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Research Article | DOI: https://doi.org/10.31579/2690-1919/406
Faculty of Earth Science and Technology, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea.
*Corresponding Author: Faculty of Earth Science and Technology, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea.
Citation: Song Guk Han, Song C. Kim, (2024), Analysis of Fluid Flow Characteristics for Stress-Sensitivity Dual-Porosity Reservoir considering Elastic Outer Boundary, J Clinical Research and Reports, 17(1); DOI:10.31579/2690-1919/406
Copyright: 2024, Song Chol Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: 12 September 2024 | Accepted: 01 November 2024 | Published: 09 November 2024
Keywords: stress sensitivity; dual-porosity reservoir; well testing; elastic outer boundary
Previous researchers have considered the influence of stress sensitivity and elastic outer boundary on the flow characteristics of fluids in reservoir, and solved the problem by considering stress sensitivity and elastic outer boundary conditions when analyzing the flow characteristics, but the stress sensitivity and elastic outer boundary conditions are not taken into account at the same time, the error still arises in the well test analysis.
In order to simplify the complex problem in solving the model, we obtained analytical solutions in Laplace space by applying inverse transformation and obtained the pressure and pressure derivative curve, and analyzed the sensitivity of the characteristic values for pressure and pressure derivative curve.
In order to simplify the complex problem in solving the model, Pedrosa’s transformation and perturbation’s transformation were applied.
Using these transformations, the nonlinear partial differential equation was linearly transformed. By Laplace transformation, we obtained analytical solutions in Laplace space and applied inverse transformation to obtain solutions in real space, and plotted the pressure and pressure derivative curves.
The model accuracy was verified by comparison with previous studies and sensitivity analysis of the characteristics for pressure and pressure derivative curves was performed.
This study is of great significance in improving the accuracy of the parameters of the well and reservoir determined by well test analysis.
During oil extraction, the permeability of the reservoir changes due to the decrease of the pressure of reservoir and the increase of effective stress. Many researchers have studied reservoir models considering the stress sensitivity and stress sensitivity of reservoir.
Jiang and Yang (2018) presented a complete coupled model of fluid flow and geomechanics characterizing stress-sensitive reservoir in hydraulic fractured coal bed methane reservoirs, and Guo etc. (2015) presented pressure unsteady analysis and flow attenuation analysis for hydraulic fracturing vertical wells with finite conductivity in shale reservoirs considering multi-flow mechanisms including desorption, diffusion, Darcy flow and stress sensitivity.
Zhao etc. (2020), Jabbari etc. (2012) studied the effect of stress sensitivity on analyzing the flow characteristics of fluids in fractured reservoirs, Wang and Wang (2016) proposed that the effect of stress sensitivity on establishing a semi-analytical model considering the sliding effect in fractured gas reservoirs, Huang etc. (2018a) presented an unsteady flow model of horizontal well in stress sensitive composite reservoirs, Li etc. (2017) developed a dual-porosity medium model of horizontal well considering the influence of stress sensitivity.
Aguilera (2008) proposed a material balance equation considering the effective compressibility of the dual-porous medium, Wang etc. (2014; 2017c) and Luo etc. (2021) and Mo etc. (2016) considered the stress sensitivity when establishing a fluid flow model in a fractured reservoirs.
Ren and Guo (2018) studied the general analytical method of unsteady flow rate considering the influence of stress sensitive permeability, Jelmert and Toverud (2017) presented a model of stress sensitive reservoir and their solution based on the outer elastic modulus and plotted the corresponding typical curve.
Li etc. (2020a) proposed a sampling model that considers stress sensitivity based on the combination of fluid flow characteristics and rock mechanics in triple porous media including kerogen, inorganic and fracture networks.
Chen etc. (2021) developed a model considering the threshold pressure gradient, gas slip and stress sensitivity when developing a mathematical model in a dual-porosity gas reservoir, and Xue etc. (2021) conducted a study of a dense sandstone gas reservoir where water is taken together and found that the influence of stress sensitivity in a dense sandstone gas reservoir is strongly influenced.
