An efficient algorithm to compute transient pressure responses of slanted wells with arbitrary inclination in reservoirs

Compared with vertical and horizontal wells, the solution and computation of transient pressure responses of slanted wells are more complex. Vertical and horizontal wells are both simplified cases of slanted wells at particular inclination, so the model for slanted wells is more general and more complex than other models for vertical and horizontal wells. Many authors have studied unsteady-state flow of fluids in slanted wells and various solutions have been proposed. However, until now, few of the published results pertain to the computational efficiency. Whether in the time domain or in the Laplace domain, the computation of integration of complex functions is necessary in obtaining pressure responses of slanted wells, while the computation of the integration is complex and time-consuming. To obtain a perfect type curve the computation time is unacceptable even with an aid of high-speed computers. The purpose of this paper is to present an efficient algorithm to compute transient pressure distributions caused by slanted wells in reservoirs. Based on rigorous derivation, the transient pressure solution for slanted wells of any inclination angle is presented. Assuming an infinite-conductivity wellbore, the location of the equivalent-pressure point is determined. More importantly, according to the characteristics of the integrand in a transient pressure solution for slanted wells, the whole integral interval is partitioned into several small integral intervals, and then the method of variable substitution and the variable step-size piecewise numerical integration are employed. The amount of computation is significantly reduced and the computational efficiency is greatly improved. The algorithm proposed in this paper thoroughly solved the difficulty in the efficient and high-speed computation of transient pressure distribution of slanted wells with any inclination angle.

Analysis on setting airshaft at mid-tunnel to reduce transient pressure variation

Setting airshafl is one of the methods efficient to control the aerodynamic effects on a train traveling through a tunnel. By using numeral simulation, analysis is carried out on the effect of airshafl on transient pressures generated in cabin. After setting airshaft, the magnitude of pressure fluctuation in cabin is reduced nearly 40%. By analyzing the role of airshaft in alleviating pressure fluctuation, a formula to determine the optimal airshaft position is deduced.

Processing and analysis of transient pressure measurements from permanent down-hole gauges

With permanent down-hole gauges （PDGs） widely installed in oilfields around the world in recent years, a continuous stream of transient pressure data in real time is now available, which motivates a new round of research interests in further developing pressure transient processing and analysis techniques. Transient pressure measurements from PDG are characterized by long term and high volume data. These data are recorded under unconstrained circumstances, so effects due to noise, rate fluctuation and interference from other wells cannot be avoided. These effects make the measured pressure trends decline or rise and then obscure or distort the actual flow behavior, which makes subsequent analysis difficult. In this paper, the problems encountered in analysis of PDG transient pressure are investigated. A newly developed workflow for processing and analyzing PDG transient pressure data is proposed. Numerical well testing synthetic studies are performed to demonstrate these procedures. The results prove that this new technique works well and the potential for practical application looks very promising.

A Transient-Pressure-Based Numerical Approach for Interlayer Identification in Sand Reservoirs

Almost all sandstone reservoirs contain interlayers. The identification and characterization of these interlayers iscritical for minimizing the uncertainty associated with oilfield development and improving oil and gas recovery.Identifying interlayers outside wells using identification methods based on logging data and machine learning isdifficult and seismic-based identification techniques are expensive. Herein, a numerical model based on seepageand well-testing theories is introduced to identify interlayers using transient pressure data. The proposed modelrelies on the open-source MATLAB Reservoir Simulation Toolbox. The effects of the interlayer thickness, position,and width on the pressure response are thoroughly investigated. A procedure for inverting interlayer parametersin the reservoir using the bottom-hole pressure is also proposed. This method uses only transient pressuredata during well testing and can effectively identify the interlayer distribution near the wellbore at an extremelylow cost. The reliability of the model is verified using effective oilfield examples.

