Analytical Solution For Transient Flow Research Articles

Two‐Dimensional Analytical Solution for Transient Flow in Unsaturated Soils Considering Hydromechanical Coupling

AbstractThis paper presents a two‐dimensional solution for unsaturated transient seepage flow in a trapezoidal region using analytical solutions for flow in a series of rectangular regions. The coupled formulation considers the interaction among fluid flow due to differences in head and displacement of soil particles. The solution in each rectangular region is derived using Laplace Transform and Discrete Fourier Series for two top boundary conditions: specified pressure head and specified flow rate. The accuracy of the proposed solution is verified by comparing the results against those attained from the finite difference method. The proposed solution is applied to two example problems, demonstrating its efficacy and accuracy. The findings show a significant difference in the coupled and uncoupled results for fine‐grained soils possessing a small Gardner's coefficient. Additionally, the uncoupled pressure heads advanced quicker to a steady state than the coupled ones for soils with a small Gardner's coefficient. The coupling effect becomes less prominent as Gardner's coefficient increases. This study is the first attempt in the literature to provide a solution for unsaturated flow in a trapezoidal or non‐rectangular domain using analytical solution techniques, which can better represent complex real‐world applications (e.g., slopes, embankments, dams).

A model for the dissipation rate in linear unsteady flow through porous media

We present a model for the volume-averaged dissipation rate in linear unsteady flow through porous media. The model is derived by blending a new small-time asymptotic expression for the dissipation rate obtained from boundary layer theory with the known large-time asymptotic expression obtained from Darcy's law. The resulting model is a second-order Volterra functional of the volume-averaged acceleration. We validate the model with an analytical solution for transient flow through a porous medium composed of circular tubes and with numerical simulations of transient and oscillatory flow through a cylinder array and through a hexagonal sphere pack.

Analytical Solution of Transient Flow of Fractional Oldroyd-B Fluid between Oscillating Cylinders

This paper investigates the fractional Oldroyd-B fluid flow across two interminable coaxial cylinders, where the fluid’s motion is generated by the oscillatory movement of cylinders and the oscillating pressure gradient. The profile of velocity and shear stress of the flow is derived with the assistance of Caputo fractional derivative utilizing an analytical technique, finite Hankel transform and Laplace transforms. The semi-analytical solution is then displayed as generalized functions of G and R satisfying fundamental constitutive equations and all initial and boundary conditions. To validate the results, some limitations have been imposed on the determined equations and the results have been contrasted against previous results. Moreover, the influence of various parameters on the flow of fractionalized Oldroyd-B fluid is investigated and depicted graphically.

A modeling and analytical solution for transient flow in natural gas pipelines with extended partial blockage

Modeling analysis of pipeline systems is a significant means of effective system management and control; however, analytical solutions to the transient flow in natural gas pipelines with partial, extended blockages remain an enduring challenge in the oil and gas industry. Based on a variable diameter pipeline in series, a physical model and a simplification suitable for a natural gas trunk pipeline, this paper proposes a novel, analytical solution to the problem. The Sturm-Liouville theory and Fourier transform have been adopted to solve the mathematical model. Finally, an analytical solution is derived, and the pressure distribution of a plugged natural gas pipeline is presented. The accuracy of the calculated, analytical solution is verified using a standard industry software package. The model and analytical solution presented in this paper can be used for additional blockage detection and characterization analysis.

Analytical Solution of Transient Flow in Pipe with Junction Coupling due to Hydro-Turbine Vane Motion

This paper is concerned with an analytical solution of transient flow on the four-equation model in fluid-structure interaction (FSI) for a reservoir-pipe-vane system subjected to a sudden closure of hydro turbine vane. The analytical solution is derived corresponding to the junction coupling based on a vane motion. The result obtained from the analytical solution is used to analyze the coupling features between pipe transient flow and hydro turbine vane motion.

Analytical Solution of Transient Flow in Pipe with Junction Coupling due to Hydro-Turbine Vane Motion

This paper is concerned with an analytical solution of transient flow on the four-equation model in fluid-structure interaction (FSI) for a reservoir-pipe-vane system subjected to a sudden closure of hydro turbine vane. The analytical solution is derived corresponding to the junction coupling based on a vane motion. The result obtained from the analytical solution is used to analyze the coupling features between pipe transient flow and hydro turbine vane motion.

Analytical solution of transient flow in a sloping soil layer with recharge

An analytical solution of planar flow in a sloping soil layer described by the linearized extended Boussinesq equation is presented. The solution consists of the sum of steady-state and transient-series solutions, the latter in a separation-of-variables form, and can satisfy an arbitrary initial condition via collocation; this feature reduces the number of series terms, making the solution efficient. Key parameter is the dimensionless linearization depth η o (R), R being the dimensionless recharge. The variable η o (R), not the slope, characterizes the flow as kinematic or diffusive, and R ≈ 0.2 demarcates the two regimes. The transient series converges rapidly for large η o (large R, near-diffusive flow) and slowly as η o → 0 (kinematic flow). The quasi-steady (QS) state method of Verhoest & Troch is also analysed and it is shown that the QS depth profiles approximate the transient ones well, only if Δt exceeds a system-dependent transition time between flow states (possibly >>1 day). In an application example for a 30-day recharge series, the QS solution differs from the transient one by as much as 20% (RMSE = 15%), does not track recharge changes as well and fails to conserve mass.

