Pressure Transient Analysis for Vertical Well Drilled in Filled-Cave in Fractured Reservoirs

Abstract

For capturing dynamic information about a filled-cave in the fractured reservoir, a novel Pressure Transient Analysis (PTA) analytical model for a well located at the filled-cave is established. In this new model, we consider the stress-sensitivity of the filled-cave and the inter-porosity flow of fracture. First, Perturbation transformation was used to obtain the pressure distribution in the filled-cave zone. Then, the Warren–Root model was applied to establish the pressure solution in the fractured reservoir. Next, the pressure and its derivative are obtained by the Laplace transformation and Steftest inversion. Lastly, the Bottomhole Pressure (BHP) and Bottomhole Pressure Derivative (BHPD) combined curve reveals the flow regimes of this novel model. The results show the composite model can be used to characterize the fractured reservoir with the filled-caves, and its flow follows the composite flow regimes. The spherical flow has an obvious slope of 0.5 on the BHPD curve, which can identify the size of the filled-caves. The boundary flow can be used to identify stress-sensitivity. Affected by the stress-sensitivity of the filled-cave, the BHPD’s slope of the boundary flow will be greater than 1. This research work provides technical support for capturing cave and fracture parameters in the fractured reservoir.

1. Introduction

In recent years, China’s marine carbonate oil reserves have been predominantly observed in the Tarim Basin, where the reservoirs are mainly characterized by fracture-cave type. The general geological framework and reserve distribution of such reservoirs have been documented in works like [1,2], while their specific characteristics and associated challenges are detailed in [3,4]. In these reservoirs, oil is mainly stored in natural fractures, caves, and the pores between cave fillings. The cave-filling space itself provides substantial storage capacity, and the complex nature of fluid flow in such fractured-vuggy carbonate reservoirs necessitates advanced modeling approaches, as demonstrated by multiphysics and multiscale methods [5]. Within this context, pressure transient analysis for wells penetrating filled-caves has been explored as a key to understanding these systems [6]. These filled-caves are not only critical storage spaces but also common targets that are easily drilled and completed [7,8].

Due to the differences in cementation and diagenesis history between the cave fillings and the surrounding rock, the filled-cave media often exhibit significant stress-sensitivity. This phenomenon, where permeability and porosity decrease with increasing effective stress, has been widely recognized in geomechanics [9] and specifically studied in carbonate reservoirs [10]. During production, a decrease in bottomhole pressure can readily alter the permeability of the near-wellbore formation, consequently directly influencing the productivity [11]. Therefore, precisely determining reservoir permeability plays a crucial role in subsequent productivity assessments. However, while the influence of stress-sensitivity on production is acknowledged, its explicit integration into well-test interpretation models for quantitative analysis remains a significant challenge. Furthermore, the discontinuous and highly heterogeneous nature of such reservoirs poses significant challenges for conventional core sampling and laboratory methods in accurately acquiring key physical parameters particularly permeability. The main reason is that due to the existence of natural fractures in the rock, the core sampling process is very easy to break, and it is difficult to obtain complete fracture rock samples under laboratory conditions, and rock parameters and permeability parameters cannot be obtained through testing.

Consequently, compared with conventional mean sandstone reservoirs, the use of pressure transient tests to obtain reservoir permeability [12,13,14] is very common in fracture-cavity reservoirs. However, the prevailing PTA (Pressure Transient Analysis) models suffer from key limitations when applied to wells drilled into filled-caves. Firstly, most existing models, including the foundational work by Gao et al. [6], do not consider the deformation characteristics of the medium filling the cave, making them unsuitable for analyzing wells drilled into the filling cave reservoir where stress-sensitive effects are pronounced. Secondly, conventional well-test models for fractured reservoirs predominantly assume radial flow geometry, as seen in studies like [15,16,17], or linearly supported radial flow [18], neglecting the distinct spherical flow characteristics [19,20] induced by the three-dimensional spatial geometry of a filled-cave. Consequently, such models are inadequate for characterizing the actual flow behavior in these complex systems, leading to potential misinterpretations, as discussed in [21,22]. This collective oversight of both the constitutive behavior of the filled-cave and its appropriate flow geometry defines a clear gap in the current modeling capabilities.

To solve these problems, we establish a novel composite model that incorporates the stress-sensitivity of filled-caves and the dual-porosity characteristics of the fractured reservoir, utilizing a spherical flow geometry. Perturbation theory and Laplace transform are used to solve the pressure solutions of each region for the inner and outer regions, respectively, and the global pressure response are obtained by coupling the pressure and flux at the composite interface. Finally, the flow regimes are identified using a combination curve of the bottomhole pressure (BHP) and its derivative, and the influence of various parameters on the pressure response and flow is analyzed. This work provides a technical foundation for characterizing cave and fracture parameters in the fractured-vuggy reservoir.

2. Methodology

2.1. Physical Model

The model idealizes a single, isolated karst cave body (a ‘bead’ in a bead-string structure [1,2,3,4]) surrounded by low-permeability matrix rock. The spherical representation is a simplification to leverage symmetry in deriving the analytical solution. As shown in Figure 1, one producing well is drilled into a filled-cave in the fractured reservoir. Compared with the surrounding fractured rock, the filled-cave formed relatively late, and its degree of compaction is much lower than that of the surrounding rock. Therefore, the filled-cave zone has high porosity, high permeability and high compressibility. The filled-cave is called the inner zone, its porosity is ϕ₁, permeability is k₁, the total compressibility is c_t1, and the distance to the inner boundary is r_f. The fractured rock is called the outer zone, its porosity is φ₂, permeability is k₂, the total compressibility is c_t2, and the distance to the outer boundary is r_e. Other assumptions are as follows.

(1)

The producing well drills into the filled-cave, and the filled-cave zone is homogeneous. At the bottom of the wellbore is located in the ball center of the filled-cave, the fluid of the filled-cave flows into the wellbore through the perforated section in the form of spherical concentric flow.

(2)

Due to the main pressure-drop occurring around the wellbore and the compaction degree of the filled-cave being lower than the peripheral rock, the deformation caused by the stress-sensitivity of the filled-cave is considered in the flow process.

(3)

The peripheral rock develops natural fractures, and it is equivalent to the dual-porous medium. In the outer zone, the natural fracture is the flow channel, and the rock matrix is the storage volume without any infiltration capacity. In the pressure drop process, the internal fluid in the matrix only flows towards the rock fracture in the form of pseudo-steady-state.