Pressure transient characteristics of non-uniform conductivity fractured wells in viscoelasticity polymer flooding based on oil-water two-phase flow

Polymer flooding in fractured wells has been extensively applied in oilfields to enhance oil recovery.In contrast to water,polymer solution exhibits non-Newtonian and nonlinear behavior such as effects of shear thinning and shear thickening,polymer convection,diffusion,adsorption retention,inaccessible pore volume and reduced effective permeability.Meanwhile,the flux density and fracture conductivity along the hydraulic fracture are generally non-uniform due to the effects of pressure distribution,formation damage,and proppant breakage.In this paper,we present an oil-water two-phase flow model that captures these complex non-Newtonian and nonlinear behavior,and non-uniform fracture characteristics in fractured polymer flooding.The hydraulic fracture is firstly divided into two parts:high-conductivity fracture near the wellbore and low-conductivity fracture in the far-wellbore section.A hybrid grid system,including perpendicular bisection(PEBI)and Cartesian grid,is applied to discrete the partial differential flow equations,and the local grid refinement method is applied in the near-wellbore region to accurately calculate the pressure distribution and shear rate of polymer solution.The combination of polymer behavior characterizations and numerical flow simulations are applied,resulting in the calculation for the distribution of water saturation,polymer concentration and reservoir pressure.Compared with the polymer flooding well with uniform fracture conductivity,this non-uniform fracture conductivity model exhibits the larger pressure difference,and the shorter bilinear flow period due to the decrease of fracture flow ability in the far-wellbore section.The field case of the fall-off test demonstrates that the proposed method characterizes fracture characteristics more accurately,and yields fracture half-lengths that better match engineering reality,enabling a quantitative segmented characterization of the near-wellbore section with high fracture conductivity and the far-wellbore section with low fracture conductivity.The novelty of this paper is the analysis of pressure performances caused by the fracture dynamics and polymer rheology,as well as an analysis method that derives formation and fracture parameters based on the pressure and its derivative curves.

Research on numerical simulation of transient pressure for high-speed train passing through the most unfavourable length tunnel

The length of high-speed railway tunnel is an important factor affecting transient pressure of high-speed train.When the tunnel length is the most unfavourable,the transient pressure changes in the tunnel and on the surface of the train are the most severe,which may affect the safe operation of the train or damage the structure in the tunnel.Based on the three-dimensional,compressible,unsteady N-S equation and finite volume method,this paper uses the CFD numerical simulation method to study the change and amplitude distribution of the transient pressure on the train surface and the tunnel when a high-speed train passes through the most unfavourable length tunnel.A fast calculation method is proposed to save the cost of calculation;it has great applicability of pressure amplitude.The results show that the pressure distribution in the tunnel and on the surface of the train is affected by the train speed,the length of the train and the position of the measuring point.The minimum negative peak value in the tunnel appears at the position where the superposition phenomenon is the most severe,and the position will change with the speed of the train.There are two negative peak waveforms of the train surface pressure,and the first waveformn is greatly affected by the train speed.It improves a reference for studying the strength requirement of the most unfavourable length tunnels and trains,and ensures the safe operation of trains in tunnels of different lengths.

Well interference evaluation considering complex fracture networks through pressure and rate transient analysis in unconventional reservoirs

Severe well interference through complex fracture networks(CFNs)can be observed among multi-well pads in low permeability reservoirs.The well interference analysis between multi-fractured horizontal wells(MFHWs)is vitally important for reservoir effective development.Well interference has been historically investigated by pressure transient analysis,while it has shown that rate transient analysis has great potential in well interference diagnosis.However,the impact of complex fracture networks(CFNs)on rate transient behavior of parent well and child well in unconventional reservoirs is still not clear.To further investigate,this paper develops an integrated approach combining pressure and rate transient analysis for well interference diagnosis considering CFNs.To perform multi-well simulation considering CFNs,non-intrusive embedded discrete fracture model approach was applied for coupling fracture with reservoir models.The impact of CFN including natural fractures and frac-hits on pressure and rate transient behavior in multi-well system was investigated.On a logelog plot,interference flow and compound linear flow are two new flow regimes caused by nearby producers.When both NFs and frac-hits are present in the reservoir,frac-hits have a greater impact on well#1 which contains frac-hits,and NFs have greater impact on well#3 which does not have frac-hits.For all well producing circumstances,it might be challenging to see divergence during pseudosteady state flow brought on by frac-hits on the logelog plot.Besides,when NFs occur,reservoir depletion becomes noticeable in comparison to frac-hits in pressure distribution.Application of this integrated approach demonstrates that it works well to characterize the well interference among different multi-fractured horizontal wells in a well pad.Better reservoir evaluation can be acquired based on the new features observed in the novel model,demonstrating the practicability of the proposed approach.The findings of this study can help for better evaluating well interference degree in multi-well systems combing PTA and RTA,which can reduce the uncertainty and improve the accuracy of the well interference analysis based on both field pressure and rate data.

