ANALYSIS OF SEEPAGE PRESSURE IN DUAL-POROSITY RESERVOIR UNDER ELASTIC BOUNDARY

In view of the large limitations of the dual-porosity media seepage model established in the early study, the elastic external boundary condition is newly presented in this paper, which can treats the idealized assumption (external boundary constant pressure, closed, infinite) in traditional model as a special case. Based on it, with considering the influence of well-bore storage, skin factors and external boundary radius on reservoir, an unstable seepage model under the elastic boundary is established first. then, the Laplace space solution of seepage model is obtained by using Laplace transform and similar structure theory in turn. Subsequently, by using Stehfest inversion transformation and the corresponding mapping software, the type curves are drawn, and the impacts of the main parameters on them are analyzed. The results indicate that the elastic coefficient has negative affect on the asymptotic rate of the type curve; And the type curves determined by different external boundary radii deviate during the later period of flow; Further more, the elastic coefficient affects the migration trajectory of the curves. Numerical simulation further verifies the scientificalness of introducing elastic external boundary conditions. The model established in this paper and the corresponding data analysis provide a more solid theoretical basis for the scientific analysis of the influence of reservoir parameters on reservoir pressure, and provide a new idea for the design and improvement of related well testing software.


INTRODUCTION
The reservoir parameters are reversely calculated after matching actual theoretical curve to measured pressure curve by using modern well test analysis methods usually. Looking back to the research results obtained via modern well test analysis methods, we get obviously that though theoretical pressure type curve is similar to the actual measured, there are large errors with the theoretical [1], which brings about the error of the obtained reservoir parameters inevitably. How can we reduce such errors?In engineering, in addition to the instrumental measurement errors, the sources of error are mostly the over-idealized assumptions [2,3]. The existing well test interpretation models are also based on specific idealized assumptions, including the ideal external boundary condition(closure, infinity and constant pressure)assumption. Based on this consideration, the authors contemplate a new external boundary condition (elastic external boundary) to decrease the errors of the obtained reservoir parameters.
What is the elastic outer boundary? In fact, the concept of elasticity has been widely used in economics, physics and engineering theory [4,5,6]. Alfred.Marshall, a famous economist, put forward the concept of elasticity: elasticity means the intensity or sensitivity of the relative change of dependent variables to the relative changes of independent variables. In 2010, Arthru P. Boresi (USA) published the "Elastic Theory" in Engineering Mechanics which describes the application of elasticity in the branches of mechanics. The significance of putting forward the concept of elasticity is that elasticity solves the rigidity problem to a great extent in engineering problems. Hence, in this paper, with the help of the definition of elasticity itself, the elastic function and the elastic external boundary condition will be defined in the seepage problem, which will transform three kinds of "rigid" external boundary conditions (closed, constant pressure, infinity) considered in the earlier studies into the elastic external boundary condition. Strictly to say, the elastic external boundary condition defined not only contains the three special boundary conditions but also extends the ideal outer boundary further, that makes it possible to reduce the error between theoretical pressure curve and the measured. This paper focuses on the problem of dual-porosity media reservoirs. The research on dualporosity media reservoirs originated in 1960, Barenblatt, Zheltov and Kochina proposed the basic concept of fluid motion in fractured rock and based on some special hypothesis, the general equation of fluid flow [7] in dual-porosity media was obtained. In1997, E.S. Choi [8], M.
Preshoand S. Wo, V. Ginting [9,10] et al. established a seepage model under infinite, constant pressure and closed external boundary conditions for the problem of dual-porosity media and dual-permeability reservoir.Although the analytical solution was not given, the boundary conditions considered by the model had always been used by contemporary reservoir researchers, so the work done by the author is still meaningful. On the basis described above, Li Shunchu, Zheng Pengshe, Chen Liya et al. [11,12,13,14] [15,16,17].
After making use of the similar structure method and Stehfest numerical inversion method, a similar structure algorithm is developed to calculate well-bore pressure and pressure derivative of reservoir percolation model, and the type curves are drawn and analyzed at last, which provides a new idea for the design of the corresponding well test software.
On the basis of the above-mentioned analysis, in this paper, the elastic outer boundary condition will be introduced firstly when solving the mathematical model of unstable double porosity permeable flow, taking the three kinds of external boundary conditions (closure, infinity and constant pressure) considered in the past as special cases. Second, the similar construction method is used to solve the seepage model solution, and the numerical inversion method is used to draw the characteristic curve of bottom hole pressure in real space. But more than those, the influence of characteristic parameters under elastic boundary condition on the characteristics of unstable seepage pressure is analyzed, so as to make a more scientific guidance for the actual situation of oil and gas reservoir in reservoir development.

