The most widely used oil and gas well test models, such as homogeneous model, dual porosity model and triple porosity model, are all continuous media models developed based on Darcy’s law. Bruce et al.^[1] established the mathematical model of single media, which solved the characterization of the seepage mechanism of continuous porous media. Warren and Root^[2] studied the seepage in fracture-porosity media reservoirs and established the Warren-Root model. Camacho et al.^[3] extended the Warren- Root model and established a triple porous media seepage model suitable for fracture-vug-matrix carbonate reservoirs. These traditional seepage models based on Darcy's seepage theory and continuity hypothesis play an important role in the development of conventional reservoirs.

However, the application of Darcy seepage law is limited in a certain range. The seepage law will become nonlinear once beyond the range^[4]. Typical formulas used to describe the nonlinear flow law include: low-speed non-Darcy formula considering the threshold pressure gradient^[5], Forchheimer high-speed non-Darcy formula considering the square term of velocity^[6], power-law non-Newtonian formula considering the rheological properties of the fluid^[7], exponential formula considering the change of permeability with pressure^[8], Darcy-Weisbach formula considering the frictional head loss of uniform flow^[9], and Navier-Stokes equation considering free flow in cavity^[10] etc. For linear flow laws, in addition to Darcy's law, commonly used description methods also include Poiseuille's law^[11], which describes the pressure loss of fluid along the thin pipe.

The above linear and nonlinear flow laws can be used individually or in combination in the reservoir, depending on the actual situation. It is easy to use when the laws are applied individually. For example, a conventional homogeneous sandstone reservoir can be directly characterized by the Darcy flow model^[1]. The power law model can be used to characterize the homogeneous heavy oil reservoir^[12]. However, the situation of combined application is relatively complex. For example, three distribution zones of polymer, water and oil would be formed around a polymer injection well in the reservoir developed by polymer flooding after water flooding. The flow law of the polymer zone is characterized by the non-Newtonian fluid model, while the flow laws of the water and oil zones are characterized by the Darcy seepage model^[13], and the flow between different zones is seepage-seepage composite flow. For another example, in the large pore path formed in the process of water flooding in the unconsolidated sandstone reservoir and the large pore path formed in the steam flooding stage of thermal recovery of the heavy oil reservoir, composite flow of tube flow and seepage exists. Naturally fracture-cavern reservoirs have complex storage space forms, large scale span and strong heterogeneity^[14,15]. In the production process of horizontal wells, pipe flow in wellbore and seepage in formation restrict mutually, influence mutually and form boundary conditions mutually, and the flow between reservoir and wellbore is coupling flow of tube flow and seepage^[16,17]. The flow of fluid in this kind of reservoir is extremely complex, and is a complex coupled flow of seepage and free flow with both porous media seepage and free flow in large spaces^[18]. The coupling tube flow and seepage as well as coupling seepage and free flow are basically solved by numerical simulation methods. Although there are relatively mature numerical algorithms at present, the solution process would meet the problem with gridding difficulty and huge computation if this kind of coupled flow model was solved directly without any simplification.

For the coupling of tube flow and seepage: Collins et al.^[19] established a simplified model of horizontal well tube flow based on equivalent permeability, and Wu et al.^[20] make the further research. In the field of hydraulic and geology engineering, Chen et al.^[21] established a coupled model of percolation-tube flow based on equivalent permeability coefficient. Chen^[22] proposed a groundwater flow model for karst tube-fissure-pore media based on the percolation-tube flow coupling model. Zhao et al.^[23] established a nonlinear percolation-tube flow coupling model of confined karst cavern water inrush based on the converted permeability coefficient. The key idea of such methods is to define the equivalent permeabilities or equivalent permeability coefficients of percolation, laminar tube flow and turbulent tube flow based on the Darcy-Weisbach formula^[9], so as to realize the unification of these three flow patterns. The advantage of these methods is that they can describe the fluid flow law in the seepage area, laminar flow area and turbulent flow area in the reservoir by a unified motion equation, which is conducive to the construction and solution of the model equation. But they have been seldom used in the coupling flow simulation of underground fluid tube flow (free flow)-seepage, and especially in the field of well test analysis, there is no literature report.

