The seepage mechanics model comprises multiple nonlinearly coupled partial differential equations. In various applications, seepage mechanics problems exhibit distinct characteristics and the corresponding solution methods are also very different. This paper focuses on the representative mathematical models used in oil and gas reservoir development. It introduces the mathematical formulation and application characteristics of multiphase multicomponent seepage mechanics equations within porous media, along with efficient techniques for solving their discretized linear equation systems, including commonly employed preconditioning methods. Additionally, this study appropriately modifies standard test cases and evaluates the shared-memory parallel efficiency of these preconditioning methods.

This paper is concerned with the turbulence at a non-turbulent/turbulent density interface in a mean shear-free stably stratified two-layer fluid using the statistical theory of turbulence, spectral analysis, and Rapid Distortion Theory. Further extended calculations are made for the non-dimensional Eulerian frequency spectra of the horizontal and vertical velocities, and the horizontal and vertical root-mean-square velocities for both arbitrary and infinite Richardson number (Ri) for Case I, the density interface thickness (h) is negligible; Case Ⅱ, h is very thin, respectively. For Case I, (1) for arbitrary and infinite Ri, the effect of a density interface on large scale eddies is more significant than on the small scale eddies; the distortion of turbulence by a density interface is more significant in the vertical direction than in the horizontal direction.(2) For arbitrary Ri, if the non-dimensional frequency is large, the non-dimensional Eulerian frequency spectra of both the horizontal and vertical velocities satisfy the -5/3 power law at the density interface and within the turbulent layer. However, they are not converged into the same line, suggesting that the turbulence at the density interface is partially transferred into internal waves. For Case Ⅱ, (i) a density interface has no the effect on non-dimensional Eulerian frequency of the horizontal velocity; a transitional zone of the non-dimensional Eulerian frequency spectrum of the vertical velocity, which does not satisfy the -5/3 power law but has an increasing power law, is present within the turbulent layer; (ii) the non-dimensional Eulerian frequency spectrum of the vertical velocity satisfies the -5/3 power law while a decreasing powerlaw is present at the density interface; (iii) a transitional zone of the non-dimensional Eulerian frequency spectrum of the vertical velocity decreases and is shifted to the left side with increasing distance from the interface, suggesting that the energy within the transitional zone decreases after taking h into account; when the non-dimensional frequency increases, the non-dimensional Eulerian frequency spectrum of the vertical velocity satisfies the -5/3 power law, suggesting that the density interface has no effect on the small scale eddies after taking h into account; far from the density interface, a transitional zone disappears; (iv) the power of the non-dimensional Eulerian frequency spectrum of the vertical velocity at the density interface decreases, suggesting that the energy is focused within the low non-dimensional frequency zone after taking h into account; (v) when h increases, the non-dimensional Eulerian frequency spectrum of the vertical velocity deceases with the same amplitude within the whole non-dimensional frequency range of the linear internal waves at the density interface, while the non-dimensional Eulerian frequency spectrum of the vertical velocity only decreases within the linear low non-dimensional frequency range and its decreasing amplitude decreases with increasing non-dimensional frequency; furthermore, the vertical range within which the density interface affects the horizontal and vertical root mean square velocities decreases with increasing Richardson number.

With a model of foam wellbore flow and a model of foam seepage flow,a mathematical model of foam plug removal is given.The model is solved with numerical method.Distributions of foam pressure,foam quality,foam density along wellbore and wellhead and bottom pressure are discussed.Furthermore,variation of bottom hole differential pressure is given as wellhead back pressure is fixed.It shows that as foam pressure and density increases,foam quality decreases with increase of well depth and bottom hole differential pressure declines.

Foam microscopic seepage experiments show that foam present two seepage states: Flowing foam and trapped foam.Foam plugging experiment show that foam plugging ability has accumulating effect.Plugging pressure gradient near core inlet is less than that of downstream of the core when foam flooding reaches steady state.A multi-component mathematical model characterizing foam flooding is established based on bubble population balance theory and foam properties.The model is solved numerically with fully implicit method.Validity of the model is verified with foam plugging experiment.Meanwhile,foam microscopic seepage characteristic parameters such as foam texture,pressure and aqueous phase saturation in foam flooding are studied.

With Langevin equation of particle motion,PDF equations in two levels are obtained in which the fluid fluctuation velocity is expanded in dimension and one-dimensional colored noise is transformed into two-dimensional white noise.They are two-phase PDF transport equation of turbulence and particle-phase PDF transport equation.With particle motion equations and assumption of Gaussian distribution,a closure problem of PDF equation is solved and a second-order-moment model is obtained.It is simplified into an algebra stress model [ASM].A finite analytic numerical method for convective diffusion equation is applied to solve the two-order moment model of two-phase flows.A wall-jet-flow loaded with solid particles is studied in the ASM by finite analytic numerical method.Numerical results are compared with experimental results.

Based on a colored noise model with general Gaussian process,by functional derivation and small correlation time expansion,high-dimensional colored noises are calculated approximately and an effective Fokker-Planck equation(FPE) is derived.The effective FPE is applied to obtain probability density function transport equations of particles in two-phase flows and a second order moment model of two-phase flows.It is simplified into an algebra stress model(ASM).The finite analytic numerical method used in convective diffusion equation is applied to the second-order moment model.A wall-jet-flow loaded with solid particles is calculated with ASM and finite analytic numerical method.Numerical results are compared with experimental results.