Either your web browser doesn't support Javascript or it is currently turned off. In the latter case, please turn on Javascript support in your web browser and reload this page. Shale gas reservoirs can be divided into three regions, including hydraulic fracture regions, stimulating reservoir volume regions SRV regions , and outer stimulating reservoir volume regions OSRV regions. Due to the impact of hydraulic fracturing, induced fractures in SRV regions are often irregular.

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Either your web browser doesn't support Javascript or it is currently turned off. In the latter case, please turn on Javascript support in your web browser and reload this page. Shale gas reservoirs can be divided into three regions, including hydraulic fracture regions, stimulating reservoir volume regions SRV regions , and outer stimulating reservoir volume regions OSRV regions.

Due to the impact of hydraulic fracturing, induced fractures in SRV regions are often irregular. In addition, a precise description of secondary fractures in SRV regions is of critical importance for production analysis and prediction. In this work, the following work is achieved: 1 the complex fracture network in the SRV region is described with fractal theory; 2 a dual inter-porosity flow mechanism with sorption and diffusion behaviors is considered in both SRV and OSRV regions; and 3 both multi-rate and multi-pressure solutions are proposed for history matching based on fractal models and Duhamel convolution theory.

Compared with previous numerical and analytic methods, the developed model can provide more accurate dynamic parameter estimates for production analysis in a computationally efficient manner. In this paper, type curves are also established to delineate flow characteristics of the system.

It is found that the flow can be classified as six stages, including a bi-linear flow regime, a linear flow regime, a transition flow regime, an inter-porosity flow regime from the matrix to the fractures in the inner region, inter-porosity flow regime from matrix to fractures in the outer region, and a boundary dominant flow regime.

The effects of the fracture and matrix properties, fractal parameters, inter-porosity flow coefficients, and sorption characteristics on type curves and production performance were studied in detail.

Finally, production performance was analyzed for Marcellus and Fuling shale gas wells, in the U. The successful exploitation of shale gas heavily depends on the combination of horizontal drilling, completions, and fracturing technology 1 , 2. It has been proven that multi-stage fracturing horizontal wells constitute a very effective way to exploit low-permeability shale gas reservoirs. Due to the complex fracture network created in stimulated reservoir volume regions, it is a great challenge to accurately analyze production performance and evaluate post-fracture performance in such complex fractured reservoirs 3.

Since , many analytical and semi-analytical methods on production and flow behaviors in conventional gas and shale gas reservoirs have been utilized to simulate pressure transient behavior for horizontal wells with hydraulic fractures.

Soliman et al. Larsen and Hegre 5 presented two kinds of models for fractured horizontal wells with multiple fractures with finite conductivity: circular fractures with radial flow and vertical fractures with linear flow. The fracture size is given by the width, the well radius r w and fracture outer radius r f.

The fracture size is given by the width, the fracture half-length x f away from the wellbore, and the fracture half-length y f along the wellbore. Guo and Evans 6 presented a randomly distributed vertical fractures model to predict production performance, and generated type curves to estimate reservoir properties and fracture characteristics.

Bin et al. However, due to constant rate assumption, the model cannot be applied to interpret production data. Wang et al. Since sorption characteristics were not incorporated into their model, the model cannot consider the effect of sorption on production. A technique was introduced by Rbeawi and Tiab 9 to interpret the pressure transient behavior of fractured horizontal wells.

The distribution of hydraulic fractures could be longitudinal or transverse, vertical or inclined, symmetrical or asymmetrical. Brown et al.

However, complex geometries and sorption characteristics were not taken into account in these studies, so that they were not suitable for production performance analysis in shale gas reservoirs. The above studies 4 — 11 mainly focused on fluid flow in primary fractures, and ignored fluid flow inside secondary fractures and matrix.

Due to the influence of induced fractures, uniform fracture models were not applicable to describe heterogeneous SRV zones. Some scholars attempted to utilize numerical models with a discrete fracture network to describe secondary fractures in SRV zones 12 — Although the production performance of a complex fracture network can be discerned through the use of numerical methods, numerical simulation is less attractive and pragmatic due to the large amount of requisite knowledge and time-consumption during the history matching process 3.

In contrast, the analytical model is an alternative for accurately forecasting flow behaviors and calculation efficiency. Fractal geometry has shown potential in the analysis of flow and transport properties in porous media.

Katz and Thompson 16 were the pioneers to find experimental evidence that indicated that the pore spaces of a set of sandstone samples were fractals and self-similar over three to four orders of magnitude in length. Fractal theory can also describe complex fracture shape in shale gas and is easy to apply to obtain analytical solutions. Chang and Yortsos 17 introduced fractal theory into petroleum engineering, and reported that the permeable fractures embedded within the matrix would be arranged in a fractal dimension.

Cossio et al. Yao et al. Kong et al. Zhang et al. The proposed fluid flow models with fractal geometry 3 , 16 — 21 are convenient for conducting pressure transient analysis for understanding flow in complex reservoirs 22 — Random fractures will be produced during the process of hydraulic fracturing.

