Volume 35, Issue 6

Hydraulic fracturing technology is widely used in the oil and gas industry. A part of the technology consists in injecting a mixture of proppant and fluid into the fracture. Proppant significantly increases the viscosity of the injected mixture and can cause plugging of the fracture. In this paper we propose a numerical model of hydraulic fracture propagation within the framework of the radial geometry taking into account the proppant transport and possible plugging. The finite difference method and the singularity subtraction technique near the fracture tip are used in the numerical model. Based on the simulation results it was found that depending on the parameters of the rock, fluid, and fluid injection rate, the plugging can be caused by two reasons. A parameter was introduced to separate these two cases. If this parameter is large enough, then the plugging occurs due to reaching the maximum possible concentration of proppant far from the fracture tip. If its value is small, then the plugging is caused by the proppant reaching a narrow part of the fracture near its tip. The numerical experiments give an estimate of the radius of the filled with proppant part of the fracture for various injection rates and leakages into the rock.

In the present paper, we propose a new combined kernel-projective statistical estimator of the two-dimensional distribution density, where the first ‘main’ variable is processed with the kernel estimator, and the second one is processed with the projective estimator for the conditional distribution density. In this case, statistically estimated coefficients of some orthogonal expansion of the conditional distribution density are used for each ‘kernel’ interval defined by a micro-sample. The root-mean-square optimization of such an estimator is performed under the assumptions concerning the convergence rate of the used orthogonal expansion. The numerical study of the constructed estimator is implemented for angular distributions of the radiation flux forward-scattered and backscattered by a layer of matter. A comparative analysis of the results is performed for molecular and aerosol scattering.

This paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

We present algorithms for fast computation of phase functions of atmospheric clouds and for stochastic simulation of such optical phenomena as fogbows, glories, coronas and halos. Using the developed numerical algorithms and software, we analyze optical phenomena for several cloud and fog models.