A Similarity Solution for Imbibition Process and its Adaptation in Finite Difference Simulation of Fractured Reservoirs

The imbibition process due to capillary force is an important mechanism that controls fluid flow between the two domains, matrix and fracture, in naturally or hydraulically fractured reservoirs. Many simulation studies have been done in the past decades to understand the multi-phase flow in the tight and shale formation. Although significant advances have been made in large-scale modeling for both unconventional and conventional fields, the imbibition processes in the fractured reservoirs remains underestimated in numerical simulation, that limits confidence in long-term field production predictions. In the meanwhile, to simulate the near-fracture imbibition process, traditionally very-fine simulation grids have to be applied so that the physical phenomena of small-length scale could be captured. However, this leads to expensive computation cost to simulate full-field models with a large number of fractures. To improve numerical efficiency in field-scale modeling, we propose a similarity solution for the imbibition process that can be incorporated into the traditional finite difference formulation with coarse grid cells. The semi-analytical similarity solutions are validated by comparing with numerical simulation results with fine-scale grids. The comparison clearly indicates that the proposed algorithm accurately represents the flow behaviors in complex fracture models. Furthermore, we adopt the semi-analytical study to hydraulic fracture models using Embedded Discrete Fracture Model (Lee et al., 2001) in our numerical studies at different scales to represent hydraulic fractures that are interconnected. We demonstrate: 1) the imbibition is critical in determining flow behavior in a capillary force dominant model, 2) conventional EDFM has its limitation in capturing sub-cell flow behaviors near fractures, 3) combining the proposed similarity solution and EDFM, we can accurately represent the multi-phase flow near fractures with coarser grids, and 4) it is straightforward to adapt the similarity solution concept in finite-difference simulations for fractured reservoirs

Lie Symmetries, Painlev\’{e} analysis and global dynamics for the temporal equation of radiating stars

We study the temporal equation of radiating stars by using three powerful methods for the analysis of nonlinear differential equations. Specifically, we investigate the global dynamics for the given master ordinary differential equation to understand the evolution of solutions for various initial conditions as also to investigate the existence of asymptotic solutions. Moreover, with the application of Lie’s theory, we can reduce the order of the master differential equation, while an exact similarity solution is determined. Finally, the master equation possesses the Painlev\’{e} property, which means that the analytic solution can be expressed in terms of a Laurent expansion.

Mathematical Modelling of Water-Based Fe3O4 Nanofluid Due to Rotating Disc and Comparison with Similarity Solution

The current research demonstrates the revolving flow of water-based Fe3O4 nanofluid due to the uniform rotation of the disc. This flow of nanofluid is investigated using CFD Module in COMSOL Multiphysics. However, the similarity solution for this flow is also obtained after transforming the given equation into a non-dimensional form. In the CFD Module, streamlines and surface plots are compared with the similarity solution for the magnitude of the velocity, radial velocity, tangential velocity, and axial velocity. The results from the direct simulation in the CFD Module and the solution of dimensionless equations represent a similar solution of velocity distribution. The derived results show that increasing the volume concentration of nanoparticles and effective magnetic parameters decrease the velocity distribution in the flow. Results in the CFD Module are important for monitoring the real-time particle tracing in the flow and, on the other hand, the dimensionless solution is also significant for the physical interpretation of the problem. Both methods of solution empower each other and present the physical model without sacrificing the relevant physical phenomena.

Magnetogasdynamic shock wave propagation using the method of group invariance in rotating medium with the flux of monochromatic radiation and azimuthal magnetic field

The propagation of a cylindrical shock wave in rotating medium with azimuthal magnetic field under the action of monochromatic radiation using a method of group invariance is investigated. To derive similarity solutions as well as exact solutions, the group invariance technique is used. All classes of the solutions depending on the absorption coefficient are discussed by considering absorption coefficient to be variable or constant. A similarity solution is obtained, when the absorption coefficient is assumed to be variable. Two cases of solutions with a power law shock path are obtained by the different choices of arbitrary constants involving in the infinitesimal generators of the Lie group of transformations. To obtain the similarity solution in the case of the power law shock path, the density, magnetic field, axial and azimuthal velocity components are assumed to be varying and obeying power laws in the undisturbed medium. It is observed that with increase in the values of Alfven Mach number, adiabatic exponent and rotational parameter, shock strength decreases. The effects of variation of magnetic field strength, adiabatic exponent, rotational parameter and initial magnetic field variation index on the flow variables and on shock waves are analyzed graphically. Also, all classes of exact solutions are obtained by considering a constant absorption coefficient.

Cross-Shelf Transport Through the Interaction among a Coastal Jet, a Topographic Wave, and Tides

Shelf break flows are often characterized by along-isobath jets with cross-shelf currents associated with tides and waves guided by variable topography. Here, we address the question: Can a superposition of such flows produce significant aperiodic cross-shelf transport? To answer this question, we use a barotropic analytic model for the jet based on a similarity solution of the shallow water equations over variable topography, a wave disturbance determined by the topography, and a diurnal tidal disturbance. We use standard Lagrangian methods to assess the cross-shelf transport, presenting the results, however, in a Eulerian frame, so as to be amenable to oceanographic observations. The relative roles of the different flow components in cross-shelf transport are assessed through an extensive parameter study. We find that a superposition of all three flow components can indeed produce consequential background aperiodic transport. An application of the model using recent observations from the Texas Shelf demonstrates that a combination of these background mechanisms can produce significant transport under realistic conditions.