Finite-Element Model of Filtration Liquid to a Well in a Deformable Formation

In this work we defined the stress-strain state of anisotropic (transversely-isotropic) formation at a liquid filtration in it. There were established connection between voltage of formation and pressure of are filtrated. Numerical solution of the problem was received based on the finite element method with application of an isoparametric element of the first order.


Introduction
A detailed literature review on the research of filtration processes shows that modern technologies and methods of influence on deposits with hardly removable stocks did not find adequate justification of the theory and practice of fluid filtration, taking into account change in structure low productivity of the collector. Also this requires further development of theoretical position on the non-stationary spatial filtering fluids in deformable low productivity porous-fissured, in medium, taking into account the sharp oscillation, hydraulic conductivities and permeability coefficients of energy in multilayer formations, which have the most significant impact on the process of hardly removable stocks. As practice of operation of multi sheeted deposits, an increase in oil production by improving of technology of production of reserves from low-productive formations, tantamount opens a new hydrocarbon fields.

Statement of the Problem
Mountain pressure upon formation is compensated as voltages solid skeleton of array, and fluid pressure. Change of the last perturbs tensely-deformed state of formation (VAT), i.e. changing of the pore pressure at one point causes reorganization of all the VAT of system and including the deformation throughout the formation.
When in the study of the stress-strain state of the formation is assumed that under the influence of the applied external forces, deformation of the formation proceeds without violation of its continuity [1]. Therefore it is necessary to impose restrictions on size component of deformation.
We will consider an elastic static condition of the horizontal well, the longitudinal axis of which is an arbitrary angle with the line of the strike plan of an isotropy of the rock massiv ( fig. 1).
We introduce the rectangular cartesian coordinate system Oxyz in such a way that the axis is Oz directed vertically up, horizontal axis Ox and Оy coincide with the lines accordingly transverse to and along the strike plane of isotropy.
Elastic condition of transversaly-isotropic of array described by the equation of generalized Hooke's law in the coordinate system of the Ox΄y΄z΄ obtained by turning of  Components of deformation can also be determined through displacement u , ν and w , (by the Ox, Oy and Oz axis respectively) by means ratio of Cauchy. Boundary conditions we will set in the form of Next non-stationary filtration of liquid to a horizontal well in transversely isotropic porous medium ( Fig. 1) described by the following equation governing the pressure P spatial filtering

= +
A characteristic feature of the model is the assumption that the porous matrix deformable absolutely freely to up some hard limit 0 ε .

Finite-Element Model
Consider the three-dimensional isoparametric six-sided finite element first order [1], for which the interpolation polynomial is a linear function of local coordinates (Fig. 2). Limits of change of local coordinates for all elements is .  , Required values are defined from the decision of systems of he linear algebraic equations [5] , KU F =

Computing Experiment
The deformable state inclined at an angle ϕ the transversal-isotropic massif is executed by using of law Hooke and deformation coefficients. Numerical experiment was conducted on the following data: in the capacity of breeds of inclined a layer are taken [2].   The analysis of the results given in inclined transversal-isotropic of formation, shows that with increase in quantity of final elements in discrete model of a body, is observed coincidence of two significant figures in values of a component of movement of u, normal tension σ, and also in values of intensity of tension and deformations.

Conclusions
In this paper we defined the stress-strain state of anisotropic (transversely-isotropic) formation at a liquid filtration in it. There were established connection between voltage of formation and pressure of are filtrated. Numerical solution of the problem was received based on the FEM with application of an isoparametric element of the first order. the Presented by a finite-element model of voltage deformable condition of formation with horizontal well an arbitrary profile. There were analysed ways of determining the elastic and filtration properties of deformable heterogeneous reservoirs. Thus, by using the finite element model can obtain the change in fluid pressure in the voltage-strain state of a transversely-isotropic formation.