(4)

The fluids present in both the inner and outer zones are single-phase and exist in a compressed liquid state. Within the rock fracture system, the internal fluid moves toward the filled-cave, exhibiting a spherical flow pattern that converges inward. There is no additional pressure loss induced by flow in the interface of the inner and outer zones.

(5)

Before the well is opened, the initial pressure is uniform throughout the reservoir. The test well is then produced at a constant rate. The impact of the wellbore storage effect and the formation skin effect are considered together.

(6)

The reservoir possesses a limited distance to its boundary, which may be either a closed (no-flow) type or a constant-pressure type. Throughout the entire flow process, variations in temperature and the effects of gravity are neglected.

2.2. Mathematical Model

The flow equation mathematically expresses the relationship between flow velocity and the pressure difference. For linear flow, it can be described by Darcy’s law

$$\mathit{v}=-{\displaystyle \frac{\mathit{k}}{\mathit{\mu }}}\nabla \mathit{p}$$

(1)

where v is the flow velocity, m/s; k is permeability, m²; μ is viscosity, Pa·s; p is pressure, Pa.

In the production process with reservoir pressure decrease, reservoir fluid will be slightly compressed and formation rocks will slightly be deformed, their state equations are

$$\left\{\begin{array}{c}\mathit{\rho }={\mathit{\rho }}_{\mathit{i}}{\mathit{e}}^{{\mathit{c}}_{\mathit{L}}(\mathit{p}-{\mathit{p}}_{\mathit{i}})}\\ \mathit{\varphi }={\mathit{\varphi }}_{\mathit{i}}{\mathit{e}}^{{\mathit{c}}_{\mathit{f}}(\mathit{p}-{\mathit{p}}_{\mathit{i}})}\\ \mathit{k}={\mathit{k}}_{\mathit{i}}{\mathit{e}}^{{\mathit{\alpha }}_{\mathit{k}}(\mathit{p}-{\mathit{p}}_{\mathit{i}})}\end{array}\right.$$

(2)

where ρ_i density in the initial pressure, kg/m³; φ_i is porosity in the initial pressure, factor; k_i is permeability in the initial pressure, m²; c_L is the compressibility of liquid, 1/Pa; c_f is the compressibility of formation, 1/Pa; α_k is the deformation coefficient of permeability, 1/Pa.

Based on the principle of material balance, the continuity equation with the source item is

$$\nabla \cdot (\mathit{\rho }\mathit{v})+\mathit{\rho }\mathit{q}+{\displaystyle \frac{\partial (\mathit{\rho }\mathit{\varphi })}{\partial \mathit{t}}}=\mathbf{0}$$

(3)

where v is flow velocity, m/s; q is flowrate of source item, m³/s; t is the time, s.

Bring the flow Equation (1) and status Equation (2) into the continuous Equation (3) to obtain

$${\nabla }^{\mathbf{2}}\mathit{p}+\mathit{q}={\displaystyle \frac{\mathbf{1}}{\mathit{\eta }}}{\displaystyle \frac{\partial \mathit{p}}{\partial \mathit{t}}}$$

(4)

where the diffusivity conduction coefficient η with m²/s is

$$\mathit{\eta }={\displaystyle \frac{\mathit{k}}{\mathit{\mu }\mathit{\varphi }{\mathit{c}}_{\mathit{t}}}}$$

(5)

In the spherical coordinate system under symmetrical conditions, the differential diffusivity equation of inward spherical flow is expressed as

$${\displaystyle \frac{{\partial }^{\mathbf{2}}\mathit{p}}{\partial {\mathit{r}}^{\mathbf{2}}}}+{\displaystyle \frac{\mathbf{2}}{\mathit{r}}}{\displaystyle \frac{\partial \mathit{p}}{\partial \mathit{r}}}+\mathit{q}={\displaystyle \frac{\mathbf{1}}{\mathit{\eta }}}{\displaystyle \frac{\partial \mathit{p}}{\partial \mathit{t}}}$$

(6)

where r is the distance to the center of the ball, m.

The relationship between fluid flow and pressure is described by the Warren–Root (W-R) model, which relies on the assumption of pseudo-steady-state flow within the dual-porosity medium [23].

$$\mathit{q}=\mathit{\alpha }{\displaystyle \frac{{\mathit{k}}_{\mathit{m}}}{\mathit{\mu }}}({\mathit{p}}_{\mathit{m}}-{\mathit{p}}_{\mathit{f}})$$

(7)

where q is the flowrate m³/s; α is the shape factor of rock matrix, factor; k_m is the permeability of rock matrix, m²; p_m is the pore fluid pressure at rock matrix, Pa; and p_f is the pore fluid pressure at fracture, Pa.

The inter-porosity flow Equation (7) is brought into the diffusivity Equation (6) to obtain a differential diffusivity equation of composite inward spherical flow in stress-sensitive reservoir. To obtain a general form of diffusivity equation, the composite inward spherical flow with dimensions needs to be processed. Firstly, the dimensionless parameters are defined as shown in

Table 1. Definition of dimensionless parameters.