Fractal theory can exhaustively describe the randomness of pore sizes and gas reservoir heterogeneity However, most of the previous models are only involved in the aspects of pressure transient analysis in tight reservoirs, while shale multi-scale flow characteristics are not considered 4 — Second, gas PVT is assumed to be constant, so that the final results will lead to an assessment of errors.

However, gas PVT changes with pressure in the development of shale gas reservoirs. Third, the characteristics of fractal geometry in SRV regions are not reflected in their models 4 — In this paper, an improved tri-linear flow model in shale gas reservoirs with fractal geometry is presented to analyze and predict production performance of multi-fractured wells.

First, multi-scale flow mechanism in shale gas reservoirs was considered in the analytical model Fig. Second, the main fractures were considered into longitudinal rectangular fractures and secondary fractures in the SRV zone were described using fractal theory. Third, gas PVT changed with reservoir pressure in this paper, and a multi-rate solution with variable pressure was also proposed to interpret real well production data measured under various production systems.

Finally, the downhill Simplex method was employed to form the history matching procedure. Many authors have presented analytical models to investigate regular fracture networks, such as the single planar hydraulic fractures network and orthogonal hydraulic fractures network 4 — A schematic diagram of the presented model is shown in Fig.

Shale gas flow can be divided into three flow regions: flow in hydraulic fractures defined as region 1 hydraulic fracture region ; flow in the SRV region defined as region 2 inner region ; and flow in the OSRV region defined as region 3 outer region. To establish mathematical models, the following basic assumptions are made:. Different from conventional reservoirs, shale gas seepage often exhibits strong nonlinearity 8.

In addition, physical parameters, such as the viscosity, compressibility factor and deviation factor, are functions of pressure. However, these methods are effective for only specific problems. The viscosity, compressibility and deviation factor of shale gas change with pressure, which leads to a non-linear seepage equation. These definitions can be written in the following forms:.

The spherical element was assumed to flow in the matrix, and a comprehensive continuity equation with sorption can be formulated as:. Pseudo-time t a is defined in Eq. The total compressibility in shale has been previously presented 29 , 30 and can be expressed as:. Accounting for gas slippage and gas diffusion, gas velocity in the matrix can be expressed as:.

Substituting Eq. The initial condition and boundary conditions can be given as:. According to the above definitions of pseudo-pressure and pseudo-time, assuming one-dimensional flow in the x direction, as shown in Fig. The change in porosity and permeability in the inner region are large and heterogeneous.

Therefore, the fractal model can be utilized to describe SRV behavior. Furthermore, we work exclusively in Cartesian coordinates. Thus, the fractal relationships are given as 17 , 18 :. Assuming one-dimensional flow in the y direction, the diffusivity equation is given by:.

The dimensionless definitions of the model and dimensionless solution at a constant rate condition are given as:. In the above results, one-dimensional linear flow in the hydraulic fracture is considered. Flow choking within the fracture must be considered by using the following equation that was presented by Mukherjee and Economides 31 :.

Note that the constant-pressure solution and constant-rate solution conform to the relation suggested by Van-Everdingen and Hurst 32 :. Therefore, the rate solution at a constant pressure condition can be obtained as:. During the process of production, both the rate and pressure change with the time. Constant rate and constant pressure solutions cannot be applied to a real reservoir.

Through using convolution theory 29 , 33 , the variable rate and variable pressure solutions can be given by. The material balance equation must be established to calculate pseudo-time and pseudo-pressure. The modified gas compressibility factor and material balance equation that were proposed by Moghadam 30 for shale gas reservoirs were adopted in this paper.

The modified material balance equation can be written as:. The entire calculation process is illustrated in Fig. The procedures above could be applied to interpret production data from shale gas reservoirs. The accuracy of the coupled model was validated by comparing our results with those of an analytical solution using IHS Harmony software. The total analysis time is d. A validation of the rate solution under a constant pressure condition of 2, psi is shown in Fig.

It is apparent that the production rate and cumulative gas production obtained from the proposed model agree very well with those from the analytical solution. A validation of the pressure solution under a variable rate condition is shown in Fig. The solution of the pressure under variable rate conditions is consistent with the solution from the commercial software IHS Harmony.

Flow characteristics analysis of pressure and pressure derivative curves is shown in Fig. It is apparent that the fluid flow in shale gas reservoirs can be classified into six stages:. The flow of gas occurs simultaneously both within hydraulic fractures and near them.

In this period, linear flow around the fractures occupies the dominant position. The transition flow regime a shows an indefinite slope line on the pressure derivative curve. The first inter-porosity flow regime is a regime of supplementation from the shale matrix to the fracture system in the inner region.

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Main results We firstly discuss the existence of the equilibria of model 2. The authors would like to thank anonymous referees for their constructive comments which improve the readability of the paper. Europe PMC requires Javascript to function effectively. Your browser is no longer supported. In item ty J llr1As Two A. Many authors have investigated the autonomous epidemic leei. Berryman and Millstein [ 3 ] investigated an SVEIS epidemic model for an infectious disease that spreads in the host lwi through horizontal transmission, and they have shown that lfi model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium.

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