Lian etc. (2011) and Zhao etc. (2013) studied the stress sensitivity of permeability in the laboratory, Wang etc. (2018), Shu and Yan (2008), Selvadurai etc. (2018), Wang etc. (2018), Wang etc. (2011), Feng etc. (2021), Tian etc. (2015) conducted a laboratory study on the effect of stress sensitivity on the bed gas layer, and Zheng etc. (2015) conducted experiments using nuclear magnetic resonance technique to investigate the relationship between the permeability and its effective stress, and Zhang etc. (2019) proposed the influence of stress sensitivity on a gas reservoir, and Zhang etc. (2019) proposed the influence of stress sensitivity on the system.
Shovkun and Espinoza (2017) considered the flow of fluid taking into account the stress induced by adsorption in the stress-sensitive coal bed methane, and carried out geomechanical simulations considering shear rupture in the coal reservoir and migration of coal dust. Wu etc. (2018c) and Xu etc. (2021) proposed a well test model in a multi-hydraulic fractured horizontal well considering stress sensitivity.
Zhang etc. (2010) developed an analytical well test model considering stress sensitivity to several factors in a low permeability reservoir, and Wang etc. (2017a) considered the effect of stress sensitivity on horizontal well test analysis in a multi-view low permeability natural fracture reservoirs.
Wu etc. (2018b) and Ji etc. (2017), Huang etc. (2018b) considered stress sensitivity in a dense reservoir, Wu etc. (2018a) proposed a multi-hydraulic fractured horizontal well test model considering stress sensitivity in a dense gas reservoir, Wang etc. (2020), Liu etc. (2015), Du etc. (2020), proposed a mathematical model considering stress sensitivity in a vertical well of a hydraulic fractured coal reservoir, Chen etc. (2016), Wang etc. (2021) proposed a numerical model considering stress sensitivity in a vertical well of a hydraulic fractured coal reservoir , and Wang etc. (2021) conducted a study.
Cao etc. (2021) and Yan etc. (2021) conducted an unsteady filtration study considering the effect of sand extraction on filtration and stress sensitivity in a well where sand is taken together.
Zhang and Tong (2008) investigated the unsteady pressure response of the purge medium in stress-sensitive reservoirs.
Gao etc. (2021) proposed a well-testing model considering stress sensitivity when performing pressure recovery tests in a high-pressure complex gas reservoir at the depth.
So far, as all idealized outer boundaries related to the stress sensitivity considered by the authors are considered, it is considered that there is no strictly idealized boundary in the reservoirs, and is extremely rare, therefore, considering the external boundary as idealized boundaries, the reservoir cannot be objectively described and the error can be avoided in the analysis.
Several researchers have investigated the flow characteristics in reservoir by considering the elastic outer boundary.
Li etc. (2019a) considered the outer boundary as an elastic boundary in the filtration modeling of homogeneous reservoir, Li etc. (2020b) modeled the extended Bessel equation by introducing elastic boundary value condition, proposed the elastic boundary value problem, and proposed the filtering model of the loose homogeneous reservoir under elastic outer boundary condition, Kim etc. (2021a) considered the outer boundary condition as an elastic boundary condition, introduced the effective borehole radius, and introduced the effective borehole radius, and Zheng etc. (2021) developed a dual-medium filtration model of the shale gas reservoir with elastic outer boundary considering the adsorption and desorption process.
In the analysis of fluid flow characteristics in reservoir, both stress sensitivity and elastic outer boundary conditions have a significant effect, but previous studies have not considered this simultaneously.
In this paper, a well test analysis model of a dual-porosity reservoir considering stress sensitivity and elastic outer boundary conditions simultaneously is developed and the solution is obtained.
EOB condition
Li et al. (2019a) defined an elastic coefficient of reservoir as follows:
As shown in Figure.1, the farther the distance from a well is, the lower the pressure difference is. Thus, the change direction of pressure difference is opposite to the change direction of the position.
Figure 1: Distribution of pressure in reservoir versus radius, r
Therefore, the elastic coefficient of outer boundary in a cylinder reservoir can be defined as follows (Kim et al., 2021a):.
