical inversion, the double logarithm characteristic curve of bottom hole pressure difference curve and pressure difference derivative curve of dual-media composite reservoir model can be obtained, as shown in Figure3.Figure 3
Theoretical double logarithm characteristic curve of dual-media composite reservoir model.The basic parameters for calculating the characteristic curve are shown in Table1.Table 1
basic parameter table.
Formation parametersResultk1 (mD)4ω10.05λ12×10−5k2 (mD)3.6ω20.05λ25.6×10−6C (m3/MPa)3S-5.7R (m)30pi (MPa)50h (m)380q (m3/d)200As shown in Figure3, according to the characteristics of double logarithm curve, the bottom hole pressure response characteristics of fractured carbonate dual-media composite reservoir can be divided into six flow stages:The first stage is wellbore storage stage. The pressure difference curve and pressure difference derivative curve coincide at this stage, and the slope is 1.The second stage is skin effect stage. The pressure difference derivative curve shows the skin effect stage with the characteristics of “hump.”The third stage is the pseudo-steady-state channeling stage from the matrix system to the fracture system in the inner zone. The “concave” feature appears on the pressure difference derivative curve.The fourth stage is flow transitional stage from the outer zone to the inner zone. Due to the interaction between different flow stages of the dual-media in the inner and outer zone, the shape of the pressure difference derivative curve is complex, and the position of the high point is not fixed. Due to the earlier transition phase, the radial flow in the inner zone is covered on the curve.The fifth stage is the pseudo-steady-state channeling stage from the matrix system to the fracture system in the outer zone. The “concave” feature appears on the pressure difference derivative curve.The sixth stage is plane radial flow stage in the outer zone. The horizontal section appears in the pressure difference derivative curve, which is a typical feature of radial flow in the outer system of the formation.


## 3. Model Verification

The research method is applied to the A-21 vertical well in the fractured carbonate reservoir block. The reservoir thickness is 12.55 m, the rock compressibility is4.35×10−4 MPa-1, its porosity is 0.0862, crude oil viscosity is 0.63, volume coefficient is 2.35 m3/m3, shut in pressure drop test is 500 h, and its medium depth of reservoir is 2248.13 m. Well A-21 was analyzed by the dual-media composite reservoir model, and the fitting diagram of the double logarithmic curve is shown in Figure 4.Figure. 4
Double logarithm curve fitting diagram of well A-21.The model is used to interpret the pressure test data of the well, and the inversion reservoir parameters are shown in Table2.Table 2
Interpretation results of formation parameters of well A-21.
Formation parametersResultk1 (mD)20.5ω10.048λ11.15×10−5k2 (mD)1.8ω20.060λ28.83×10−5C (m3/MPa)1.61S-1.35R (m)108.5


## 4. Analysis on Influencing Factors of Bottom Hole Pressure Characteristic Curve

In order to study the change characteristics of the double logarithmic curve of the dual-media composite reservoir model, we analyzed the effects of the six parameters such as the radius of the inner(R), mobility ratio of the inner zone and outer zone (M), elastic storage capacity ratio of the inner zone (ω1), elastic storage capacity ratio of the outer zone (ω2), channeling coefficient of the inner zone (λ1), and channeling coefficient of the outer zone(λ2) on the shape characteristics of the differential pressure derivative curve.


### 4.1. The Radius of the Inner Zone

The radius of the inner zone is taken as 400 m, 600 m, 800 m, and 1000 m in turn. It can be seen from Figure5 that the influence of the radius of the inner zone on the well test curve is mainly in the middle and end sections of the curve, which determines the time of flow transitional stage from the outer zone to the inner zone. The larger the inner radius, the later the flow transitional stage from the outer zone to the inner zone appears, the longer the fluid flow in the inner zone, and the less the curve shape of the inner zone flow stage is affected by the outer zone flow, so the inner zone flow stage is lower on the curve. At the same time, because the first “concave” and the second “concave” are similar, the change of the flow transitional stage from the outer zone to the inner zone curve will affect the shape of the two “concaves” at the same time and depth of depression of derivative curve.Figure 5
The influence of inner zone radius on pressure difference and pressure difference derivative curve.


