Submit manuscript...
International Journal of
eISSN: 2475-5559

Petrochemical Science & Engineering

Review Article Volume 3 Issue 5

Analysis of the interface between two immiscible fluids in porous media considering special cases

Mohammad Reza Shahnazari, Ali Saberi

Mechanical Engineering Department, KN Toosi University of Technology, Tehran, Iran

Correspondence: Shahnazari MR, Associate Professor, Department of Mechanical Eng, KN Toosi University of Technology, Tehran, Iran, Tel 9821-8406-42254

Received: September 19, 2018 | Published: November 8, 2018

Citation: Shahnazari MR, Saberi A. Analysis of the interface between two immiscible fluids in porous media considering special cases. Int J Petrochem Sci Eng. 2018;3(5):182-186. DOI: 10.15406/ipcse.2018.03.00093

Download PDF

Abstract

Analysis of the interface between two immiscible fluids in porous media is considered as a classic problem with numerous industrial applications. Interface between two fluids could be unstable. In addition to defining dimensionless equations for a generalized geometry, this study developed equations for special cases. For the problem in one dimension, linear steady state analysis was done and it is indicated that steady state condition is maintained as long as the density of the upper fluid is less than the density of the lower fluid. By using the "K-Method", an analytical solution of the Boussinesq equation is presented and the results for constant and linear flux are obtained and discussed. Considering a Power-law flux at the origin and dimensional analysis, the drilling mud invasion into a permeable aquifer problems becomes a nonlinear boundary value problem that have been investigated in this paper. The obtained results show that the K-Method has a good ability for investigation of B-L problems such as this solved problem.

Keywords: interface, immiscible fluids, porous media, kourosh method, boussinesq equation

Introduction

Evolution of the interface between two immiscible fluids is one of the most complicated problems in the fluid dynamics. The major complicity of this problem is due to difficulty of an explicitly closed that describe the interface movement. Parameter h denote the interface position is to deduce the close differential equation for the function h( x, y, t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadIgajuaGdaqadaGcpaqaaKqzGeWdbiaadIhacaGG SaGaaiiOaiaadMhacaGGSaGaaiiOaiaadshaaOGaayjkaiaawMcaaa aa@4127@ , x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadIhaaaa@37C2@ and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadMhaaaa@37C3@ are the coordinates in horizontal plate, t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadshaaaa@37BD@ is the time variable and h( x,y,t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObqcfa4aaeWaaOWdaeaajugib8qacaWG4bGaaiilaiaa dMhacaGGSaGaamiDaaGccaGLOaGaayzkaaaaaa@3EB5@ is the thickness of the lower fluid.

The behavior of the stratified pair "gas-oil" or "oil-water" natural gas-oil reservoir, can be describe by the thickness of the lower fluid, on the modeling of this function h and then for an optimized recovery, it is absolutely necessary to have an appreciate modeling.1 Whitham2 has considered the case that the viscous force for both fluids can be neglected. Also the case when one of the two fluids is inviscid (like as in water-oil flow investigation in Shallow water theory or ground water flow) have been investigated by Dagan,3 Bear4 and Barenblatt.5 In ground water hydrology, Boussinesq models are involved to describe radial flow from or to well.6,7 Axisymmetric flow is a feature of ground water systems subject to pumping.8,9 Invasion of drilling fluid into a permeable bed is an example of involving pumping problems.10,12 Dussan12 has been investigated the axisymmetric Boussinesq model by similarity variable approach. An early investigation of drilling mud invasion into a permeable aquifer has been provided by Doll.13 Calugaru1 proposed a generalized model for description of evolution of the interface between two immiscible fluids in porous media.

Tang14 studied transient groundwater flow in an unconfined aquifer subject to a constant water variation at the sloping water-land boundary. To characterize the transient groundwater flow, they presented a novel approximate solution to the 1-D Boussinesq equation. They applied the proposed method to various hydrological problems and showed that it can achieve desirable precisions, even in the cases with strong nonlinearity. Bartlet15 introduced a class of solutions of the nonlinear Boussinesq equation with source/sink terms. They applied their new solution to sloping aquifers and analytical results capture hysteresis between the groundwater level and groundwater flow rate as a function of hillslope characteristics. Lu16 suggested a homotopy analysis method (HAM) to solve the generalized Boussinesq equation. Due to the two-degree approximate solution of the variable coefficient Boussinesq equation, they showed that the homotopy perturbation method is effective to solve the variable solution equations. Bansal17 developed a new analytical solution of 2-dimensional linearized Boussinesq equation for approximation of subsurface seepage flow in confined and unconfined aquifers under varying hydrological conditions. He showed that the vertical flow through the base of the aquifer is an important factor in the determination of groundwater mound and cone of depression. Telyakovskiy18 modeled water injection at a single well in an unconfined aquifer by the Boussinesq equation with cylindrical symmetry. By introducing similarity variables, they reduced the original problem to a boundary-value problem for an ordinary differential equation. Their approximate solution incorporated both a singular part to model the behavior near the well and a polynomial part to model the behavior in the far field.

The problem of invasion by drilling fluid into a permeable empty bed solved by Li.19 They used the similarity variable and reduced the axisymmetric Boussinesq equation to a nonlinear ordinary differential equation. Their solution which was a sum of a singular term and a Taylor expansion at the wetting front was demonstrated to be highly accurate. Mortensen20 considered the same problem21 in spherical coordinates with the prescribed power law point source boundary condition. They constructed an approximate similarity solution to a nonlinear diffusion equation in spherical coordinates. In this paper, first, the general dimension equation for interface position between two immiscible fluids in porous media has been presented. Then, the instability of this equation for one dimensional case by considering a self-similar solution and using the perturbation method has been investigated. And finally, an accurate solution to axisymmetric Boussinesq problem has been obtained. Also, in order to describe deformations of the interface separating two immiscible fluids in porous media, a system of partial differential equations which are true on either sides of the interface is required.

Theory

Figure 1 represents the generalized geometry of the problem in an orthogonal coordinate system. The initial position of the interface between two immiscible fluids is identified as initial assumed value of t o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiDaS WaaSbaaeaajugWaiaad+gaaSqabaaaaa@39CD@ .When the flow is initiated by a driving force; changes of the interface are specified as a function of space x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiaadIhaaaaaaa@3821@ , y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiaadMhaaaaaaa@3821@ and time t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiaadshaaaaaaa@381C@ .

Porous media is assumed homogeneous with constant porosity and isentropic. Permeability tensor for such media could be considered a diagonal matrix. By determining the characteristic length of L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGmbaaaa@376C@ for horizontal scale and characteristic height of H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGibaaaa@3768@ , the dimensionless equation of changes of the surface could be written as follow [2]:

