Single-phase Gas Flow Research Articles

A New Method for Computing Pseudo-Time for Real Gas Flow Using the Material Balance Equation

Abstract Analytical solutions in well-test analysis are oriented towards fluids with a constant viscosity and compressibility in a porous medium with a constant porosity. To use these solutions in gas-flow situations, one needs to apply the pseudo-pressure and pseudo-time transformations. Thus, the non-linear diffusivity equation for gas flow is transformed into a linear one, allowing one to use the solutions for fluids with constant compressibility, viscosity, and porosity. Although the computation of pseudopressure is reasonably accurate, the conventional computation of pseudo-time by direct integration of the compressibility-viscosity function over time can result in significant errors when modelling gas reservoirs with residual fluid saturation, rock compressibility, and a large degree of depletion. Errors in calculating the pseudo-time result in substantial errors in the material balance, which has an adverse impact on reservoir modelling and production forecasting. In this study, a new method for computing pseudo-time is presented. This method is based on the material balance equation that considers the rock and fluid compressibility. This formulation honours the material balance equation in all situations. Examples are presented to show that the problems with computing pseudo-time, using the traditional definition, can be resolved when the new method is used. Accurate computations of pseudo-time allow one to use the solutions for fluids with constant compressibility and viscosity for modelling and forecasting gas production. Introduction Analytical solutions are generally used to analyze and model well test and production data. However, these solutions have been developed for fluids with a constant viscosity and compressibility and for formations with a constant porosity. The governing diffusivity equation and its boundary conditions are linear when expressed in terms of pressure, space, and time variables. The analytical solutions to these equations are reasonably accurate for the liquid-flow situations. In contrast, the diffusivity equation for gas flow and its boundary conditions are non-linear when expressed in terms of pressure, space, and time variables. As no analytical solutions to these non-linear equations are available, one needs to apply pseudo-pressure and pseudo-time transformations in order to use the analytical solutions for liquid flow in gas-flow situations. This approach introduces two variables in the diffusivity equation for gas flow?pseudo-pressure (Ψ) as the dependent variable, and pseudo-time (ta) as an independent variable. As a result, the diffusivity equation for gas flow is transformed into a linear one, allowing one to use the slightly compressible fluid (liquid) solutions. In this study, we are considering a single-phase gas flow situation in the presence of residual fluid saturation and a compressible formation. Here gas is the only mobile phase, while oil and water phases are immobile, if there are any. The pseudo-variables (pseudo pressure and pseudo-time) can be defined as: Equation (Available In Full Paper) The rationale for defining the above pseudo-variables is demonstrated in Appendix A. Martin(3) made the first systematic attempt to define the total system compressibility in multi-phase conditions, neglecting the rock compressibility. Later, Ramey(4) followed Martin's lead and included the formation compressibility in defining the total system compressibility, ct, as:

Experimental Determination of Two-Phase Flow Rates of Hydrocarbons Through Restrictions

Accurate prediction of flow rate through a given restriction is important in the design of pressure relief and blowdown systems. While this prediction is well understood for single-phase gases and liquids, it is much less well understood for two-phase gas-liquid flows. We report here a large number of measurements conducted on highly volatile mixtures of hydrocarbons (mainly C1 to C10) at pressures up to 100 bar and flow rates of up to 4 kg s−1. For mixtures in which the liquid mass fraction is below about 0.8, we find that the homogeneous equilibrium model (HEM) provides a good approximation: the discharge coefficient varies from 0.90 for pure single-phase gas flow to about 0.98 when the upstream liquid fraction is 0.8. For flows of compressed volatile liquid, we find that the incompressible-flow model provides a good approximation, with a discharge coefficient of 0.60: however, preliminary experiments indicate that this simple model progressively breaks down as the volatile gas content increases. We also find that, in a number of circumstances, the widely-used recommendations of the American Petroleum Institute (API) can significantly over- or under-predict flow rates through restrictions if misapplied.