Parameters	Definition
Dimensionless distance	${\mathit{r}}_{\mathit{D}}={\displaystyle \frac{\mathit{r}}{{\mathit{r}}_{\mathit{w}}}}$
Dimensionless distance to inner boundary	${\mathit{r}}_{\mathit{f}\mathit{D}}={\displaystyle \frac{{\mathit{r}}_{\mathit{f}}}{{\mathit{r}}_{\mathit{w}}}}$
Dimensionless distance to outer boundary	${\mathit{r}}_{\mathit{e}\mathit{D}}={\displaystyle \frac{{\mathit{r}}_{\mathit{e}}}{{\mathit{r}}_{\mathit{w}}}}$
Dimensionless wellbore storage coefficient	${\mathit{C}}_{\mathit{D}}={\displaystyle \frac{\mathbf{0.159}\mathit{C}}{{\mathit{\varphi }}_{\mathbf{1}}{\mathit{c}}_{\mathit{t}\mathbf{1}}{\mathit{r}}_{\mathit{w}}^{\mathbf{3}}}}$
Dimensionless time	${\mathit{t}}_{\mathit{D}}={\displaystyle \frac{\mathbf{3.6}{\mathit{k}}_{\mathbf{1}}\mathit{t}}{{\mathit{\varphi }}_{\mathbf{1}}{\mathit{c}}_{\mathit{t}\mathbf{1}}\mathit{\mu }{\mathit{r}}_{\mathit{w}}^{\mathbf{2}}}}$
Dimensionless pressure of inner zone	${\mathit{p}}_{\mathbf{1}\mathit{D}}={\displaystyle \frac{{\mathit{k}}_{\mathbf{1}}{\mathit{r}}_{\mathit{w}}\left({\mathit{p}}_{\mathit{i}}-{\mathit{p}}_{\mathbf{1}}\right)}{\mathbf{1.842}\times {\mathbf{10}}^{-\mathbf{3}}\mathit{q}\mathit{\mu }\mathit{B}}}$
Dimensionless pressure of outer zone	${\mathit{p}}_{\mathbf{2}\mathit{D}}={\displaystyle \frac{{\mathit{k}}_{\mathbf{1}}{\mathit{r}}_{\mathit{w}}\left({\mathit{p}}_{\mathit{i}}-{\mathit{p}}_{\mathbf{2}}\right)}{\mathbf{1.842}\times {\mathbf{10}}^{-\mathbf{3}}\mathit{q}\mathit{\mu }\mathit{B}}}$
Mobility ratio of inner zone to outer zone	${\mathit{M}}_{\mathbf{12}}={\displaystyle \frac{{\mathit{k}}_{\mathbf{1}}/\mathit{\mu }}{{\mathit{k}}_{\mathbf{2}}/\mathit{\mu }}}$
Storability ratio of inner zone to outer zone	${\mathit{\omega }}_{\mathbf{12}}={\displaystyle \frac{{\mathit{\varphi }}_{\mathbf{1}}{\mathit{c}}_{\mathit{t}\mathbf{1}}}{{\mathit{\varphi }}_{\mathbf{2}}{\mathit{c}}_{\mathit{t}\mathbf{2}}}}$
Stress-sensitivity coefficient	$\mathit{\gamma }={\displaystyle \frac{\mathbf{1.842}\times {\mathbf{10}}^{-\mathbf{3}}\mathit{q}\mathit{\mu }\mathit{B}}{{\mathit{k}}_{\mathbf{1}}{\mathit{r}}_{\mathit{w}}}}{\mathit{\alpha }}_{\mathit{k}}$
Storability ratio of fracture	${\mathit{\omega }}_{\mathit{f}}={\displaystyle \frac{{\mathit{\varphi }}_{\mathbf{2}\mathit{f}}{\mathit{c}}_{\mathit{t}\mathbf{2}\mathit{f}}}{{\mathit{\varphi }}_{\mathbf{2}}{\mathit{c}}_{\mathit{t}\mathbf{2}}}}$
Inter-porosity flow coefficient	${\mathit{\lambda }}_{\mathit{m}}=\mathit{\alpha }{\mathit{r}}_{\mathit{w}}^{\mathbf{2}}{\displaystyle \frac{{\mathit{k}}_{\mathbf{2}\mathit{m}}}{{\mathit{k}}_{\mathbf{2}\mathit{f}}}}$

, where r_w is wellbore radius, m; r_f is composite distance, m; r_e is boundary distance, m; p_i is initial pressure, MPa; C is wellbore storage coefficient, m³/MPa; S is skin factor, factor; B is fluid volume factor, factor; α is shape factor, m⁻²; α_k is deformation coefficient of permeability, Pa⁻¹; ϕ₁ is porosity of inner zone, factor; k₁ is permeability of inner zone, m²; c_t1 is the total compressibility of inner zone, MPa⁻¹; ϕ₂ (ϕ₂ = ϕ_2m + ϕ_2f) is porosity of outer zone, factor; ϕ_2m is matrix porosity of outer zone, factor; φ_2f is fracture porosity of outer zone, factor; k₂ is permeability of outer zone, m²; k_2m is the permeability of rock matrix of the outer zone, m²; k_2f is the permeability of fracture of the outer zone, m²; c_t2 (ϕ₂c_t2 = ϕ_2mc_t2m + ϕ_2fc_t2f) is the total compressibility of inner zone, MPa⁻¹; c_t2f is the fracture compressibility of inner zone, MPa⁻¹; c_t2m is the matrix compressibility of inner zone, MPa⁻¹.

Then, the mathematical problems corresponding to the physical model are obtained by the initial condition and the inner and outer boundary conditions.

(1)

Dimensionless diffusivity equation of inward spherical flow in the stress-sensitive filled-cave (inner zone) is

$${\displaystyle \frac{{\partial }^{\mathbf{2}}{\mathit{p}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}^{\mathbf{2}}}}+{\displaystyle \frac{\mathbf{2}}{{\mathit{r}}_{\mathit{D}}}}{\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}-\mathit{\gamma }{({\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}})}^{\mathbf{2}}={\mathit{e}}^{\mathit{\gamma }{\mathit{p}}_{\mathbf{1}\mathit{D}}}{\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{t}}_{\mathit{D}}}}$$

(8)

(2)

See Appendix A, dimensionless diffusivity equation of inward spherical flow in the dual-pore media rock (outer zone) is

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial }^{\mathbf{2}}{\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}^{\mathbf{2}}}}+{\displaystyle \frac{\mathbf{2}}{{\mathit{r}}_{\mathit{D}}}}{\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}}}+{\mathit{\lambda }}_{\mathit{m}}({\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{m}}-{\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}})={\displaystyle \frac{{\mathit{\omega }}_{\mathbf{12}}}{{\mathit{M}}_{\mathbf{12}}}}{\mathit{\omega }}_{\mathit{f}}{\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{t}}_{\mathit{D}}}}\\ -{\mathit{\lambda }}_{\mathit{m}}({\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{m}}-{\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}})={\displaystyle \frac{{\mathit{\omega }}_{\mathbf{12}}}{{\mathit{M}}_{\mathbf{12}}}}\left(\mathbf{1}-{\mathit{\omega }}_{\mathit{f}}\right){\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{m}}}{\partial {\mathit{t}}_{\mathit{D}}}}\end{array}\right.$$

(9)

(3)

Corresponding initial condition is

$${\mathit{p}}_{\mathbf{1}\mathit{D}}({\mathit{t}}_{\mathit{D}}=\mathbf{0})={\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}}({\mathit{t}}_{\mathit{D}}=\mathbf{0})=\mathbf{0}$$