Eq. 5 or 6 is an elastic outer condition.
Methodology
Physical model
1) Reservoir consists of the natural fracture and matrix, the fluid flows through the nature fracture. That is, the reservoir a dual-porosity and a single permeability reservoir.
2) It is a natural fractured reservoir considering stress sensitivity in permeability.
3) The reservoir has a constant thickness h.
4) The fluid flow is assumed to be isotropic and a single phase.
5) The effects of the gravity and capillary are neglected.
Mathematical model
SSP
In order to describe the degree of the SSP of reservoir and its influence, Pedrosa (1986) introduced the concept of a permeability modulus g for the low-permeability homogenous reservoir. As the reservoir pressure decreases, the opening of natural fracture decreases. Therefore, the permeability of the natural fracture also decreases.
The permeability modulus is defined as follows:
Motion equation
It is assumed that the fluid flow in natural fracture system obeys Darcy's law. The following equation may be obtained, which shows the fluid flow in natural fracture.
State equation
To describe the changes of oil density in the natural fracture and matrix during production of oil, the following equations are applied, respectively.
To describe the changes of porosity of the natural fracture and matrix during production of oil, the following equations are applied, respectively.
Interporosity flow equation between matrix and fracture
It is assumed that the interporosity flow is pseudo-steady state. Then, the following equation should be satisfied (Warren and Root, 1963):
Seepage differential equation
In order to obtain a seepage differential equation for matrix system, Eqs. 15, 17 and 18 are substituted into Eq. 10.
Eq. 19 is a seepage differential equation for the matrix system in the low-permeability natural fracture reservoir.
In order to obtain a seepage differential equation for natural fracture system, Eqs. 13, 14, 16 and 18 are substituted into 9.
Equation 21 is a seepage differential equation for the natural fracture system considering the SSP in the low-permeability natural fracture reservoir.
Initial and boundary conditions
The reservoir has the uniform pressure. Therefore, initial condition is as follows.
Considering the outer boundary as the elastic boundary, outer boundary condition can be obtained as follows.
For convenience, the following dimensionless variables are defined. Dimensionless pressure
The seepage differential equation, Eq. 35 for fracture system and inner boundary condition, Eq. 37 have strong nonlinearity. Therefore, to linearize these equations, Pedrosa ’s transformation and perturbation transformation are applied
Pedrosa ’s transformation is as follows (Pedrosa, 1986):
Perturbation transformation
The perturbation transformation is as follows (Pedrosa, 1986):
As the value of 
is small, the solution of zero-order perturbation can satisfy the accuracy requirement.
Laplace transformation
The Laplace transformation of Eqs. 36 and 52~57 is taken with respect to TD.
The Laplace transformation is defined as follows:
When 
the values of 0th and 1st order Bessel functions of the first kind converge to infinity. Therefore, these values cannot be calculated by using computer. In order to avoid this, the following asymptotic formula is used.
is obtained by numerically inverting Laplace transformation, Eqs. 76 and 77. In order to numerically invert Laplace transformation, the algorithm by Kim et al. (2021b) is used, where n=25, a=6.5, k=2, σ=0.
Pressure is calculated with Eq. 42 and pressure derivative with the following equation.
Verification of model
Comparison verification
To verify our model, this is compared to the model for the DPR with EOB without consideration of the stress sensitivity. When stress sensitivity is not considered in this model, as there is permeability modulus in denominator of Eq. 42 and it converges into infinity, (here). Other parameters are as follows: RDe=2000,. Figure.2 shows the result of comparison of two models. The solid lines represent our model and circle mark the dual-porosity without consideration of stress sensitivity (Kim et al., 2021a). From Figure.2, it can be seen that two models agree well, which shows that our model is valid.