Numerical Modelling Of The Riemann ProblemFor A Mathematical Two-phase Flow Model

We report preliminary results obtained using Godunov methods of a centred type for hyperbolic systems of conservation laws for which the analytical solution of the Riemann problem is too difficult to develop. For this purpose, we consider a mathematical model used for modelling an unsteady compressible non-equilibrium mixture two-phase flow which results in a well-posed initial value problem in a conservative form. The mathematical two-phase flow model consists of six first-order partial differential equations that represent one-dimensional mass, momentum and energy balances for a mixture of gas-liquid and gas volume concentration, gas mass concentration and the relative velocity balances. This system of six partial differential equations has the mathematical property that its six characteristic roots are all real with a complete set of independent eigenvectors which form the basic structure of the solution of the Riemann problem. The construction of the solution of the Riemann problem for the model poses several difficulties. Since the model possesses a large number of non-linear waves, it is not easy to consider each wave separately to derive a single non-linear algebraic equation for the unknown region, the star region, between the left and right waves. We propose numerical techniques of a centred type specifically developed for high speed single-phase gas flows. A main feature of centred methods is that they do not explicitly require the solution of the Riemann problem. This is a desirable property which guarantees that the methods can handle the solution of the Riemann problem numerically and resolve both rarefaction and shock waves for the model in a simple way with good accuracy. Finally, we present some numerical simulations for the gas-liquid two-phase flow Riemann problem to illustrate the efficiency of the proposed schemes.

Experimental Study of Gas Slippage in Two-Phase Flow

Summary Gas slippage in single-phase gas flow (the Klinkenberg effect) has been investigated extensively. Few papers, however, have been published on the gas slippage in gas/liquid two-phase flow. The gas relative permeabilities at water saturations close to the residual value have been found to be significantly greater than 1 in both nitrogen/water and steam/water flow through rocks. These values became less than 1 after the calculation was calibrated by taking the two-phase gas slip effect into consideration. The gas relative permeabilities have been measured at different mean pore pressures, and the values of two-phase gas slip factors have been computed at different water saturations. The effects of temperature on both nitrogen and steam slip factors also have been studied and compared. These data have then been used to conduct a calibration to obtain intrinsic gas relative permeabilities that do not vary with the test pressures. It has been found from the present work that neglecting the two-phase gas slip effect may overestimate gas relative permeabilities. It is the intrinsic gas relative permeabilities instead of those measured at low test pressures that should be used in numerical simulation or other reservoir-engineering calculations.

A research into thermal field in fluid-saturated porous media

Until now, the theory, methodology of investigations, and interpretation of thermometry data have been most completely developed for single-phase (oil, water, or gas) flows in formations. However, multiphase (oil+gas, oil+water, and oil+water+gas) flows in formations are more common in practice. This is primarily typical for fields featuring a high value of gas factor and saturation pressure, as well as for cases of formation tests at low values of bottom-hole pressure. Analysis of actual thermograms under these conditions has shown that the earlier-developed techniques for the cases of single-phase flows in the formation and the well cannot be applied here. This paper presents research data on the influence of the adiabatic and Joule–Thomson effects and the heat of fluid degassing on temperature field in porous medium.

Fluidelastic instability of flexible tubes subjected to two-phase internal flow

The fluidelastic instability behaviour of flexible tubes subjected to internal single-phase (liquid or gas) flows is now reasonably well understood. Although many piping systems operate in two-phase flows, so far very little work has been done to study their dynamic behaviour under such flows. This paper presents the results of a series of experiments to study the fluidelastic instability behaviour of flexible tubes subjected to two-phase internal flow. Several flexible tubes of different diameters, lengths and flexural rigidities were tested over a broad range of flow velocities and void fractions in an air–water loop to simulate two-phase flows. Well-defined fluidelastic instabilities were observed in two-phase flows. The existing theory to formulate the fluidelastic behaviour under internal flow was developed further to take into account two-phase flow. The agreement between the experimental results and the modified theory is remarkably good. However, it depends on using an appropriate model to formulate the characteristics of the two-phase flows.

