Isothermal Fluid Flow Research Articles

A combined Markov Chain Monte Carlo and Levenberg–Marquardt inversion method for heterogeneous subsurface reservoir modeling

In this study, we systematically investigate an inverse method for heterogeneous porous media to obtain porosity and permeability fields considering only production well data. The forward modeling of the input data, namely pressure and saturation fields, is based on the motion equations of a coupled Darcy flow system involving two phases of isothermal fluid flow. We discretize these equations using a multiscale finite volume simulation technique in the spatial domain, and the backward Euler method in time domain. In the inversion procedure, we combine a global optimization method, the Markov Chain Monte Carlo (MCMC) method, with a local optimization method, the Levenberg–Marquardt (LM) method. The MCMC was implemented as the Random Walk algorithm, and to generate the samples of the porosity and permeability fields, we employed Karhunen–Loève (KL) expansion of second-order stationary fields with Gaussian covariance. The coefficients of the KL expansion are estimated by minimizing the norm of the residual dependent on these fields. At the end of the MCMC iterations, we refine the KL coefficients associated with the porosity and permeability fields using the LM method. We verify in the numerical experiments that the accuracy of the porosity and permeability fields is improved by the LM refinement step.

A Hermite-type collocation mesh-free approach for simulating incompressible viscous fluid flows

In this work, a novel Hermite-type mesh-free model for incompressible viscous fluid flows is proposed, as a way, to overcome the numerical difficulties experienced by solving under weak or strong formulation the pure stream-function Navier–Stokes equation. In this model a dimensionless virtual domain is utilized, in order to make an accurate dimensionless shape-functions of the recently developed Hermite-type collocation Weighted Least Square approximation, by using dimensionless scale parameters. Then, a Jacobian transformation matrix is built to express the shape-functions spatial derivatives from the virtual to the considered domain. The resolution of the obtained nonlinear algebraic system is performed by the High Order Continuation Method. Numerical tests were investigated on incompressible viscous isothermal fluid flows in lid-driven cavity, backward-facing step, sudden expansions and around various obstacle shapes. They show the advantages of the proposed model compared to the reference Hermite-type collocation models used for pure stream-function equation and the weak Galerkin procedures as Finite Element Method and Element Free Galerkin method, in term of computation accuracy, number of degree of freedom, implementation simplicity and computation time. The numerical and experimental results of literature confirm the validity and capability of the proposed dimensionless model to simulate fluid flows in a expended-lengths inflow–outflow geometry and around obstacles of circular, inclined elliptical and Joukowski airfoil shapes.

A very high order Flux Reconstruction (FR) method for the numerical simulation of 1-D compositional fluid flow model in petroleum reservoirs

Compositional reservoir simulation is an extremely important tool for modeling the fluid flow in complex petroleum reservoirs, as in cases where the reservoir fluid is volatile and composed of several pseudo-components with distinct characteristics, or reservoirs that require the use of enhanced oil recovery process. For these cases, simpler black-oil models are not suitable. The compositional model involves the solution of a large system of partial differential equations, that comprises the mass conservation, the Darcy’s law and the fugacity constraints. However, the high number of equations and constraints render the compositional problems extremely complex and computationally demanding solutions. To improve accuracy and reduce the high computational costs, one can use higher than first or even second-order methods to approximate the advective flux terms in the hyperbolic conservation laws that describe the multicomponent transport in the reservoir. Those very high-order schemes can replace low-order approaches, such as the First Order Upwind (FOU) method, which is traditionally used in commercial petroleum reservoir simulators (CMG, 2019; ECLIPSE, 2022), obtaining more accurate solutions with reduced computational cost. In this work, we consider a three-phase and isothermal fluid flow of water, oil and gas and assume that there is no mass transfer between the water and the hydrocarbon phases. Physical dispersion and capillary effects are neglected. To solve the system of Partial Differential Equations (PDEs), we used the classical IMPEC (Implicit Pressure Explicit Composition) approach and, to model the complex phase behavior, we used the Peng–Robinson equation of state (PR-EOS). The classical Two Point Flux Approximation (TPFA) method is used to spatially discretize the diffusion terms in the pressure equation. For the first time in literature the very high order Flux Reconstruction (FR) method is adapted for the numerical simulation of the compositional model in petroleum reservoirs. The FR method is a numerical scheme employed to discretize the hyperbolic conservation laws using general grids. To avoid spurious oscillations in the vicinity of solution discontinuities, the h-MLP (hierarchical Multi-dimensional Limiting Process) is applied in the reconstruction stage. For the time integration, we have used the third-order Runge–Kutta approach. To appraise the robustness of our formulation, we have solved some one-dimensional benchmark problems found in literature and compared the FR solution with the FOU method and with the second-order Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) method. From our results, the FR method has shown improved accuracy, efficiency and capability of detecting sharper shocks when compared to the other methods that we have considered for the numerical modeling of the compositional flow in the present paper.