(10)

(4)

Corresponding flux condition in the wellbore, see Appendix B, is

$${\mathit{C}}_{\mathit{D}}{\displaystyle \frac{\mathit{d}{\mathit{p}}_{\mathit{w}\mathit{D}}}{\mathit{d}{\mathit{t}}_{\mathit{D}}}}-{\left(\mathit{S}{\mathit{e}}^{-\mathit{\gamma }{\mathit{p}}_{\mathbf{1}\mathit{D}}}{\mathit{r}}_{\mathit{D}}{\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{w}\mathit{D}}}=\mathbf{1}$$

(11)

(5)

Corresponding pressure condition in the wellbore, see Appendix B, is

$${\mathit{p}}_{\mathit{w}\mathit{D}}={\left({\mathit{p}}_{\mathbf{1}\mathit{D}}+\mathit{S}{\mathit{e}}^{-\mathit{\gamma }{\mathit{p}}_{\mathbf{1}\mathit{D}}}{\mathit{r}}_{\mathit{D}}{\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{w}\mathit{D}}}$$

(12)

(6)

Pressure condition in the interface of filled-cave and fractured rock is

$${\left({\mathit{p}}_{\mathbf{1}\mathit{D}}\right)}_{{\mathit{r}}_{\mathit{f}\mathit{D}}}={\left({\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}}\right)}_{{\mathit{r}}_{\mathit{f}\mathit{D}}}$$

(13)

(7)

Flux condition in the interface of filled-cave and fractured rock is

$${\left({\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{f}\mathit{D}}}={\displaystyle \frac{\mathbf{1}}{{\mathit{M}}_{\mathbf{12}}}}{\left({\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{f}\mathit{D}}}$$

(14)

(8)

If the boundary type is constant-pressure, the boundary condition can be written as

$${\left({\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}}\right)}_{{\mathit{r}}_{\mathit{e}\mathit{D}}}=\mathbf{0}$$

(15)

(9)

If the boundary type is no-flow, the boundary condition can be written as

$${\left({\displaystyle \frac{\partial {\mathit{p}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{e}\mathit{D}}}=\mathbf{0}$$

(16)

2.3. Solution Approach

2.3.1. Inner Diffusivity Equation

The inner diffusivity equation contains a stress-sensitivity coefficient with an index form, which has strong non-linearity according to Pedrosa’s work [24], using Perturbation transformation to solve the non-linear flow equation. Perturbation transformation of the dimensionless is

$${\mathit{p}}_{\mathit{D}}=-{\displaystyle \frac{\mathbf{1}}{\mathit{\gamma }}}\mathit{L}\mathit{n}(\mathbf{1}-\mathit{\gamma }{\mathit{\zeta }}_{\mathit{D}})$$

(17)

By Perturbation transformation, the outer diffusivity equation can be written as follows:

$${\displaystyle \frac{{\partial }^{\mathbf{2}}{\mathit{\zeta }}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}^{\mathbf{2}}}}+{\displaystyle \frac{\mathbf{2}}{{\mathit{r}}_{\mathit{D}}}}{\displaystyle \frac{\partial {\mathit{\zeta }}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}={\displaystyle \frac{\mathbf{1}}{\mathbf{1}-\mathit{\gamma }{\mathit{\zeta }}_{\mathbf{1}\mathit{D}}}}{\displaystyle \frac{\partial {\mathit{\zeta }}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{t}}_{\mathit{D}}}}$$

(18)

Based on Perturbation transformation, the ζ_D be also expanded as follows ζ_D0 + γζ_D1 + γ²ζ_D2 + ⋯. Hence, the term on the right-hand side of Equation (18) can also be expanded as follows:

$${\displaystyle \frac{\mathbf{1}}{\mathbf{1}-\mathit{\gamma }{\mathit{\zeta }}_{\mathit{D}}}}=\mathbf{1}+\mathit{\gamma }{\mathit{\zeta }}_{\mathit{D}}+{\mathit{\gamma }}^{\mathbf{2}}{\mathit{\zeta }}_{\mathit{D}}^{\mathbf{2}}+\dots$$

(19)

Pedrosa [24] and Kikani [25] gave 0-, 1-, and 2-order approximation solutions of flow problems in the infinite-acting tress-sensitivity reservoir, respectively. Then the 0-order approximation of the inner flow equation is

$${\displaystyle \frac{{\partial }^{\mathbf{2}}{\mathit{\zeta }}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}^{\mathbf{2}}}}+{\displaystyle \frac{\mathbf{2}}{{\mathit{r}}_{\mathit{D}}}}{\displaystyle \frac{\partial {\mathit{\zeta }}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}={\displaystyle \frac{\partial {\mathit{\zeta }}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{t}}_{\mathit{D}}}}$$

(20)

By the Laplace transferring, the diffusivity equation of inner region is

$${\displaystyle \frac{{\partial }^{\mathbf{2}}{\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}^{\mathbf{2}}}}+{\displaystyle \frac{\mathbf{2}}{{\mathit{r}}_{\mathit{D}}}}{\displaystyle \frac{\partial {\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}-\mathit{u}{\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}=\mathbf{0}$$

(21)

The diffusivity Equation (21) is a class of semi-level transformer Bessel Equations. The form of solution is

$${\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}={\displaystyle \frac{\mathbf{1}}{{\mathit{r}}_{\mathit{D}}}}\left[{\mathit{A}}_{\mathbf{1}}\mathbf{sinh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}})+{\mathit{B}}_{\mathbf{1}}\mathbf{cosh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}})\right]$$

(22)

The pressure gradient of distance to the center of ball can be written as

$${\displaystyle \frac{\partial {\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}={\displaystyle \frac{-\mathbf{1}}{{{\mathit{r}}^{\mathbf{2}}}_{\mathit{D}}}}\left[{\mathit{A}}_{\mathbf{1}}\mathbf{sinh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}})+{\mathit{B}}_{\mathbf{1}}\mathbf{cosh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}})\right]+{\displaystyle \frac{\sqrt{\mathit{u}}}{{\mathit{r}}_{\mathit{D}}}}\left[{\mathit{A}}_{\mathbf{1}}\mathbf{cosh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}})+{\mathit{B}}_{\mathbf{1}}\mathbf{sinh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}})\right]$$

(23)

