Transient Pressure Performance Analysis of Hydraulically Fractured Horizontal Well in Tight Oil Reservoir

: Utilizing the discrete fracture model (DFM) , a transient ﬂow model is established for fractured horizontal wells in tight oil reservoirs, accounting for threshold pressure gradient (TPG), stress sensitivity eﬀect , hydraulic fracture parameters, and fracture distribution pattern . This model is solved using the ﬁnite-volume method (FVM), and an important sensitivity analysis is conducted. The ﬁndings reveal that the models incorporat ed by the threshold pressure gradient result in an upward trend in the pressure - derivative curve. As the threshold pressure gradient increases, this upward trend becomes more pronounced, rendering the distinction between ﬂow regimes more challenging. The stress sensitivity eﬀect predominantly impacts the pressure-derivative curve during the later ﬂow period. Additionally, as the fracture half-length increases, the pressure p erfor-mance of both fracture radial ﬂow and formation radial ﬂow becomes more diﬃcult. Fracture conductivity has a signiﬁcant inﬂuence during the early ﬂow period, facilitating the identiﬁcation of ﬂow regimes with the trend of increasing fracture conductivity.


Introduction
As the key technology of tight oil reservoir development, multiple-hydraulic-fracture stimulation [1,2] can enlarge the accessible reservoir volume and improve well productivity significantly.To gain a deeper understanding of the impact of the fracturing and formation characteristics, it is imperative to research the transient flow model of fractured horizontal wells.Over the past few decades, numerous studies have been conducted in this field.Guo and Evans [3][4][5][6] utilized the source, Green's function, and the superposition principle to derive solutions for fractured horizontal wells.Raghavan et al. [7] further explored the distinct response patterns of such wells by developing a mathematical model.Additionally, Zerzar et al. [8] introduced an analytical model specifically tailored for fractured horizontal wells in which semi-infinite, or anisotropic closed, homogenous, or naturally fractured systems have been considered for finite-conductivity, infinite-conductivity, and uniform-flux models.Al-Kobaisi et al. [9] introduced a hybrid model that integrated numerical and analytical techniques, dynamically linking a numerical fracture model with an analytical reservoir model.This approach allowed for a comprehensive analysis of the pressure-transient response in a horizontal well intersecting a finite-conductivity fracture.In addition, Wei et al. [10] examined the influence of turbulence on the pressure variation behavior of fluid flow and specifically focused on the fluid movement from the formation to the horizontal well with transverse fractures.Medeiros et al. [11] delved into the diagnostic analysis of pressure and pressure-derivative curves for fractured horizontal wells in inhomogeneous, locally fractured, and globally fractured reservoirs.Their findings revealed that the flow patterns in horizontal wells with transverse and longitudinal hydraulic fractures diverged primarily during the early and intermediate stages.Ozkan [12] employed the simplified trilinear flow model to simulate the transient pressure behavior and production patterns in fractured horizontal wells within unconventional reservoirs.Yao et al. [13] introduced a semi-analytical model leveraging Green's function and the source/sink method, to present transient pressure analysis for multi-stage fractured horizontal wells in closed box-shaped reservoirs.Rbeawi et al. [14] derived solutions for fractured horizontal wells accounting for various fracture orientations, symmetries, and fracture propagation behaviors in isotropic or anisotropic formations with varying dimensions and spacing.Zhao et al. [15] utilized the source function to examine the transient pressure response of multiple-fractured horizontal wells in shale gas reservoirs, providing crucial insights into their performance.Yao et al. [16] derived an analytical solution for pressure dynamics in horizontal wells with multiple hydraulic fractures, leveraging the mirror-injection and superposition theorem given by the source/Green's function method.Fan et al. [17] constructed a numerical composite-formation model for multiple-fractured horizontal wells, taking into account the stimulated reservoir volume (SRV).In their paper, three key zones were divided for the reservoir: hydraulic fracture zone, stimulated reservoir volume zone, and unstimulated matrix zone.Ren et al. [18] set up a transient pressure model of tight oil considering stress sensitive effect and double porosity model.In addition, the Laplace transform, perturbation transform, and superposition have been applied to solve the mathematical model.
The threshold pressure gradient (TPG) and stress sensitivity effect play pivotal roles in regulating fluid flow within tight oil reservoirs [19,20].Song et al. [21] devised an experimental framework using a static approach to quantitatively assess the TPG within distinct wettability surface microchannels.Their findings demonstrated the TPG's existence in microchannels.Zeng et al. [22] constructed an experimental apparatus to probe single-phase oil/water flow in ultra-low-permeability cores, leveraging a capillary flow meter for precise fluid volume measurements.Their study revealed that single-phase oil/water flow in these cores deviated from Darcy's law, and a minimum threshold pressure gradient was widely observed for various fluids.Zhu et al. [23] conducted experimental research on the TPG's occurrence conditions for gas transport in water-bearing tight gas reservoirs.Additionally, Davies et al. [24] presented several cases with a conceptual model of tight sandstone gas reservoirs, including stress-dependent permeability.Their results show that stress-dependent permeability should be accurately forecast to predict optimum production conditions.Shi et al. [25] explored the stress sensitivity effect of the Changqing tight gas reservoir, and a close exponential relationship between formation stress and rock permeability/porosity was obtained.In another study, Dautriat et al. [26] utilized an integrated approach combining hydrostatic and triaxial compression experiments, petrophysical analysis, high-resolution CMT imaging, and pore network simulations to simulate the mechanical impact of hydrostatic stress on rock properties in the linear deformation regime.Hence, the existence of the threshold pressure gradient (TPG) and stress sensitivity effect poses significant challenges in establishing a reliable well testing model for fractured horizontal wells in tight oil reservoirs because Darcy's law fails to accurately describe fluid flow in these unique geological formations.Diwu et al. [27] proposed a new pressure superposition principle for shale and tight oil by considering TPG and different shut-in periods.Wu et al. [28] provided a simplified multi-linear transient pressure mathematical model considering TPG and the stress sensitivity effect.In their paper, six flow regimes were given, but wellbore storage effects were ignored.
Drawing upon the discrete fracture model, this study constructed a transient flow model specifically tailored for fractured horizontal wells within tight oil reservoirs.This model took into account crucial factors such as the threshold pressure gradient, stress sensitivity effects, fracture parameters, and their distribution patterns.Subsequently, the transient flow model was solved using the finite-volume method, and a comprehensive sensitivity analysis was conducted.This approach provides a more meaningful understanding of pressure performance for complex fluid flow behavior in tight oil reservoirs.