For the coupled flow of seepage-free flow, Wan et al.^[24] used areas (blocks) with high permeability and high porosity to describe large-scale fractures or karst caves. Due to the fluid flow in fractures or caves still conforms to Darcy flow, a semi-analytical flow simulation model for fracture-vuggy reservoirs is established. The advantage of this method is that it can solve the model quickly and is suitable for large-scale reservoir calculation, but the disadvantage is that the calculation accuracy of caves reduces greatly. Duan et al.^[25] considered that karst caves could be regarded as isopotential bodies, and established fracture and cavity well test analysis models under this assumption, in which fluid flow in fractures was pseudo-steady interporosity flow. Wu et al.^[26] divided the fracture-vuggy reservoirs into rock block system (including bedrock, micro-fracture and micro-karst caves), fractures and karst caves, with fractures and karst caves embedded in the rock block system and interconnected into a network. They characterized the flow of fluid in the rock block system by the triple porous medium model, the fluid flow in fracture by Darcy seepage model, and the fluid flow in the karst caves as a pseudo-steady state flow by the isopotential body model. The advantages of this method of treating the karst cave as an isopotential body are convenient for model construction and solution, fast calculation speed, and the capability to calculate volume of cracks and caves. The shortcoming is that it is impossible to determine the geometric size of the fracturs and caves, and has poor adaptability to long strip or strip flow systems. Considering that karst caves were often accompanied by different degrees of filling, Popov et al.^[27] considered that karst caves are often accompanied by different degrees of filling, so that Stokes-Brinkman equation^[28] has been proposed to describe the flow in fractures and caves. In the process of model solving, Stokes-Brinkman equation can be simplified to Stokes equation or Darcy seepage equation by selecting appropriate parameters. At the same time, because of the unified governing equation, the coupling calculation at the interface of free flow zone and seepage zone is avoided, making the solution simpler. Considering that Stokes-Brinkman equation was applicable to both steady and unsteady flows, Jie et al.^[29] worked out a simulation analysis method of tube flow-seepage coupling based on the generalized mass conservation equation and Stokes-Brinkman equation. Huang et al.^[30] established a mathematical model for macro flow of discrete fracture-cavity network based on Stokes-Brinkman equation.

In this paper, "generalized mobility"^[31] is introduced to realize the unification of tube flow and seepage in form, and the unification of commonly used linear and nonlinear flow laws in form. The same form of equations of motion can be used to construct a unified governing equation for different regions or different scales in oil and gas reservoirs, and the complex coupling flow of linear and nonlinear as well as nonlinear and nonlinear can be changed into a composite flow. Under the definition of generalized mobility, there is no difference in the form between laminar tube flow and Darcy seepage. On this basis, a model system based on the generalized mobility has been proposed, and the basic governing equation of the generalized tube flow and seepage coupling reservoir has been established. Two sample models of well test analysis are constructed, and the Laplace transform method is used to analyze and solve the models. The typical characteristic plots corresponding to the changes of different model parameters are given. The correctness and rationality of the method in this paper are verified by field examples.

Fig. 1 shows a typical physical model coupling tube flow and seepage in a complex reservoir. The whole reservoir space Ω contains four subspaces: wellbore space Ω₀, porous media space Ω₁, pore channel space Ω₂ and karst cave space Ω₃. The flow of fluid in the four subspaces conforms to different flow laws.

In order to solve the coupling of tube flow and seepage as shown in Fig. 1, a completely unified motion equation is defined in the reservoir Ω:

where

$\lambda =\text{diag}\left[ {{\lambda }_{X}}(x,t),{{\lambda }_{Y}}(x,t),{{\lambda }_{Z}}(x,t) \right]$ $x\in \Omega $ $t\in \left( 0,T \right]$

In Eq. (1), λ is called generalized mobility, which is a diagonal matrix. λ_X(x,t), λ_Y(x,t), λ_Z(x,t) are generalized mobility components in X, Y and Z directions respectively. The generalized mobility component is a continuous function of x inside the subspace Ω₀, Ω₁, Ω₂, Ω₃. But it doesn't have to be a continuous function of x on the subspace boundaries$\partial {{\Omega }_{0}}$,$\partial {{\Omega }_{2}}$and$\partial {{\Omega }_{3}}$.

Based on Eq. (1), a unified governing equation of fluid flow coupling tube flow and seepage can be constructed:

The initial conditions are:

The boundary conditions are:

Eqs. (2)-(4) are usually solved by numerical method: First, the entire reservoir space Ω is divided into a series of non-overlapping tetrahedrons. Then, the finite volume method is used to discretize Eqs. (2)-(4) to form discrete equations. Finally, the pressure distribution at any moment in the solved region can be obtained by solving the discrete equations.