 ( t el t xy )( h 2 + δ 1 2 t r t 3 K z h 2 h t + δ 2 P r )= t r t er h t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaGGGcqcfa4aaeWaaOWdaeaajugib8qacaWG0bWcpaWaaSba aeaajugWa8qacaWGLbGaamiBaaWcpaqabaqcfa4dbmaalaaak8aaba qcLbsapeGaeyOaIylak8aabaqcLbsapeGaeyOaIyRaamiDaaaacqGH sislcqGHhis0l8aadaWgaaqaaKqzadWdbiaadIhacaWG5baal8aabe aaaOWdbiaawIcacaGLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaadIga juaGpaWaaWbaaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSscfa 4aaSaaaOWdaeaajugib8qacqaH0oazl8aadaqhaaqaaKqzadWdbiaa igdaaSWdaeaajugWa8qacaaIYaaaaKqzGeGaamiDaKqba+aadaWgaa WcbaqcLbmapeGaamOCaaWcpaqabaqcLbsapeGaamiDaaGcpaqaaKqz GeWdbiaaiodacaWGlbWcpaWaaSbaaeaajugWa8qacaWG6baal8aabe aaaaqcLbsapeGaamiAaSWdamaaCaaabeqaaKqzadWdbiaaikdaaaqc fa4aaSaaaOWdaeaajugib8qacqGHciITcaWGObaak8aabaqcLbsape GaeyOaIyRaamiDaaaacqGHRaWkcqaH0oazjuaGpaWaaSbaaSqaaKqz adWdbiaaikdaaSWdaeqaaKqzGeWdbiaadcfal8aadaWgaaqaaKqzad WdbiaadkhaaSWdaeqaaaGcpeGaayjkaiaawMcaaKqzGeGaeyypa0Ja eyOeI0IaamiDaSWdamaaBaaabaqcLbmapeGaamOCaaWcpaqabaqcLb sapeGaamiDaKqba+aadaWgaaWcbaqcLbsapeGaamyzaKqzadGaamOC aaWcpaqabaqcfa4dbmaalaaak8aabaqcLbsapeGaeyOaIyRaamiAaa GcpaqaaKqzGeWdbiabgkGi2kaadshaaaaaaa@8C32@ (1)

( t el μ r t xy )( 2 + δ 1 2 ρ r t r t 3 K z μ r δ o 2 t + δ 2 δ o ρ r P )= ρ r δ o t r t el μ r h t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaOWdaeaajuaGpeWaaSaaaOWdaeaajugib8qacaWG0bWc paWaaSbaaeaajugWa8qacaWGLbGaamiBaaWcpaqabaaakeaajugib8 qacqaH8oqBjuaGpaWaaSbaaSqaaKqzadWdbiaadkhaaSWdaeqaaaaa juaGpeWaaSaaaOWdaeaajugib8qacqGHciITaOWdaeaajugib8qacq GHciITcaWG0baaaiabgkHiTiabgEGirNqba+aadaWgaaWcbaqcLbma peGaamiEaiaadMhaaSWdaeqaaaGcpeGaayjkaiaawMcaaKqbaoaabm aak8aabaqcLbsapeGaeyOeI0IaeyybIy8cpaWaaWbaaeqabaqcLbma peGaaGOmaaaajugibiabgUcaRKqbaoaalaaak8aabaqcLbsapeGaeq iTdq2cpaWaa0baaeaajugWa8qacaaIXaaal8aabaqcLbmapeGaaGOm aaaajugibiabeg8aYLqba+aadaWgaaWcbaqcLbmapeGaamOCaaWcpa qabaqcLbsapeGaamiDaSWdamaaBaaabaqcLbmapeGaamOCaaWcpaqa baqcLbsapeGaamiDaaGcpaqaaKqzGeWdbiaaiodacaWGlbqcfa4dam aaBaaaleaajugWa8qacaWG6baal8aabeaajugib8qacqaH8oqBjuaG paWaaSbaaSqaaKqzadWdbiaadkhaaSWdaeqaaKqzGeWdbiabes7aKL qba+aadaWgaaWcbaqcLbmapeGaam4BaaWcpaqabaaaaKqzGeWdbiab gwGigVWdamaaCaaabeqaaKqzadWdbiaaikdaaaqcfa4aaSaaaOWdae aajugib8qacqGHciITcqGHfiIXaOWdaeaajugib8qacqGHciITcaWG 0baaaiabgUcaRiabes7aKLqba+aadaWgaaWcbaqcLbmapeGaaGOmaa WcpaqabaqcLbsapeGaeqiTdq2cpaWaaSbaaeaajugWa8qacaWGVbaa l8aabeaajugib8qacqaHbpGCjuaGpaWaaSbaaSqaaKqzadWdbiaadk haaSWdaeqaaKqzGeWdbiaadcfaaOGaayjkaiaawMcaaKqzGeGaeyyp a0JaeyOeI0scfa4aaSaaaOWdaeaajugib8qacqaHbpGCl8aadaWgaa qaaKqzadWdbiaadkhaaSWdaeqaaKqzGeWdbiabes7aKTWdamaaBaaa baqcLbmapeGaam4BaaWcpaqabaqcLbsapeGaamiDaSWdamaaBaaaba qcLbmapeGaamOCaaWcpaqabaqcLbsapeGaamiDaKqba+aadaWgaaWc baqcLbmapeGaamyzaiaadYgaaSWdaeqaaaGcbaqcLbsapeGaeqiVd0 wcfa4damaaBaaaleaajugWa8qacaWGYbaal8aabeaaaaqcfa4dbmaa laaak8aabaqcLbsapeGaeyOaIyRaamiAaaGcpaqaaKqzGeWdbiabgk Gi2kaadshaaaaaaa@BB1F@ (2)

= δ o h+ δ o +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHfiIXcqGH9aqpcqGHsislcqaH0oazjuaGpaWaaSbaaSqa aKqzadWdbiaad+gaaSWdaeqaaKqzGeWdbiaadIgacqGHRaWkcqaH0o azl8aadaWgaaqaaKqzadWdbiaad+gaaSWdaeqaaKqzGeWdbiabgUca Riaaigdaaaa@4796@ (3)

Where dimensionless parameters are specified as below:

x= x ¯ L , y= y ¯ L , h= h ¯ h o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG4bGaeyypa0tcfa4aaSaaaOWdaeaajuaGdaqdaaGcbaqc LbsacaWG4baaaaGcbaqcLbsapeGaamitaaaacaGGSaGaaiiOaiaadM hacqGH9aqpjuaGdaWcaaGcpaqaaKqbaoaanaaabaGaamyEaaaaaOqa aKqzGeWdbiaadYeaaaGaaiilaiaacckacaWGObGaeyypa0tcfa4aaS aaaOWdaeaajuaGdaqdaaqaaiaadIgaaaaakeaajugib8qacaWGObWc paWaaSbaaeaajugWa8qacaWGVbaal8aabeaaaaaaaa@4ED0@

δ o = h ¯ 0 H ¯ h ¯ o , δ 1 = h ¯ 0 L  ,  δ 2 = 2 P ¯ 0 ρ ¯ g h ¯ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazl8aadaWgaaqaaKqzadWdbiaad+gaaSWdaeqaaKqz GeWdbiabg2da9Kqbaoaalaaak8aabaqcfa4aa0aaaOqaaKqzGeGaam iAaaaajuaGdaWgaaWcbaqcLbmapeGaaGimaaWcpaqabaaakeaajuaG peWaa0aaaeaajugibiaadIeaaaGaeyOeI0scfa4aa0aaaeaacaWGOb aaaKqzadGaam4BaaaajugibiaacYcacqaH0oazl8aadaWgaaqaaKqz adWdbiaaigdaaSWdaeqaaKqzGeWdbiabg2da9Kqbaoaalaaak8aaba qcfa4aa0aaaeaajugibiaadIgaaaWcdaWgaaqaaKqzadWdbiaaicda aSWdaeqaaaGcbaqcLbsapeGaamitaaaacaGGGcGaaiilaiaacckacq aH0oazjuaGpaWaaSbaaSqaaKqzGeWdbiaaikdaaSWdaeqaaKqzGeWd biabg2da9Kqbaoaalaaak8aabaqcLbsapeGaaGOmaKqbaoaanaaaba Gaamiuaaaal8aadaWgaaqaaKqzadWdbiaaicdaaSWdaeqaaaGcbaqc fa4dbmaanaaabaGaeqyWdihaaKqzGeGaam4zaKqbaoaanaaabaGaam iAaaaapaWaaSbaaSqaaKqzadWdbiaaicdaaSWdaeqaaaaaaaa@6C44@