PRELIMINARIES
This paper is based on the following assumptions.

ANALYSIS OF SEEPAGE PRESSURE IN DUAL-POROSITY RESERVOIR UNDER ELASTIC BOUNDARY 319
(1) Consider the shape of the matrix block to be a spherical block.
(2) The flow of a fluid is isothermal.
(3) There is no direct fluid flow between the pores and the well, and the flow sequence is:pore → crack → well.
(4) Fluid flow in the seepage field obeys Darcy's law.
(5) Do not take the influence of gravity into account.
2.1 Elasticity in seepage problem. In the process of fluid seepage, the pressure, affected by time and space, can be expressed as p = p(x 1 , x 2 , x 3 ,t), so as to give the pressure drop expression where p 0 is the original formation pressure. In particular, the pressure drop expressed as P = P(r,t) = p 0 − p(r,t) is considered to be a function of time and radial radius in the plane radial flow process.
In solving the issue of unstable seepage pressure dynamics, the state at every moment of the unstable seepage process can be considered stable. This method is referred to as the steady state replacement method [18]. Therefore, when defining the concept of elasticity below, the pressure drop can be considered as a function P(r)related only to the radial radius.
Definition 1:the pressure drop P(r), a function of the fluid for radial radius r, is differentiable during seepage, then we call ε P r is the elastic function of ∂ lnr , and the value of ε P r at the point r 0 is the elastic coefficient of ε P r at point r 0 , recorded as ε P r 0 .
ε P r reflects the relationship between the relative rate of change of the radial radius and the relative rate of change of the pressure drop, also known as the sensitivity of P to r. For instance, the value of elastic function (elastic coefficient) is ε P r (r 1 ) when r = r 1 at a certain time , Then the pressure drop is also approximately changed ε P r (r 1 )% if the radial radius changes 1%.
2.2 Elastic external Boundary in the seepage problem of dual-porosity media. In this paper, we only need to establish the elastic function of the pressure drop P 1 (r) related to the radial radius r in the fracture system in view that fluid flows into the well-bore only through fractures in the process of formation seepage of dual-porosity media.
Let the external boundary Γ : r = R, according to the definition of elastic coefficient, the elastic coefficient of its boundary can be described as Three kinds of external boundary conditions (closed, constant pressure, infinite) are considered in the study of early petroleum engineering as follows.
(1) The equation ∂ P 1 ∂ r | r=R = 0 is established when the outer boundary is closed, and we get ε P 1 Γ = 0 at this time.
(2) The equation P 1 | r=R = 0 is established when the outer boundary pressure is constant and we get ε P 1 Γ → +∞ at this time. (3) The equation P 1 | r→+∞ = 0 is established when the outer boundary is infinite and we get R → +∞ at this time.
Therefore the following definition can be given.
Definition 2:Let the pressure drop of the fracture system be P(r) as a function of the radial radius r and the outer boundary radius is R, then is called the elastic outer boundary condition of the fluid.

MAIN RESULTS
3.1 basic assumptions and mathematical models. Combined with the initial condition and boundary condition, the seepage model of dual-porosity medium can be established as follows [19,20,21] (3.1.1) Based on the assumption of 3.1.1, the dimensionless seepage model of matrix system is obtained by dimensionless treatment of the variables in the seepage model of matrix system.
For (3.2.1), taking the Laplace transform as follows on dimensionless time t D .
Where z is a parameter, and then we get .Its easy to obtained the answer(derivation details are included 3.2.2 Dimensional seepage model of fracture system under elastic boundary. On the basis of section 3.2.1, the following dimensionless quantities are introduced.
The dimensionless mathematical model of fluid seepage in fracture system is obtained as follows.