Figure 2: Result of comparison of our model to the model without consideration of stress sensitivity
Analysis of Sensitivity
Effect of permeability modulus on the PPD curves
Figure.3 shows the effect of the permeability modulus (=0.001, 0.1, 0.2, 0.3) on the PPD curves. Other parameters are as follows: CDe2S=100, ,
l=10-3, w=10-5, RDe=2 000. As shown in Figure.3, the larger value of, the higher PPD curves. As time increase, the effect of the permeability modulus on PPD curves increases. When, horizontal line of value 0.5 does not appear in the middle stage of pressure derivative curve. This means that the SSP affects analysis of the well-testing data.
Figure 3. Effect of gamma on the PPD curves
Effect of elastic coefficient of outer boundary on the PPD curves
According to Li et al. (2019a), reflects the closed boundary, the constant pressure boundary, the infinite boundary and between closed boundary and constant pressure boundary.
Figure.4 shows the effect of elastic coefficient (=0, 0.1, 1, 1000) of outer boundary on the PPD curves. Other parameters are as follows: CDe2S=100, l=10-3, w=10-5, RDe=2 000. As shown in Figure.5, does not affect the early and middle stages of the pressure derivative curves, but affects the late stage of the pressure derivative curves. When
the value of is zero, it reflects the closed boundary. While, when the value of is 1 000, it reflects that is close to the closed boundary. Moreover, when, it reflects the infinite boundary. This means that with the value of, the middle stage of PPD curves are between closed boundary and constant pressure boundary. However, the outer boundary condition between closed boundary and constant pressure boundary has not been considered at the model of reservoir with the ideal outer boundary (infinite boundary, closed boundary and constant pressure boundary). Thus, our model has generality in comparison to the model with ideal outer boundary. Therefore, this means that the accuracy of well-testing analysis can be enhanced. 
Figure.4. Effect of elastic coefficient of outer boundary on the PPD curves
Effect of CDe2S on the PPD curves
CDe2S is a dimensionless quantity, which considers wellbore storage and skin simultaneously, and shows degrees of improvement or damage of well.
Figure.5 shows the effect of CDe2S (CDe2S=1000, 2 000, 5 000, 10 000) on the PPD curves. Other parameters are as follows:, =0.01, , l=10-5, w=10-5, RDe=2 000. CDe2S affects the middle stage of pressure curve. The lager the value of CDe2S increases, the upper the pressure derivative curves are in the middle stage and the earlier the effect of boundary appears on the pressure derivative curves.
Figure 5: Effect of CDe2S on the PPD curves
Effect of skin S on the PPD curves
Figure.6 shows the effect of skin S (S=-3, -1, 0, 1, 3) on the PPD curves. Other parameters are as follows: =0.01, l=10-3, w=10-4, CDe2S=10, RD=5 000. Skin S affects the middle and late stages of the PPD. The
smaller value of S decreases, the upper the PPD curves are, the higher “hump” of the pressure derivative curve gets, and the later it appears.
Figure 6: Effect of skin S on the PPD curves
Effect of dimensionless radius RD on the PPD curves
Figure.7 shows effect of dimensionless radius RD (RD =100, 1 000, 5 000, 10 000) on the PPD curves. Other parameters are as follows: =0.01,
, l=10-3, w=10-5, CDe2S=1 000,. RD affects only the late stage of the PPD curves. The larger the value of RD increases, the later the effect of boundary appear
Figure 7: Effect of dimensionless radius RD on the PPD curves
Effect of inter-porosity flow coefficient on the PPD curves
Figure.8 shows effect of inter-porosity flow coefficient (=10-2, 10-3, 10-4) on the PPD curves. Other parameters are as follows: =0.01, w=10-4, CDe2S=10, RD=5 000. The inter-porosity flow coefficient affects the middle stage of the PPD curves. The larger the value of increases, the upper the PPD curves are, the higher “hump” of the pressure derivative curve gets, the deeper the concave of the pressure derivative curve are and the later the concave appears
Figure 8: Effect of inter-porosity flow coefficient 
on the PPD curves
Effect of storage ratio on the PPD curves
Figure.9 shows effect of storage ratio (=10-1, 10-2, 10-3, 10-4) on the PPD curves. Other parameters are as follows: =0.01, =10-4, CDe2S=10, , RD=5 000. affects both the early and middle period of flow. The smaller
the upper the pressure curves are, the higher “hump” of the pressure derivative curve gets, the deeper the concave are and the earlier the concave appears
Figure 9: Effect of storage ratio on the PPD curves
Both the SSP and the condition of elasticity of the outer boundary greatly affect the well-testing analysis, but the previous papers have considered these effects individually and have not considered simultaneously, which may result in the considerable errors in the well-testing analysis. The well-testing model proposed in our paper considers the effect of the SSP and the condition of elasticity of the outer boundary simultaneously. Thus, this model may improve the accuracy of the well-testing analysis for the DPR.