Physical Model
Drawing upon the fractured horizontal well configuration depicted in Figure 1 within a tight oil reservoir, several key assumptions should be given, as follows.The porous medium is assumed to be homogeneous, isotropic, and isothermal, ensuring consistency in its physical properties.Both the fluid and the rock are considered to be slightly compressible, and constant individual compressibility factors are considered.Moreover, the influence of stress sensitivity and wellbore storage is taken into account, capturing the dynamic interactions between the reservoir and the wellbore.The effects of gravity and capillary force are neglected to simplify the analysis.Meanwhile, it is assumed that the fractures penetrate the reservoir and are symmetric around the wellbore with a regular geometric distribution.

Mathematical Model for the Matrix System
The motion equation of a matrix porous medium in tight oil reservoirs, including TPG and the stress sensitivity effect, is as follows [27]: where vm is the fluid flow velocity in the matrix porous medium, cm/s; k0 is the initial formation permeability, µm 2 ; α is the permeability modulus, 10 MPa −1 ; G is the TPG, 10 −1 MPa/cm; µ is the fluid viscosity, mPa•s; P0 is the initial formation pressure, 10 −1 MPa; and P is the formation pressure, 10 −1 MPa.
The rock state equation of elastic porous media is as follows [29]: ( ) where φ is the porosity of the matrix porous medium; 0 φ is the initial porosity of the matrix porous medium; and Cf is the rock compressibility, 10 MPa −1 .The state equation of the formation fluid is the following [29]: where ρ is the fluid density, g/cm 3 ; ρ0 is the initial fluid density, g/cm 3 ; and Cl is the fluid compressibility, 10 MPa −1 .
The mass conservation equation is the following [29]: Equation ( 5) is obtained based on Equations (2)-( 4): By substituting Equation ( 1) into the second term on the left of Equation ( 5), one can obtain the following: where k is the effective permeability,  6) can be ignored as they are too small and have a different sign, which means that the l m 6) can also be ignored.Then, the governing equation of the matrix porous medium in a tight oil reservoir becomes the following:

Mathematical Model for Fracture System Based on DFM
We take the fractures as one-dimensional entities based on the discrete fracture model [30][31][32][33][34] (Figure 2).Based on the above assumptions, the motion equation of the fracture porous medium is as follows: where vf is the velocity in the fracture, cm/s; kf is the fracture permeability, µm 2 ; and Pf is the fracture pressure, 10 −1 MPa.
Some researchers [35,36] have proven that the compressibility of the fracture porous medium can be ignored; hence, the governing equation is the following:

Mathematical Model for the Inner Boundary
The fluid flows from the fractures into the horizontal wellbore.The relationship between the surface production rate and the sandface rate when considering the wellbore storage is as follows: Production well: Injection well: where q is the surface production rate (std), cm 3 /s; qb is the sandface rate, cm 3 /s; C is the wellbore storage coefficient, cm 3 /10 −1 MPa; B is the formation volume factor; and Pwf is the bottomhole pressure, 10 −1 MPa.

Numerical Solution for the Well Test Model
The new well test model can be solved numerically by FVM for fractured horizontal wells in tight oil reservoirs.Figure 3  Figure 2 represents the notion that, if the global region is Ω, the matrix region is Ωm, the fracture region is Ωf, then the global region can be presented as follows: where a is the fracture aperture, cm.
(1) The matrix region We integrate Equation (7) to obtain Equation ( 13) for the matrix zone: We then transform the left term in Equation ( 13) as the integration over the control volume (CV) surface based on Gauss' Divergence Theorem.
For CVM, the physical meaning of the left term of Equation ( 14) is the total flux from neighboring CVs.The schematic of unit vector n and unit vector eMi of vm is represented in Figure 4.

T hl =
, h is the reservoir height, lAB is the side length of CVi and CVM contact, lMi is the distance between the CVM centroid and the CVi centroid, n is the unit normal vector to the control interface, and Mi e is the unit vector along the direction of the line joining the centroids of CVM and CVi.
Governing Equation ( 16) can be obtained by discretizing Equation (15) when where N is the number of neighboring CVs of CVM, ( )  16) is as follows: The flux from CVi into CVM is zero when The fracture region We integrate Equation (9) (fracture region) into Equation ( 18): ( ) We transform the term on the left terms of Equation ( 18) as the integration over the fracture segments of the control volume surface, which can be shown as Equation (19): For CVM, the physical meaning of the left term in Equation ( 19) is the total flux from the neighboring CVs through fractures.
The flux from CVi into CVM is as follows: where The following numerical scheme of Equation ( 21) of CVM, which contains a fracture, can be obtained using Equations ( 17)- (20): where Nf is the number of CVs connected to CVM through the fracture.
(3) The inner boundary's treatment Taking a production well as an example, the centroid of CVM, which contains a fracture, is on the inner boundary.This numerical scheme can be given as follows: where ∑ λfM is the mobility of the fracture in CVM, Nfw is the number of intersections of wellbore and fractures, and the sandface rate (std), cm 3 /s.
When considering the effect of wellbore storage, 1 b n q + is as follows:

Accuracy Verification and Flow Regime Analysis
In this paper, we take a fractured horizontal well in tight oil reservoirs as an example [18,27,28,[37][38][39]: the reservoir has a length of 3000 m, a width of 2000 m, a thickness of 10 m, an initial formation pressure of 30 MPa, an initial formation permeability of 2 mD, a permeability modulus of 0.02 MPa −1 , three fractures with a fracture half-length of 30 m, fracture spacing of 200 m, fracture conductivities (FCD) of 30 cm•µm 2 , a formation volume factor of 1.2, a fluid viscosity of 2 mPa•s, a total compressibility of 4 × 10 −4 MPa −1 , a TPG of 0.02 MPa/m, a productivity of 50 m 3 /d, a wellbore storage coefficient of 0.1 m 3 /MPa, and its outer boundary is closed.

