Fractured Vertical Wells Research Articles

Rate decline analysis for finite conductivity vertical fractured gas wells produced under a variable inner boundary condition

This paper presents a new approximate semi-analytical method for predicting performance of a finite conductivity vertical fractured gas well produced at a variable production system. The proposed solution is based on the gas diffusivity equation with a variable inner boundary condition. The variable inner boundary condition in this study represents that the type of the inner boundary conditions changes from constant rate to constant pressure during the production time. The calculations of the two-stage performance for a finite vertical fractured gas well are accomplished. The two-stage performance is influenced by the limiting bottom-hole pressure, the initial rate and the fracture conductivity. A high initial rate reduces the extents of both the constant-rate period and the total production life. However, the initial rate has no effect on the cumulative gas production when the production period is sufficiently long. Moreover, the initial rate may be appropriately adjusted to achieve the optimum cumulative gas production in a specific production period. The cumulative gas production can be effectively improved by establishing a lower limiting bottom-hole pressure. The higher the fracture conductivity is, the longer the constant rate period elapses, and the shorter the constant pressure period remains. A higher fracture conductivity is helpful to achieve a larger cumulative gas production in a shorter production period. Thus, the operation can be optimized by setting a low limiting bottom-hole pressure, a suitable initial rate and fracture conductivity. The proposed method may be applied to other typical well performance prediction in a variable production system.

A Semianalytical Productivity Model for a Vertically Fractured Well With Arbitrary Fracture Length Under Complex Boundary Conditions

Summary Vertically fractured wells are used widely to achieve efficient development of low-permeability reservoirs and unconventional tight reservoirs. Currently, there are some commonly adopted productivity models for a fractured vertical well, including the Giger (1985) model, the Economides model (Economides et al. 1989), the Joshi (1991) model, the Guo model (Guo and Evans 1993), the Zhang (1999) model, and others. However, these models are only suitable for either short fractures or long fractures, and thus cannot satisfy the need for arbitrary-length fracture selection. In addition, these models have certain requirements for boundary conditions, and few models consider the nonuniform flow in actual hydraulic factures. Therefore, it is desirable to establish a productivity model for vertically fractured wells with arbitrary fracture length under complex boundary conditions. In this paper, first, we evaluate the existing fractured-vertical-well productivity models in detail. After that, on the basis of the potential superposition principle and mirror-image method, we develop the productivity model for a fractured vertical well with arbitrary fracture length under different boundary conditions. Grounded in this model, further considering the frictional pressure drop caused by fracture-flow nonuniformity, we establish the productivity equation of a finite-conductivity-fracture vertical well with nonuniform flow in the fracture. Finally, after comparing with the existing model under certain conditions, the reliability of the proposed model is verified successfully. Results show that the Giger model assumes that the boundary in the fracture-extension direction is impermeable, and the pressure wave reaches the boundary earlier than the constant-pressure boundary, which is perpendicular to the extension direction of the fracture, whereas both the Economides model and the Joshi model assume that the boundary in the fracture-extension direction is far, and is always farther than the boundary perpendicular to the direction of the fracture extension. The results forecast by the previous models will deviate greatly from actual well-production performance, if the real field case cannot meet the requirements of these models. The proposed model is not only in good agreement with the previous models at given conditions, but is also suitable for fractured vertical wells under arbitrary-length fractures and complex boundary conditions. Compared with the existing models, the correctness and reasonableness of the model are shown, and the application of the proposed model is broadened. For the development of low-permeability and unconventional tight oil and gas reservoirs with fractured vertical wells or fractured horizontal wells, the establishment and application of this model have great theoretical significance and application value for production prediction. This model can be a foundation or reference for productivity prediction of low-permeability and unconventional tight oil and gas reservoirs with fractured vertical wells or fractured horizontal wells.

Performance analysis for a model of a multi-wing hydraulically fractured vertical well in a coalbed methane gas reservoir

This paper proposes a composite model to investigate the pressure behavior and production performance of a multi-wing hydraulically fractured multiple fractured vertical well (MFVW) in a coalbed gas reservoir with a stimulated reservoir volume (SRV). The model simultaneously considers adsorption–desorption, diffusion, and viscosity flow. The MFVW model combines line-source method with Laplace transformation and discretization method. Then, the Stehfest numerical inversion and the Gauss elimination are utilized to obtain the transient pressure responses and production dynamics.The results show that the main flow regimes for an MFVW in a coal seam gas reservoir are linear flow between hydraulic fractures, pseudo radial flow in the SRV, transition flow from the inner to the outer region, and radial flow in the outer region. Further, a slight “hump” is observed during the linear flow period, which could not be observed in the curves of MFVWs with two-wing hydraulic fractures. The effects on MFVW performance of permeability and size of the SRV, properties of hydraulic fractures (fracture angle asymmetry, different fracture angle, and fracture number), Langmuir volume, and Knudsen diffusion coefficient are discussed. The conclusions obtained in this paper will be helpful to understand the transient performance of MFVWs in coalbed gas reservoirs.

