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Multi-wing and asymmetrical hydraulic fractures around wellbore are usually observed in hydraulic fracturing and refracturing treatments. Therefore, it is significant for us to study pressure transient analysis. In this paper, by employing Laplace transform and finite cosine Fourier transform, a new semi-analytical model was deviated and solved under condition of constant rate for multi-wing fractured wells (MWFWs) of finite conductivity in hydrocarbon reservoirs with dual-permeability behavior in Laplace domain. Stehfest inversion algorithm can be adopted to obtain wellbore pressure of MWFWs in real domain. The solution of the simplified model in this paper was validated with dual porosity dual-permeability and asymmetrical hydraulic fracture, and results reach a good agreement. Type curve, according to pressure derivative characteristic, can be divided into six regimes (bilinear flow, linear flow, radial flow of natural fractures, cross flow and radial flow of system). The influence of some vital parameters (dimensionless conductivity, storage coefficient, crossflow coefficient, permeability ratio et al.) on dimensionless pressure and its derivative curves were analyzed in details. The presented model can be used to understand pressure transient characteristic of MWFWs in hydrocarbon reservoirs with dual-permeability behavior.

Shale gas, as an unconventional gas resource that has great exploitation potential, is stored as absorbed gas and free gas. Development of shale gas by multi-stage hydraulic fracturing has become a key technology for releasing this stored gas. We considered several flow mechanisms through both the multi-scaled porous media and the complex fracture networks caused by stimulated reservoir volume (SRV) fracturing. We developed a five-linear-flow model for fractured horizontal wells in dual-porosity formations, considering adsorption-desorption, diffusion, and percolation behavior. An analytical solution was obtained by means of a Laplace transformation. The number of fractures, the fracture half-length, the fracture conductivity, the width of the SRV, the inter-porosity flow coefficient, the volume of adsorbed gas, and the Langmuir pressure were considered as sensitive factors having significant influences on the production behavior of gas wells. Our analysis shows that the number of fractures, fracture half-length, fracture conductivity, and the width of the SRV are the main influencing factors controlling well productivity. The longer the fracture half-length, the longer the stable production period will be. An increase in the number of fractures also leads to a longer stable production period, but there is an optimum value for a given well. The time of the stable production period will level out once the dimensionless fracture conductivity (the fracture conductivity times the distance from the well to the tip of the fracture divided by reservoir permeability) increases to 100. The optimum value of the dimensionless fracture conductivity is between 10 and 100. The stable production period increases as the width of the SRV approaches the half distance of two fractures.

Selectively completed horizontal wells (SCHWs) can significantly reduce cost of completing wells and delay water breakthrough and prevent wellbore collapse in weak formations. Thus, SCHWs have been widely used in petroleum development industry. SCHWs can shorten the effective length of horizontal wells and thus have a vital effect on production. It is significant for SCHWs to study their rate decline and flux distribution in naturally fractured reservoirs. In this paper, by employing motion equation, state equation, and mass conservation equation, three-dimension seepage differential equation is established and corresponding analytical solution is obtained by Laplace transform and finite cosine Fourier transform. According to the relationship of constant production and wellbore pressure in Laplace domain, dimensionless rate solution is gotten under constant wellbore pressure in Laplace domain. Dimensionless pressure and pressure derivate curves and rate decline curves are drawn in log-log plot and seven flow regimes are identified by Stehfest numerical inversion. We compared the simplified results of this paper with the results calculated by Saphir for horizontal wells in naturally fractured reservoirs. The results showed excellent agreement. Some parameters, such as outer boundary radius, storativity ratio, cross-flow coefficient, number and length of open segments, can obviously affect the rate integral and rate integral derivative log-log curves of the SCHWs. The proposed model in this paper can help better understand the flow regime characteristics of the SCHWs and provide more accurate rate decline analysis of the SCHWs data to evaluate formation.

The use of multiple hydraulically fractured horizontal wells has been proven to be an efficient and effective way to enable shale gas production. Meanwhile, analytical models represent a rapid evaluation method that has been developed to investigate the pressure-transient behaviors in shale gas reservoirs. Furthermore, fractal-anomalous diffusion, which describes a sub-diffusion process by a non-linear relationship with time and cannot be represented by Darcy’s law, has been noticed in heterogeneous porous media. In order to describe the pressure-transient behaviors in shale gas reservoirs more accurately, an improved analytical model based on the fractal-anomalous diffusion is established. Various diffusions in the shale matrix, pressure-dependent permeability, fractal geometry features, and anomalous diffusion in the stimulated reservoir volume region are considered. Type curves of pressure and pressure derivatives are plotted, and the effects of anomalous diffusion and mass fractal dimension are investigated in a sensitivity analysis. The impact of anomalous diffusion is recognized as two opposite aspects in the early linear flow regime and after that period, when it changes from 1 to 0.75. The smaller mass fractal dimension, which changes from 2 to 1.8, results in more pressure and a drop in the pressure derivative.