Eq. (1) describes both tube flow and seepage laws. For one-dimensional flow, the generalized mobility can be defined as different forms listed in Table 1 according to different fluid flow conditions. In order to facilitate understanding and application, some one-dimensional generalized mobility formulas corresponding to commonly used single-phase flow laws are listed in Table 1. These one-dimensional generalized mobilities can be extended directly to two or three dimensions, as well as to multiphase flow. Different forms of formulas in Table 1 can be used in different parts of the physical model coupling tube flow and seepage showed in Fig. 1 to form a composite model.

The complex model coupling tube flow and seepage constructed by generalized mobility has exactly the same form in different regions, so it is convenient to discretize the model by using the numerical calculation methods well known in the industry, which can reduce the complexity of the coupling issue and make the solution of the issue simple and unified.

For the issue of transient flow, in order to obtain the analytical solution, two typical example models for the transient flow of the slightly compressible single-phase fluid in the reservoir which conforms to the linear flow law have been established based on the above model coupling tube flow and seepage.

In a pipe-shaped composite reservoir (Fig. 2) with one-dimensional transient flow coupling laminar tube flow and seepage flow, the wellbore is produced or injected at a fixed rate, and the pressure in the reservoir is the uniform initial formation pressure before opening the well. Meanwhile, it is assumed that the reservoir and fluid meet the following requirements: (1) The reservoir is composed of two pipe-shaped reservoirs with coupled tube flow and seepage. (2) The left end of reservoir 1 is connected with the wellbore, and the right end is connected with reservoir 2, and the fluid flow in both reservoirs conforms to the linear flow law. (3) The rock is slightly compressible. (4) The fluid in reservoirs 1 and 2 is slightly compressible single-phase flow (the compressible single-phase flow and multiphase flow issues are transformed into slightly compressible single-phase flow issues by using pseudo-pressure functions^[32,33]). (5) The wellbore storage effect and skin effect are considered. (6) The flow in pipe is considered as one-dimensional flow, regardless of the specific well type; and the outflow (inflow) flow rate at the left end of reservoir 1 is equal to the output (injection) of the wellbore.

According to the above assumptions, the following dimensionless model is established.

where

where

$\delta =\frac{{{\lambda }_{1}}/\left( {{\phi }_{1}}{{C}_{\text{t1}}} \right)}{{{\lambda }_{2}}/\left( {{\phi }_{2}}{{C}_{\text{t2}}} \right)} {{p}_{\text{2D}}}=\frac{{{\lambda }_{1}}{{A}_{1}}}{qL}\left[ M\left( {{p}_{\text{i}}} \right)-M\left( {{p}_{2}} \right) \right]$

where

The Laplace transform of Eqs. (5)-(12) is made based on t_D, and the dimensionless bottomhole pressure solution in Laplace space is obtained:

where

${{p}_{\text{wD}}}=\frac{{{\lambda }_{1}}{{A}_{1}}}{qL}\left[ M\left( {{p}_{\text{i}}} \right)-M\left( {{p}_{\text{w}}} \right) \right]$

For Eq. (13), the dimensionless bottomhole pressure solution considering wellbore storage effect and skin effect in Laplace space can be obtained by using Duhamel principle^[4]:

where

${{C}_{\text{D}}}=\frac{C}{{{\phi }_{1}}{{C}_{\text{t1}}}{{A}_{1}}L}$

Euler numerical inversion algorithm^[34] was used to simulate Eq. (14) inversely to obtain the bottomhole pressure in real space, and the pressure drawdown curves (Figs. 3 and 4) and pressure build-up curves (Fig. 5) with the change of some parameters were drawn. The pressure build-up curves were drawn according to the following calculation formula^[35]:

where

${{p}_{\text{sD}}}=\frac{{{\lambda }_{1}}{{A}_{1}}}{qL}\left[ M\left( {{p}_{\text{i}}} \right)-M\left( {{p}_{\text{s}}} \right) \right]$

Fig. 3 shows the response characteristics (pressure drawdown) of different combined parameter mδ^1/2. Generally, it can be divided into four flow regimes. I is the well-bore storage regime and the early transition regime, in which the curve of pressure and its derivative basically appear as a straight line with a slope of 1, and then a hump occurs (positive skin effect). II and IV are linear flow regime of reservoirs 1 and 2, when the curves of the pressure and its derivative in the later regime are basically parallel lines with a slope of 0.5. III is the transition section of the linear flow between reservoirs 1 and 2. With the increase of mδ^1/2, the positions of pressure and its derivative curves move down. When mδ^1/2=1, there is no transition regime response between the two linear flow sections.