Pressure transient analysis for vertical fractured wells with fishbone fracture patterns

Many diagnostic technologies have proven that complex fracture pattern would be generated during the operation of fracture treatment for vertical well. However, there had been very little research reported on fishbone fracture pattern for vertical well, which is one of the typical fracture geometries. To analysis the pressure behavior of this kind fracture pattern, a semi-analytical method is proposed and some typical fracture patterns are compared in this work.Based on the theory of superposition, the physical model is divided into two sub-systems: (a) a formation/fracture sub-system and (b) a fracture sub-system. A flexible discretization and coupling method is developed to investigate irregular fracture geometry and non-orthogonal between two fractures. The mathematical model is solved in the Laplace domain and the Stehfest inversion algorithm is applied to determine the corresponding pressure distribution in the real-time domain. Type curves are generated and flow regimes are identified. Sensitivity analysis and actual field case are conducted.It is found that there are five flow regimes, they are bilinear flow, flow feed, fracture linear flow, formation linear flow, bi-radial flow and pseudoradial flow. The pressure depletion decreases with increase of the number, length and conductivity of secondary fractures, and their effect on contribution of primary fracture have a similar tendency. Effects of Inclined angle between secondary and primary fractures on pressure transit behavior occurs during flow feed and later regimes, while fractures pattern with different geometries can influence all the flow regimes. The derived analytical pressure transient solution may provide a theoretical basis for the analysis of the pressure transient behavior of the fishbone fracture pattern in a vertical well.

Production analysis for fractured vertical well in rectangular coal reservoirs

As one kind of unconventional natural gas, coalbed methane is an important energy resource that is subject to active research. Gas exists in coalbed methane reservoirs in two forms: free gas and adsorbed gas. In the course of coalbed methane production, the reservoir experiences pressure decrease, desorption, diffusion, and seepage. Previous models of coalbed methane production were mainly concerned with circular boundaries. However, field tests revealed that some fractured wells possess the characteristics of rectangular boundaries. For fractured rectangular coalbed methane reservoirs, it is necessary to deal with the four boundaries with mirror image theory, which complicates calculations. In addition, the desorption and adsorption process of coalbed methane exerts a strong effect on the seepage process. Furthermore, the complexity of the rectangular coal seam embedded with the finite-conductivity fracture results in a significant computational challenge. For the first time, this paper presented a fast analytical solution for a finite-conductivity fractured vertical well model with either rectangular closed or constant-pressure boundaries in the coal seam. On the basis of the Fick diffusion law and the Darcy seepage law, a mathematical model that considers diffusion in matrix and seepage within natural fractures was established. Then, we integrated the fracture conductivity function method with the hydraulic fracture model to greatly increase computational efficiency. The analytical solutions were validated against a numerical simulation. Parameter sensitivity analysis reveals that interporosity coefficient and storage coefficient, respectively, affect the appearance time and degree of desorption and diffusion. Desorption coefficient mainly describes the capacity of desorption and diffusion. Well storage coefficient, conductivity factor, and skin factor mainly affect the early stage of production. Finally, the proposed solutions were applied to field history match. The model developed is applicable to production analysis and well testing for coalbed methane reservoirs. The new proposed model extended flow mechanism of coalbed methane, and provided a better production and pressure forecast for coalbed methane reservoirs. In addition, the analytical solutions can be used to generate type curves for fractured vertical wells with finite conductivity and in the rectangular boundary, and provide a sound theoretical basis for well tests in the coal seam. The model is also applicable to other types of unconventional gas reservoirs, such as gas shales, in which the same processes are present.

Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs

Abstract The mathematical model of coupled flow and geomechanics for a vertical fractured well in tight gas reservoirs was established. The analytical modeling of unidirectional flow and radial flow was achieved by Laplace transforms and integral transforms. The results show that uncoupled flow would lead to an overestimate in performance of a vertical fractured well, especially in the later stage. The production rate decreases with elastic modulus because porosity and permeability decrease accordingly. Drawdown pressure should be optimized to lower the impact of coupled flow and geomechanics as a result of permeability decreasing. Production rate increases with fracture half-length significantly in the initial stage and becomes stable gradually. This study could provide a theoretical basis for effective development of tight gas reservoirs.

