Singular integral equation for an electric dipole taking into account the finite metal conductivity from which it is made

A singular integral equation for an electric dipole has been obtained, which makes it possible to take into account the finite conductivity of the metal from which it is made. The derivation of the singular integral equation is based on the application of the Greens function for free space, written in a cylindrical coordinate system, taking into account the absence of the dependence of the field on the azimuthal coordinate, on a point source located on the surface of an electric dipole. Methods for its solution are proposed. In contrast to the well-known mathematical models of an electric dipole, built in the approximation of an ideal conductor, the use of the singular integral equation obtained in this work makes it possible to take into account heat losses and calculate the efficiency.

Production Analysis for Fractured Vertical Well in Coal Seam Reservoirs with Stimulated Reservoir Volume

The development and utilization of coalbed methane is of great significance to reduce carbon dioxide emission. Through the research, this paper presents a fast analytical solution method for the productivity of coalbed methane reservoir with finite-conductivity fractured well and stimulated reservoir volume region. Based on the dual-porosity flowing mechanism, combined with the Langmuir adsorb equation, Fick diffusion law, and Darcy law, a mathematical model considering diffusion in matrix and transport in natural fracture system is established, using spherical matrix to describe the transient steady-state sorption, and using cubic matrix to describe the pseudosteady-state sorption. Then, combined with the inner system and outer system, the analytical solution was obtained. Furthermore, the accuracy of the solution was validated against a numerical simulation. According to the Duhamel principle, the effect of wellbore storage and skin factor was got. Due to the SRV region, the linear flow and radial flow will appear before the pressure wave reach the outer region. And then, based on the pressure analysis result, we will have made the sensitivity analysis with different influence parameter. The result reveals that storage coefficient and conductivity factor mainly influence the early time; the permeability ratio and dimensionless SRV region radius mainly influence the property of SRV region. Finally, the analytical solution of the new model was applied to field history match. This model takes into account the adsorption and desorption characteristics of coalbed methane, as well as the SRV zones generated during fracturing. The calculation speed of the new model is increased while the calculation accuracy is retained, and the intensity of software application is reached. The model achieves the purpose of rapid evaluation and accurate prediction of gas well productivity and obtains a set of productivity evaluation method suitable for coalbed methane reservoir with fractured vertical well, which provides a basis for the development and productivity evaluation of coalbed methane reservoir in domestic and international cooperation.

New Analytical Approach to Operational Assessment of Fractured Well Productivity with Variable Permeability

Requirements of targeted optimization are imposed on the hydraulic fracturing operations carried out in the conditions of borderline economic efficiency of fields taking into account geological and technological features. Consequently, the development of new analytical tools foranalyzing and planning the productivity of fractured wells, taking into account the structuralfeatures of the productive reservoir and inhomogeneous distribution of the fracture conductivity, is becoming highly relevant. The paper proposes a new approach of assessing the vertical hydraulic fracture productivityin a rectangular reservoir in a pseudo-steady state, based on reservoir resistivity concept described in the papers of Meyer et al. However, there is a free parameter in the case of modeling the productivity of a hydraulic fracture by the concept. The parameter describes the distribution of the inflow along the plane of the fracture. This paper presents a systematic approach to determining of the parameter. The resulting model allows to conduct an assessment of the influence of various complications in the fracture on the productivity index. During the research a method of determining the free parameter was developed,it was based on the obtained dependence of the inflow distribution on the coordinate along the fracture of finite conductivity. The methodology allowed to refine existent analytical solution of the Meyer et al. model, which, in turn, allowed to assess the influence of different fracture damages in the hydraulic fracture on the productivity index of the well. The work includes the cases of the presence of fracture damages at the beginning and at the end of the fracture. A hydraulic fracture model was built for each of the types of damages, it was based on the developed method, and also the solution of dimensionless productivity ratio was received. The results of the obtained solution were confirmed by comparison with the numerical solutions of commercial simulators and analytical models available in the literature. The advantage of the methodology is the resulting formulas for well productivity are relatively simple, even for exotic cases ofvariable conductivity fractures. The approaches and algorithms described in the paper assume the calculation of the productivity of a hydraulic fracture with variable conductivity and the presence of other complicatingfactors.The methodology of the paper can be used for analysis and diagnosis problems with formation hydraulic fracturing. The efficiency of the calculations allows using the presented methodology to solve inverse problems of determining the efficiency of the hydraulic fracturing operation.