Fig. 4 is the response characteristics (pressure drawdown) of length variation of reservoir 1. Obviously, the longer the reservoir 1, the longer the response time of its linear flow is.

Fig. 5 shows the effect of production time on pressure build up curve. The pressure buildup derivative in early wellbore storage regime coincides with that of pressure drawdown derivative, and there is no obvious difference between them. With the prolongation of production time, the pressure buildup derivative curve approaches infinitely to the pressure drawdown derivative curve. The shorter the production time, the earlier the pressure buildup derivative curve starts to deviate from the pressure drawdown derivative curve, and the greater the deviation from the pressure drawdown curve will be. At given production time, as long as the test time is long enough, the derivative curve of pressure buildup falls in the late regime.

In a cylindrical reservoir with three-dimensional transient flow coupled by laminar tube flow and seepage (Fig. 6), the production well is produced or injected at a fixed flow rate, and before opening the well the reservoir pressure is equal to the uniform initial formation pressure. Meanwhile, it is assumed that the reservoir and fluid meet the following requirements: (1) The reservoir is a cylindrical reservoir with coupled single tube flow and seepage; the generalized mobility of the reservoir is different in the vertical direction and radial direction; the reservoir is of equal thickness horizontally, and the fluid flow in the reservoir conforms to linear flow law. (2) The rock is slightly compressible. (3) The fluid in the reservoir is a slightly compressible single-phase flow (the compressible single-phase flow and multiphase flow issues are transformed into a slightly compressible single-phase flow issues by using a pseudo-pressure function^[32,33]). (4) The wellbore is located in the central axis of the cylindrical reservoir, and only opened partially. (5) The skin effect and wellbore storage effect are considered.

According to the model assumptions, the following dimensionless model can be established in cylindrical coordinates:

where

where

where

where

${{r}_{\text{eD}}}=\frac{{{r}_{\text{e}}}}{L}$

Laplace transform of Eqs. (16)-(22) is made based on t_D, and the dimensionless bottomhole pressure solution in Laplace space is obtained by using the method of separating variables:

where

${{a}_{i}}=\left\{ \begin{array}{*{35}{l}} 1 & \quad i=0 \\ 8{{\left( \frac{\sin \frac{i\pi {{\varepsilon }_{\text{D}}}}{2{{h}_{\text{D}}}}\cos \frac{i\pi {{z}_{\text{wD}}}}{{{h}_{\text{D}}}}}{\frac{i\pi {{\varepsilon }_{\text{D}}}}{{{h}_{\text{D}}}}} \right)}^{2}} & \quad i=1,2,\cdots,\infty \\ \end{array} \right.$

${{b}_{i}}=\sqrt{u+{{\lambda }_{\text{z}}}/{{\lambda }_{\text{r}}}{{\left( i\pi /{{h}_{\text{D}}} \right)}^{2}}}$

For Eq. (23), the dimensionless bottomhole pressure solution considering wellbore storage effect and skin effect in Laplace space can be obtained by using Duhamel principle^[4]:

where

${{C}_{\text{D}}}=\frac{C}{2\pi \phi {{C}_{\text{t}}}h{{L}^{2}}}$

Euler numerical inversion algorithm^[34] is used to simulate Eq. (24) inversely to obtain the bottomhole pressure in real space, and the pressure drawdown plot (Figs. 7-10) and pressure buildup plot (Figs. 11-12) with some parameters change are drawn. The pressure buildup plot is drawn according to Eq. (15), at this point:

Fig. 7 shows the response characteristics (pressure drawdown) of radius change of the cylindrical reservoir. When the vertical generalized mobility and radial generalized mobility are equal, and the thickness of opened interval is smaller than that of the reservoir, and under fixed reservoir thickness, the pseudo steady flow in the later regime of the pressure derivative curve appears late with the increase of the reservoir radius. In addition, when r_eD is smaller than h_D, spherical flow and linear flow may appear in the middle of the pressure derivative curve (red curve in Fig. 7). With the increase of r_eD, the spherical flow features in the middle of the pressure derivative curve become more and more obvious, while the linear flow features gradually disappear (black curve in Fig. 7). When r_eD increases further, spherical flow and radial flow may both appear on the pressure derivative curve (blue curve in Fig. 7).