P r = P ¯ P o ,  ρ r = ρ ¯ I ρ ¯ II ,  μ r = μ ¯ I μ ¯ II  ,  K r = K ¯ z k ¯ x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGqbWcpaWaaSbaaeaajugWa8qacaWGYbaal8aabeaajugi b8qacqGH9aqpjuaGdaWcaaGcpaqaaKqbaoaanaaakeaajugibiaadc faaaaakeaajugib8qacaWGqbqcfa4damaaBaaaleaajugWa8qacaWG Vbaal8aabeaaaaqcLbsapeGaaiilaiaacckacqaHbpGCl8aadaWgaa qaaKqzadWdbiaadkhaaSWdaeqaaKqzGeWdbiabg2da9Kqbaoaalaaa k8aabaqcfa4aa0aaaeaajugibiabeg8aYbaajuaGdaWgaaWcbaqcLb sapeGaamysaaWcpaqabaaakeaajuaGdaqdaaqaaKqzGeGaeqyWdiha aKqbaoaaBaaaleaajugib8qacaWGjbGaamysaaWcpaqabaaaaKqzGe WdbiaacYcacaGGGcGaeqiVd0wcfa4damaaBaaaleaajugWa8qacaWG Ybaal8aabeaajugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqbaoaana aabaqcLbsacqaH8oqBaaWcdaWgaaqaaKqzadWdbiaadMeaaSWdaeqa aaGcbaqcfa4aa0aaaeaajugibiabeY7aTbaajuaGdaWgaaWcbaqcLb sapeGaamysaKqzadGaamysaaWcpaqabaaaaKqzGeWdbiaacckacaGG SaGaaiiOaiaadUeal8aadaWgaaqaaKqzadWdbiaadkhaaSWdaeqaaK qzGeWdbiabg2da9Kqbaoaalaaak8aabaqcfa4aa0aaaeaajugibiaa dUeaaaqcfa4aaSbaaSqaaKqzadWdbiaadQhaaSWdaeqaaaGcbaqcfa 4aa0aaaeaajugibiaadUgaaaWcdaWgaaqaaKqzadWdbiaadIhaaSWd aeqaaaaaaaa@8048@ (4)

t ¯ el = μ ¯ L L ¯ 2 K ¯ x ,  t gr =2 μ I ε L ¯ 2 K ¯ x ρ ¯ I g h ¯ 0 ,  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaaba qcLbsacaWG0baaaKqbaoaaBaaabaWcdaWgaaqcfayaaKqzadaeaaaa aaaaa8qacaWGLbGaamiBaaqcfa4daeqaaaqabaqcLbsapeGaeyypa0 tcfa4aaSaaaOWdaeaajuaGdaqdaaqaaKqzGeGaeqiVd0gaaKqbaoaa Baaaleaajugib8qacaWGmbaal8aabeaajuaGdaqdaaqaaKqzGeGaam itaaaalmaaCaaabeqaaKqzadWdbiaaikdaaaaak8aabaqcfa4aa0aa aeaajugibiaadUeaaaqcfa4aaSbaaSqaaKqzadWdbiaadIhaaSWdae qaaaaajugib8qacaGGSaGaaiiOaiaadshajuaGpaWaaSbaaSqaaKqz GeWdbiaadEgacaWGYbaal8aabeaajugib8qacqGH9aqpcaaIYaGaeq iVd02cpaWaaSbaaeaajugWa8qacaWGjbaal8aabeaajugib8qacqaH 1oqzjuaGdaWcaaGcpaqaaKqbaoaanaaabaqcLbsacaWGmbaaaSWaaW baaeqabaqcLbmapeGaaGOmaaaaaOWdaeaajuaGdaqdaaqaaKqzGeGa am4saaaalmaaBaaabaqcLbmapeGaamiEaaWcpaqabaqcfa4aa0aaae aajugibiabeg8aYbaalmaaBaaabaqcLbmapeGaamysaaWcpaqabaqc LbsapeGaam4zaKqbaoaanaaabaqcLbsacaWGObaaaSWdamaaBaaaba qcLbmapeGaaGimaaWcpaqabaaaaKqzGeWdbiaacYcacaGGGcaaaa@74BB@ t el = t ¯ el t ¯ c ,  t r = 2ε ρ ¯ g h ¯ 0 = t ¯ gr t ¯ el MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG0bWcpaWaaSbaaeaajugWa8qacaWGLbGaamiBaaWcpaqa baqcLbsapeGaeyypa0tcfa4aaSaaaOWdaeaajuaGdaqdaaGcbaqcLb sacaWG0baaaKqbaoaaBaaaleaajugWa8qacaWGLbGaamiBaaWcpaqa baaakeaajuaGdaqdaaGcbaqcLbsacaWG0baaaKqbaoaaBaaaleaaju gib8qacaWGJbaal8aabeaaaaqcLbsapeGaaiilaiaacckacaWG0bqc fa4damaaBaaaleaajugib8qacaWGYbaal8aabeaajugib8qacqGH9a qpjuaGdaWcaaGcpaqaaKqzGeWdbiaaikdacqaH1oqzaOWdaeaajuaG peWaa0aaaOqaaKqzGeGaeqyWdihaaKqzadGaam4zaKqbaoaanaaake aajugibiaadIgaaaWcpaWaaSbaaeaajugWa8qacaaIWaaal8aabeaa aaqcLbsapeGaeyypa0tcfa4aaSaaaOWdaeaajuaGdaqdaaGcbaqcLb sacaWG0baaaKqbaoaaBaaaleaajugWa8qacaWGNbGaamOCaaWcpaqa baaakeaajuaGdaqdaaGcbaqcLbsacaWG0baaaSWaaSbaaeaajugWa8 qacaWGLbGaamiBaaWcpaqabaaaaaaa@6B1D@

In these equations subscripts I and II stand for fluid one and fluid two, respectively. In addition, superscript "-" stands for vector quantities and subscript r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGYbaaaa@3792@ stands for the ratio of the two similar quantities. t el MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG0bqcfa4damaaBaaaleaajugWa8qacaWGLbGaamiBaaWc paqabaaaaa@3B90@ and t gr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG0bqcfa4damaaBaaaleaajugWa8qacaWGNbGaamOCaaWc paqabaaaaa@3B98@ indicate the time of propagation of an elastic disturbance, respectively, and the time needed for the media to be filled with gravity flow of the fluid one and t c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG0bWcpaWaaSbaaeaajugWa8qacaWGJbaal8aabeaaaaa@3A0E@ is the characteristic time.

