Measurement and Modelling of Gas Condensate Flow Around Rock Perforation

In the petroleum production industry wells are mostly cased and perforated in the producing formation. Perforation characteristics such as size, length, number of perforation tunnels and their arrangements as well as fluid and rock properties determine fluid flow behaviour in the wellbore region, hence, well productivity. Flow of gas and condensate around a perforation tunnel (including the damaged zone) has been studied by performing steady-state core experiments and simulating the results numerically, using a finite element modelling approach. The model allows for the changes in fluid properties and accounts for the coupling of the two phases and the inertial effect using a fractional flow based correlation. The results indicated that different sets of thickness-permeability (h−k) values obtained from matching single-phase flow performance could be assigned to the damaged zone around perforation to represent the two-phase flow performance. The status of the tip of the perforation for two extreme cases of totally closed and fully open was investigated and found to have a minimal effect on the performance of the system.     

Acoustic sounding of perforated wellbores by short waves

The reflection and transmission of harmonic waves and waves of finite duration through the boundary of the perforated zone of a cased wellbore filled with a fluid are studied. A model of the plane fluid flow in the well (in a quasi-one-dimensional approximation) and filtration absorption of the fluid by the porous medium surrounding the well is proposed. The effect of the quality of the perforation (length of perforation tunnels) on the evolution of the waves reflected from the boundary of the perforated zone of the well is studied.     

High Velocity Flow in and Around Long Perforation Tunnels

In oil industries, wells are mostly cased and perforated rather than completing the producing formation as open-hole. This study concentrates on the steady-state flow behaviour of a single-phase fluid in and around perforated completion tunnels (up to 50 inches long) including inertial effects (Non-Darcy flow). It is shown that the pressure drop inside long perforated tunnels under high flow velocity conditions is negligible compared to that around the perforated region within the porous medium. The results also indicate that the impact of perforation parameters varies with increasing fluid velocity but approaches an asymptotic value at very high flow velocity. The perforation length is the most important parameter whereas perforation radius, rock and fluid properties have little impact on the perforation performance.     

An experimental investigation of the static pressure fluctuation mechanism for porous transonic wind tunnel wall configurations

An experimental investigation of the static pressure fluctuation generation mechanism was performed for different transonic wind tunnel test section perforated wall configurations. Different hole diameters and geometrical configurations were investigated. Most tests were carried out with isolated perforations, while some were done with a three hole, 16° perforation pattern. To suppress the oscillation amplitudes generated by perforations, splitter-plates as flow conditioning devices along the perforations were implemented on a large number of perforated transonic test section wall samples. It was found that all the hole configurations tested, regardless of diameter or shape, resonate at discrete frequencies which order themselves along several modes.     

Enhancement of natural convection heat transfer from a fin by triangular perforation of bases parallel and toward its tip

This study examines the heat transfer enhancement from a horizontal rectangular fin embedded with triangular perforations （their bases parallel and toward the fin tip） under natural convection. The fin＇s heat dissipation rate is compared to that of an equivalent solid one. The parameters considered are geometrical dimensions and thermal properties of the fin and the perforations. The gain in the heat transfer enhancement and the fin weight reduction due to the perforations are considered. The study shows that the heat dissipation from the perforated fin for a certain range of triangular perforation dimensions and spaces between perforations result in improvement in the heat transfer over the equivalent solid fin. The heat transfer enhancement of the perforated fin increases as the fin thermal conductivity and its thickness are increased.     

Analytical Solutions to Gas Flow Problems in Low Permeability Porous Media

In most of conventional porous media the flow of gas is basically controlled by the permeability and the contribution of gas flow due to gas diffusion is ignored. The diffusion effect may have significant impact on gas flow behavior, especially in low permeability porous media. In this study, a dual mechanism based on Darcy flow as well as diffusion is presented for the gas flow in homogeneous porous media. Then, a novel form of pseudo pressure function was defined. This study presents a set of novel analytical solutions developed for analyzing steady-state and transient gas flow through porous media including effective diffusion. The analytical solutions are obtained using the real gas pseudo pressure function that incorporates the effective diffusion. Furthermore, the conventional assumption was used for linearizing the gas flow equation. As application examples, the new analytical solutions have been used to design new laboratory and field testing method to determine the porous media parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs. Then, permeability (k) and effective diffusion coefficient (D _e) was determined; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Approximate Semi-analytical Solution for Injection–Falloff–Production Well Test: An Analytical Tool for the In Situ Estimation of Relative Permeability Curves