The transient elliptic flow of power-law fluid in fractal porous media

The steady oil production and pressure distribution formulae of vertically fractured well for power-law non-Newtonian fluid were derived on the basis of the elliptic flow model in fractal reservoirs. The corresponding transient flow in fractal reservoirs was studied by numerical differentiation method: the influence of fractal index to transient pressure of vertically fractured well was analyzed. Finally the approximate analytical solution of transient flow was given by average mass conservation law. The study shows that using elliptic flow method to analyze the flow of vertically fractured well is a simple method.

An analytical solution for transient flow of Bingham viscoplastic materials in rock fractures

We present below an analytical solution to model the one-dimensional transient flow of a Bingham viscoplastic material in a fracture with parallel walls (smooth or rough) that is subjected to an applied pressure gradient. The solution models the acceleration and the deceleration of the material as the pressure gradient changes with time. Two cases are considered: a pressure gradient applied over a finite time interval and an applied pressure gradient that is constant over time. The solution is expressed in dimensionless form and can therefore be used for a wide range of Bingham viscoplastic materials. The solution is also capable of capturing the transition that takes place in a fracture between viscoplastic flow and rigid plug flow. Also, it shows the development of a rigid central layer in fractures, the extent of which depends on the fluid properties (viscosity and yield stress), the magnitude of the pressure gradient, and the fracture aperture and surface roughness. Finally, it is shown that when a pressure gradient is applied and kept constant, the solution for the fracture flow rate converges over time to a steady-state solution that can be defined as a modified cubic law. In this case, the fracture transmissivity is found to be a non-linear function of the head gradient. This solution provides a tool for a better understanding of the flow of Bingham materials in rock fractures, interfaces, and cracks.

Analytical solution for transient flow into a hemispherical auger hole

An explicit analytic solution for a transient groundwater flow to an auger hole of hemispherical shape is found on the base of ideas and under the same assumptions as in the original paper of Kirkham and van Bavel (1948). Theory of seepage into auger holes. Soil Sci. Soc. Am. Proc., 13, 75–82). The Legendre polynomial expansions are used to solve the Laplace equation with a combination of seepage face and equipotential conditions along the hole contour. A type curve for increasing water level in an initially empty hole is plotted, which allows for a fast determination of the hydraulic conductivity from the observation of filling of a small hole tapping a shallow water table. Time-decreasing total flow rates, velocities, particle trajectories, a capture zone of the hole are calculated using the Wolfram Mathematica software.

Soil Water Dynamics Under Drip Irrigation: Transient Flow and Uptake Models

Advances in drip irrigation management rely on knowledge of the distribution and dynamics of soil water within the wetted soil volume. Part of the information can be obtained from numerical or analytical models for water flow from point or line sources. However, a more complete picture of soil water dynamics in cropped fields requires explicit consideration of plant water extraction patterns. In this study we propose to combine an analytical solution for transient flow from cyclic point sources representing drippers with a parametric model for uptake by plant roots. The linearized flow and uptake problems are amenable for spatial superposition. The transient uptake pattern is represented by an array of buried line sinks of different strengths determined by a new parametric uptake model. In addition, a point approximation is proposed (for monitoring purposes) based on a localized water balance which considers changes in water content due to transient flow from the dripper coupled with plant uptake only from the soil volume under consideration. Comparisons with greenhouse (and field) measurements show very good agreement with the localized approximation for one- and two-day irrigation intervals. The superposition solution of point source with an array of line sinks tended to under-estimate uptake. The two methods offer reasonable means to describe soil water dynamics within wetted soil volumes under cropped conditions for various irrigation management scenarios.

Flow of groundwater in fractured rocks

A pervasive problem in dealing with fractured rocks is the importance of the flow of ground water through the discontinuities. This paper describes the results of recent work in this laboratory to investigate this problem. A much better understanding of the physics of fluid flow in a natural fracture from a sample of granite has been obtained from metal casts of the complex topography of the surfaces of the fracture as it is subjected to normal stresses up to 85 MPa. Contact area within the deforming aperture increases up to 30 percent and produces a flow regime that cannot be described by the cubic law. An investigation of flow in a network of fractures using a new numerical technique has been carried out to determine the effect of length and density of fractures on permeability. Networks with shorter fracture lengths and higher density will have lower permeabilities and will behave less like porous media than networks with longer fracture lengths and lower density. As fracture length increases, permeability approaches a maximum that can be predicted on the basis of infinite length fractures. A new analytical solution for transient flow to a borehole that penetrates a fracture dominated rock mass is summarized. A new derivative method of analyzing pressure transients from this solution is discussed and enables one to distinguish a fracture dominated system from one that exhibits double-porosity behavior.

Analytical solution for transient flow of energy in a one-dimensional radiating fin.

Analytical solution for transient flow of energy in one dimensional semiinfinite fin in zero irradiation environment