### 4.2. Mobility Ratio of the Inner Zone to the Outer Zone

Mobility ratio of the inner zone to the outer zone is taken as 0.4, 1.4, 2.4, and 3.4. It can be seen from Figure6 that mobility ratio of the inner zone to the outer zone reflects the ratio of fluid flow capacity of the inner zone to the outer zone. The influence of mobility ratio of the inner zone to the outer zone on the well test curve is mainly in the middle and later stages of the curve. The larger the mobility ratio of the inner and outer zones, the smaller the depression formed in the later transition section of the pressure difference derivative curve and the higher the horizontal section, the worse the morphological characteristics of the second concave, and the higher the horizontal section of radial flow in the outer zone, indicating that the fluid flow performance of the outer zone is poor.Figure 6
The influence of mobility ratio of the inner zone to the outer zone on pressure difference and pressure difference derivative curve.


### 4.3. Elastic Storage Capacity Ratio of the Inner Zone (ω1)

The elastic storage capacity ratio of the inner zone is taken as 0.01, 0.05, 0.09, and 0.14. It can be seen from Figure7 that the influence of the elastic storage capacity ratio of the inner zone on the shape of the well test curve is mainly reflected in the middle section of the pressure difference derivative curve, which determines the concave degree of the first “concave” of the pressure difference derivative curve. The greater the elastic storage capacity ratio of the inner zone, the smaller the concave degree of the first “concave.”Figure 7
The influence of the elastic storage capacity ratio of the inner zone on pressure difference and pressure difference derivative curve.


### 4.4. Elastic Storage Capacity Ratio of the Outer Zone (ω2)

The elastic storage capacity ratio of the outer zone is taken as 0.05, 0.2, 0.35, and 0.5. It can be seen from Figure8 that the influence of the elastic storage capacity ratio of the outer zone on the shape of the well test curve is mainly reflected in the middle section of the pressure difference derivative curve, which determines the concave degree of the second “concave” of the pressure difference derivative curve. The greater the elastic storage capacity ratio of the inner zone, the smaller the concave degree of the second “concave.”Figure 8
The influence of the elastic storage capacity ratio of the outer zone on pressure difference and pressure difference derivative curve.


### 4.5. Channeling Coefficient of the Inner Zone (λ1)

The channeling coefficient of the inner zone is taken as3.0×10−5, 2.0×10−5, 1.0×10−5, and 9.0×10−6. It can be seen from Figure 9 that the channeling coefficient of the inner zone determines the time when the first “concave” of the pressure difference derivative curve appears. The smaller the channeling coefficient of the inner zone value, the later the first “concavity” will appear. If the channeling coefficient of the inner zone continues to decrease, the quasiradial flow stage of the internal system may not appear and even the shape of the second “concave” is covered.Figure 9
The influence of channeling coefficient of the inner zone on pressure difference and pressure difference derivative curve.


### 4.6. Channeling Coefficient of the Outer Zone (λ2)

The channeling coefficient of the outer zone is taken as3.0×10−6, 5.0×10−6, 7.0×10−6, and 9.0×10−6. It can be seen from Figure 10 that the influence of the channeling coefficient in the outer zone on the well test curve is similar to the channeling coefficient in the inner zone. The value of the channeling coefficient of the outer zone determines the occurrence time of the second “concave.” The smaller the value of the channeling coefficient, the more right the position of the second “concave.”Figure 10
Influence curve of channeling coefficient of the outer zone on pressure difference and pressure difference derivative curve.


## 4.1. The Radius of the Inner Zone

The radius of the inner zone is taken as 400 m, 600 m, 800 m, and 1000 m in turn. It can be seen from Figure5 that the influence of the radius of the inner zone on the well test curve is mainly in the middle and end sections of the curve, which determines the time of flow transitional stage from the outer zone to the inner zone. The larger the inner radius, the later the flow transitional stage from the outer zone to the inner zone appears, the longer the fluid flow in the inner zone, and the less the curve shape of the inner zone flow stage is affected by the outer zone flow, so the inner zone flow stage is lower on the curve. At the same time, because the first “concave” and the second “concave” are similar, the change of the flow transitional stage from the outer zone to the inner zone curve will affect the shape of the two “concaves” at the same time and depth of depression of derivative curve.Figure 5
The influence of inner zone radius on pressure difference and pressure difference derivative curve.