Analysis of Two-Phase Homogeneous Bubbly Flows Including Friction and Mass Addition

The present work is a theoretical investigation of two-phase bubbly flows. The main objective is to get a better insight of the basic phenomena associated with such flows through nozzles via physical modeling and mathematical formulation. Introducing Mach number into the flow equations, we find novel, closed-form analytical solutions and expressions for homogeneous bubbly flows including the influence of wall friction and mass addition. The expressions obtained demonstrate an analogy to those of classical, single-phase gas flows. The study deals with homogeneous flows, however, its approach and results can also be applied to investigate flows with unequal phase velocities, to study instability phenomena, as well as to design and analyze water jet propulsion systems.

Experimental Characterization and Computational Fluid Dynamics Simulation of Gas Distribution Performance of Liquid (Re)Distributors and Collectors in Packed Columns

The usability of a state of the art computational fluid dynamics simulation package as a tool for analyzing the fluid-dynamic performance of internals encountered in packed distillation columns, such as initial gas distributors, liquid distributors and liquid collectors is demonstrated. A 1.4 m internal diameter column hydraulics simulator was used to provide detailed experimental evidence on the gas distribution pattern imposed by various types of column internals. The comparison of measured and predicted profiles for single-phase gas flow conditions indicates a strikingly good agreement.

Tailoring the pressure drop of structured packings through CFD simulations

A computational fluid dynamic methodology is proposed to breakdown into elementary dissipation mechanisms the overall single-phase gas flow bed pressure drop in towers containing corrugated sheet structured packings. The goal behind was to allow piecewise geometry optimization of such packings in terms of capacity enlargement and efficiency enhancement. The dissipations sorted in order of decreasing importance were the collision losses by jet streams at criss-crossing junctions within corrugated channels, elbow loss by form drag at interlayer transition, elbow loss by jets striking wall and subsequent flow redirection to upper channels, and elbow loss in bed entrance. Replacement of sharp bends at the interlayer junctions by progressive direction change was beneficial for the reduction of the dissipations at the wall and the interlayer junction thus stretching capacity of the structured packing. However, this improvement was not spectacular because the most energy-intensive component (criss-crossing) remained unaffected by such modifications. Computational fluid dynamics is foreseen to be a successful, rapid and economic tool to theoretically explore new geometries coping with this limitation.

A Numerical Model Coupling Reservoir and Horizontal Well-Flow Dynamics: Transient Behavior of Single-Phase Liquid and Gas Flow

Summary A fully implicit, 3D simulator with local refinement around the wellbore is developed to solve reservoir and horizontal well flow equations simultaneously for single-phase liquid and gas cases. The model consists of conservation of mass and Darcy's law in the reservoir, and mass and momentum conservation in the wellbore for isothermal conditions. The coupling requirements are satisfied by preserving the continuity of pressure and mass balance at the sandface. The proposed simulator is tested against and verified with the results obtained from a commercial code ECLIPSE-100, and available public domain simulators and semianalytical models. The model can be used to simulate the transient pressure and flow rate behavior of both the reservoir and the horizontal wellbore. Traditional decoupling of the wellbore transients from the reservoir transients does not capture the horizontal well pressure and flow rate transients at early times because the interaction between the reservoir and wellbore is inherently neglected. Simulation runs with the proposed model reveal the actual characteristics of horizontal wellbore storage and unloading, as well as flow pattern determination during transient well testing using pressure derivative curves. The effects of permeability, formation thickness, well length, and fluid compressibility are also studied.

Gas flow in a microdevice with a mixing layer configuration

We have designed an integrated microsystem with a mixing layer configuration, fabricated using standard micromachining techniques, in order to study fluid flows, in complex microfluidic systems, that are certain to find numerous important applications, especially in biomedical or chemical analysis. The device features two narrow and parallel channels merging smoothly into a wide channel downstream of a splitter plate, all 1 μm in height, integrated with distributed pressure sensors. The characterization of the device included measurements of flow rate and pressure distribution for single-phase gas flow. Argon gas was passed either through one of the inlet channels, while the other was blocked, or through both inlet channels. Simple flow models of either a single straight microchannel or a pair of microchannels with different widths, connected in series, have been found to provide reasonable predictions of the evolving flow fields.