Transient pressure analysis of a volume fracturing well in fractured tight oil reservoirs

This paper presents a semi-analytical model to simulate transient pressure curves for a vertical well with a reconstructed fracture network in fractured tight oil reservoirs. In the proposed model, the reservoir is a composite system and contains two regions. The inner region is described as a formation with a finite conductivity hydraulic fracture network and the flow in the fracture is assumed to be linear, while the outer region is modeled using the classical Warren–Root model where radial flow is applied. The transient pressure curves of a vertical well in the proposed reservoir model are calculated semi-analytically using the Laplace transform and Stehfest numerical inversion. As shown in the type curves, the flow is divided into several regimes: (a) linear flow in artificial main fractures; (b) coupled boundary flow; (c) early linear flow in a fractured formation; (d) mid radial flow in the semi-fractures of the formation; (e) mid radial flow or pseudo steady flow; (f) mid cross-flow; (g) closed boundary flow. Based on our newly proposed model, the effects of some sensitive parameters, such as elastic storativity ratio, cross-flow coefficient, fracture conductivity and skin factor, on the type curves were also analyzed extensively. The simulated type curves show that for a vertical fractured well in a tight reservoir, the elastic storativity ratios and crossflow coefficients affect the time and the degree of crossflow respectively. The pressure loss increases with an increase in the fracture conductivity. To a certain extent, the effect of the fracture conductivity is more obvious than that of the half length of the fracture on improving the production effect. With an increase in the wellbore storage coefficient, the fluid compressibility is so large that it might cover the early stage fracturing characteristics. Linear or bilinear flow may not be recognized, and the pressure and pressure derivative gradually shift to the right. With an increase in the skin effect, the pressure loss increases gradually.

Effects of fractures on the well production in a coalbed methane reservoir

A water-gas two-phase flow model of a fractured vertical well in a coalbed methane (CBM) reservoir is developed based on the Buckley-Leverett equation. A program is developed based on the proposed model to predict the production of a CBM well. Compared with the monitoring results of an actual vertical fractured well, the accuracy and reliability of the analytical model are approved. The impact of reservoir parameters and fracture parameters on the CBM production are quantitatively studied. The results indicate that the production is affected by the three major factors: gas supply capacity of the matrix (viz., CBM volume released by unit volume of matrix during unit time), supply capacity of the fracture network to the major fracture, and gas exhausting ability of the major fracture to the wellbore. The permeability, matrix radius, and coal seam thickness with larger values are good for the gas supply capacity of the matrix. Higher flow conductivity of major fracture shows a stronger impact on the gas exhausting ability of the major fracture to the wellbore. A smaller fracture interval is beneficial to the supply capacity of the fracture network to the major fracture to obtain higher production. To improve the efficiency of the actual well, the flow conductivity of the major fracture should be no more than 200 mD m and the crack interval should be between 20 and 30 m. The achievements of this study provide a case reference for the production prediction of vertical fractured wells and the optimization design of hydraulic fracture parameters of CBM reservoirs.

Research for Unsteady Seepage Flow of Asymmetrical Fractured Vertical Wells in Coalbed

In this study, based on Green function and Laplace transformation, a pressure transient analysis semi-analytical evaluation model which can be used for coalbed reservoirs of finite conductivity asymmetrically fractured vertical well was established. The mathematical model considers asymmetrically fracture and finite conductivity. Fick law was used to describe gas diffusion in spherical matrix and Lagrange function was used to depict the unsteady desorption of coalbed gas. The influences of related parameters, such as artificial fracture conductivity, asymmetry factor, storativity ratio, cross-flow factor, dimensionless drainage radius and so forth, on the seepage flow were analyzed by using the established model. The results show that The percolation process include 6 stages (a) artificial fracture flow with the effect of wellbore storage effect; (b) transition flow; (c) linear flow between matrix and artificial fracture; (d) cross-flow between matrix and fracture in double medium fractures; (e) the pseudoradial flow stage of whole system; (f) closed boundary flow. Artificial fracture conductivity has great impact on whole production cycle. With the artificial fracture conductivity increases, the raise of capacity is very significant, especially for the production period after the influence of wellbore storage effect. Asymmetry factor is also important to whole production cycle. The bigger asymmetry factor is, the lower capacity is. Storativity ratio and cross-flow factor influence the degree and occurrence time of cross-flow between matrix and fractures respectively. The smaller storativity ratio is, the more observer cross-flow is. The smaller cross-flow factor is, the earlier cross-flow will happen. The dimensionless drainage radius has only an effect on the later production. The smaller dimensionless drainage radius is, the earlier the closed boundary flow will happen.