2.3.2. Outer Diffusivity Equation

For the outer region, the corresponding diffusivity equation in the Laplace domain has the following mathematical representation:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial }^{\mathbf{2}}{\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}^{\mathbf{2}}}}+{\displaystyle \frac{\mathbf{2}}{{\mathit{r}}_{\mathit{D}}}}{\displaystyle \frac{\partial {\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}}}-\mathit{u}\mathit{f}{\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}=\mathbf{0}\\ \mathit{f}={\displaystyle \frac{{\mathit{\omega }}_{\mathbf{12}}}{{\mathit{M}}_{\mathbf{12}}}}\left[{\displaystyle \frac{{\mathit{\lambda }}_{\mathit{m}}\left(\mathbf{1}-{\mathit{\omega }}_{\mathit{f}}\right)}{{\mathit{\lambda }}_{\mathit{m}}+\mathit{u}\left(\mathbf{1}-{\mathit{\omega }}_{\mathit{f}}\right)}}+{\mathit{\omega }}_{\mathit{f}}\right]\end{array}\right.$$

(24)

Equation (24) belongs to the Bessel Equation; its solution can be expressed as follows:

$${\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}={\displaystyle \frac{\mathbf{1}}{{\mathit{r}}_{\mathit{D}}}}\left[{\mathit{A}}_{\mathbf{2}}\mathbf{sinh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}\mathit{f}})+{\mathit{B}}_{\mathbf{2}}\mathbf{cosh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}\mathit{f}})\right]$$

(25)

The pressure guidance of spherical distance to the center of the ball can be written as

$${\displaystyle \frac{\partial {\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}}}={\displaystyle \frac{-\mathbf{1}}{{{\mathit{r}}^{\mathbf{2}}}_{\mathit{D}}}}\left[{\mathit{A}}_{\mathbf{2}}\mathbf{sinh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}\mathit{f}})+{\mathit{B}}_{\mathbf{2}}\mathbf{cosh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}\mathit{f}})\right]+{\displaystyle \frac{\sqrt{\mathit{u}\mathit{f}}}{{\mathit{r}}_{\mathit{D}}}}\left[{\mathit{A}}_{\mathbf{2}}\mathbf{cosh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}\mathit{f}})+{\mathit{B}}_{\mathbf{2}}\mathbf{sinh}({\mathit{r}}_{\mathit{D}}\sqrt{\mathit{u}\mathit{f}})\right]$$

(26)

2.3.3. Boundary Conditions

The pressure and flux in the wellbore are

$$\left\{\begin{array}{l}{\left({\mathit{C}}_{\mathit{D}}\mathit{u}{\overline{\mathit{\zeta }}}_{\mathit{w}\mathit{D}}-\mathit{S}{\mathit{r}}_{\mathit{D}}{\displaystyle \frac{\partial {\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{w}\mathit{D}}}={\displaystyle \frac{\mathbf{1}}{\mathit{u}}}\\ {\overline{\mathit{\zeta }}}_{\mathit{w}\mathit{D}}={\left({\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}-\mathit{S}{\mathit{r}}_{\mathit{D}}{\displaystyle \frac{\partial {\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{w}\mathit{D}}}\end{array}\right.$$

(27)

The pressure and flux in the inner boundary are

$$\left\{\begin{array}{c}{\left({\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}-{\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}\right)}_{{\mathit{r}}_{\mathit{f}\mathit{D}}}=\mathbf{0}\\ {\left({\displaystyle \frac{\partial {\overline{\mathit{\zeta }}}_{\mathbf{1}\mathit{D}}}{\partial {\mathit{r}}_{\mathit{D}}}}-{\displaystyle \frac{\mathbf{1}}{{\mathit{M}}_{\mathbf{12}}}}{\displaystyle \frac{\partial {\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{f}\mathit{D}}}=\mathbf{0}\end{array}\right.$$

(28)

The pressure or flux in the outer boundary is

$$\left\{\begin{array}{l}{\left({\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}\right)}_{{\mathit{r}}_{\mathit{e}\mathit{D}}}=\mathbf{0}\\ {\left({\displaystyle \frac{\partial {\overline{\mathit{p}}}_{\mathbf{2}\mathit{D}\mathit{f}}}{\partial {\mathit{r}}_{\mathit{D}}}}\right)}_{{\mathit{r}}_{\mathit{e}\mathit{D}}}=\mathbf{0}\end{array}\right.$$

(29)

2.3.4. Solution of BHP

According to the inner boundary condition Equation (27), composite interface condition Equation (28), and outer boundary conditions Equation (29), the coefficients A and B of Equations (22) and (25) are obtained

$$\left[\begin{array}{cccc}{\mathit{a}}_{\mathbf{11}}& {\mathit{a}}_{\mathbf{12}}& & \\ {\mathit{a}}_{\mathbf{21}}& {\mathit{a}}_{\mathbf{22}}& {\mathit{a}}_{\mathbf{23}}& {\mathit{a}}_{\mathbf{24}}\\ {\mathit{a}}_{\mathbf{31}}& {\mathit{a}}_{\mathbf{32}}& {\mathit{a}}_{\mathbf{34}}& {\mathit{a}}_{\mathbf{34}}\\ & & {\mathit{a}}_{\mathbf{43}}& {\mathit{a}}_{\mathbf{44}}\end{array}\right]\cdot \left[\begin{array}{c}{\mathit{A}}_{\mathbf{1}}\\ {\mathit{B}}_{\mathbf{1}}\\ {\mathit{A}}_{\mathbf{2}}\\ {\mathit{B}}_{\mathbf{2}}\end{array}\right]=\left[\begin{array}{c}-{\displaystyle \frac{\mathbf{1}}{\mathit{u}}}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\end{array}\right]$$