561 砕波を伴う風波乱流場の数値シミュレーション(OS18 気液および液液界面を含む流れの数値シミュレーション)(OS18 気液および液液界面を含む流れの数値シミュレーション)

In this study, numerical investigation of spatial developing breaking air-water interface turbulent flow with heat transfer was conducted. The Reynolds number is about 3300 based on the gas layer height and the free stream velocity. The computational domain consists of a driver region used for Direct Numerical Simulation (DNS) of single phase gas flow of turbulent boundary layer and a main region for the two phase flow simulation. Grids number is 128×82×256 (in the driver region) and 256×157×256 (in the main region) in streamwise, vertical and spanwise directions, respectively. In the main region, the calculation of wind-driven turbulent flow was carried out by means of the direct numerical solution procedure (MARS method) for a coupled gas-liquid flow.

Pressure gradient and holdup in horizontal two-phase gas–liquid flows with low liquid loading

Pressure gradient and holdup data are presented for air–water and air–oil flows in a horizontal, 0.079 m diameter pipe. Addition of a very small liquid flow was found to result in a considerable increase in the pressure gradient compared with single phase gas flow. The pressure gradient and the holdup data were compared with predictions of the ‘apparent rough surface’ (ARS) and the ‘double-circle’ models. The ARS model generally gave better predictions for the holdup over the experimental range. Both models predicted the pressure gradient for air–water flows at high gas flowrates reasonably well. However, the predictions of both methods were unsatisfactory for air–water experiments at low gas flowrates and for air–oil experiments. Overall the ARS model was judged to be the more robust.

A slip with entrained liquid fraction approach to compressible two-phase flow in nozzles

Ideal flow theory adequately predicts pressure drops, critical mass flowrates and critical pressure ratios for single-phase gas flows in nozzles. A model based on these principles has been developed for two-phase, gas-liquid flows using a slip with entrained liquid fraction approach. The method relies on being able to establish the momentum flux of the fluid in the upstream supply pipe and therefore allows pipe flow correlations for slip ratios and entrained liquid fractions to be used at the nozzle inlet. Choking conditions are established from an isentropic pressure pulse approach. This produces a speed of sound for the two-phase mixture that gives choking conditions that are compatible with the end-limits of the momentum and energy equations used to estimate the pressure drops for non-choked compressible flows. This allows a consistency in approach between non-choked compressible and choked flows. The model predictions of pressure drops, critical pressure ratios and critical mass flowrates compare well with data sources and are an improvement on those made by other models available in the open literature.

Gas-wall shear stress distribution in horizontal stratified two-phase flow

Gas-wall shear stresses in the stratified gas-liquid flow in pipes are obtained using Preston's method for measuring skin friction in the turbulent boundary layer. The non-dimensional relationship between the Preston tube reading and wall shear stress over a wide range of single-phase gas flow rates is reported. The wall shear stresses up to positions close to the gas-liquid interface, for various interface conditions, are obtained for the two-phase flow experiment. The distribution of the gas-wall shear stress and the effect of diameter on those distributions are investigated. The friction factors obtained from the experiments are also compared with those reported in the literature.

A slip with entrained liquid fraction approach to compressible two-phase flow in nozzles

Ideal flow theory adequately predicts pressure drops, critical mass flowrates and critical pressure ratios for single-phase gas flows in nozzles. A model based on these principles has been developed for two-phase, gas-liquid flows using a slip with entrained liquid fraction approach. The method relies on being able to establish the momentum flux of the fluid in the upstream supply pipe and therefore allows pipe flow correlations for slip ratios and entrained liquid fractions to be used at the nozzle inlet. Choking conditions are established from an isentropic pressure pulse approach. This produces a speed of sound for the two-phase mixture that gives choking conditions that are compatible with the end-limits of the momentum and energy equations used to estimate the pressure drops for non-choked compressible flows. This allows a consistency in approach between non-choked compressible and choked flows. The model predictions of pressure drops, critical pressure ratios and critical mass flowrates compare well with data sources and are an improvement on those made by other models available in the open literature.