Investigation of eigenfunctions of perturbation of the radial component of flow velocity

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of a generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The influence of taking into account the exponential dependence of fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of incompressible fluid flow in an annular channel is investigated. The eigenvalue spectra were obtained numerically. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity disturbances, their possible growth or decay over time. Graphs of the eigenfunctions of the generalized flow stability equation in an annular channel have been constructed. It is shown that the structure of the spectra largely depends on both the properties of the liquid, determined by the functional dependence of viscosity, and on the geometry of the channel. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It has been established that at small values of the thermoviscosity parameter the spectrum is comparable to the spectrum for isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are a greater number of points at which the problem has non-zero amplitudes of transverse velocity disturbances. It is worth noting that the diversity of eigenvalue spectra corresponds to the diversity of eigenfunctions, which have a nontrivial distribution of the oscillation amplitude over the cross section in each case. Smooth curves are obtained for a narrow channel, as for the case of a flat channel. However, as the ratio of the channel radii increases, “jumps” appear. Also, note that the eigenfunctions do not have the property of symmetry; this follows from the fact that the velocity profile in the unperturbed state also does not have symmetry. The maximum values of the eigenfunctions are shifted to the right from the center of the channel, which corresponds to the fact that disturbances arise and grow intensively near the hot wall.

Well and reservoir characterization using combined temperature and pressure transient analysis: A case study

Significant temperature variations are measured in oil and gas production wells with large pressure drawdown such as high production rate wells and tight formations. The considerable heating or cooling effects in the wellbore sandface invalidates the assumption of isothermal fluid flow in porous media. While this assumption satisfies most of the classic reservoir engineering analyses, the potential of slight thermal effects are attracting extensive interest and numerous thermal models have been proposed recently.The pitfalls and weaknesses of pressure transient analysis have been extensively investigated since 1930's which is still persistence despite developments of more accurate analytical and numerical models. Some of the weaknesses are rooted in the assumption of isothermal fluid flow in well test analysis methods.In this paper, we show the power and added value of combining temperature transient analysis to complement pressure transient analysis. Interestingly, the characteristics of skin effect such as radius and permeability of the damaged zone can be determined from temperature measurements. We study various parameters such as rock and fluid heat capacity, rock thermal conductivity, rock porosity, rock permeability and skin properties to illustrate the effect of each parameter on wellbore temperature data and hence the application of TTA for reservoir characterization. Using an accurate thermal flow mode, a case study has been done to investigate the importance of temperature transient analysis to better characterize an oil field well.

Investigation of the spectral characteristics of a thermoviscous fluid flow in an annular channel

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of the generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The effect of taking into account the exponential dependence of the fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of an incompressible fluid flow in a flat channel for various wall temperatures is studied. Spectral patterns of eigenvalues of the generalized equation are constructed. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity perturbations, their possible growth or decay with time. It is shown that the structure of the spectra largely depends both on the properties of the liquid, determined by the index of the functional dependence of viscosity, and on the geometry of the channel. It has been established that for small values of the thermoviscosity parameter, the spectrum is comparable to the spectrum for an isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has nonzero amplitudes of transverse velocity perturbations. The stability of a thermoviscous fluid flow depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It is shown that, at fixed values of the Reynolds number and wave number, the flow can become unstable with an increase in the thermoviscosity parameter.