(30)

where

$$\left\{\begin{array}{c}{\mathit{a}}_{\mathbf{11}}=-(\mathbf{1}+{\mathit{C}}_{\mathit{D}}\mathit{S}\mathit{u}+{\mathit{C}}_{\mathit{D}}\mathit{u})\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\sqrt{\mathit{u}}\right)+\sqrt{\mathit{u}}\left(\mathbf{1}+{\mathit{C}}_{\mathit{D}}\mathit{S}\mathit{u}\right)\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left(\sqrt{\mathit{u}}\right)\hfill \\ {\mathit{a}}_{\mathbf{12}}=-(\mathbf{1}+{\mathit{C}}_{\mathit{D}}\mathit{S}\mathit{u}+{\mathit{C}}_{\mathit{D}}\mathit{u})\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left(\sqrt{\mathit{u}}\right)+\sqrt{\mathit{u}}\left(\mathbf{1}+{\mathit{C}}_{\mathit{D}}\mathit{S}\mathit{u}\right)\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\sqrt{\mathit{u}}\right)\hfill \\ {\mathit{a}}_{\mathbf{21}}={\displaystyle \frac{\mathbf{1}}{{\mathit{r}}_{\mathit{f}\mathit{D}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}}\right)\hfill \\ {\mathit{a}}_{\mathbf{22}}={\displaystyle \frac{\mathbf{1}}{{\mathit{r}}_{\mathit{f}\mathit{D}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}}\right)\hfill \\ {\mathit{a}}_{\mathbf{23}}={\displaystyle \frac{-\mathbf{1}}{{\mathit{r}}_{\mathit{f}\mathit{D}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)\hfill \\ {\mathit{a}}_{\mathbf{24}}={\displaystyle \frac{\mathbf{1}}{{\mathit{r}}_{\mathit{f}\mathit{D}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)\hfill \\ {\mathit{a}}_{\mathbf{31}}={\displaystyle \frac{-\mathbf{1}}{{{\mathit{r}}_{\mathit{f}\mathit{D}}}^{\mathbf{2}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}}\right)+{\displaystyle \frac{\sqrt{\mathit{u}}}{{{\mathit{r}}_{\mathit{f}\mathit{D}}}^{\mathbf{2}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}}\right)\hfill \\ {\mathit{a}}_{\mathbf{32}}={\displaystyle \frac{-\mathbf{1}}{{{\mathit{r}}_{\mathit{f}\mathit{D}}}^{\mathbf{2}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}}\right)+{\displaystyle \frac{\sqrt{\mathit{u}}}{{{\mathit{r}}_{\mathit{f}\mathit{D}}}^{\mathbf{2}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}}\right)\hfill \\ {\mathit{a}}_{\mathbf{33}}={\displaystyle \frac{-\mathbf{1}}{{\mathit{M}}_{\mathbf{12}}}}\left[{\displaystyle \frac{-\mathbf{1}}{{{\mathit{r}}_{\mathit{f}\mathit{D}}}^{\mathbf{2}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)+{\displaystyle \frac{\sqrt{\mathit{u}}}{{{\mathit{r}}_{\mathit{f}\mathit{D}}}^{\mathbf{2}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)\right]\hfill \\ {\mathit{a}}_{\mathbf{34}}={\displaystyle \frac{-\mathbf{1}}{{\mathit{M}}_{\mathbf{12}}}}\left[{\displaystyle \frac{-\mathbf{1}}{{{\mathit{r}}_{\mathit{f}\mathit{D}}}^{\mathbf{2}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)+{\displaystyle \frac{\sqrt{\mathit{u}}}{{{\mathit{r}}_{\mathit{f}\mathit{D}}}^{\mathbf{2}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{f}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)\right]\hfill \\ {\mathit{a}}_{\mathbf{43}}={\displaystyle \frac{\mathbf{1}}{{\mathit{r}}_{\mathit{e}\mathit{D}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{e}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right),\hfill \\ {\mathit{a}}_{\mathbf{44}}={\displaystyle \frac{\mathbf{1}}{{\mathit{r}}_{\mathit{e}\mathit{D}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{e}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)\hfill \\ \mathit{o}\mathit{r}\left\{\begin{array}{l}{\mathit{a}}_{\mathbf{43}}={\displaystyle \frac{-\mathbf{1}}{{{\mathit{r}}_{\mathit{e}\mathit{D}}}^{\mathbf{2}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{e}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)+{\displaystyle \frac{\sqrt{\mathit{u}}}{{{\mathit{r}}_{\mathit{e}\mathit{D}}}^{\mathbf{2}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{e}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)\\ {\mathit{a}}_{\mathbf{44}}={\displaystyle \frac{-\mathbf{1}}{{{\mathit{r}}_{\mathit{e}\mathit{D}}}^{\mathbf{2}}}}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left({\mathit{r}}_{\mathit{e}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)+{\displaystyle \frac{\sqrt{\mathit{u}}}{{{\mathit{r}}_{\mathit{e}\mathit{D}}}^{\mathbf{2}}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left({\mathit{r}}_{\mathit{e}\mathit{D}}\sqrt{\mathit{u}\mathit{f}}\right)\end{array}\right.\hfill \end{array}\right.$$

(31)

A₁ and B₁ of Equation (30) can be calculated through the Crammer law

$$\left\{\begin{array}{l}{\mathit{A}}_{\mathbf{1}}={\displaystyle \frac{{\mathit{a}}_{\mathbf{22}}\left({\mathit{a}}_{\mathbf{34}}{\mathit{a}}_{\mathbf{44}}-{\mathit{a}}_{\mathbf{43}}{\mathit{a}}_{\mathbf{34}}\right)-{\mathit{a}}_{\mathbf{32}}\left({\mathit{a}}_{\mathbf{23}}{\mathit{a}}_{\mathbf{44}}-{\mathit{a}}_{\mathbf{43}}{\mathit{a}}_{\mathbf{24}}\right)}{-\mathit{u}\Delta }}\\ {\mathit{B}}_{\mathbf{1}}={\displaystyle \frac{{\mathit{a}}_{\mathbf{21}}\left({\mathit{a}}_{\mathbf{34}}{\mathit{a}}_{\mathbf{44}}-{\mathit{a}}_{\mathbf{43}}{\mathit{a}}_{\mathbf{34}}\right)-{\mathit{a}}_{\mathbf{31}}\left({\mathit{a}}_{\mathbf{23}}{\mathit{a}}_{\mathbf{44}}-{\mathit{a}}_{\mathbf{43}}{\mathit{a}}_{\mathbf{24}}\right)}{\mathit{u}\Delta }}\end{array}\right.$$

(32)

where Δ is a value calculated by Det(a_ij).