## 4.2. Mobility Ratio of the Inner Zone to the Outer Zone

Mobility ratio of the inner zone to the outer zone is taken as 0.4, 1.4, 2.4, and 3.4. It can be seen from Figure6 that mobility ratio of the inner zone to the outer zone reflects the ratio of fluid flow capacity of the inner zone to the outer zone. The influence of mobility ratio of the inner zone to the outer zone on the well test curve is mainly in the middle and later stages of the curve. The larger the mobility ratio of the inner and outer zones, the smaller the depression formed in the later transition section of the pressure difference derivative curve and the higher the horizontal section, the worse the morphological characteristics of the second concave, and the higher the horizontal section of radial flow in the outer zone, indicating that the fluid flow performance of the outer zone is poor.Figure 6
The influence of mobility ratio of the inner zone to the outer zone on pressure difference and pressure difference derivative curve.


## 4.3. Elastic Storage Capacity Ratio of the Inner Zone (ω1)

The elastic storage capacity ratio of the inner zone is taken as 0.01, 0.05, 0.09, and 0.14. It can be seen from Figure7 that the influence of the elastic storage capacity ratio of the inner zone on the shape of the well test curve is mainly reflected in the middle section of the pressure difference derivative curve, which determines the concave degree of the first “concave” of the pressure difference derivative curve. The greater the elastic storage capacity ratio of the inner zone, the smaller the concave degree of the first “concave.”Figure 7
The influence of the elastic storage capacity ratio of the inner zone on pressure difference and pressure difference derivative curve.


## 4.4. Elastic Storage Capacity Ratio of the Outer Zone (ω2)

The elastic storage capacity ratio of the outer zone is taken as 0.05, 0.2, 0.35, and 0.5. It can be seen from Figure8 that the influence of the elastic storage capacity ratio of the outer zone on the shape of the well test curve is mainly reflected in the middle section of the pressure difference derivative curve, which determines the concave degree of the second “concave” of the pressure difference derivative curve. The greater the elastic storage capacity ratio of the inner zone, the smaller the concave degree of the second “concave.”Figure 8
The influence of the elastic storage capacity ratio of the outer zone on pressure difference and pressure difference derivative curve.


## 4.5. Channeling Coefficient of the Inner Zone (λ1)

The channeling coefficient of the inner zone is taken as3.0×10−5, 2.0×10−5, 1.0×10−5, and 9.0×10−6. It can be seen from Figure 9 that the channeling coefficient of the inner zone determines the time when the first “concave” of the pressure difference derivative curve appears. The smaller the channeling coefficient of the inner zone value, the later the first “concavity” will appear. If the channeling coefficient of the inner zone continues to decrease, the quasiradial flow stage of the internal system may not appear and even the shape of the second “concave” is covered.Figure 9
The influence of channeling coefficient of the inner zone on pressure difference and pressure difference derivative curve.


## 4.6. Channeling Coefficient of the Outer Zone (λ2)

The channeling coefficient of the outer zone is taken as3.0×10−6, 5.0×10−6, 7.0×10−6, and 9.0×10−6. It can be seen from Figure 10 that the influence of the channeling coefficient in the outer zone on the well test curve is similar to the channeling coefficient in the inner zone. The value of the channeling coefficient of the outer zone determines the occurrence time of the second “concave.” The smaller the value of the channeling coefficient, the more right the position of the second “concave.”Figure 10
Influence curve of channeling coefficient of the outer zone on pressure difference and pressure difference derivative curve.


## 5. Conclusion

Some fractured carbonate reservoirs have different reservoir conditions in the near-well zone and far-well zone. Therefore, we can establish a dual-media composite reservoir model explain and analyze the reservoir.(1)
The establishment of the theoretical curve chart found that the double logarithmic curve of the model can be divided into six stages: wellbore storage stage, skin effect stage, the pseudo-steady-state channeling stage, flow transitional stage from the outer zone to the inner zone, the pseudo-steady-state channeling stage from the matrix system to the fracture system in the outer zone, and plane radial flow stage(2)
Dual-media composite reservoir model is used to explain and analyze the reservoir conditions around Well A-21 of the fractured carbonate reservoir. The double logarithmic curve fits well, indicating that it is feasible to apply the model to explain the reservoir(3)
Through the sensitivity analysis of six parameters such as the radius of the inner(R), mobility ratio of the inner zone and outer zone (M), elastic storage capacity ratio of the inner zone (ω1), elastic storage capacity ratio of the outer zone (ω2), channeling coefficient of the inner zone (λ1), and channeling coefficient of the outer zone (λ2), the influence on the pressure difference derivative curve at each stage is obtained



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*Source: 6785373-2022-04-01.xml*