Accretion of matter onto black holes in massive gravity with Lorentz symmetry breaking

In this paper we study the accretion of matter onto the black holes and the shadow images obtained by an infalling accretion flow of gas in a theory of massive gravity (MG) with a spontaneously breaking Lorentz symmetry. This black hole solution is characterized by mass $M$, scalar charge $S$ and the parameter $\lambda$. In order to extract the astrophysical results of our analyses, firstly, we have assumed a specific range for the parameter $\lambda$ to constrain the scalar charge $S$ using the EHT result. To this end, we have studied the effect of the scalar charge on the intensity of the electromagnetic radiation from the black hole. Moreover, we investigate the behavior of polytropic as well as the isothermal fluid flow onto massive gravity black hole and notice that accretion starts from supersonic/subsonic flow, passes through the critical point using particular model parameter and ends near the horizon. We also analyzed the mass accretion rate in the presence of various fluids which indicates important signatures. We also elaborate on the possibility to analyze the phase transition and the stability of the black hole using the shadow formalism.

Diffusive terms applied in smoothed particle hydrodynamics simulations of incompressible and isothermal Newtonian fluid flows

Diffusive terms applied in smoothed particle hydrodynamics simulations of incompressible and isothermal Newtonian fluid flows

The impact of isothermal flow assumption on accuracy of pressure transient analysis results

AbstractAdiabatic expansion and viscous dissipation of fluid flow in porous media result in considerable heating or cooling of hydrocarbon fluids when pressure gradients are very large. Traditionally, fluid flow in porous media has been assumed isothermal in hydrocarbon reservoirs. While this assumption helps with simplifying the physics of fluid flow by neglecting the thermal effects, it imposes a dominant impact on fluid properties in situations where flow undergoes fast paced and abrupt changes in flow rate and or pressure. The analytical solutions of fluid flow for pressure transient analysis rely on the isothermal flow assumption. Since the variations of fluid properties are not accounted for in temperature fluctuations, there is an inherent error in standard methods of well test analysis. In this paper, the errors in pressure transient analysis as a consequence of the assumption of isothermal fluid flow are investigated. The magnitude of error is quantified as a function of flow rate and pressure drop. Error analysis shows that at higher flow rates, the thermal impacts are elevated such that the application of isothermal flow models becomes erroneous and invalid. It can be shown that oil well tests are more influenced by thermal effects compared to gas well tests in conventional dry gas reservoirs. This study highlights an important thermal impact that is often neglected and should be considered in the design, execution, and analysis stages of well testing. Also, it suggests that isothermal analytical models should be avoided for well test analysis with large flow rates.

Riemann problem and limits of solutions to the isentropic relativistic Euler equations for isothermal gas with flux approximation

Abstract We are concerned with the vanishing flux-approximation limits of solutions to the isentropic relativistic Euler equations governing isothermal perfect fluid flows. The Riemann problem with a two-parameter flux approximation including pressure term is first solved. Then, we study the limits of solutions when the pressure and two-parameter flux approximation vanish, respectively. It is shown that, any two-shock-wave Riemann solution converges to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between these two shocks tends to a weighted δ-measure that forms a delta shock wave. By contract, any two-rarefaction-wave solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations, and the intermediate state in between tends to a vacuum state.

Experimental validation of computational fluid dynamics for solving isothermal and incompressible viscous fluid flow

In order to validate a computational method for solving viscous fluid flows, experiments are carried out in an eccentric cylindrical cavity showing various flow formations over a range of Reynolds numbers. Especially, in numerical solution approaches for isothermal and incompressible flows, we search for simple experimental data for evaluating accuracy as well as performance of the computational method. Verification of different computational methods is arduous, and analytic solutions are only obtained for simple geometries like a channel flow. Clearly, a method is expected to predict different flow patterns within a cavity. Thus, we propose a configuration generating different flow formations depending on the Reynolds number and make the experimental results freely available in order to be used as an assessment criterion to demonstrate the reliability of a new computational approach.