For situations in which the time of propagation of the elastic wave, compared to t gr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG0bqcfa4damaaBaaaleaajugWa8qacaWGNbGaamOCaaWc paqabaaaaa@3B97@ , is very short, general Equation 3 could be written as follows:

xy ( h+ δ 1 2 3 K z h 2 h t + δ 2 P )= h t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHhis0l8aadaWgaaqaaKqzadWdbiaadIhacaWG5baal8aa beaajuaGpeWaaeWaaOWdaeaajugib8qacaWGObGaey4kaSscfa4aaS aaaOWdaeaajugib8qacqaH0oazl8aadaqhaaqaaKqzadWdbiaaigda aSWdaeaajugWa8qacaaIYaaaaaGcpaqaaKqzGeWdbiaaiodacaWGlb qcfa4damaaBaaaleaajugWa8qacaWG6baal8aabeaaaaqcLbsapeGa amiAaSWdamaaCaaabeqaaKqzadWdbiaaikdaaaqcfa4aaSaaaOWdae aajugib8qacqGHciITcaWGObaak8aabaqcLbsapeGaeyOaIyRaamiD aaaacqGHRaWkcqaH0oazl8aadaWgaaqaaKqzadWdbiaaikdaaSWdae qaaKqzGeWdbiaadcfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4a aSaaaOWdaeaajugib8qacqGHciITcaWGObaak8aabaqcLbsapeGaey OaIyRaamiDaaaaaaa@6631@ (4)

xy ( 2 + δ 1 ρ r 3 K z μ r δ o 2 t + δ 2 δ o ρ r P )= ρ r δ o μ r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHhis0l8aadaWgaaqaaKqzadWdbiaadIhacaWG5baal8aa beaajuaGpeWaaeWaaOWdaeaajugib8qacqGHsislcqGHfiIXl8aada ahaaqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSscfa4aaSaaaOWd aeaajugib8qacqaH0oazjuaGpaWaaSbaaSqaaKqzadWdbiaaigdaaS WdaeqaaKqzGeWdbiabeg8aYLqba+aadaWgaaWcbaqcLbmapeGaamOC aaWcpaqabaaakeaajugib8qacaaIZaGaam4saSWdamaaBaaabaqcLb mapeGaamOEaaWcpaqabaqcLbsapeGaeqiVd0wcfa4damaaBaaaleaa jugWa8qacaWGYbaal8aabeaajugib8qacqaH0oazjuaGpaWaaSbaaS qaaKqzadWdbiaad+gaaSWdaeqaaaaajugib8qacqGHfiIXl8aadaah aaqabeaajugWa8qacaaIYaaaaKqbaoaalaaak8aabaqcLbsapeGaey OaIyRaeyybIymak8aabaqcLbsapeGaeyOaIyRaamiDaaaacqGHRaWk cqaH0oazjuaGpaWaaSbaaSqaaKqzadWdbiaaikdaaSWdaeqaaKqzGe Wdbiabes7aKTWdamaaBaaabaqcLbmapeGaam4BaaWcpaqabaqcLbsa peGaeqyWdi3cpaWaaSbaaeaajugWa8qacaWGYbaal8aabeaajugib8 qacaWGqbaakiaawIcacaGLPaaajugibiabg2da9iabgkHiTKqbaoaa laaak8aabaqcLbsapeGaeqyWdixcfa4damaaBaaaleaajugWa8qaca WGYbaal8aabeaajugib8qacqaH0oazjuaGpaWaaSbaaSqaaKqzadWd biaad+gaaSWdaeqaaaGcbaqcLbsapeGaeqiVd0wcfa4damaaBaaale aajugWa8qacaWGYbaal8aabeaaaaqcfa4dbmaalaaak8aabaqcLbsa peGaeyOaIyRaeyybIymak8aabaqcLbsapeGaeyOaIyRaamiDaaaaaa a@9683@ (5)

= δ o h+ δ o +1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHfiIXcqGH9aqpcqGHsislcqaH0oazjuaGpaWaaSbaaSqa aKqzadWdbiaad+gaaSWdaeqaaKqzGeWdbiaadIgacqGHRaWkcqaH0o azl8aadaWgaaqaaKqzadWdbiaad+gaaSWdaeqaaKqzGeWdbiabgUca Riaaigdaaaa@4797@

Time and considering the fact that t el t gr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG0bWcpaWaaSbaaeaajugWa8qacaWGLbGaamiBaaWcpaqa baqcLbsapeGaeSOAI0JaamiDaSWdamaaBaaabaqcLbmapeGaam4zai aadkhaaSWdaeqaaaaa@4169@ . Relative time of t r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG0bWcpaWaaSbaaeaajugWa8qacaWGYbaal8aabeaaaaa@3A1D@ would be significantly less than 1. By substituting h for   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiiOaKqzGeGaeyybIymaaa@3937@ and eliminating P from the two equations, the system could be revised as a single equation of the coefficient h.

xy ( t el μ r t xy )( h 2 2( δ o +1 ) δ o + ρ r h+ δ 1 2 ρ r 3 K z ( δ o + ρ r ) [ h 2 + 1 μ r δ o ( 1+ δ o δ o h ) 2 ] h t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHhis0l8aadaWgaaqaaKqzadWdbiaadIhacaWG5baal8aa beaajuaGpeWaaeWaaOWdaeaajuaGpeWaaSaaaOWdaeaajugib8qaca WG0bWcpaWaaSbaaeaajugWa8qacaWGLbGaamiBaaWcpaqabaaakeaa jugib8qacqaH8oqBjuaGpaWaaSbaaSqaaKqzadWdbiaadkhaaSWdae qaaaaajuaGpeWaaSaaaOWdaeaajugib8qacqGHciITaOWdaeaajugi b8qacqGHciITcaWG0baaaiabgkHiTiabgEGirNqba+aadaWgaaWcba qcLbmapeGaamiEaiaadMhaaSWdaeqaaaGcpeGaayjkaiaawMcaaKqb aoaabmaak8aabaqcLbsapeGaamiAaSWdamaaCaaabeqaaKqzadWdbi aaikdaaaqcLbsacqGHsisljuaGdaWcaaGcpaqaaKqzGeWdbiaaikda juaGdaqadaGcpaqaaKqzGeWdbiabes7aKTWdamaaBaaabaqcLbmape Gaam4BaaWcpaqabaqcLbsapeGaey4kaSIaaGymaaGccaGLOaGaayzk aaaapaqaaKqzGeWdbiabes7aKLqba+aadaWgaaWcbaqcLbmapeGaam 4BaaWcpaqabaqcLbsapeGaey4kaSIaeqyWdi3cpaWaaSbaaeaajugW a8qacaWGYbaal8aabeaaaaqcLbsapeGaamiAaiabgUcaRKqbaoaala aak8aabaqcLbsapeGaeqiTdq2cpaWaa0baaeaajugWa8qacaaIXaaa l8aabaqcLbmapeGaaGOmaaaajugibiabeg8aYLqba+aadaWgaaWcba qcLbmapeGaamOCaaWcpaqabaaakeaajugib8qacaaIZaGaam4saKqb a+aadaWgaaWcbaqcLbmapeGaamOEaaWcpaqabaqcfa4dbmaabmaak8 aabaqcLbsapeGaeqiTdqwcfa4damaaBaaaleaajugWa8qacaWGVbaa l8aabeaajugib8qacqGHRaWkcqaHbpGCjuaGpaWaaSbaaSqaaKqzad WdbiaadkhaaSWdaeqaaaGcpeGaayjkaiaawMcaaaaajuaGdaWadaGc paqaaKqzGeWdbiaadIgal8aadaahaaqabeaajugWa8qacaaIYaaaaK qzGeGaey4kaSscfa4aaSaaaOWdaeaajugib8qacaaIXaaak8aabaqc LbsapeGaeqiVd02cpaWaaSbaaeaajugWa8qacaWGYbaal8aabeaaju gib8qacqaH0oazjuaGpaWaaSbaaSqaaKqzadWdbiaad+gaaSWdaeqa aaaajuaGpeWaaeWaaOWdaeaajugib8qacaaIXaGaey4kaSIaeqiTdq 2cpaWaaSbaaeaajugWa8qacaWGVbaal8aabeaajugib8qacqGHsisl cqaH0oazl8aadaWgaaqaaKqzadWdbiaad+gaaSWdaeqaaKqzGeWdbi aadIgaaOGaayjkaiaawMcaaSWdamaaCaaabeqaaKqzadWdbiaaikda aaaakiaawUfacaGLDbaajuaGdaWcaaGcpaqaaKqzGeWdbiabgkGi2k aadIgaaOWdaeaajugib8qacqGHciITcaWG0baaaaGccaGLOaGaayzk aaaaaa@C35B@ = ρ r ( δ o + μ r ) μ r ( δ o + ρ r ) h t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiabeg8aYTWdamaa BaaabaqcLbmapeGaamOCaaWcpaqabaqcfa4dbmaabmaak8aabaqcLb sapeGaeqiTdqwcfa4damaaBaaaleaajugib8qacaWGVbaal8aabeaa jugib8qacqGHRaWkcqaH8oqBl8aadaWgaaqaaKqzadWdbiaadkhaaS WdaeqaaaGcpeGaayjkaiaawMcaaaWdaeaajugib8qacqaH8oqBl8aa daWgaaqaaKqzadWdbiaadkhaaSWdaeqaaKqba+qadaqadaGcpaqaaK qzGeWdbiabes7aKTWdamaaBaaabaqcLbmapeGaam4BaaWcpaqabaqc LbsapeGaey4kaSIaeqyWdi3cpaWaaSbaaeaajugWa8qacaWGYbaal8 aabeaaaOWdbiaawIcacaGLPaaaaaqcfa4aaSaaaOWdaeaajugib8qa cqGHciITcaWGObaak8aabaqcLbsapeGaeyOaIyRaamiDaaaaaaa@62B4@ (6)