Bring A₁ and B₁ into the solution of Equation (22) to obtain the dimensionless inner pressure solution after Perturbation transformation. In the Laplace domain, the dimensionless bottomhole pressure (BHP) solution after Perturbation transformation is

$${\overline{\mathit{\zeta }}}_{\mathit{w}\mathit{D}}={\mathit{A}}_{\mathbf{1}}\mathbf{sinh}(\sqrt{\mathit{u}})+{\mathit{B}}_{\mathbf{1}}\mathbf{cosh}(\sqrt{\mathit{u}})$$

(33)

To obtain the dimensionless bottomhole pressure (BHP) in the time domain, the numerical inversion was performed using the method proposed by Stehfest [26].

$${\mathit{p}}_{\mathit{w}\mathit{D}}=-{\displaystyle \frac{\mathbf{1}}{\mathit{\gamma }}}\mathit{L}\mathit{n}\left[\mathbf{1}-\mathit{\gamma }{\mathit{L}}^{-\mathbf{1}}({\overline{\mathit{\zeta }}}_{\mathit{w}\mathit{D}})\right]$$

(34)

3. Results

3.1. Model Verification

Prior to analyzing the novel flow behaviors, the correctness of the proposed analytical model was verified by degenerating it to established classical cases. The comprehensive model presented in this study can be reduced to simpler, well-validated models under specific parameter constraints. Firstly, by setting the stress-sensitivity coefficient γ = 0, the inner zone model simplifies to a homogeneous, spherical flow model without permeability alteration. Secondly, by setting the inter-porosity flow coefficient to a very large value (λ_m = 10⁸), the outer dual-porosity zone approximates an equivalent homogeneous system with pseudo-steady-state flow being instantaneous. Figure 2 demonstrates the excellent agreement between the degenerate solution of our model (γ = 0, λ_m = 10⁸, and M₁₂ = 0) and the classic spherical flow solution for a homogeneous reservoir presented by Chatas [19]. The match across all flow regimes, including the wellbore storage unit-slope line, the transition, and the characteristic −0.5 slope of the spherical flow derivative, is nearly perfect. This confirms the accuracy of our solution methodology for the spherical flow geometry.

3.2. BHP Response and Flow Regimes

In this part, the BHP response curve and the sensitivity of each parameter are analyzed. The dimensionless wellbore storage coefficient C_D is 10, the skin factor S is 0.3, and the other parameter values are shown in

Table 2. Parameter values in the type model.

Dimensionless Parameters	Type Curve Part	Sensitivity Analysis Part
Composite distance, rfD	10.0	10~100
Stress-sensitivity factor, γ	0.1	0~0.5
Mobility ratio, M12	2.0	1~100
Storability ratio, ω12	1.0	1~10
Inter-porosity flow coefficient, λm	0.5 × 10−5	(0.5~10) × 10−5
Storability of fracture, ωf	0.01	0.05~0.9
Distance to outer boundary, reD	2 × 103	(1~100) × 103

.

According to the characteristics of the BHP and its derivative (BHPD) curves, the bottomhole pressure response reflects eight flow regimes (Figure 3). The type curve presented in Figure 2 reveals a complex sequence of flow regimes, which is a direct consequence of the integrated physical processes considered in our novel model. In the following sections, we will not only describe the sensitivity of the pressure response to key parameters but also discuss the underlying physical mechanisms and contrast our findings with what would be predicted by conventional models that omit stress-sensitivity or use simplified flow geometries. This comparative analysis highlights the practical significance and novelty of our work.

3.3. Flow Regimes

As shown in Figure 3, ① is a wellbore storage regime, it is influenced by the wellbore storage coefficient. ② is the transient flow regime, which is controlled by the skin factor. ③ is a spherical flow regime for the inner zone, it is a steady-state flow with −0.5 slope in the BHPD curve. ④ Transitional flow caused by the mobility and storability difference between the inner zone and outer zone. ⑤ is a spherical flow regime for the outer zone, it is a steady-state flow with −0.5 slope in the BHPD curve. ⑥ is the flow regime induced by pseudo-steady-state inter-porosity flow with an obvious V-shape in the BHPD curve. ⑦ In the flow regime dominated by boundary effects, the BHPD curve exhibits a unit slope, a −0.5 slope, or a sharp drop in response to closed, infinite-acting, and constant-pressure boundaries, respectively. ⑧ is the flow regime controlled by the stress-sensitivity of the filled-cave with a slope greater than 1 in the BHPD curve. All flow regimes and curve features can be listed and found in

Table 3. Flow regimes and curve feature of the type model.

Flow Stage	Flow Regime	HPD Curve Feature p’wD
Early	① wellbore storage	Slope = 1
	② transitional flow
	③ inner spherical flow	Slope = −1/2
Middle	④ transitional flow
	⑤ outer spherical flow	Slope = −1/2
	⑥ inter-porosity flow	V-shape
Late	⑦ boundary control flow	Closed boundary, slope = 1 Infinite-acting boundary, slope = −1/2,
	⑧ stress-sensitivity control flow	Slope > 1

.

4. Discussion

Taking the closed boundary as an example, the sensitivity of each parameter to the typical curve of BHP response is analyzed according to each flow stage.

4.1. Composite Distance

Denoted as the inner zone, the filled-cave has a composite radius equal to its own radius, defined by the distance to the inner-outer zone interface. It follows that the composite radius increases with the filled-cave radius. Moreover, Figure 4 demonstrates that the radius of the filled-cave governs the timing at which the transitional flow regime (④) emerges. The larger the scope of the filled-cave, the longer the duration of the inner spherical flow (③), and the later the transitional flow regime (④) appears.

4.2. Stress-Sensitivity Factor

Since the inner zone was formed later, the degree of compaction is lower than that of the surrounding rocks. Therefore, the rock in the filled-cave zone has a stress-sensitive phenomenon in the process of pressure reduction, i.e., the porosity and permeability of the inner zone decrease with the depletion of pressure. Here, the stress-sensitivity coefficient (γ) is introduced to represent the coupling between medium deformation and permeability. The magnitude of γ directly reflects the extent of this influence, with a larger value indicating a more significant effect. Figure 5 shows the effect of the stress-sensitivity factor on BHP. Stress-sensitivity affects the entire flow stage. The stress-sensitivity factor increases and the early-middle stage pressure curve moves up as a whole, which indicates that the pressure loss increases under the same production rate, indicating that the reservoir permeability becomes worse. In the later stage, especially in the stage where stress-sensitivity control flow regime ⑧, the pressure upturn occurs earlier, and the degree of the upwarp is intensified, indicating that the influence of the formation deformation on the flow is gradually increasing.