Accuracy Verification
To assess the precision of the aforementioned transient flow model, we simplify the fractured horizontal well into a fractured vertical configuration [37,40].A reservoir radius of 500 m, a fracture half-length of 100 m, and a daily production rate of 20 cubic meters have been given, and the effects of TPG and stress sensitivity are disregarded.The numerical solution derived in this study is compared to an analytical solution, as presented in Figure 5.This comparison reveals a strong concordance between the two solutions, which indicates well the accuracy of the transient flow model.

Flow Regime Analysis
In this paper, we analyze the flow regimes considering the effects of TPG and stress sensitivity, which can be categorized into seven distinct flow periods.Pressure-derivative curves are chosen to present the prominent characteristics of the different situations.The flow regimes are outlined in Figure 6 and described as follows: (1) Pure wellbore storage effect: This flow behavior occurs when the fluid has not yet penetrated into the wellbore.During this flow regime, the pressure and pressurederivative curves completely overlap, displaying a unit slope line.(2) Channel flow: This situation represents the transitional flow regime between the pure wellbore storage effect and the fracture linear flow.(3) Fracture linear flow (Figure 7a): The fluid flows perpendicular to the fracture, resulting in a positive slope in the pressure-derivative curve.The main feature of this flow regime is that both the dimensionless pressure and the pressure-derivative curves are straight lines with a slope of 1/2.(4) Fracture radial flow (Figure 7b): When the fracture half-length is short and the spacing between fractures is wide, the fluid flows radially into each fracture before they begin to interact.This flow regime is characterized by a horizontal pressure-derivative curve.During this period, fluid flows radially from the fracture to the wellbore, and the main characteristic of this flow stage is a horizontal line on the dimensionless pressure-derivative curve.(5) Formation linear flow (Figure 7c): As the fractures begin to interact, the fluid flows parallel to the fractures, leading to a positive slope in the pressure-derivative curve.
During this flow regime, the pressure-derivative curve shows a straight line with a slope of 1/2.This flow regime is mainly influenced by the dimensionless fracture conductivity and dimensionless fracture half-length.(6) Formation radial flow (Figure 7d): In this scenario, the reservoir is sufficiently large, and the production time is long enough, but the pressure wave has not yet reached the outer boundary; hence, the fluid flows radially into the fractured zone.This flow regime is also characterized by a horizontal pressure-derivative curve.During this flow regime, the dimensionless pressure-derivative curve shows a horizontal line but no longer adheres to the "0.5 slope line rule" when considering the TGP and the stress sensitivity.

Sensitivity Analysis
Figure 8 demonstrates that the pressure-derivative curve exhibits an upward shift when the total pressure gradient (TPG) is considered and the stress sensitivity effects are neglected.This upward trend becomes more pronounced as the TPG increases, leading to increased difficulty in distinguishing between different flow regimes.Notably, the sensitivity of the pressure-derivative curve to TPG diminishes as the TPG value rises, but it increases over time.Figure 9 illustrates that, as the permeability modulus increases, the pressure-derivative curve shifts upward.This upward trend becomes more prominent with higher permeability modulus values, particularly during later flow periods.The sensitivity of the pressure-derivative curve to the permeability modulus can be intensified over production time.Notably, as the permeability modulus increases, it becomes increasingly challenging to distinguish the fracture linear flow and the subsequent flow periods.Interestingly, when the permeability modulus remains constant, the upward shift initially intensifies but subsequently diminishes.This observation can be attributed to the exponential relationship between matrix permeability and pressure drop, which means that its variations become negligible, and the significance of the stress sensitivity effects can be reduced when matrix permeability decreases to a certain level.with the fracture half-length increasing.In addition, the fractur half-length primarily influences the fracture linear flow and subsequent flow periods, while the formation of fracture radial flow and the formation radial flow become increasingly challenging as the fracture half-length increases.Interestingly, during later flow periods, a decrease in the fracture half-length leads to a more severe upward shift in the pressure-derivative curve.This observation is attributed to the increasing influence of the matrix porous medium as the fracture half-length decreases, which enhances the effects of stress sensitivity.Figure 11 illustrates that, as the fracture conductivity increases, the pressure-derivative curve exhibits a downward trend.The sensitivity of this curve to changes diminishes with fracture conductivity increasing.Notably, the fracture conductivity has a significant impact on the early flow period.In addition, a situation with an increase in fracture conductivity can be given to distinguish between various flow regimes, including transient flow, fracture linear flow, and fracture radial flow.Figure 12 illustrates the effect of secondary fractures on formation pressure and pressure-derivative curve.In this case, two secondary fractures can be connected into each main crack, and the secondary fractures have a fracture half-length of 30 m [28], fracture conductivities (FCD) of 30 cm•µm 2 [37], and a fracture orientation of 45° to the main fracture.We can find that secondary fractures have a significant impact on the late flow period.Compared with the case without considering secondary fractures, the pressure-derivative curve of secondary fractures increases in the late flow period, which means that the pressure propagates faster to the boundary of the oil reservoir.