By substituting   μ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaGGGcGaeqiVd0wcfa4damaaBaaaleaajugib8qacaWGYbaa l8aabeaajugib8qacqGHsgIRcqGHEisPaaa@3FEA@ , ρ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCl8aadaWgaaqaaKqzadWdbiaadkhaaSWdaeqaaKqz GeWdbiabgkziUkabg6HiLcaa@3EE1@ and δ o 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazl8aadaWgaaqaaKqzadWdbiaad+gaaSWdaeqaaKqz GeWdbiabgkziUkaaicdaaaa@3E0C@ in a problem containing one fluid with free boundary, Equation (6) could be written as follows:

xy ( h 2 + δ 1 2 ρ r 3 K z ( δ o + ρ r ) h 2 h t )= h t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHhis0l8aadaWgaaqaaKqzadWdbiaadIhacaWG5baal8aa beaajuaGpeWaaeWaaOWdaeaajugib8qacaWGObWcpaWaaWbaaeqaba qcLbmapeGaaGOmaaaajugibiabgUcaRKqbaoaalaaak8aabaqcLbsa peGaeqiTdq2cpaWaa0baaeaajugWa8qacaaIXaaal8aabaqcLbmape GaaGOmaaaajugibiabeg8aYTWdamaaBaaabaqcLbmapeGaamOCaaWc paqabaaakeaajugib8qacaaIZaGaam4saSWdamaaBaaabaqcLbmape GaamOEaaWcpaqabaqcfa4dbmaabmaak8aabaqcLbsapeGaeqiTdqwc fa4damaaBaaaleaajugWa8qacaWGVbaal8aabeaajugib8qacqGHRa WkcqaHbpGCl8aadaWgaaqaaKqzadWdbiaadkhaaSWdaeqaaaGcpeGa ayjkaiaawMcaaaaajugibiaadIgajuaGpaWaaWbaaSqabeaajugWa8 qacaaIYaaaaKqbaoaalaaak8aabaqcLbsapeGaeyOaIyRaamiAaaGc paqaaKqzGeWdbiabgkGi2kaadshaaaaakiaawIcacaGLPaaajugibi abg2da9Kqbaoaalaaak8aabaqcLbsapeGaeyOaIyRaamiAaaGcpaqa aKqzGeWdbiabgkGi2kaadshaaaaaaa@74E4@ (7)

One of the applications of Equation (7) is in cases where the height of the porous layer can be considered negligible in comparison to the horizontal length. In such a case δ 1 1  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazl8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiablQMi9iaaigdakiaacckaaaa@3E6F@ and as a result h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObaaaa@3788@ can be expressed as a series based on δ 1 2 /3 K z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazl8aadaqhaaqaaKqzadWdbiaaigdaaSWdaeaajugW a8qacaaIYaaaaKqzGeGaai4laiaaiodacaWGlbWcpaWaaSbaaeaaju gWa8qacaWG6baal8aabeaaaaa@41EA@ powers.

In limit conditions, by ignoring the term ( δ 1 2 ρ r /3 K z ) h 2 ( h/t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaOWdaeaajugib8qacqaH0oazl8aadaqhaaqaaKqzadWd biaaigdaaSWdaeaajugWa8qacaaIYaaaaKqzGeGaeqyWdi3cpaWaaS baaeaajugWa8qacaWGYbaal8aabeaajugib8qacaGGVaGaaG4maiaa dUeal8aadaWgaaqaaKqzadWdbiaadQhaaSWdaeqaaaGcpeGaayjkai aawMcaaKqzGeGaamiAaSWdamaaCaaabeqaaKqzadWdbiaaikdaaaqc fa4aaeWaaOWdaeaajugib8qacqGHciITcaWGObGaai4laiabgkGi2k aadshaaOGaayjkaiaawMcaaaaa@551D@ , the Equation (7) will turn into the Boussinesq equation.

xy ( h 2 )= h t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaey4bIe Dcfa4aaSbaaSqaaKqzadaeaaaaaaaaa8qacaWG4bGaamyEaaWcpaqa baqcfa4dbmaabmaak8aabaqcLbsapeGaamiAaSWdamaaCaaabeqaaK qzadWdbiaaikdaaaaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaa laaak8aabaqcLbsapeGaeyOaIyRaamiAaaGcpaqaaKqzGeWdbiabgk Gi2kaadshaaaaaaa@4A7E@ (8)

Figure 2 Schematic of diffusion of drilling fluid.

Equation (8) is known as the classic equation of "Shallow Water Flow" theory in porous media. To model the diffusion of drilling fluid in porous media surrounding a circular well (Figure 2), Eq. (8)is given by:

1 r r ( rh h r )= h t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaaOWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaamOC aaaajuaGdaWcaaGcpaqaaKqzGeWdbiabgkGi2cGcpaqaaKqzGeWdbi abgkGi2kaadkhaaaqcfa4aaeWaaOWdaeaajugib8qacaWGYbGaamiA aKqbaoaalaaak8aabaqcLbsapeGaeyOaIyRaamiAaaGcpaqaaKqzGe WdbiabgkGi2kaadkhaaaaakiaawIcacaGLPaaajugibiabg2da9Kqb aoaalaaak8aabaqcLbsapeGaeyOaIyRaamiAaaGcpaqaaKqzGeWdbi abgkGi2kaadshaaaaaaa@536D@ (9)

where r= r ¯ / r o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGYbGaeyypa0tcfa4aa0aaaeaacaWGYbaaaKqzGeGaai4l aiaadkhajuaGpaWaaSbaaSqaaKqzGeWdbiaad+gaaSWdaeqaaaaa@3EDD@ and r o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGYbWcpaWaaSbaaeaajugWa8qacaWGVbaal8aabeaaaaa@3A19@ is considered to be the characteristic radius. Equation (6) for porous layer with negligible thickness compared to horizontal characteristic length could be indicated as follows:

xy ( h 2 2( δ o +1 ) ( δ o + ρ r ) h )= h t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHhis0juaGpaWaaSbaaSqaaKqzadWdbiaadIhacaWG5baa l8aabeaajuaGpeWaaeWaaOWdaeaajugib8qacaWGObqcfa4damaaCa aaleqabaqcLbmapeGaaGOmaaaajugibiabgkHiTKqbaoaalaaak8aa baqcLbsapeGaaGOmaKqbaoaabmaak8aabaqcLbsapeGaeqiTdq2cpa WaaSbaaeaajugWa8qacaWGVbaal8aabeaajugib8qacqGHRaWkcaaI XaaakiaawIcacaGLPaaaa8aabaqcfa4dbmaabmaak8aabaqcLbsape GaeqiTdq2cpaWaaSbaaeaajugWa8qacaWGVbaal8aabeaajugib8qa cqGHRaWkcqaHbpGCjuaGpaWaaSbaaSqaaKqzadWdbiaadkhaaSWdae qaaaGcpeGaayjkaiaawMcaaaaajugibiaadIgaaOGaayjkaiaawMca aKqzGeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacqGHciITcaWGOb aak8aabaqcLbsapeGaeyOaIyRaamiDaaaaaaa@6704@ (10)