Numerical RTA in Tight Unconventionals

We extend the numerically-assisted RTA workflow proposed by Bowie and Ewert (2020) to (a) all fluid systems and (b) finite conductivity fractures. The simple, fully-penetrating planar fracture model proposed is a useful numerical symmetry element model that provides the basis for the work presented in this paper. Results are given for simulated and field data. The linear flow parameter (LFP) is modified to include porosity (LFPꞌ=LFP√φ). The original (surface) oil in place (OOIP) is generalized to represent both reservoir oil and reservoir gas condensate systems, using a consistent initial total formation volume factor definition (Bti) representing the ratio of a reservoir HCPV containing surface oil in a reservoir oil phase, a reservoir gas phase, or both phases. With known (a) well geometry, (b) fluid initialization (PVT and water saturation), (c) relative permeability relations, and (d) bottomhole pressure (BHP) time variation (above and below saturation pressure), three fundamental relationships exist in terms of LFPꞌ and OOIP. Numerical reservoir simulation is used to define these relationships, providing the foundation for numerical RTA, namely that wells: (1) with the same value of LFPꞌ, the gas, oil and water surface rates will be identical during infinite-acting (IA) behavior; (2) with the same ratio LFPꞌ/OOIP, producing GOR and water cut behavior will be identical for all times, IA and boundary dominated (BD); and (3) with the same values of LFPꞌ and OOIP, rate performance of gas, oil, and water be identical for all times, IA and BD. These observations lead to an efficient, semi-automated process to perform rigorous RTA, assisted by a symmetry element numerical model. The numerical RTA workflow proposed by Bowie and Ewert solves the inherent problems associated with complex superposition and multiphase flow effects involving time and spatial changes in pressure, compositions and PVT properties, saturations, and complex phase mobilities. The numerical RTA workflow decouples multiphase flow data (PVT, initial saturations and relative permeabilities) from well geometry and petrophysical properties (L, xf, h, nf, φ, k), providing a rigorous yet efficient and semi-automated approach to define production performance for many wells. Contributions include a technical framework to perform numerical RTA for unconventional wells, irrespective of fluid type. A suite of key diagnostic plots associated with the workflow is provided, with synthetic and field examples used to illustrate the application of numerical simulation to perform rigorous RTA. Semi-analytical models, time, and spatial superposition (convolution), pseudopressure and pseudotime transforms are not required.

Pressure and reciprocal rate transient analysis for finite-conductivity vertical fractured hydrocarbon wells under trilinear flow

Hydraulic well fracturing has been an important practice for well stimulation along the history of the hydrocarbon industry. Therefore, accurate and practical assessment of the fracturing job should be of importance. Interpretation of well pressure and rate test data in vertical finite-conductivity fractured hydrocarbon wells by the TDS methodology and conventional analysis are presented in this paper. The accuracy and practicality of the developed equations are successfully tested with synthetical examples and a field case.

A Practical Solution Model for Transient Pressure Behavior of Multistage Fractured Horizontal Wells with Finite Conductivity in Tight Oil Reservoirs

Fractured horizontal wells have been widely used to develop unconventional oil and gas reservoirs. In previous studies, most studies on the transient pressure behavior of multistage horizontal wells were based on the assumption of single porosity medium, in which the coupling relationship of natural fractures and artificial fractures was not taken into account or artificial fractures were assumed to be infinitely conductive. In this paper, the fracture is finite conductive, which means that there is flow resistance in the fracture. Based on point-source method and superposition principle, a transient model for multistage fractured horizontal wells, which considers the couple of fracture flow and reservoir seepage, is built and solved with the Laplace transformation. The transient pressure behavior in multistage fractured horizontal wells is discussed, and effects of influence factors are analyzed. The result of this article can be used to identify the response characteristic of fracture conductivity to pressure and pressure differential and provide theoretical basis for effective development of tight oil reservoirs. The findings of this study can help for better understanding of transient pressure behavior of multistage fractured horizontal wells with finite conductivity in tight oil reservoirs.

Productivity Analysis of Volume Fractured Wells under Different Working Systems

The volume fracturing technique has been widely used to improve the productivity of ultralow-permeability reservoirs. This paper presents a new semianalytical model to simulate the pressure transient and production behaviour of finite conductivity vertical fractured wells with stimulated reservoir volume (SRV) in heterogeneous reservoirs. The model is based on the five-linear flow model, the Warren-Root model, and fracture conductivity influence function. The model is validated by comparing its results with a numerical model. One novelty of this model is its consideration of three different kinds of production prediction models. Constant rate, constant pressure, and compound working systems are taken into account. This paper illustrates the effects of the SRV size and shape, mobility ratio, initial flow rate, limiting wellbore pressure, and hydraulic fracture parameters under different working systems. Results show that the SRV and parameters of fractures have a significant influence on long-term well performance. Moreover, the initial rate can extend the constant rate period by 418%, and limiting wellbore pressure can effectively improve the cumulative recovery rate by 23%. Therefore, this model can predict long-term wells’ behaviour and provide practical guiding significance for hydraulic fracturing design.

Unique Fall Off Signatures for Stage Fracture Characterization, Actual Field Cases

Multistage hydraulic fracturing is the common stimulation technique for shale formations. The treatment design, formation in-situ stress, and reservoir heterogeneity govern the fracture network propagation. Different techniques have been used to evaluate the fracture geometry and the completion efficiency including Chemical Tracers, Microseismic, Fiber Optics, and Production Logs. Most of these methods are post-fracture as well as time and cost intensive processes. The current study presents the use of fall-off data during and after stage fracturing to characterize producing surface area, permeability, and fracture conductivity. Shut-in data (15-30 minutes) was collected after each stage was completed. The fall-off data was processed first to remove the noise and water hammer effects. Log-Log derivative diagnostic plots were used to define the flow regime and the data were then matched with an analytical model to calculate producing surface area, permeability, and fracture conductivity. Diagnostic plots showed a unique signature of flow regimes. A long period of a spherical flow regime with negative half-slope was observed as an indication for limited entry flow either vertically or horizontally. A positive half-slope derivative represents a linear flow regime in an infinitely conductive tensile fracture. The quarter-slope derivative was observed in a bilinear flow regime that represents a finite conductivity fracture system. An extended radial flow regime was observed with zero slope derivative which represents a highly shear fractured network around the wellbore. For a long fall-off period, formation recharge may appear with a slope between unit and 1.5 slopes derivative, especially in over-pressured dry gas reservoirs. Analyzing fall-off data after stages are completed provides a free and real-time investigation method to estimate the fracture geometry and a measure of completion efficiency. Knowing the stage properties allows the reservoir engineer to build a simulation model to forecast the well performance and improve the well spacing.