Fig. 8 shows the response characteristics (pressure drawdown) of reservoir thickness variations. Under the conditions that the vertical generalized mobility and radial generalized mobility are equal, the opened interval thickness is smaller than the reservoir thickness, the ratio of the two is constant, and under fixed conditions of reservoir radius, the concavity in the middle of the pressure derivative curve becomes shallow with the increase of the reservoir thickness.

Fig. 9 shows the response characteristics (pressure drawdown) of different vertical and radial generalized mobility ratios. Under fixed conditions of reservoir thickness, reservoir radius and opened interval thickness, with the increase of generalized mobility ratio, the concavity in the middle of the pressure derivative curve becomes deeper, and the linear flow duration becomes shorter.

Fig. 10 shows the response characteristics (pressure drawdown) of the opened interval proportion variations. Under fixed conditions of reservoir thickness, reservoir radius and ratio of vertical generalized mobility and radial generalized mobility, with the increase of opened interval thickness, the position of transition regime on pressure derivative curve becomes lower, and early radial flow around opened interval may appear around the transition regime (rarely appear). In a special case of ε_D=h_D, there are no spherical flow and linear flow characteristic responses in the middle of the pressure derivative curve.

Fig. 11 and Fig. 12 show the influence of production time on the pressure buildup curve. Fig. 11 shows the case with larger vertical generalized mobility and Fig. 12 shows the case with smaller vertical generalized mobility. The curves of pressure buildup derivative and pressure drawdown derivative in early wellbore storage coincide, with no obvious difference. As the production time gets longer, the pressure buildup derivative curve approaches to the pressure drawdown derivative curve. But when the production time increases to a certain value, the shape of the pressure buildup derivative curve does not change anymore. The shorter the production time, the earlier the pressure buildup derivative curve starts to deviate from the pressure drawdown derivative curve, and the greater the deviation from the pressure drawdown derivative curve is. When the production time is constant, as long as the test time is long enough, the derivative curve of pressure buildup turns downward falls downward in the late regime.

This example is a well test analysis of shut-in pressure falloff in a water injection well of a low permeability reservoir. Well X1 is a water injection well located in a fractured sandstone reservoir in Ordos Basin. Before the shut-in test, water was injected at an average daily rate of 10.0 m³ continuously for 1401.5 h, and the effective shut-in test time was 455.5 h. The well radius is 0.062 1 m, the reservoir has an effective thickness of 10.4 m, a porosity of 30.0%, a fluid viscosity of 0.6 mPa•s, and a comprehensive compressibility coefficient of 0.001 5 MPa^-1. The production wells around the water injection well X1 show unidirectional water flooding, and the log-log plots of pressure and its derivative have obvious unidirectional or linear flow characteristics (Fig. 13a). Based on comprehensive consideration, it is speculated that there are probably flow channels with coupled tube flow and seepage or dominant flow channels or high permeability channels around the well X1.

The well test interpretation model for pipe-shaped composite reservoir was applied for the fitting analysis, and the fitting result is shown in Fig. 13, which shows a good fitting effect. From the well test interpretation results in Table 2, it can be seen that the flow capacity and cross flow section area of the near-wellbore reservoir 1 are both larger than those of the reservoir 2 far from the well. The generalized mobility values of reservoirs 1 and 2 are higher in this well region, and the mobility in this well block mainly ranges from 1.66×10^-3 μm²/(MPa·s) to 33.33×10^-3 μm²/(MPa·s), indicating that there may be dominant flow channels around the water injection well. The calculated flow channel characteristic parameters are shown in Table 2, in which the extrapolated formation pressure was calculated by using Kuchuk's linear flow formation pressure calculation method^[36]. By comparing and analyzing the interpretation results of conventional models, it is concluded that the interpretation results of this model can better describe the actual situation.