This equation points out the position of the interface between two immiscible fluids in porous media with insignificant height compared to its characteristic length, in x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG4baaaa@3798@ and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG5baaaa@3799@ directions.

Figure 1 Generalized geometry of the problem in an orthogonal coordinate system.

Analyzing the equation and instability

For a one-dimensional flow, Equation (10) changes into a parabolic equation. By considering new independent variable η=x/ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH3oaAcqGH9aqpcaWG4bGaai4laKqbaoaakaaak8aabaqc LbsapeGaamiDaaWcbeaaaaa@3D56@ , following ordinary differential equation (ODE) could be substituted:

d 2 d η 2 ( h 2 2( δ o +1 ) δ o + ρ r h )= h 2 ρ r ( δ o + μ r ) μ r ( δ o + ρ r ) h η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaaOWdaeaajugib8qacaWGKbWcpaWaaWbaaeqabaqcLbma peGaaGOmaaaaaOWdaeaajugib8qacaWGKbGaeq4TdGwcfa4damaaCa aaleqabaqcLbmapeGaaGOmaaaaaaqcfa4aaeWaaOWdaeaajugib8qa caWGObWcpaWaaWbaaeqabaqcLbmapeGaaGOmaaaajugibiabgkHiTK qbaoaalaaak8aabaqcLbsapeGaaGOmaKqbaoaabmaak8aabaqcLbsa peGaeqiTdqwcfa4damaaBaaaleaajugWa8qacaWGVbaal8aabeaaju gib8qacqGHRaWkcaaIXaaakiaawIcacaGLPaaaa8aabaqcLbsapeGa eqiTdq2cpaWaaSbaaeaajugWa8qacaWGVbaal8aabeaajugib8qacq GHRaWkcqaHbpGCl8aadaWgaaqaaKqzadWdbiaadkhaaSWdaeqaaaaa jugib8qacaWGObaakiaawIcacaGLPaaajugibiabg2da9iabgkHiTK qbaoaalaaak8aabaqcLbsapeGaamiAaaGcpaqaaKqzGeWdbiaaikda aaqcfa4aaSaaaOWdaeaajugib8qacqaHbpGCjuaGpaWaaSbaaSqaaK qzadWdbiaadkhaaSWdaeqaaKqba+qadaqadaGcpaqaaKqzGeWdbiab es7aKTWdamaaBaaabaqcLbmapeGaam4BaaWcpaqabaqcLbsapeGaey 4kaSIaeqiVd02cpaWaaSbaaeaajugWa8qacaWGYbaal8aabeaaaOWd biaawIcacaGLPaaaa8aabaqcLbsapeGaeqiVd0wcfa4damaaBaaale aajugWa8qacaWGYbaal8aabeaajuaGpeWaaeWaaOWdaeaajugib8qa cqaH0oazjuaGpaWaaSbaaSqaaKqzadWdbiaad+gaaSWdaeqaaKqzGe WdbiabgUcaRiabeg8aYTWdamaaBaaabaqcLbmapeGaamOCaaWcpaqa baaak8qacaGLOaGaayzkaaaaaKqbaoaalaaak8aabaqcLbsapeGaey OaIyRaamiAaaGcpaqaaKqzGeWdbiabgkGi2kabeE7aObaaaaa@93D6@ (11)

The first point about Equation (11) is attained by investigating the linear stability. By considering h= h ¯ +ε δ h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObGaeyypa0tcfa4aa0aaaeaacaWGObaaaKqzGeGaey4k aSIaeqyTduMaeqiTdq2cpaWaaSbaaeaajugWa8qacaWGObaal8aabe aaaaa@4156@ where h ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaa0aaaOqaaKqzGeGaamiAaaaaaaa@3830@ a non-disturbance response of the equation is, it could simply be given as:

2( hγ ) δ h " = 1 2 η δ h ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaaIYaqcfa4aaeWaaOWdaeaajugib8qacaWGObGaeyOeI0Ia eq4SdCgakiaawIcacaGLPaaajugibiabes7aKTWdamaaDaaabaqcLb mapeGaamiAaaWcpaqaaKqzadWdbiaackcaaaqcLbsacqGH9aqpcqGH sisljuaGdaWcaaGcpaqaaKqzGeWdbiaaigdaaOWdaeaajugib8qaca aIYaaaaiabeE7aOjabes7aKTWdamaaDaaabaqcLbmapeGaamiAaaWc paqaaKqzadWdbiaacEcaaaaaaa@520C@ (12)

H = η 4( 1γ ) δ h ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qaceWGibWdayaagaWdbiabg2da9iabgkHiTKqbaoaalaaak8aa baqcLbsapeGaeq4TdGgak8aabaqcLbsapeGaaGinaKqbaoaabmaak8 aabaqcLbsapeGaaGymaiabgkHiTiabeo7aNbGccaGLOaGaayzkaaaa aKqzGeGaeqiTdq2cpaWaa0baaeaajugWa8qacaWGObaal8aabaqcLb mapeGaai4jaaaaaaa@4AC5@ (13)

The domain of disturbance and is equal to ( δ o +1 )/( δ o + ρ r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaOWdaeaajugib8qacqaH0oazjuaGpaWaaSbaaSqaaKqz adWdbiaad+gaaSWdaeqaaKqzGeWdbiabgUcaRiaaigdaaOGaayjkai aawMcaaKqzGeGaai4laKqbaoaabmaak8aabaqcLbsapeGaeqiTdqwc fa4damaaBaaaleaajugib8qacaWGVbaal8aabeaajugib8qacqGHRa WkcqaHbpGCl8aadaWgaaqaaKqzadWdbiaadkhaaSWdaeqaaaGcpeGa ayjkaiaawMcaaaaa@4DEB@ . Therefore, to limit the amount of δ h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazl8aadaWgaaqaaKqzadWdbiaadIgaaSWdaeqaaaaa @3ABF@ for infinite time period, γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHZoWzaaa@3841@ should possess measures below one, in other words, ρ II MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCjuaGpaWaaSbaaSqaaKqzadWdbiaadMeacaWGjbaa l8aabeaaaaa@3C17@ should be less than ρ I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCl8aadaWgaaqaaKqzadWdbiaadMeaaSWdaeqaaaaa @3ABB@ , ( ρ II ρ I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCl8aadaWgaaqaaKqzadWdbiaadMeacaWGjbaal8aa beaajugib8qacqGHKjYOcqaHbpGCl8aadaWgaaqaaKqzadWdbiaadM eaaSWdaeqaaaaa@41FE@ ). As a result, if the upper fluid is heavier than the lower fluid, instability phenomenon occurs.