The horizontal well incorporated with massive hydraulic fracturing has become a key and necessary technology to develop shale gas reservoirs efficiently, and transient pressure analysis is a practical method to evaluate the effectiveness of the fracturing. Until now, however, the related studies on the pressure of such wells have mainly focused on regular outer-boundaries, such as infinite, circular and rectangular boundary shapes, which do not always fulfill the practical conditions and, of course, could cause errors. By extending the boundary element method (BEM) into the application of multi-staged fractured horizontal wells, this paper presents a way of analyzing the transient pressure in arbitrary shaped shale gas reservoirs considering ad-/de-sorption and diffusion of the shale gas with the 'tri-porosity' mechanism model. The boundary integral equation can be obtained by coupling the fundamental solution of the Helmholtz equation with the dimensionless diffusivity equation. After discretizing the outer-boundaries and the fractures, the boundary integral equations are linearized and the coefficient matrix of the pressure on the boundaries is assembled, after which bottom-hole pressure can be calculated conveniently. Comparing the BEM solution with semi-analytical solution cases, the accuracy of the new solution can be validated. Then, the characteristic curves of the dimensionless pseudo pressure, as well as its derivative for a well in shale gas reservoirs, are drawn, based on which the parameters' sensitivity analyses are also conducted. This paper not only enriches the well testing theory and method in shale gas reservoirs, but also provides an effective method to solve problems with complex inner- and outer-boundaries.

The development of shale gas reservoir has becoming tremendously thrived around the world in recent years. The horizontal well drilling, advanced completion and massive hydraulic fracturing are the key technologies. How to accurately evaluate the production performance of the wells in these reservoirs, especially for multi-fractured horizontal wells, has always been the hot topic for the engineers and relevant researchers. In this paper, we presented a mathematical model to describe a horizontal well with multiple fractures in rectangular outer-boundary reservoir, which uses the Knudsen diffusion to describe the gas flow in nano-pores and also takes the slippage effect into account. By the method of point source function, Laplace transform and numerical discrete fracture methods, a semi-analytical solution of the model is obtained. Thereafter, the well test type curves and the production performance curves are plotted through Gauss elimination and Stehfest numerical inversion methods. Finally, the effects of related influential factors on well performance are also analyzed. It can be concluded that the Knudsen diffusion coefficient and slippage factor mainly affect the interporosity flow period when well producing with constant rate. When well producing with constant bottom hole pressure, the bigger the values of Knudsen diffusion coefficient and slippage factor are, the larger production rate will be. The research results summarized in this paper have significant theoretical meanings for the development, well test interpretations and production performance analysis of such unconventional gas reservoirs.

For hydraulic fractured well, how to evaluate the properties of fracture and formation are always tough jobs and it is very complex to use the conventional method to do that, especially for partially penetrating fractured well. Although the source function is a very powerful tool to analyze the transient pressure for complex structure well, the corresponding reports on gas reservoir are rare. In this paper, the continuous point source functions in anisotropic reservoirs are derived on the basis of source function theory, Laplace transform method and Duhamel principle. Application of construction method, the continuous point source functions in bi-zonal gas reservoir with closed upper and lower boundaries are obtained. Sequentially, the physical models and transient pressure solutions are developed for fully and partially penetrating fractured vertical wells in this reservoir. Type curves of dimensionless pseudo-pressure and its derivative as function of dimensionless time are plotted as well by numerical inversion algorithm, and the flow periods and sensitive factors are also analyzed. The source functions and solutions of fractured well have both theoretical and practical application in well test interpretation for such gas reservoirs, especial for the well with stimulated reservoir volume around the well in unconventional gas reservoir by massive hydraulic fracturing which always can be described with the composite model.