An injection–falloff–production test (IFPT) was originally proposed in Chen et al. (in: SPE conference paper, 2006. doi: 10.2118/103271-MS, SPE Reserv Eval Eng 11(1):95–107, 2008) as a well test for the in situ estimation of two-phase relative permeability curves to be used for simulating multiphase flows in porous media. Hence, we develop an approximate semi-analytical solution for the two-phase saturation distribution in an oil–water system during the flowback period of an IFPT according to the mathematical theory of waves. In fact, we show that the weak solution we construct for the saturation equation for the flowback period satisfies the Oleinik entropy condition and hence is unique. In addition, we allow the governing relative permeabilities during the flowback period to be different from the relative permeabilities during injection. Using the saturation solution with the steady-state pressure theory of Thompson and Reynolds, we obtain a solution for the wellbore pressure during the flowback period. By comparing results from our solution with those from a commercial numerical simulator, we show that our approximate semi-analytical solution yields accurate saturation profiles and bottom hole pressures history. The use of very small time steps and a highly refined radial grid is necessary to generate a good solution from a reservoir simulator. The approximate analytical pressure solution developed is used as a forward model to match pressure and water flow rate data from an IFPT in order to estimate reservoir rock absolute permeability and skin factor in conjunction with in situ imbibition and drainage water–oil relative permeabilities.     

Forced convective heat transfer enhancement with perforated pin fins

Increasing miniaturization of high speed multi-functional electronics demands ever more stringent thermal management. The present work investigates experimentally and numerically the use of staggered perforated pin fins to enhance the rate of heat transfer in these devices. In particular, the effects of the number of perforations and the diameter of perforation on each pin are studied. The results show that the Nusselt number for the perforated pins is 45 % higher than that for the conventional solid pins and it increases with the number of perforation. Pressure drop with perforated pins is also reduced by 18 % when compared with that for solid pins. Perforations produce recirculations in the x–y as well as the x–z planes downstream of the pins which effectively increase convective heat transfer. However, thermal dissipation decreases significantly when the ratio of pin diameter to perforation diameter exceeds 0.375. This is due to both a reduction in the number of perforation per pin and the decrease in the axial heat conduction along the pin.     

Gas Flow in Porous Media With Klinkenberg Effects

Gas flow in porous media differs from liquid flow because of the large gas compressibility and pressure-dependent effective permeability. The latter effect, named after Klinkenberg, may have significant impact on gas flow behavior, especially in low permeability media, but it has been ignored in most of the previous studies because of the mathematical difficulty in handling the additional nonlinear term in the gas flow governing equation. This paper presents a set of new analytical solutions developed for analyzing steady-state and transient gas flow through porous media including Klinkenberg effects. The analytical solutions are obtained using a new form of gas flow governing equation that incorporates the Klinkenberg effect. Additional analytical solutions for one-, two- and three-dimensional gas flow in porous media could be readily derived by the following solution procedures in this paper. Furthermore, the validity of the conventional assumption used for linearizing the gas flow equation has been examined. A generally applicable procedure has been developed for accurate evaluation of the analytical solutions which use a linearized diffusivity for transient gas flow. As application examples, the new analytical solutions have been used to verify numerical solutions, and to design new laboratory and field testing techniques to determine the Klinkenberg parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs from The Geysers geothermal field. We show that this new approach and the traditional method of Klinkenberg yield similar results of Klinkenberg constants for the laboratory tests; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Shock wave attenuation in partially confined channels

An approximate analytical solution is presented for the attenuation of planar shock waves in channels with perforated walls. The problem is considered as quasi-one-dimensional. Good agreement is found between the theoretical results and available experimental data regarding the rate of shock wave attenuation within the range of initial shock Mach numbers between 1.1 and 4 and perforation ratios between 4.5 × 10^–3 and 0.53. A correlation for the discharge coefficient of a single hole perforation is presented which gives quantitatively good agreement with particular experimental observations.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

考虑诱导缝多段压裂水平井非均质改造压力动态模型研究

为了准确模拟致密油藏水平井大规模压裂形成复杂裂缝网络系统和非均质储层井底压力变化,建立考虑诱导缝矩形非均质储层多段压裂水平井不稳定渗流数学模型,耦合裂缝模型与储层模型得到有限导流裂缝拉普拉斯空间井底压力解,对两种非均质储层模型分别利用数值解、边界元和已有模型验证其准确性.基于压力导数曲线特征进行流动阶段划分和参数敏感性分析,得到以下结果:和常规压裂水平井井底压力导数曲线相比较,理想模式下,考虑诱导缝影响时特有的流动阶段是综合线性流阶段、诱导缝向压裂裂缝“补充”阶段、储层线性流动阶段和拟边界控制流阶段.诱导缝条数的增加加剧了综合线性流阶段的持续时间,降低了流体渗流阻力,早期阶段压力曲线越低;当诱导缝与压裂裂缝导流能力一定时,裂缝导流能力越大,线性流持续时间越长;当所有压裂裂缝不在一个区域时,沿井筒方向两端区域低渗透率弱化了低渗区域诱导缝流体向压裂裂缝“补充”阶段,因此,沿井筒方向两端区域渗透率越低,早期阶段压力曲线越高;当所有压裂裂缝在一个区域时,渗透率变化只影响径向流阶段之后压力曲线形态,外区渗透率越低,早期径向流阶段之后压力曲线越高.通过实例验证,表明该模型和方法的实用性和准确性.     