Similarity Construction
Theory for Boundary value problems of differential equations. Theorem 1: In the form of (3.2.8) this boundary value problem of ordinary differential systems have the following form of solution where a, b, A, E, F are the real constant and A > 0, E 2 + F 2 = 0, 0 < a < b:  Step1:The function is constructed from the parameter A and the independent variable x in the solution equation and its value a, b on the boundary are as follows: then calculate the value of ϕ 1,0 (a, b, √ A) and ϕ 1,1 (a, b, √ A) Step2 Therefore, a similar kernel function, which is easy to obtain the solution of the (3.2.6) model, is as follows.
Γ ϕ 1,0 (1, R D , z f (z)) + R D z f (z)ϕ 1,1 (1, R D , z f (z))] and the Laplace space solution of dimensionless pressure of the crack system can be obtained as follows .
Owing to P wD = [P 1D − Sr 1D dP 1D dr 1D ]| r 1D =1 , the Laplace space solution of dimensionless bottomhole pressure can be obtained as follows.   4.1 Effect of ε P 1D Γ and C D for the type curves. Fig.1 shows the variation of dimensionless bottom hole pressure with time under the action of well-bore reservoir C D . The figure firstly shows that the well-bore reservoir mainly affects the initial stage of fluid flow.Secondly, the larger the well-bore reservoir is, the smaller the dimensionless bottom hole pressure well be. Thirdly, the dimensionless bottom hole pressure increases at a relatively stable rate for a period of time. When the matrix system fluid flows to the fracture system, the growth rate of the curve appears to be transiently stable, and then continues to increase. The characteristic curve determined by different well-bore reservoir coefficients eventually asymptotically follows a curve.    ative. It can be seen from the figure that the dimensionless pressure derivative increases continuously in the initial stage of production. When the matrix system fluid flows to the fracture system, the dimensionless pressure derivative becomes smaller in a short time and then continues to increase, and finally characteristic curve is asymptotic to a curve. The value of the well-bore storage coefficient is positively correlated with the time it takes for the peak to appear, that is, the smaller the C D , the earlier the peak appears.       The elastic coefficient is positively correlated with the total length of the significant difference, negatively correlated with the asymptotic velocity of the curve, and negatively correlated with the pressure derivative when the curve is asymptotic.   that the elastic outer boundary condition does not affect the overall trend of the characteristic curve determined by R D . characteristic curve in the later stage of flow appears to be offset.

SENSITIVITY ANALYSIS
When the outer boundary radius is the same, the larger the inclination of the characteristic curve is with the larger the elastic coefficient.     between the curves determined by the elastic coefficient ε P 1D Γ = 0 and ε P 1D Γ = in f , which further verifies the scientificity of introducing the elastic outer boundary condition.

Conclusions
Considering the influence of well-bore storage, skin factors and external boundary radius on reservoir pressure, an unsteady seepage model under elastic boundary is established. The unsteady seepage systems in dual-porosity media is obtained by using Laplace transformation and initial boundary conditions. Using the similar construction theory, the expression of the dimensionless pressure solution of the matrix system and the fracture system under the elastic boundary is obtained.
In view of the limitation of the traditional model in describing the percolation law, the elastic coefficient and the elastic outer boundary conditions are defined based on the steady-state displacement method. The variation law of bottom hole pressure and pressure derivative is obtained through comprehensive analysis of well-bore reservoir, skin factor, outer boundary radius and elastic boundary. The research shows that under the wellbore reservoir effect, the characteristic curve will have extreme points. When the value of C D is smaller, picture shows that the inflection point and extreme point appear earlier. The smaller the S value in the middle phase of the flow is, the smaller the bottom hole pressure and pressure derivative in the same period. The smaller the R D in the later stage of fluid flow, the earlier migration occurs in the characteristic curve. When the R D is fixed, the elastic coefficient affects the asymptotic time of the characteristic curve. The larger the elastic coefficient, the longer the asymptotic time is. The greater the medium-term elastic coefficient, the more gradual the pressure changes. When C D and S are fixed, the elastic coefficient affects the deviation of the curve steady state trajectory. the elastic external boundary condition defined not simply take the external boundary conditions (closed, infinite and constant pressure) considered in the past are regarded as special cases but can solve more complex boundary seepage problem with taking difference elastic coefficient. Therefore, the introduction of elastic outer boundary makes it possible to reduce the error looking from the type curve between theoretical pressure curve and the measured. At the same time, the model and corresponding data analysis established in this paper provide a solid theoretical basis for the scientific analysis of the influence of reservoir coefficient on reservoir pressure, and provide a new idea for the design and improvement of the corresponding well testing software.