Permeability is one of the most important petrophysical parameters in the design of reasonable portion of production, but is the most difficult to obtain. It is vital to predict the value of permeability correctly for which has a significant effect on the oil/gas recovery. Well test analysis is a dynamic method to recognize the characteristics of reservoirs including permeability. According to the seepage problems of high condensed fluid, the continuous point source functions in bi-zonal gas reservoir under three kinds of boundaries are presented by constructing function method, which are useful but reported rarely. Physical models and pressure responses of fully/partial penetrating vertical well in bi-zonal composite reservoir are obtained on the basis of the source functions. Type curves of the wells are plotted through the Stehfest numerical inversion and the flow periods are analyzed. The authors validate their approach with a field case and demonstrate its application.

This paper established a triple porosity physical model in rectangular closed reservoirs to understand the complex fluid flowing mechanism and production behavior of multifractured horizontal wells in shale gas reservoirs, which is more appropriate for practical situation compared with previous ones. According to the seepage theory considering adsorption and desorption process in stable state, the gas production rate of a well producing at constant wellbore pressure was obtained by utilizing the methods of Green’s and source function theory and superposition principle. Meanwhile, the volume of adsorbed gas (GL) and the number of hydraulic fractures (M) as well as permeabilities of matrix system (km) and microfractures (kf) were discussed in this paper as sensitive factors, which have significant influences on the production behavior of the wells. The bigger the value ofGLis, the larger the well production rate will be in the later flowing periods, and the differences of production rate with the increasing ofMare small, which manifest that there is an optimumMfor a given field. Therefore, the study in this paper is of significant importance to understand the dynamic production declining performance in shale gas reservoirs.

Kickoff pressure gradient occurs when gas flows in a low permeability reservoir. In order to describe the issue of unsteady seepage flow in this kind of reservoir, an unsteady-state seepage flow mathematical model which takes kickoff pressure gradient into consideration is built up based on previous research and the characteristic of start-up pressure gradient existing in low velocity non-Darcy percolation flow (the fluid flow boundary expands outwards constantly). Combining the Laplace space analytic solution by using Green function method with the numerical approximation, the nonlinear mathematical model can be resolved. An equation for calculating single-well control radius is established, and a single-well control radius chart board is drawn up which can be used to calculate single well control radius. Case study indicates that the unsteady seepage flow model taking kickoff pressure gradient into account can correctly present the percolation mechanism and real production performance of a low-permeability gas reservoir. Key words: starting pressure gradient, low-permeability gas reservoir, unsteady seepage, well control radius

Low-permeability, low-production, low-abundance are the major features of the low-permeable gas reservoir. Commingling production has been widely used as an important method to improve the producing ability of gas reservoirs, however, the dynamic analysis and technical countermeasures for the multi-commingled wells are still not enough. Started from the multi-commingled filtration theory model, the pressure characteristics and the contribution rate of production of each layer, which are under the different multi-commingled well conditions such as reservoir properties differences, interlayer pressure difference and different damage degree, are studied in this paper. The results show that the contribution rate is fixed on formation coefficient before the pressure wave reaches the boundary; and after that, it is fixed on the storage ratio, it could be fixed on the formation coefficient, storage ratio and kickoff pressure gradient when considering the kickoff pressure gradient. Although there are significant differences in permeability between layers, the well test pressure curves have little difference. Taking the Da-niudi gas field for example, the single layer production rate, recovery and total recovery are studied by numerical simulation at different permeability ratio in multi-commingled well. The results show that the different permeability ratio contrast ( All the reservoirs in the multi-commingled well had better to be put into operation at one time. In this way not only the operation to succeed the late fill-holes can be saved, but also overall commingled layer recovery could increase.

This paper presents a mathematical model for the analysis of the transient pressure behaviour of a horizontal well in naturally fractured low permeability reservoirs, which takes the threshold pressure gradient (TPG) into consideration. Then, the solution of this model is obtained by using the method of Laplace transform and Fourier cosine transform, and the type curves are plotted by the Stehfest numerical inversion method. Pressure behaviour is analysed by examining the pressure drawdown curves, the derivative plots and the effect of the characteristic parameters. The typical pressure response of this reservoir is presented by the following five flow regimes: (1) wellbore storage and transition flow; (2) early radial flow in the vertical plane; (3) line flow in the horizontal plane; (4) matrix-fracture system transition flow; and (5) later pseudo-radial flow affected by the TPG. At the end, a field application manifests the correctness of the solutions derived in this paper, and the results have both theoretical and practical significance in predicting the production behaviour of carbonate reservoirs and evaluating fluid flow and transport in such a formation.