TRANSIENT PRESSURE BEHAVIOR FOR HORIZONTAL WELL IN DOUBLE-POROUS MEDIA

This paper presents the solutions of three -dimensional unsteady flow in Laplace domain for horizontal well in double porous media with vertical feed, using the method both integral transforms and variable separation. The influences of parameters for double porous media ω,λ, attitudes of the horizontal well ZWD. LD and the coefficient of vertical feed a, on the dimensionless pressure behaviors of horizontal well are discussed and calculated by means of the algorithm of Laplace numerical inversion. The diagnosis and specialized graphs are plotted and analysed.

Experimental investigation into transient pressure pulses during pneumatic conveying of fine powders using Shannon entropy

This paper presents the results of an ongoing investigation into transient pressure pulses using Shan- non entropy. Pressure fluctuations （produced by gas-solid two-phase flow during fluidized dense-phase conveying） are recorded by pressure transducers installed at strategic locations along a pipeline. This work validates previous work on identifying the flow mode from pressure signals （Mittal, Mallick, ＆ Wypych, 2014）. Two different powders, namely fly ash （median particle diameter 45 μm, particle den- sity 1950 kg/m3. loosely poured bulk density 950 kg/m3） and cement （median particle diameter 15 p,m, particle density 3060 kg/m3, loosely poured bulk density 1070 kg/m3）, are conveyed through different pipelines （51 mm I.D. × 70 m length and 63 mm I.D. × 24 m length）. The transient nature of pressure fluc- tuations （instead of steady-state behavior） is considered in investigating flow characteristics. Shannon entropy is found to increase along straight pipe sections for both solids and both pipelines. However, Shannon entropy decreases after a bend. A comparison of Shannon entropy among different ranges of superficial air velocity reveals that high Shannon entropy corresponds to very low velocities （i.e. 3-5 m/s） and very high velocities （i.e. 11-14 m/s） while low Shannon entropy corresponds to mid-range velocities （i.e. 6-8 m/s）.

Characterizing hydraulic fractures in shale gas reservoirs using transient pressure tests

Hydraulic fracturing combined with horizontal drilling has been the technology that makes it possible to economically produce natural gas from unconventional shale gas or tight gas reservoirs.Hydraulic fracturing operations,in particular,multistage fracturing treatments along with horizontal wells in unconventional formations create complex fracture geometries or networks,which are difficult to characterize.The traditional analysis using a single vertical or horizontal fracture concept may be no longer applicable.Knowledge of these created fracture properties,such as their spatial distribution,extension and fracture areas,is essential information to evaluate stimulation results.However,there are currently few effective approaches available for quantifying hydraulic fractures in unconventional reservoirs.This work presents an unconventional gas reservoir simulator and its application to quantify hydraulic fractures in shale gas reservoirs using transient pressure data.The numerical model incorporates most known physical processes for gas production from unconventional reservoirs,including two-phase flow of liquid and gas,Klinkenberg effect,non-Darcy flow,and nonlinear adsorption.In addition,the model is able to handle various types and scales of fractures or heterogeneity using continuum,discrete or hybrid modeling approaches under different well production conditions of varying rate or pressure.Our modeling studies indicate that the most sensitive parameter of hydraulic fractures to early transient gas flow through extremely low permeability rock is actually the fracture-matrix contacting area,generated by fracturing stimulation.Based on this observation,it is possible to use transient pressure testing data to estimate the area of fractures generated from fracturing operations.We will conduct a series of modeling studies and present a methodology using typical transient pressure responses,simulated by the numerical model,to estimate fracture areas created or to quantity hydraulic fractures with traditional well testing technology.The type curves of pressure transients from this study can be used to quantify hydraulic fractures in field application.