Pressure transient analysis for off-centered fractured vertical wells in arbitrarily shaped gas reservoirs with the BEM

Because transient pressure analysis is important for postfracture evaluation and reservoir characteristic assessment, and hydraulic fracturing has been widely applied in recent oil/gas development, physical and mathematical models are important for petroleum engineers. Although there numerous reports relate to fractured wells in the centers of reservoirs, research on off-centered fractured wells is rare, and it is difficult to obtain a semianalytical solution with conventional methods. In this paper, a mathematical model of an off-centered, infinite-conductivity fractured well in an arbitrarily shaped gas reservoir is established. The boundary element method (BEM) is implemented to analyze the well transient pressure response. Two validation cases of center wells with circular and rectangular outer boundaries are used to verify the correctness of our method. The pressure response of an off-centered fractured well differs from that of the conventional case at the early stage of the boundary reflection flow period. Furthermore, if the well is close enough to the boundary, the pressure derivative curves will show another step before the boundary-dominated flow period. In the end, the model is extended to dual- and triple-porosity reservoirs. The pressure type curves and their sensitivity analyses are also determined. From this study, it is clear that the BEM can be an effective and precise method to cope with the fluid flow problems in such a complex situation.

Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs

The paper developed a new constant-rate solution for non-planar asymmetric fracture connected to a vertical wellbore in hydrocarbon reservoirs. The non-planar asymmetric fracture has been modeled by combination of two unequal length wings with varied intersection angles. The type curves were drawn in the case of varied fracture conductivities, non-planar angles, and asymmetric factors. The results indicated that fracture conductivity, non-planar angle, and asymmetric factor have coefficient effects on the pressure response, which makes the dimensionless wellbore pressure and pressure derivative curves deviate from the conventional type curves of vertical fractured wells. The dimensionless wellbore pressure and pressure derivative curves are divided into five flow periods. The non-planar angle and asymmetry factor have their unique influences on the shape of type curves during different periods under the different fracture conductivities. The termination of bilinear flow and formation linear flow happened increasingly early with the decrease of non-planar angle or the increase of asymmetry factor. As the fracture conductivity increases, the effect of non-planar angle on the curves moves from the early flow period to the late flow period while the trend of effect of asymmetric factor is quite opposite. The new method lays the foundation of the pressure analysis of the complicated fracture system in developing hydrocarbon reservoirs.

Geothermal exploitation from hot dry rocks via recycling heat transmission fluid in a horizontal well

A new method for geothermal exploitation from hot dry rocks by recycling heat transmission fluid in a horizontal well via a closed loop is proposed, in which the costly and complex hydro-fracturing can be avoided. In this paper, numerical simulation models were established to calculate the heat mining rate for the new technology to assess its technical and economic feasibility. Sensitivity studies were performed to analyze the effects of various parameters on heat mining rate, including the injection rate, the horizontal segment length and the thermal conductivity of the tubing. The results show that a high heat mining rate over 1.7 MW can be obtained using a 3000 m long horizontal well to extract geothermal energy from a typical hot dry rock of 235 °C with a water circulation rate of 432 m3/d. For low-temperature geothermal reservoirs, higher injection rate, longer horizontal wells and better thermal insulation of tubing can be applied to increase the heat mining rate. The cost of geothermal power generation using a single horizontal well is estimated as 0.122 $/kWh, and this could be further reduced to 0.084 $/kWh when the multi-branch horizontal well pattern was adopted, slightly lower than a fractured vertical well case.

Productivity Analysis of Volume Fractured Vertical Well Model in Tight Oil Reservoirs

This paper presents a semianalytical model to simulate the productivity of a volume fractured vertical well in tight oil reservoirs. In the proposed model, the reservoir is a composite system which contains two regions. The inner region is described as formation with finite conductivity hydraulic fracture network and the flow in fracture is assumed to be linear, while the outer region is simulated by the classical Warren-Root model where radial flow is applied. The transient rate is calculated, and flow patterns and characteristic flowing periods caused by volume fractured vertical well are analyzed. Combining the calculated results with actual production data at the decline stage shows a good fitting performance. Finally, the effects of some sensitive parameters on the type curves are also analyzed extensively. The results demonstrate that the effect of fracture length is more obvious than that of fracture conductivity on improving production in tight oil reservoirs. When the length and conductivity of main fracture are constant, the contribution of stimulated reservoir volume (SRV) to the cumulative oil production is not obvious. When the SRV is constant, the length of fracture should also be increased so as to improve the fracture penetration and well production.