THE PERFORATION PARAMETER OPTIMIZATION FOR MULTI FRACTURE INITIATION IN HYDRAULIC FRACTURING

针对复杂地质近井筒多裂缝起裂的难题, 结合岩体力学与损伤力学相关理论, 建立了近井筒地层流固——损伤力学模型, 采用数值求解获得了射孔壁面的应力及损伤状态, 计算表明:初始地应力最小值越小, 地层的起裂压力越低;射孔方位角越小, 起裂压力越低. 随着射孔方位角增大, 近井筒微裂缝为45° 夹角的拐折裂缝, 起裂压力增大, 基于此, 对于正断层及顺滑断层地应力类型, 取射孔方位角小于30°, 螺旋射孔夹角30°或45°, 对于逆断层地应力类型, 螺旋射孔夹角取45°, 可有效地实现多裂缝起裂.     

A perturbation method for mixed boundary-value problems in pressure transient testing

Data recorded during a well test is interpreted for formation parameters, such as permeability, by comparing the measured pressure transient with that predicted by a mathematical model of the system. In single-phase homogeneous situations, this model is based on the linear diffusion equation. Despite the simplicity of this equation, it is often difficult or laborious to construct exact solutions, in some cases because the relevant problem is of mixed boundary-value character. Here we describe an approximate calculational method which, with little effort, gives good results in a variety of well test problems of this type.We first prove a rather general perturbation theorem, and then apply it in three illustrative cases. The first example provides a test of our basic result for the case of a fully penetrating vertical fracture, for which an exact (though nonelementary) solution is known. The second example is that of partially penetrating or horizontal wells; it is shown that our general result can provide a mathematical basis for the so-called pressure-averaging technique in which the pressure-flux relationship is approximated by assuming that the flux is uniform along the well and then computing the spatial average of the wellbore pressure. Our third example is a new result for the steady-state pressure drop due to flow into a small circular hole on the surface of an impermeable cylinder. This example has relevance for the Schlumberger RFT tool, which withdraws fluid from the formation via a circular probe which penetrates the mudcake surrounding the borehole wall. Numerical results for the shape factor which represents the effect of the borehole curvature are provided.     

Time-dependent numerical simulation of viscoelastic flow and stability

The 4×4 mixed finite-element method is extended in order to calculate time-dependent viscoelastic flow. The basic algorithm uses a fully implicit technique with a predictor-corrector control of the time step for monitoring the accuracy. Excellent agreement is found with analytical test cases. Several decoupled schemes are also developed with a view to reducing the cost of the time-dependent calculations. However, none of them reaches that goal on actual flow problems because of a drastic reduction of the time step.The time-dependent algorithm is used for verifying the stability of steady-state viscoelastic flow solutions. The approach consists of using a steady-state solution as a set of initial conditions for a time-dependent algorithm with the hope that, in case of instability, the system will lead to a stable solution. More precisely, we consider the flow of an Oldroyd-B fluid through a four-to-one contraction. Starting from a steady-state flow, we impose a pressure impulse in the entry section and calculate the approach toward another steady-state. For planar contractions, we do not constrain the flow to be symmetric while, for circular contractions, the flow is endowed with swirling capabilities. It is found that the stable or unstable character of the flow depends upon the mesh as well as upon the method of discretization.     

Forced Vibration of Continuous System with Unsymmetrical Collision Characteristics

This paper deals with steady-state response of a continuous system with nonlinear boundary conditions which are motion-limiting constraint. An analytical method of approximate solution for the continuous system with unsymmetrical collision characteristics in which the beam end collides with a stop once in one period of its vibration is presented. Some numerical results of the approximate solution are shown. Contrary to the case of continuous system with symmetrical collision characteristics, the resonance curves of nonlinear response of approximate solution are shown as discontinuous line. Some numerical results of a continuous system with no hysteresis damping are compared with those of a continuous system with hysteresis damping and a single-degree-of-freedom system.     