There are also several models including the triple-porosity model. However, this model is limited to the dual-porosity model for vertical well. We are going to study the triple-porosity dual-permeability model for horizontal and inclined well considering EOB.
Based on the consideration of stress sensitivity and elastic outer boundary conditions, we first formulated the seepage differential equation in the dual-porosity reservoir considering wellbore storage and skin factor, and defined the dimensionless variable to facilitate the derivation of the solution.
This mathematical model is strong nonlinear, the Pedrosa’ s transformation is applied.
However, after the transformation of Pedrosa is applied, the mathematical model is still nonlinear, So the perturbation transformation is performed to eliminate the nonlinearity of the model, and then the Laplace transformation is performed to obtain the analytical solution of the well test model in the dual-porosity reservoir considering elastic outer boundary and stress sensitivity in the Laplace space, and the Laplace numerical inversion (Kim etc., 2021b) is applied to obtain the dimensionless pressure and pressure derivative model curve in the wellbore.
To verify the accuracy of the model, we verified that the model is valid compared with the dual-porosity reservoir model (Kim etc., 2021a) with the elastic boundary without considering stress sensitivity, we performed a parametric sensitivity analysis on the pressure and pressure derivative curve and showed that the elastic outer boundary and stress sensitivity affect the well- test data analysis.
The model presented in this paper can improve the accuracy of well-test data analysis compared to the well test model in the previous dual-porosity reservoir.
B-Volume factor, dimensionless
C- wellbore storage coefficient, m3/Pa
CD-Dimensionless wellbore storage, dimensionless
Cm, Cf-Compressibility of matrix system and natural fracture system, respectively, Pa-1
Cmt-Total compressibility of matrix system and oil, Pa-1
Cft-Total compressibility of natural fracture system and oil, Pa-1
Co-Oil compressibility, Pa-1
km, kf-Permeability of matrix system and natural fracture system, respectively, m2
kfi-Permeability of natural fracture under initial condition, m2
h- Reservoir thickness, m
p-Reservoir pressure, Pa
R- outer boundary radius, m
RD- dimensionless outer boundary radius, dimensionless
pm, pf-Pressure in matrix system and natural fracture system, respectively, Pa
pi-Initial reservoir pressure, Pa
pw-Wellbore pressure, Pa
q-Well rate, m3/s
q*-Mass flow velocity between matrix system and fracture system in the reservoir of unit volume, kg/(m3s)
r-Radius, m rw-Well radius, m
S- skin, dimensionless
t-Time, s
tD-Dimensionless time, dimensionless
z-Laplace variable, dimensionless
-Shape factor between matrix and fracture, m-2
-Oil density at pressure pm in the matrix system and at pressure pf in natural fracture system, respectively, kg/m3
-Oil density under initial condition, kg/m3
- Permeability modulus, Pa-1
-Dimensionless permeability modulus, dimensionless
-Dimensionless inter-porosity flow coefficient, dimensionless
- Storage ratio, dimensionless
-Viscosity of oil, Pa∙s.
-Porosity of matrix system and natural fracture system, respectively, fraction
-Porosity of matrix system and natural fracture system under initial condition, respectively, fraction
-Flow velocity of fluid in matrix system and natural fracture system, respectively, m/s
Superscripts
- Laplace transformation
Subscripts
D-Dimensionless
m-Matrix
f-Fracture
w-Well
We gratefully acknowledge all professors of Petroleum Institute, Faculty of Geoscience and Technology, Kim Chaek University of Technology for her valuable suggestions and discussions.
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