A comprehensive productivity equation for multiple fractured vertical wells with non-linear effects under steady-state flow

In Ordos Basin low-permeability reservoirs, taking large-scale fracturing (LSF) measures on vertical wells is being a common practice. Thus, it is a significant problem that should be solved urgently about how to estimate productivity of those wells. During the LSF treatments, micro-seismic monitoring (MSM) strongly shows multiple fractures (MF), with arbitrary length and arbitrary angle, develop along the wellbore. However, most of published works on the productivity of vertically fractured wells assumed that the wells contained a single fracture, and few work has been done to focus on MF.In this paper, a comprehensive steady-state productivity equation for a multiple fracture vertical well (MFVW) is proposed under a constant pressure drawdown. First, a mathematical model of the MFVW is established with consideration of multiple factors including stress-sensitivity effect of permeability (SEP), threshold pressure gradient (TPG), fracture angle asymmetry (FAA), fracture length asymmetry (FLA), and fracture conductivity asymmetry (FCA). Then, using conformal mapping and Pedrosa's perturbation, the productivity equation of MFVW is solved. After that, based on real cases from Ordos Basin reservoir, error analysis of the proposed equation is performed. Equation validation is also conducted by comparing the proposed equation with those of conventional wells in literatures. Finally, the effects of some critical parameters on productivity of MFVW are analyzed.Equation validation shows there are good agreements between the proposed productivity equation and those of conventional wells. The results from sensitivity analysis demonstrate the productivity of MFVW increases with the increase of fracture length, fracture conductivity, fracture number, and degree of FLA; it decreases with the increase of TPG value, SEP coefficient, degree of FCA, and degree of FAA.It provides significant references for reservoir engineers in productivity estimations as well as fracturing evaluations of vertically fractured wells in the low-permeability oil reservoirs.

A numerical approach for pressure transient analysis of a vertical well with complex fractures

A new well test model for a vertical fractured well is developed based on a discrete-fracture model in which the fractures are discretized as one dimensional (1-D) entities. The model overcomes the weakness of complex meshing, a large number of grids, and instability in conventional stripe-fracture models. Then, the discrete-fracture model is implemented using a hybrid element finite-element method. Triangular elements are used for matrix and line elements for the fractures. The finite element formulation is validated by comparing with the semi-analytical solution of a single vertical fractured well. The accuracy of the approach is shown through several examples with different fracture apertures, fracture conductivity, and fracture amount. Results from the discrete-fracture model agree reasonably well with the stripe-fracture model and the analytic solutions. The advantages of the discrete-fracture model are presented in mesh generation, computational improvement, and abilities to handle complex fractures like wedge-shaped fractures and fractures with branches. Analytical results show that the number of grids in the discrete-fracture model is 10 % less than stripe-fracture model, and computational efficiency increases by about 50 %. The more fractures there are, the more the computational efficiency increases.

A MIXED-FRACTAL FLOW MODEL FOR STIMULATED FRACTURED VERTICAL WELLS IN TIGHT OIL RESERVOIRS

Stimulated reservoir volume (SRV) with large fracture networks can be generated near hydraulic fractured vertical wells (HFVWs) in tight oil reservoirs. Statistics show that natural microfractures and fracture networks stimulated by SRV were self-similar in statistical sense. Currently, various analytical models have been presented to study pressure behaviors of HFVWs in tight oil reservoirs. However, most of the existing models did not take the distribution and self-similarity of fractures into consideration. To account for stimulated characteristic and self-similarity of fractures in tight oil reservoirs, a mixed-fractal flow model was presented. In this model, there are two distinct regions, stimulated region and unstimulated region. Dual-porosity model and single porosity model were used to model stimulated and unstimulated regions, respectively. Fractal geometry is employed to describe fractal permeability and porosity relationship (FPPR) in tight oil reservoirs. Solutions for the mixed-fractal flow model were derived in the Laplace domain and were validated among range of the reservoir parameters. The pressure transient behavior and production rate derivative were used to analyze flow regimes. The type curves show that the fluid flow in HFVWs can be divided into six main flow periods. Finally, effect of fractal parameters and SRV size on flow periods were also discussed. The results show that the SRV size and fractal parameters of fracture network have great effect on the former periods and fractal parameters of matrix mainly influence the later flow periods.