Assessment of Stimulated Area Growth During High-Pressure Hydraulic Stimulation of Fractured Subsurface Reservoir

Hydraulic stimulation is performed by high-pressure fluid injection, which permanently increases the permeability of a volume of rock, typically transforming it from the microdarcy into the millidarcy range. After a period of stimulation, fluid injection and recovery boreholes are introduced into the stimulated rock volume, and heat is extracted by water circulation. In the present study a simplified mathematical model of non steady-state hydraulic stimulation is proposed and analyzed. Fluid flow is assumed to be radial, injected flow rate constant; and fluid density, rock porosity, and permeability depend on fluid pressure. The conventional boundary of the growing stimulated rock volume is introduced as a surface where the porosity and permeability of the stimulated rock exhibit a sharp decline and remain constant within the undisturbed area. The problem is solved analytically by a modified method of integral correlations. As a result, approximate close-form solutions for pressure distributions in the stimulated and nonstimulated (undisturbed) areas are obtained, and an equation for the moving boundary of the stimulated volume is derived. The correctness of the approximate solution is validated by comparison to an exact self-similar solution of the problem obtained for the particular case when the well’s radius is assumed to be equal to zero.     

Three-point bending of honeycomb sandwich beams with facesheet perforations

A novel square honeycomb-cored sandwich beam with perforated bottom facesheet is investigated under threepoint bending,both analytically and numerically.Perforated square holes in the bottom facesheet are characterized by the area ratio of the hole to intact facesheet(perforation ratio).While for large-scale engineering applications like the decks of cargo vehicles and transportation ships,the perforations are needed to facilitate the fabrication process(e.g.,laser welding)as well as service maintenance,it is demonstrated that these perforations,when properly designed,can also enhance the resistance of the sandwich to bending.For illustration,fair comparisons among competing sandwich designs having different perforation ratios but equal mass is achieved by systematically thickening the core webs.Further,the perforated sandwich beam is designed with a relatively thick facesheet to avoid local indention failure so that it mainly fails in two competing modes:(1)bending failure,i.e.,yielding of beam cross-section and buckling of top facesheet caused by bending moment;(2)shear failure,i.e.,yielding and buckling of core webs due to shear forcing.The sensitivity of the failure loads to the ratio of core height to beam span is also discussed for varying perforation ratios.As the perfo-ration ratio is increased,the load of shear failure increases due to thickening core webs,while that of bending failure decreases due to the weakening bottom facesheet.Design of a sandwich beam with optimal perforation ratio is realized when the two failure loads are equal,leading to significantly enhanced failure load(up to 60%increase)relative to that of a non-perforated sandwich beam with equal mass.     

Approximate analytical solutions and approximate value of skin friction coefficient for boundary layer of power law fluids

This paper presents a theoretical analysis for laminar boundary layer flow in a power law non-Newtonian fluids.The Adomian analytical decomposition technique is presented and an approximate analytical solution is obtained.The approximate analytical solution can be expressed in terms of a rapid convergent power series with easily computable terms.Reliability and efficiency of the approximate solution are verified by comparing with numerical solutions in the literature.Moreover,the approximate solution can be successfully applied to provide values for the skin friction coefficient of the laminar boundary layer flow in power law non-Newtonian fluids.     

孔眼流入对水平井中流动影响的实验研究

设计并建立了水平井筒变质量流动模拟实验装置。对具有射孔完井的水平井筒变质量流动进行了实验研究,利用先进的数据采集系统获取了大量的实验数据。对水平井井筒单孔眼段流动阻力损失分析可知,流动阻力由三部分组成：管壁摩擦阻力、加速损失和混合损失。对实验数据进行处理得出了混合损失和主流流速与孔眼流速之间的关系。对于一定的主流流速,混合损失随孔眼流速的增大而增大,同时,对于一定的孔眼流速,混合损失随主流流速的增大而增大,此时孔眼的流入对水平井筒中的流动造成的影响比较显著。孔眼的流入增加了井筒中流动的复杂性。对实验数据进行的回归结果表明,在本实验条件和范围内,主流流速和孔眼流速是造成混合损失的主要因素。     

Boundary-layer flow characteristics on perforated plates and a method of approximately computing their temperature under combined cooling

Results of an experimental investigation of the heat elimination from perforated plates to cooling air during its motion along walls and partial drainage through the perforations are elucidated. Boundary-layer characteristics on perforated plates, their thermal state, and their efflux coefficients are presented for a set of orifices. An approximate method to compute the temperature of the permeable wall under combined cooling is given and its domain of application is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 182–187, March–April, 1977.