Pressure Transient Analysis of Arbitrarily Shaped Fractured Reservoirs

Reservoir boundary shape has a great influence on the transient pressure response of oil wells located in arbitrarily shaped reservoirs. Conventional analytical methods can only be used to calculate transient pressure response in regularly shaped reservoirs. Under the assumption that permeability varies exponentially with pressure drop, a mathematical model for well test interpretation of arbitrarily shaped deformable reservoirs was established. By using the regular perturbation method and the boundary element method, the model could be solved. The pressure behavior of wells with wellbore storage and skin effects was obtained by using the Duhamel principle. The type curves were plotted and analyzed by considering the effects of permeability modulus, arbitrary shape and impermeable region.

Influence of transient intraocular pressure elevation during laser in situ keratomileusis on rabbit retina thickness

AIMTo utilize tissue micro measurement to study the effect of transient high intraocular pressure (IOP) induced by different durations of suction during laser in situ keratomileusis (LASIK) on rabbit retina thickness.METHODSSixty healthy New Zealand white rabbits were randomly divided into a control group, and 3 negative-pressure suction groups (20s group, 45s group, and 3min group) and each group was comprised of 15 rabbits (30 eyes); the latter 3 groups were the transient high IOP models. The retinal tissue around the papilledema was separated. Hematoxylin and eosin (HE) staining was carried out to generate slices for light microscopy. The changes in the retina thickness values of each layer were measured for all animals in each group at different postoperative recovery periods and compared with the values recorded for the animals in the control group. The thickness of the retinal tissue showed a normal distribution. The ANOVA was performed by using SPSS13.0 statistic software.RESULTSIn the comparison between the 20s and 45s negative-pressure suction groups and the control group, no significant differences were observed, except at 14d. Significant difference was observed between the 3min negative-pressure suction group and the control group, and the retina thickness value of each layer reached a peak at 14d after repair.CONCLUSIONConventional negative suction during LASIK may not lead to significant changes in retinal tissue thickness; however, if the suction duration is increased to 3min, it will cause significant changes in retinal tissue thickness.

A review of pressure transient analysis in reservoirs with natural fractures,vugs and/or caves

A review of the pressure transient analysis of flow in reservoirs having natural fractures,vugs and/or caves is presented to provide an insight into how much knowledge has been acquired about this phenomenon and to highlight the gaps still open for further research.A comparison-based approach is adopted which involved the review of works by several authors and identifying the limiting assumptions,model restrictions and applicability.Pressure transient analysis provides information to aid the identification of important features of reservoirs.It also provides an explanation to complex reservoir pressuredependent variations which have led to improved understanding and optimization of the reservoir dynamics.Pressure transient analysis techniques,however,have limitations as not all its models find application in naturally fractured and vuggy reservoirs as the flow dynamics differ considerably.Pollard’s model presented in 1953 provided the foundation for existing pressure transient analysis in these types of reservoirs,and since then,several authors have modified this basic model and come up with more accurate models to characterize the dynamic pressure behavior in reservoirs with natural fractures,vugs and/or caves,with most having inherent limitations.This paper summarizes what has been done,what knowledge is considered established and the gaps left to be researched on.

THE TRANSIENT TWO-DIMENSIONAL FLOW THROUGH DOUBLE POROUS MEDIA

This paper presents the analytical solutions in Laplace domain for two-dimensional nonsteady flow of slightly compressible liquid in porous media with double porosity by using the methods of integral transforms and variables separation. The effects of the ratio of storativities to , interporosity flow parameter on the pressure behaviors for a vertically fractured well with infinite conductivity are investigated by using the method of numerical inversion. The new log-log diagnosis graph of the pressures is given and analysed.

An approximate analytical solution for transient gas flows in a vertically fractured well of finite fracture conductivity

An analytical solution in physical variable space is presented for transient gas flows during constant-rate production from a vertically-fractured well in an infinite homogeneous reservoir with finite fracture conductivity.The solution is based on the short-time asymptotic solution and a new approximate transient elliptical flow solution,which covers transient flows from the bilinear flow regime to the pseudo-radial flow regime.The solution covers the well-known asymptotic solutions in both short-and long-time limits of bilinear and pseudo-radial flows.The analytical model provides a practical and reliable engineering tool to evaluate the fractured reservoir properties,which can be programmed using a spreadsheet.