Improved model of isothermal and incompressible fluid flow in pipelines versus the Darcy–Weisbach equation and the issue of friction factor

In this article, we consider the modelling of stationary incompressible and isothermal one-dimensional fluid flow through a long pipeline. The approximation of the average pressure in the developed model by the arithmetic mean of inlet and outlet pressures leads to the known empirical Darcy–Weisbach equation. Most importantly, we also present another improved approach that is more accurate because the average pressure is estimated by integrating the pressure along the pipeline. Through appropriate transformation, we show the difference between the Darcy–Weisbach equation and the improved model that should be treated as a Darcy–Weisbach model error, in multiplicative and additive form. This error increases when the overall pressure drop increases. This symptomatic phenomenon is discussed in detail. In addition, we also consider four methods of estimating the coefficient of friction, assess the impact of pressure difference on the estimated average flow velocity and, based on experimental data, we show the usefulness of new proposals in various applications.

Shock-wave propagation in an isothermal slightly-compressible bubbly viscoelastic fluid flow

In this work, weakly nonlinear wave equation in mono-dispersed, isothermal, bubbly, slightly compressible viscoelastic liquid is derived, using reduction perturbation method. The constitutive equation is based on the second-grade fluid model. Viscosity, relaxation, retardation and surface tension are considered under isothermal condition, which modifies the Rayleigh–Plesset equation for compressible liquid. A kink traveling wave solution is obtained using the tangent hyperbolic method combined with the Riccati equation. Graphical representation of the solution is given and analyzed with different parameter values.

Modeling and Simulation of Top and Bottom Lid Driven Cavity using Lattice Boltzmann Method

Simulation of computational fluid dynamic (CFD) in isothermal fluid flow is very important in engineering and modeling problem. The simulation can explain the closed-flow behavior as complement with experimental field. Lid Driven Cavity as the tool for CFD benchmark is important to investigate. Lattice Boltzmann Method (LBM) as non-conventional CFD is used as present method investigation. The keys parameters of lid driven cavity problem are Reynolds number and the parts of moving boundaries that driven the cavity. In the present investigation was used the variations of Reynolds numbers: 500, 1000 and 2000. The lid that driven the cavity are top and bottom part. The results presented in visual flow pattern and in two dimensional velocity plot.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Isothermal laminar fluid flow in spiral tube coils

An experimental study is performed to measure pressure drop for Newtonian fluid flow through copper spirals of different geometries. The experimental friction factors obtained are presented and correlated with the different geometrical parameters of the spiral coils. Four spiral coils with $$ \left(\frac{D_i}{D}\right) $$ ratio ranging from 0.0178 to 0.028 and $$ \left(\frac{L}{D_i}\right) $$ ratio ranging from 527.5 to 2110.169 are investigated. A new dimensionless number, the R number is introduced which is found to characterize the fluid flow phenomenon in spiral coil tubes. An innovative approach to correlate Dean and R numbers with friction factor data of variable curvature coils for laminar flow regime is presented for the first time. The study will prove useful to bridge the gap between the straight tube flow and curved coil flow based on a single dimensionless number.

Introducing a variable speed of sound in single-component lattice Boltzmann simulations of isothermal fluid flows

To simulate the hydrodynamics and mixing characteristics of chemical reactors by means of a lattice Boltzmann method (LBM), it is essential to consider components with varying molecular weights (and therefore speeds of sound). This option requires modification of the standard equilibrium distribution function and the use of an extended velocity set. In this paper, we show that, for isothermal incompressible single-component non-reactive flows, tuning the speed of sound with a modified equilibrium distribution and an extended velocity set allows for reproducing the proper flow characteristics with strongly reduced errors (compared to LBM simulations on standard lattices). This is done for two isothermal benchmarks, viz. a damped standing pressure wave and a decaying viscous Taylor–Green Vortex. The convergence as a function of the number of lattice nodes used improves substantially for varying values of the speed of sound.