Succession-of-States Model for Calculating Long-Time Performance of Depletion Reservoirs

Summary This study presents a new finite-element approach for directly calculating pseudosteady-state flow behavior for wells in depletion systems. The approach allows for spatially dependent reservoir properties, complex reservoir geometries, and multiple wells. Results are verified against long-time transient solutions reported in the literature for several regularly shaped systems. The paper also demonstrates application of the approach to field-scale problems. Results show that this approach provides a fast and accurate method for modeling the long-time behavior of depletion reservoirs. The approach is particularly applicable to single-phase volumetric gas reservoirs. A bounded reservoir with wells producing at constant rate will exhibit pseudo steady-state behavior after the end of typically short-lived infinite-acting and transition flow periods. This study develops a new approach for directly calculating pseudosteady-state flow behavior without solving the full time-dependent form of the diffusivity equation. This approach can be applied to the linearized forms of the diffusivity equation for either single-phase liquid or gas flow. A finite-element method is used that allows for spatially dependent reservoir properties, complex reservoir geometries, and multiple wells. The first part of this paper presents a verification of the approach by comparing results for some regularly shaped systems against full-transient solutions reported in the literature. For the simulation of field-scale problems with multiple wells of differing production rates, a well model based on a near-wellbore approximation of the pseudopressure distribution during pseudosteady-state is introduced to reduce the concentration of elements near wells. The second part of the paper demonstrates application of the direct pseudosteady-state concept to actual reservoir problems. To account for rate changes during extended production periods, the pseudosteady-state equation was solved successively for each flow period and combined with an overall reservoir material balance analysis. Results from this study show that this approach provides a fast and accurate method for modeling the long-time behavior of various types of reservoirs under depletion conditions. The approach is particularly applicable to single-phase volumetric gas reservoirs.

Two-phase momentum flux in pipes and its application to incompressible flow in nozzles

The pressure drop produced when a single-phase gas or liquid flows through a nozzle can be reasonably well estimated from inviscid flow theory. A similarly based model, taking the slip with entrained liquid fraction approach, has been developed for two-phase gas—liquid flows. Fundamental to the model is the constraint that the nozzle flow depends on the entry conditions, particularly the momentum flux. At low qualities the tendency was for fractions of the liquid flow to be accelerated with the gas phase to maintain uniform momentum flux across the nozzle flow area. At higher qualities the gas core of the annular flow, along with its associated droplets, was found to contract with negligible interaction with the surrounding liquid film. The model has been compared with data sources and other models available in the literature and shown to perform well.

Stability of small diameter airlift pumps

In an airlift pumping process, air is injected into the pipe containing the fluid to be transferred. Small diameter airlift pumps are, in particular, used for corrosive or radioactive liquids. However, for certain combinations of the geometrical parameters and air flow rate, they may become unstable. In this case, the flow at the riser outlet pulsates strongly, which cannot be accepted for many applications. An airlift pump involves three different regions, e.g. a single phase liquid flow and a separate single phase gas flow upstream of the air injection device and a two-phase flow downstream. The instabilities are due to density wave oscillations in the two-phase flow. Depending on the liquid flow inertia, friction effects and gas flow compressibility, the density waves are sustained or not. The present study is based upon a detailed description of the steady state flow in a small diameter airlift pump. A linear stability analysis is performed and assessed against an extensive set of experimental data. Both the experimental and analytical results show that the influencing parameters have complex effects and strongly interact: the same variation of a parameter may have opposite effects, i.e. stabilizing or destabilizing, depending on the values of the other parameters. The effect of the compressibility of the gas flow between the regulating valve and the air-injection device is shown to be very important. The analysis presented leads to a numerical model that can be considered as a practical tool for airlift performance and stability analysis.

Numerical simulation of multi-dimensional two-phase flow based on flux vector splitting

This paper describes a new approach to the numerical simulation of transient, multidimensional two-phase flow. The development is based on a fully hyperbolic two fluid model of two-phase flow using separated conservation equations for the two phases. Features of the new model include the existence of real eigenvalues, and a complete set of independent eigenvectors which can be expressed algebraically in terms of the major dependent flow parameters. This facilitates the application of numerical techniques specifically developed for high speed single-phase gas flows which combine signal propagation along characteristic lines with the conservation property with respect to mass, momentum and energy. Advantages of the new model for the numerical simulation of 1- and 2-dimensional two-phase flow are discussed.