Well X2 is an oil production well located in a volcanic reservoir in the northwest margin of Junggar Basin. It was completed by casing, and put into production after perforation and fracturing. The well is drilled in a fracture-pore volcanic reservoir block with rich high-angle structural fractures which has better vertical flow capacity than radial flow capacity. In the reservoir, the fractures have an average length of 2.26 m, and an average density of 1.87 fracture/m. The volcanic rock body thickness is 22-299.8 m, the effective thickness of the producing interval is 20-113 m, with a porosity of 8.0%-22.30%, and a permeability of (0.02-468.00)× 10^-3 μm², showing strong heterogeneity. Before the shut-in well test, the Well X2 had continuously produced for 2000 h at an average daily liquid production rate of 3.1 m³, and the effective shut-in well test time was 234.35 h. The comprehensive water cut of the well was 69% due to rising bottom water. The wellbore radius of the well is 0.062 m. The opened interval in this well is 5 m. The oil of this well has a volume factor of 1.686 m³/m³ and viscosity of 0.33 mPa·s, the water in the well has a volume factor of 1.023 m³/m³ and viscosity of 0.29 mPa·s. The reservoir in the well has a comprehensive compressibility of 0.002 1 MPa^-1, a porosity of 15.0%, a temperature of 101.47 °C and a depth of 4247.5 m in the middle. First, this well started production 3 years ago in natural flow with 3.0 mm nozzle, and produced 32.54 t of oil per day with no water, and had fast production decline in the initial stage. After that, the well started to produce water due to the rise of bottom water. In the last year, it had a stable liquid production by natural flow. This well has low average daily liquid production, and relatively high water cut, and no gas produced in the whole production process, which indicates the flow in this well is oil-water two-phase flow. Second, the volcanic reservoir has abundant fractures, mainly high angle structural fractures, and vertical flow capacity better than radial flow capacity. In addition, the reservoir in the well is partially perforated, so the log-log plot also shows the characteristics of spherical flow. Finally, by comparing the geological background and dynamic production data, it is concluded that this reservoir probably has coupling of tube flow and seepage.

The cylindrical reservoir well test interpretation model in this paper was used for fitting analysis, and the fitting result is shown in Fig. 14, which shows a good fitting effect. According to the characteristic parameters of the reservoir around the well in Table 3, the calculated generalized mobility value and the further estimated generalized permeability value (the radial generalized permeability and vertical generalized permeability of the reservoir are 0.087 78×10^-3 μm² and 423.0×10^-3 μm² respectively, and the comprehensive viscosity of the oil-water two-phase fluid is approximately 0.3 mPa·s) in general tally with the physical property level of this well area, and the calculated reservoir thickness parameter is in accordance with the above reservoir conditions. In Table 3, the extrapolated formation pressure was calculated by using Kuchuk^[36] radial flow formation pressure calculation method.

Well X3 is an exploration well in a fracture-vuggy reservoir in the northern margin of Tarim Basin. It had leakage during drilling (when drilling to 6162.5 m, and a total of 807 m³ drilling fluid with a density of 1.15 g/cm³ leaked during forced drilling to 6177 m). The well was finished (in open hole completion) on March 17. The well was tested for production from March 18 to April 5 for 343.5 h with an average daily oil production of 80.5 m³.

After that, it was tested for buildup pressure for 134.0 h in effective test time. The well radius is 0.074 9 m. The reservoir has an effective thickness of 72.21 m and a porosity of 21.0%. The fluid in the well has a viscosity of 0.67 mPa·s and fluid volume factor of 1.4 m³/m³. The reservoir has a comprehensive compressibility factor of 0.002 85 MPa^-1, and a temperature of 135.22 °C and depth of 6140.9 m in the middle.

The minimum bottomhole flowing pressure of this well is 66.06 MPa, which is higher than the reservoir saturation pressure (31.53 MPa). Before shut-in well test, it had an average water cut of 0.317%, indicating that the reservoir in this well had mainly single-phase oil flow during the test period. The producing interval of 6104.79- 6177.00 m in the well had drilling fluid leakage during drilling, and the available seismic data of the reservoir show the feature of "bead string", which indicate that the reservoir has abundant fractures and cavities, and stronger radial flow ability and weaker vertical flow ability.

The cylindrical reservoir well test interpretation model presented in this paper was adopted to do the fitting analysis. The fitting results are shown in Fig. 15. The pressure and its derivative show a good fitting results on the log-log plot. The early regime is characterized by variable wellbore storage effect, and the lower concave regime in the later regime represents the flow characteristics of the cylindrical reservoir. The interpretation results in Table 4 are consistent with the above reservoir block situation. In Table 4, the extrapolated formation pressure was calculated by using the radial flow formation pressure calculation method of Kuchuk^[36].