PRESSURE TRANSIENT ANALYSIS OF HETEROGENOUS RESERVOIRS WITH IMPERMEABILITY BARRIER USING PERTURBATION BOUNDARY ELEMENT METHOD

The transient flow mathematical model of arbitrary shaped heterogeneous reservoirs with impermeability barrier is proposed in this paper. In order to establish this model, the perturbation method is employed and the solution of model is expanded into a series in powers of perturbation parameter. By using the Boundary Element Method (BEM) and Duhamel principle, wellbore pressure with effects of skins and wellbore storage is obtained. The type curves are plotted and analyzed considering effects of heterogeneity, arbitrary shape and impermeable barriers. Finally, the results obtained by perturbation boundary element method is compared with the analytical solution and is available for the transient pressure analysis of arbitrary shaped reservoirs.

Water hammer prediction and control:the Green's function method

By Green＇s function method we show that the water hammer （WH） can be analytically predicted for both laminar and turbulent flows （for the latter, with an eddy vis- cosity depending solely on the space coordinates）, and thus its hazardous effect can be rationally controlled and mini- mized. To this end, we generalize a laminar water hammer equation of Wang et al. （J. Hydrodynamics, B2, 51, 1995） to include arbitrary initial condition and variable viscosity, and obtain its solution by Green＇s function method. The pre- dicted characteristic WH behaviors by the solutions are in excellent agreement with both direct numerical simulation of the original governing equations and, by adjusting the eddy viscosity coefficient, experimentally measured turbulent flow data. Optimal WH control principle is thereby constructed and demonstrated.

Frequency-dependent friction in pipelines

A comprehensive study of modeling the frequency-dependent friction in a pipeline during pressure transients following a sudden cut-off of the flow is presented. A new method using genetic algorithms（GAs） for parameter identification of the weighting function coefficients of the frequency-dependent friction model is described. The number of weighting terms required in the friction model is obtained. Comparisons between simulation results and experimental data of transient pressure pulsations close to the valve in horizontal upstream and downstream pipelines are carried out respectively.The validity of the parameter identification method for weighting function coefficients and the calculation method for the number of weighting terms in the friction model is confirmed.

A STUDY ON TRANSIENT FLUID FLOW OF HORIZONTAL WELLS IN DUAL-PERMEABILITY MEDIA

To adopt horizontal wells in dual media reservoirs, a good understanding of the related fluid flows is necessary. Most of the recent studies focus on dual porosity media instead of dual permeability media. In this article, through both integral transformation and sink-source superposition, a new Laplace-domain solution is obtained for the slightly-compressible fluid flow in the 3-D dual-permeability media in which the horizontal well is operating in a constant rate of production. Major asymptotic characteristics of diagnosis curves of dimensionless downhole pressure are analyzed by the limited analysis. Effects of parameters of dual-permeability media including mobility ratio k, storativity ratio ω and inter-porosity flow parameter k on the downhote pressure are studied by using the Laplace numerical inversion. The new solution obtained in this article includes and improves the previous results and then can be used as a basis for either pressure transient analysis or formation behavior evaluation for the typical reservoir with horizontal wells.

Exact analytical solutions for moving boundary problems of one-dimensional flow in semi-infinite porous media with consideration of threshold pressure gradient

By defining new dimensionless variables, nonlinear mathematical models for one-dimensional flow with unknown moving boundaries in semi-infinite porous media are modified to be solved analytically. The exact analytical solutions for both constant-rate and constant-pressure inner boundary constraint problems are obtained by applying the Green＇s function. Two transcendental equations for moving boundary problems are obtained and solved using the Newton-Raphson iteration. The exact analytical solutions are then compared with the approximate solutions. The Pascal＇s approximate formula in reference is fairly accurate for the moving boundary development under the constant-rate condition. But another Pascal＇s approximate formula given in reference is not very robust for constant-pressure condition problems during the early production period, and could lead to false results at the maximum moving boundary distance. Our results also show that, in presence of larger TPG, more pressure drop is required to maintain a constant-rate production. Under the constant-pressure producing condition, the flow rate may decline dramatically due to a large TPG. What＇s more, there exists a maximum distance for a given TPG, beyond which the porous media is not disturbed.