Prediction of sanding in subsurface hydrocarbon reservoirs.

Sand production in oil and gas wells can occur if the fluid velocity exceeds a certain value. Due to drilling operations, the mechanical stresses can exceed the load bearing capacity of the rock. As the local stresses exceed certain level, a certain amount of rock is fractured into sand. Then, the sand is carried by the fluid through the wellbore depending on the flow rate. The amount of the solids can be less than a few grams per cubic meter of reservoir fluid or an essential amount. In the later case erosion of the rock and removing sufficient quantities of rock can occur. This can produce subsurface cavities which collapse and destroy the well. When sanding is unavoidable it is necessary to estimate the characteristics of the process. Our aim was to generate a simple one-dimensional local model, which predicts the volume of sanding, the radius and the porosity of the yielded zone. Such model will help the company in the development of complex 3D models

Orbital Stability of Solitary Waves to Fourth Order Dispersive Equations with Quadratic Nonlinearity

[Kolkovska N.; Кольковска Н.]; [Dimova M.; Димова М.]; [Kutev N.; Kutev Nikolai; Кутев Николай]2010 Mathematics Subject Classification: 35B44, 35L75

Necessary and Sufficient Condition for Finite Time Blow up of the Solutions to Sixth Order Double Dispersive Equations

[Kutev N.; Kutev Nikolai; Кутев Николай]; [Kolkovska N.; Кольковска Н.]; [Dimova M.; Димова М.]The nonlinear double dispersive equation of sixth order with linear restoring force is investigated. Necessary and sufficient condition for finite time blow up of the solution with arbitrary positive energy is obtained. New very general sufficient conditions for blow up of the solution are proved. Explicit choice of initial data with arbitrary positive initial energy, satisfying all conditions of the theorems, are given. 2010 Mathematics Subject Classification: 35B44, 35L75

Global Solvability to Double Dispersion Equation with Bernoulli Type Nonlinearity via One Parametric Family of Potential Wells

[Kutev N.; Kutev Nikolai; Кутев Николай]; [Dimova M.; Димова М.]; [Kolkovska N.; Кольковска Н.]One parametric family of potential wells for double dispersion equation with Bernoulli type nonlinearity is introduced. Sign preserving properties of the Nehari functionals are obtained. Global existence of the weak solution to the Cauchy problem is proved for wider class of initial data than the corresponding ones in the classical potential well method. 2010 Mathematics Subject Classification: 35L30, 35L75

Application of the Improved Concavity Method to Sixth Order Boussinesq Equations with Arbitrary High Initial Energy

[Kutev N.; Kutev Nikolai; Кутев Николай]; [Dimova M.; Димова М.]; [Kolkovska N.; Кольковска Н.]Finite time blow up of the solutions to sixth order Boussinesq equation with arbitrary positive initial energy is proved. An improved variant of the concavity method of Levine is applied. This new method allows us to derive nonexistence of global solutions under conditions on the initial data which are more general than the assumptions used in the literature. 2010 Mathematics Subject Classification: 35L30; 35L75; 35B44

Finite Time Blow up of the Solutions to Nonlinear Klein-Gordon Equation with Arbitrary High Positive Initial Energy

[Kutev N.; Кутев Н.]; [Kolkovska N.; Колковска Н.]; [Dimova M.; Димова М.]2010 Mathematics Subject Classification: 35L05, 35L15

Global Behavior of the Solutions to Sixth Order Boussinesq Equation with Linear Restoring Force

[Kutev N.; Kutev Nikolai; Кутев Николай]; [Kolkovska N.; Кольковска Н.]; [Dimova M.; Димова М.]Potential well method is established to sixth order Boussinesq equation with linear restoring force and subcritical initial energy. For supercritical initial energy finite time blow up of the solutions is proved under general structural conditions on the initial data. Numerical experiments, illustrating the theoretical results, are presented. 2000 Mathematics Subject Classification: 35L30,76B15,65M06

Обобщен метод на Левин и приложения за нелинейни дисперсни уравнения

[Kutev N.; Кутев Н.]; [Kolkovska N.; Кольковска Н.]; [Dimova M.; Димова М.]A new ordinary differential inequality without global solutions is proposed. Comparison with similar differential inequalities in the well-known concavity method is performed. As an application, finite time blow up of the solutions to nonlinear Klein-Gordon equation is proved. The initial energy is arbitrary high positive. The structural conditions on the initial data generalize the assumptions used in the literature for the time being