Different fluid motion equations are unified by defining generalized mobility. A general well test analysis model for single phase flow of coupled tube flow and seepage is proposed, which can be used to simulate complex coupled tube flow and seepage.In the sense of generalized mobility, the same form of motion equation can be used to construct a unified governing equation for the entire reservoir in different regions or at different scales. Therefore, it is more convenient to solve the discretized model, reducing the complexity of coupling issue, and making the solution of the model simple and unified.Two types of well test models for coupling tube flow and seepage have been constructed. The pressure drawdown plot of the pipe-shaped composite reservoir model can have two linear flow characteristics. The middle of the pressure derivative curve of the pressure drawdown plot of the cylindrical reservoir model could have spherical flow and linear flow features or spherical flow and radial flow features. The practicability and reliability of the models are verified by application examples. It should be noted that for reservoirs with strong heterogeneity, it is often difficult to obtain accurate values of basic parameters such as reservoir thickness and porosity. In practical application, it is necessary to adjust some basic parameters within a reasonable value range in combination with specific conditions to ensure the rationality of interpretation results.

The two case models presented here are analytical solutions. In order to obtain more abundant reservoir parameters, the future work will focus on the general numerical solution method for the case that the generalized mobility is nonlinear, to achieve a more accurate interpretation and description of the fine characteristics of the reservoir.

A₁, A₂—cross flow areas of reservoirs 1 and 2, m²;

c₁—coefficient of pressure function item, Pa^-1;

c₂—coefficient of pressure gradient function item, s/m;

C—wellbore storage coefficient, m³/Pa;

C_t—comprehensive compressibility factor, Pa^-1;

C_t1, C_t2—comprehensive compressibility factors of reservoirs 1 and 2, Pa^-1;

d—hydraulic diameter, m;

e—roughness, m;

f(·)—function of Re and e/d;

G—starting pressure gradient, Pa/m;

h—height of cylindrical reservoir, m;

h₀—initial pressure distribution function, Pa;

h₁—boundary condition function, dimensionless;

I₀(·), I₁(·)—0 and 1 order of class 1 modified Bessel function;

K—permeability, m²;

K₀(·), K₁(·)—0 and 1 order class 2 modified Bessel function;

L—reference length (selected randomly), m;

m—product of cross flow area and generalized mobility ratio;

M(·)—pseudo-pressure function, Pa;

n—flow characteristic index;

n—normal vector of reservoir boundary, m;

p—pressure, Pa;

p_i—initial pressure, Pa;

p_s—buildup of bottomhole pressure during shut-in test, Pa;

p_w—bottomhole pressure during production, Pa;

$\bar{p}_{w}$—bottomhole pressure of Laplace space, Pa;

p₁, p₂—pressure distribution of reservoirs 1 and 2 along horizontal direction, Pa;

q—flow rate, m³/s;

r—radial distance, m;

r_e, r_w—radii of cylindrical reservoir and wellbore, m;

Re—Reynolds number;

S—skin factor at the connection of wellbore and reservoir, the cross flow area of reservoir 1 is taken as reference in the pipe-shaped complex reservoir, the area of the opened section of wellbore is taken as reference in cylindrical reservoir;

t—time, s;

t_a(·)—pseudo-time function, s;

t_pD—dimensionless production time, s;

T—maximum value of time, s;

u—Laplace variable corresponding to dimensionless time;

v—velocity, m/s;

v—velocity vector, m/s;

x—distance, m;

x—distance vector, m;

x₁—length of reservoir 1, m;

X, Y, Z—rectangular coordinate system, m;

z—vertical distance, m;

z_w—vertical position of the opened interval center, m;

β—high speed non-Darcy coefficient, m^-1;

γ—permeability modulus, Pa^-1;

δ—pressure conductivity coefficient ratio;

ε—thickness of the opened interval, m;

λ—generalized mobility, m²/(Pa•s);

λ—generalized mobility matrix, m²/(Pa•s);

λ_r,λ_z—radial and vertical generalized mobility of the reservoir, m²/(Pa•s);

λ_X(x,t), λ_Y(x,t), λ_Z(x,t)—generalized mobility components in X, Y, Z directions respectively, m²/(Pa•s);

λ₁, λ₂—generalized mobility of reservoirs 1 and 2, m²/(Pa•s);

μ—fluid viscosity, Pa•s;

μ_e—effective viscosity, (Pa•s)•(m/s)^1-n;

ρ—density, kg/m³;

ϕ—porosity, %;

ϕ₁, ϕ₂—porosities of reservoirs 1 and 2, %.

Subscript: D—dimensionless.