The upward deviation in the late-time pressure derivative, characterized by a slope greater than 1, is a unique signature of stress-sensitivity that cannot be captured by any of the existing models for filled-caves, including Ref. [6]. This has critical implications for well test interpretation: misinterpreting this late-time rise as a closed boundary effect in a homogeneous reservoir would lead to a grossly underestimated drainage volume. Our model provides the first analytical framework to correctly identify this phenomenon as formation damage due to permeability impairment near the wellbore, allowing for a more realistic assessment of well performance and reserves.

4.3. Mobility Ratio

The marked permeability contrast between the inner zone and the outer reservoir results in a mobility ratio M₁₂ > 1, a key parameter affecting the bottomhole pressure (BHP). Figure 6 shows the mobility ratio greatly affects the entire flow stage. The transitional flow regime becomes more distinct as the mobility in the inner zone increases, particularly when compared to cases where the outer zone mobility remains constant (④). When the mobility of the inner and outer regions differs greatly (M₁₂ > 10), the outer spherical flow (⑤) is easily covered up by the transitional flow (④) or even disappears. On the other hand, the large mobility ratio in the inner zone indicates that the physical properties of the filled-caves are excellent (that is, the degree of compaction is extremely low), and the reservoir in the inner area is greatly affected by stress-sensitivity. Therefore, for a filled-cave with a large difference in mobility, such as M₁₂ = 100, the outer zone is equivalent to a closed boundary.

A higher M₁₂ signifies that the filled-cave is a high-permeability “sweet spot” relative to the surrounding fractured rock. The discussion of these findings is two-fold. Firstly, the pronounced transitional flow regime at high M₁₂ values is a direct result of this sharp contrast. It represents the time when the pressure transient encounters the lower-mobility outer zone, acting as a flow barrier. This is a critical period for accurately estimating the cave’s size from the derivative curve. Secondly, and more importantly from an engineering perspective, a high M₁₂ has a direct and adverse coupling with stress-sensitivity. The excellent flow capacity of the inner zone implies that a larger pressure drop is concentrated across a smaller volume near the wellbore, thereby amplifying the stress-sensitive permeability damage. Consequently, for a well with excellent initial productivity (high M₁₂), the productivity decline due to stress-sensitivity can be more severe. This insight, provided by our model, is vital for production forecasting and designing appropriate drawdown management strategies to mitigate formation damage in high-deliverability filled-cave wells.

4.4. Storability Ratio

Since the inner zone possesses greater porosity and compressibility than the outer zone, the storability ratio ω12 leads to a value exceeding 1. The magnitude of ω12 is a critical factor affecting the behavior of the bottomhole pressure (BHP), Figure 7, reveals the larger the storability ratio coefficient, the later the boundary control flow occurs. Compared with the influence of the mobility ratio M₁₂, while the storability ratio has a minimal impact on BHP in the middle flow stage, discernible only through analysis of the BHPD curves, its influence grows progressively more evident during the late flow period. With the increase in the storability ratio, the time when the BHPD curve rises is delayed.

4.5. Diffusivity Ratio

The mobility ratio and the storability ratio reflect the difference between the inner and outer zones from the perspective of fluid flow and reservoir properties, respectively. In the process of pressure transmission, diffusivity η = k/(μφct) can represent the overall difference in flow process. In this part, the diffusivity ratio η₁₂ is the ratio of the diffusivity η₁ to the diffusivity η₂, that is, η₁₂ = M₁₂/ω₁₂. Figure 8 shows the influence of mobility ratio (storage ratio) on BHP under the same pressure propagation velocity (η₁₂ = 1). It can be seen that under the parameter form of diffusivity ratio, the influence of mobility ratio M₁₂ on BHP has obvious quantitative characteristics.

In the outer spherical flow stage (⑤), firstly, according to the pressure derivative value p’_wD(M₁₂) under different M₁₂, and then the characteristic value of the pressure derivative under any M₁₂ is calculated D = p’_wD(M₁₂)/p’_wD(M₁₂ = 1). The scatter data for D vs. M₁₂, as shown in Figure 9, have a good linear relationship, i.e., D ≈ M₁₂. This relationship can be used to calculate the mobility ratio M₁₂ from the BHPD curve, intuitively.

4.6. Inter-Porosity Flow Coefficient

The inter-porosity flow coefficient, which quantifies the difficulty of matrix fluid transfer into fractures, governs the timing of the inter-porosity flow regime (⑥), as illustrated in Figure 10. A larger value of this coefficient consequently leads to an earlier appearance of this flow regime.

4.7. Storability Ratio of Fracture

While the fracture storability ratio represents the fracture storage fraction in the outer zone (a larger value meaning more fracture fluid), Figure 11 reveals that a smaller ratio makes the inter-porosity flow regime (⑥) more evident and the V-shape on the BHPD curve deeper.

4.8. Boundary Distance

As illustrated in Figure 12, the boundary distance determines the onset timing of the boundary-dominated flow regime (⑦). An increase in this distance not only delays the occurrence of regime (⑦) but also diminishes the prominence of the stress-sensitivity controlled flow (⑧). This is because the larger the boundary distance, the more sufficient the reservoir fluid, the smaller the pressure drop under the same production conditions, and the smaller the deformation of the formation caused by the stress-sensitivity of the reservoir.

5. Conclusions

For capturing the cave and fracture dynamic information of a fracture-cave reservoir, a novel PTA analytical model for a well located at the filled-cave, considering the stress-sensitivity of the cave and inter-porosity flow of fracture, is established. According to the BHP response and parameter sensitivity, the following conclusions should be highlighted.

(1)

The composite model can be used to characterize the fractured reservoir with the filled-caves, and its flow follows the composite flow regimes The stress-sensitivity effect of the filled-cave area and the inter-porosity flow phenomenon within the fractured reservoir should be considered. In this model, the corresponding analytical solution can be obtained by using Perturbation transformation.

(2)

There are eight flow regimes for a well drilled into filled-caves in fractured reservoirs. Among them, the spherical flow and boundary flow can be used to identify the size of the filled-caves and the stress-sensitivity, respectively. The spherical flow has an obvious slope of 0.5 on the BHPD curve.

(3)

Affected by the stress-sensitivity of the filled-cave, the BHPD’s slope of the boundary flow will be greater than 1. The greater the stress-sensitivity coefficient, the greater the deviation of the boundary flow. At the same time, other flow stages (inter-porosity flow) will also deviate.