Effect of Non‐Darcy Flow on the Integral Fracturing Design of Low‐Permeability Petroleum Reservoirs

AbstractAn integral fracturing plan for low‐permeability petroleum reservoirs is commonly designed by using a black oil model; however, significant error may exist when non‐Darcy flow effects are not taken into account. Thus a three‐dimensional, two‐phase numerical reservoir model was established for vertical fractured wells by considering the characteristics of low‐permeability reservoirs. Based on the new model, a field case in the Zhao Ao oilfield was studied. The results show that the optimal well spacing of the five‐spot well network is 300 m for oil well, and the optimal half fracture length is 60 m with a fracture conductivity of 45 μm2 cm. In 2007, the recommended development plan was adopted and 18 oil wells and 11 water‐injection wells were implemented using the integral fracturing technique. Until June 2009, the total oil increase was 9373.4 ton. Successful field applications verify that the integral fracturing technique could greatly improve the oil recovery rate and recovery efficiency, and that the new model is reliable for new applications.

Performance of vertical fractured wells with multiple finite-conductivity fractures

Multiple fractures originating from a vertical wellbore are usually observed in hydraulic fracturing treatments. Most of the previous models for vertical fractured wells are based on single-planar fractures that may be symmetric or asymmetric about the wellbore, but few studies have been devoted to multiple-planar fractures. This paper presents a new semi-analytical solution in Laplace space for the pressure responses of a vertical fractured well, producing at a constant flow rate through multiple finite-conductivity fractures. The solution is presented in Laplace space so that the effects of the wellbore storage and skin factor can be easily incorporated by Duhamel’s principle, and then the solution in real space can be obtained using the numerical inversion algorithm proposed by Stehfest. Pressure response curves are plotted and the effects of the relevant parameters on these are analyzed. It is found that with the increase of the fracture number, smaller pressure depletion will appear under the same conditions, and the interaction between fractures takes place much earlier and becomes much stronger. We find that with the increase of the fracture number, the duration of the wellbore storage period becomes shorter, and the bilinear flow appears earlier. Decreasing the angles between fractures and increasing the fracture asymmetry coefficient will lead to a stronger interaction between them, and will then affect the bilinear flow and linear flow behaviors. The present model can be used to interpret the pressure data of the vertical fractured wells with multiple fractures and to provide more accurate dynamic parameters.

Productivity Prediction Model of Vertical Artificial Fracture Well and Analysis of Influence Factor in Low Permeability Reservoir

Based on the analysis of seepage mechanism of fracturing wells in low permeability reservoir, this paper establishes the capacity model of the vertical fractured well production under the factors of Start-up pressure gradient, pressure sensitive effect and the artificial fracture length. The numerical simulation is compiled and software calculates the capacity model by using numerical simulation. This simulation technique verifies the validity of the model and numerical method. On this basis, we study the influence of the included angle of artificial fracture and well array direction, artificial fracture length, start-up pressure gradient and production pressure difference to the capacity of the oil well.

A semi-analytical mathematical model for transient pressure behavior of multiple fractured vertical well in coal reservoirs incorporating with diffusion, adsorption, and stress-sensitivity

Vertically fractured wells have been widely used in coal gas reservoirs. Investigating transient pressure behaviors (TPB) of these wells is a considerably significant task for fracturing evaluation as well as productivity prediction. Although some studies have been done in this area, most of them focus on two-wing fractured wells. Both micro-seismic fracture imaging and samples testing strongly showed that multiple fractures can develop during hydraulic fracturing in naturally fractured reservoirs like coal seam. However, little work has been done to study the TPB of multiple fractured vertical wells (MFVW) in coal reservoirs.In this paper, to study this problem thoroughly, a well testing model of MFVW at a constant production rate was established with consideration of multiple mechanisms including diffusion, adsorption, and stress-sensitivity effect. Applying line source function, superposition principle, Pedrosa's substation, Gauss elimination, and Stehfest numerical inversion, the transient pressure solution of the mathematical model was solved. Based on this solution, model validation was conducted by selecting a case in literature. In addition, the influences of some critical parameters on TPB were studied, containing fracture length, fracture conductivity, fracture number, fracture length asymmetry, fracture conductivity asymmetry, and fracture angle asymmetry.This work provides important references for some reservoir engineers in performance forecast and completion design of MFVW in coal gas reservoirs.