Investigation of the Hydraulic Characteristics of Capillary Elements of the Injector Head of Jet Engines under Conditions of Isothermal Flow of A Liquid

The article presents the results of an experimental study of the hydraulic characteristics of capillary elements of the injector head of jet engines in isothermal fluid flow and the proposed method of their calculation. The main geometric dimensions of the capillaries in the experiment were changed in the following range: Inner diameter from 0.16 to 0.36 mm, length from 4.3 to 158 mm and relative length from 25 to 614 and the inlet edge of the capillaries: sharp or smooth the leading edge. As the working fluid during the tests were distilled water, acetone and ethyl alcohol. Based on the results of a study of the dependences for calculation of ultimate losses in laminar and turbulent flow regimes in capillary tubes with smooth and sharp edges input. The influence of surface tension forces on loss of input on a sharp cutting edge. Experimentally confirmed the possibility of calculating the linear coefficient of hydraulic resistance of capillary tubes with a diameter of 0.16-0.36 mm in isothermal stable during the known dependencies that are valid for hydrodynamically smooth round tube.

Application of Regularized Hydrodynamic Equations for Direct Numerical Simulation of Micro-Scale Flows in Core Samples

The paper is devoted to numerical simulation of three-dimensional isothermal two-phase two-component viscous fluid flows with surface effects in the pore space of core samples. The voxel representation of the flow domain is used suitable for digital rock physics applications. The flow is described by viscous compressible Navier-Stokes-Cahn-Hilliard equations. In order to use simple and computationally efficient explicit numerical algorithms with central difference approximations of spatial terms, a quasi-hydrodynamic regularization of equations is used. Simulation results of fluid displacement in pore space of realistic core sample (sandstone) are presented. The results demonstrate the applicability and good prospects of quasi-hydrodynamic regularization technique to solve the problem.

Multiple stable isotope fronts during non-isothermal fluid flow

Stable isotope signatures of oxygen, hydrogen and other elements in minerals from hydrothermal veins and metasomatized host rocks are widely used to investigate fluid sources and paths. Previous theoretical studies mostly focused on analyzing stable isotope fronts developing during single-phase, isothermal fluid flow. In this study, numerical simulations were performed to assess how temperature changes, transport phenomena, kinetic vs. equilibrium isotope exchange, and isotopic source signals determine mineral oxygen isotopic compositions during fluid-rock interaction. The simulations focus on one-dimensional scenarios, with non-isothermal single- and two-phase fluid flow, and include the effects of quartz precipitation and dissolution.If isotope exchange between fluid and mineral is fast, a previously unrecognized, significant enrichment in heavy oxygen isotopes of fluids and minerals occurs at the thermal front. The maximum enrichment depends on the initial isotopic composition of fluid and mineral, the fluid-rock ratio and the maximum change in temperature, but is independent of the isotopic composition of the incoming fluid. This thermally induced isotope front propagates faster than the signal related to the initial isotopic composition of the incoming fluid, which forms a trailing front behind the zone of transient heavy oxygen isotope enrichment. Temperature-dependent kinetic rates of isotope exchange between fluid and rock strongly influence the degree of enrichment at the thermal front. In systems where initial isotope values of fluids and rocks are far from equilibrium and isotope fractionation is controlled by kinetics, the temperature increase accelerates the approach of the fluid to equilibrium conditions with the host rock. Consequently, the increase at the thermal front can be less dominant and can even generate fluid values below the initial isotopic composition of the input fluid. As kinetics limit the degree of isotope exchange, a third front may develop in kinetically limited systems, which propagates with the advection speed of the incoming fluid and is, therefore, traveling fastest. The results show that oxygen isotope signatures at thermal fronts recorded in rocks and veins that experienced isotope exchange with fluids can easily be misinterpreted, namely if bulk analytical techniques are applied. However, stable isotope microanalysis on precipitated minerals may – if later isotope exchange is kinetically limited – provide a valuable archive of the transient thermal and hydrological evolution of a system.