Conclusions
Drawing upon the discrete fracture model, this paper constructs a transient flow model specifically tailored to fractured horizontal wells in tight oil reservoirs.This model takes into account the intricate effects of total pressure gradient (TPG), stress sensitivity, fracture parameters, and fracture distribution.In addition, the finite-volume method is employed to solve this model.The precision of the aforementioned transient flow model is verified well with simplified numerical and analytical solutions.Finally, the sensitivity analysis of the obtained model has been discussed with such critical parameters.Some important conclusions have been summarized as follows.
As for fractured horizontal wells, there are typically seven distinct flow regimes, and the characteristic parameters for these have been given.However, incorporating the effects of TPG and stress sensitivity complicates the division of these flow regimes, because these factors contribute to an upward trend in the pressure-derivative curve.The pressure-derivative curve exhibits a more pronounced upward trend as the TPG and permeability modulus increase, which means that the pressure is more likely to reach the boundary as TPG increases.Notably, stress sensitivity primarily impacts the flow in later periods, and it initially enhances the upward trend but subsequently causes it to taper off, and its variations become negligible, and the significance of the stress sensitivity effects can be reduced when matrix permeability decreases to a certain level.Additionally, as the fracture half-length increases, the formation of fracture radial flow and formation radial flow becomes increasingly challenging because hydraulic fractures have a higher conductivity compared to the matrix and have an important role in pressure performance in tight oil development.Conversely, the fracture conductivity has a significant influence on the early flow period.Notably, accurately distinguishing flow regimes becomes more difficult with the increase in TPG and the permeability modulus, as well as the decrease in fracture conductivity.Compared with the case without considering secondary fractures, the pressurederivative curve of secondary fractures increases in the late flow period, which means that the pressure propagates faster to the boundary of the oil reservoir.
In future works, the fracture network will be verified in the tight oil reservoir by microseism or fiber optic testing, and the discrete fracture model (DFM) or the embedded discrete fracture model (EDFM) will be employed to treat the fracture network when meaningful field data can be obtained [41,42].In addition, after hydraulic fracturing, the fracturing liquid cannot flow back totally; hence, in the early flow period, the fluid flow should be a two-phase flow, which would affect the transient pressure performance in new models [43,44].

Figure 1 .
Figure 1.The schematic of a fractured horizontal well in tight oil reservoirs.

Figure 2 .
Figure 2. The schematic figure of the discrete fracture model.

Figure 3 .
Figure 3.The schematic of the cell vertex network model.
scheme of Equation (

Figure 5 .
Figure 5.Comparison between numerical solution and analytical solution.

( 7 )
Pseudo-steady flow: This regime emerges when the pressure wave reaches the closed outer boundary, resulting in a positive slope in the pressure-derivative curve.During this flow regime, the pressure and pressure-derivative curves still overlap but no longer adhere to the "unit slope line rule" when considering the TGP and the stress sensitivity.

Figure 6 .Figure 7 .
Figure 6.Pressure performance behavior of fracturing horizontal wells in tight oil reservoirs.

Figure 8 .
Figure 8.Effect of TPG on formation pressure and pressure-derivative curve.

Figure 9 .
Figure 9.Effect of permeability modulus on formation pressure and pressure-derivative curve.

Figure 10 .
Figure 10.Effect of the fracture half-length on formation pressure and pressure-derivative curve.

Figure 11 .
Figure 11.Effect of fracture conductivity on formation pressure and pressure-derivative curve.

Figure 12 .
Figure 12.Effect of secondary fractures on formation pressure and pressure-derivative curve.