Similarity solution of boussinesq equation

By introducing the similarity variable η=r/t 1+α 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiabeE7aOjabg2da9iaadkhacaGGVaGaamiDaSWaaSaa aOqaaKqzadGaaGymaiabgUcaRiabeg7aHbGcbaqcLbmacaaI0aaaaa aa@429E@ and relation  h=t α1 2 f(η) N 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCaeaaaaaa aaa8qacaGGGcqcLbsacaWGObGaeyypa0JaamiDaSWaaSaaaKqbagaa jugWaiabeg7aHjabgkHiTiaaigdaaKqbagaajugWaiaaikdaaaqcLb sacaWGMbGaaiikaiabeE7aOjaacMcacaWGobWcpaWaaWbaaeqabaqc LbmapeGaaGOmaaaaaaa@49F2@ ,5 Equation (9) could be written as follows:

( α+1 4 ) η 2 df dη ( α1 ) 2 f+ 1 η d dη ( ηf df dη ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaOWdaeaajuaGpeWaaSaaaOWdaeaajugib8qacqaHXoqy cqGHRaWkcaaIXaaak8aabaqcLbsapeGaaGinaaaaaOGaayjkaiaawM caaKqzGeGaeq4TdG2cpaWaaWbaaeqabaqcLbmapeGaaGOmaaaajuaG daWcaaGcpaqaaKqzGeWdbiaadsgacaWGMbaak8aabaqcLbsapeGaam izaiabeE7aObaacqGHsisljuaGdaWcaaGcpaqaaKqba+qadaqadaGc paqaaKqzGeWdbiabeg7aHjabgkHiTiaaigdaaOGaayjkaiaawMcaaa Wdaeaajugib8qacaaIYaaaaiaadAgacqGHRaWkjuaGdaWcaaGcpaqa aKqzGeWdbiaaigdaaOWdaeaajugib8qacqaH3oaAaaqcfa4aaSaaaO Wdaeaajugib8qacaWGKbaak8aabaqcLbsapeGaamizaiabeE7aObaa juaGdaqadaGcpaqaaKqzGeWdbiabeE7aOjaadAgajuaGdaWcaaGcpa qaaKqzGeWdbiaadsgacaWGMbaak8aabaqcLbsapeGaamizaiabeE7a ObaaaOGaayjkaiaawMcaaaaa@6A39@ (14)

A power-law flux has been considered at the origin where the power is related to the parameter α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiabeg7aHbaa@3864@ [5]:

q=2πrh ( h r ) r=0 =2π t α1 [ ηf f η ] η=0 N 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGXbGaeyypa0JaaGOmaiabec8aWjaadkhacaWGObqcfa4a aeWaaOWdaeaajuaGpeWaaSaaaOWdaeaajugib8qacqGHciITcaWGOb aak8aabaqcLbsapeGaeyOaIyRaamOCaaaaaOGaayjkaiaawMcaaSWd amaaBaaabaqcLbmapeGaamOCaiabg2da9iaaicdaaSWdaeqaaKqzGe Wdbiabg2da9iaaikdacqaHapaCcaWG0bWcpaWaaWbaaeqabaqcLbma peGaeqySdeMaeyOeI0IaaGymaaaajuaGdaWadaGcpaqaaKqzGeWdbi abeE7aOjaadAgacaWGMbWcpaWaaSbaaeaajugWa8qacqaH3oaAaSWd aeqaaaGcpeGaay5waiaaw2faaKqba+aadaWgaaWcbaqcLbmapeGaeq 4TdGwcLbsacqGH9aqpcaaIWaaal8aabeaajugib8qacaWGobWcpaWa aWbaaeqabaqcLbmapeGaaGinaaaaaaa@67F7@ (15)

Dussan12 disclosed that state α=1.2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiabeg7aHjabg2da9iaaigdacaGGUaGaaGOmaaaa@3B93@ indicates filtrate invasion state in special physical problem.  α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaacckacqaHXoqycqGH9aqpcaaIXaaaaa@3B48@ Discusses the problem with constant flux, while α=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIYaaaaa@39FB@ is linearly dependent to the increase in flux with respect to time. With respect to front position ( r o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcfa4aae WaaOqaaKqzGeaeaaaaaaaaa8qacaWGYbqcfa4damaaBaaaleaajugi b8qacaWGVbaal8aabeaaaOGaayjkaiaawMcaaaaa@3C5C@ , N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaad6eaaaa@3798@ is given by:5

N= r o t α+1 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGobGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaWGYbqc fa4damaaBaaaleaajugWa8qacaWGVbaal8aabeaaaOqaaKqzGeWdbi aadshajuaGpaWaaWbaaSqabeaapeWaaSaaa8aabaqcLbmapeGaeqyS deMaey4kaSIaaGymaaWcpaqaaKqzadWdbiaaisdaaaaaaaaaaaa@4701@ (16)

In this paper the "K-method" has been applied to solve the Equation (14) by adding and removing the term ( α+1/2 )ηf MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcfaieaa aaaaaaa8qadaqadaGcpaqaaKqzGeWdbiabeg7aHjabgUcaRiaaigda caGGVaGaaGOmaaGccaGLOaGaayzkaaqcLbsacqaH3oaAcaWGMbaaaa@40E0@ to this equation. The new form of equation would be obtained as follows:21

( α+1 4 ) ( η 2 f) ' ( ηf f ) ' =αηf MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaOWdaeaajuaGpeWaaSaaaOWdaeaajugib8qacqaHXoqy cqGHRaWkcaaIXaaak8aabaqcLbsapeGaaGinaaaaaOGaayjkaiaawM caaKqzGeGaaiikaiabeE7aOTWdamaaCaaabeqaaKqzadWdbiaaikda aaqcLbsacaWGMbGaaiykaSWdamaaCaaabeqaaKqzadWdbiaacEcaaa qcLbsacqGHsisljuaGdaqadaGcpaqaaKqzGeWdbiabeE7aOjaadAga ceWGMbWdayaafaaak8qacaGLOaGaayzkaaWcpaWaaWbaaeqabaqcLb mapeGaai4jaaaajugibiabg2da9iabeg7aHjabeE7aOjaadAgaaaa@57B9@ (17)

[ ( ηf ) f + α+1 4 η( ηf ) ] 1 η = 1 η αηfdη MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaamWaaOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH3oaA caWGMbaakiaawIcacaGLPaaajugibiqadAgapaGbauaapeGaey4kaS scfa4aaSaaaOWdaeaajugib8qacqaHXoqycqGHRaWkcaaIXaaak8aa baqcLbsapeGaaGinaaaacqaH3oaAjuaGdaqadaGcpaqaaKqzGeWdbi abeE7aOjaadAgaaOGaayjkaiaawMcaaaGaay5waiaaw2faaSWdamaa DaaabaqcLbmapeGaaGymaaWcpaqaaKqzadWdbiabeE7aObaajugibi abg2da9Kqbaoaawahakeqal8aabaqcLbsapeGaaGymaaWcpaqaaKqz GeWdbiabeE7aObqdpaqaaKqzGeWdbiabgUIiYdaacqaHXoqycqaH3o aAcaWGMbGaamizaiabeE7aObaa@62EE@ (18)

And as for η 1 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiabeE7aOTWdamaaBaaabaqcLbmapeGaaGymaaWcpaqa baqcLbsapeGaeyypa0JaaGymaaaa@3D1F@ , f equals 0:

{ f ' +( α+1 4 )η= K ηf α( ηf )dη K 1                                                MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaiqaaOWdaeaajugibuaabeqaceaaaOqaaKqzGeWdbiaadAga lmaaCaaameqabaGaai4jaaaajugibiabgUcaRKqbaoaabmaak8aaba qcfa4dbmaalaaak8aabaqcLbsapeGaeqySdeMaey4kaSIaaGymaaGc paqaaKqzGeWdbiaaisdaaaaakiaawIcacaGLPaaajugibiabeE7aOj abg2da9Kqbaoaalaaak8aabaqcLbsapeGaam4saaGcpaqaaKqzGeWd biabeE7aOjaadAgaaaqcfa4damaavacakeqaleqabaqcLbsacaaMb8 oaneaajugib8qacqGHRiI8aaGaeqySdewcfa4aaeWaaOWdaeaajugi b8qacqaH3oaAcaWGMbaakiaawIcacaGLPaaajugibiaadsgacqaH3o aAaOWdaeaajugib8qacaWGlbqcfa4damaaxabakeaajugib8qacqGH sgIRaSWdaeaaaeqaaKqzGeWdbiaaigdacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaaaaaOGaay5Eaaaaaa@990F@ (19)

Where K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadUeaaaa@3795@ is the K-method parameter and possesses measures between 0 and 1. For K1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadUeacqGHsgIRcaaIXaaaaa@3A3D@ , response of the Equation (19) and (15) would be the same. With regard to the series 1/ηf MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaaigdacaGGVaGaeq4TdGMaamOzaaaa@3ACA@ , it could be indicated that:

1 ηf = 1 η f o K f 1 η f o 2 + K 2 f 1 2 f o f 2 η f o 3 + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaaOWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaeq4T dGMaamOzaaaacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaaigdaaO Wdaeaajugib8qacqaH3oaAcaWGMbWcpaWaaSbaaeaajugWa8qacaWG Vbaal8aabeaaaaqcLbsapeGaeyOeI0Iaam4saKqbaoaalaaak8aaba qcLbsapeGaamOzaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaaa keaajugib8qacqaH3oaAcaWGMbWcpaWaa0baaeaajugWa8qacaWGVb aal8aabaqcLbmapeGaaGOmaaaaaaqcLbsacqGHRaWkcaWGlbWcpaWa aWbaaeqabaqcLbmapeGaaGOmaaaajuaGdaWcaaGcpaqaaKqzGeWdbi aadAgal8aadaqhaaqaaKqzadWdbiaaigdaaSWdaeaajugWa8qacaaI YaaaaKqzGeGaeyOeI0IaamOzaSWdamaaBaaabaqcLbmapeGaam4Baa WcpaqabaqcLbsapeGaamOzaKqba+aadaWgaaWcbaqcLbmapeGaaGOm aaWcpaqabaaakeaajugib8qacqaH3oaAcaWGMbWcpaWaa0baaeaaju gWa8qacaWGVbaal8aabaqcLbmapeGaaG4maaaaaaqcLbsacqGHRaWk cqGHMacVaaa@721E@ Kα ηf 1 η ηfdη=K( 1 η f o ) 1 η αη f o dη+ K 2 [ 1 η f o 1 η αη f 1 dη f 1 η f o 2 1 η αη f o dη ]+ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaaOWdaeaajugib8qacaWGlbGaeqySdegak8aabaqcLbsa peGaeq4TdGMaamOzaaaalmaawahakeqal8aabaqcLbmapeGaaGymaa WcpaqaaKqzadWdbiabeE7aObqdpaqaaKqzadWdbiabgUIiYdaajugi biabeE7aOjaadAgacaWGKbGaeq4TdGMaeyypa0Jaam4saKqbaoaabm aak8aabaqcfa4dbmaalaaak8aabaqcLbsapeGaaGymaaGcpaqaaKqz GeWdbiabeE7aOjaadAgajuaGpaWaaSbaaSqaaKqzadWdbiaad+gaaS WdaeqaaaaaaOWdbiaawIcacaGLPaaalmaawahakeqal8aabaqcLbma peGaaGymaaWcpaqaaKqzadWdbiabeE7aObqdpaqaaKqzadWdbiabgU IiYdaajugibiabeg7aHjabeE7aOjaadAgajuaGpaWaaSbaaSqaaKqz adWdbiaad+gaaSWdaeqaaKqzGeWdbiaadsgacqaH3oaAcqGHRaWkca WGlbqcfa4damaaCaaaleqabaqcLbmapeGaaGOmaaaajuaGdaWadaGc paqaaKqba+qadaWcaaGcpaqaaKqzGeWdbiaaigdaaOWdaeaajugib8 qacqaH3oaAcaWGMbqcfa4damaaBaaaleaajugWa8qacaWGVbaal8aa beaaaaWdbmaawahakeqal8aabaqcLbmapeGaaGymaaWcpaqaaKqzad WdbiabeE7aObqdpaqaaKqzadWdbiabgUIiYdaajugibiabeg7aHjab eE7aOjaadAgal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqzGe WdbiaadsgacqaH3oaAcqGHsisljuaGdaWcaaGcpaqaaKqzGeWdbiaa dAgajuaGpaWaaSbaaSqaaKqzadWdbiaaigdaaSWdaeqaaaGcbaqcLb sapeGaeq4TdGMaamOzaSWdamaaDaaabaqcLbmapeGaam4BaaWcpaqa aKqzadWdbiaaikdaaaaaaSWaaybCaOqabSWdaeaajugWa8qacaaIXa aal8aabaqcLbmapeGaeq4TdGgan8aabaqcLbmapeGaey4kIipaaKqz GeGaeqySdeMaeq4TdGMaamOzaSWdamaaBaaabaqcLbmapeGaam4Baa WcpaqabaqcLbsapeGaamizaiabeE7aObGccaGLBbGaayzxaaqcLbsa cqGHRaWkcqGHMacVaaa@B33F@ (20)

or

{ f o ' +( α+1 4 )η=0                f 1 ( 1 )=0                                   f o ( 1 )=0 f 1 ' =( 1 η f o ) 1 η αη f o dη      f 1 ( 1 )=0   f 2 ' = 1 η f o 1 η αη f 1 dη f 1 η f o 2 1 η αη f o dη       f 2 ( 1 )=0    MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaju gibqaabeGcbaqcLbsaqaaaaaaaaaWdbiaadAgal8aadaqhaaqaaKqz adWdbiaad+gaaSWdaeaajugWa8qacaGGNaaaaKqzGeGaey4kaSscfa 4aaeWaaOWdaeaajuaGpeWaaSaaaOWdaeaajugib8qacqaHXoqycqGH RaWkcaaIXaaak8aabaqcLbsapeGaaGinaaaaaOGaayjkaiaawMcaaK qzGeGaeq4TdGMaeyypa0JaaGimaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaadAgal8aadaWgaaqaaKqzadWdbiaaigda aSWdaeqaaKqba+qadaqadaGcpaqaaKqzGeWdbiaaigdaaOGaayjkai aawMcaaKqzGeGaeyypa0JaaGimaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcaakeaajugibiaacckacaWGMbqcfa4d amaaBaaaleaajugib8qacaWGVbaal8aabeaajuaGpeWaaeWaaOWdae aajugib8qacaaIXaaakiaawIcacaGLPaaajugibiabg2da9iaaicda caWGMbWcpaWaa0baaeaajugWa8qacaaIXaaal8aabaqcLbmapeGaai 4jaaaajugibiabg2da9Kqbaoaabmaak8aabaqcfa4dbmaalaaak8aa baqcLbsapeGaaGymaaGcpaqaaKqzGeWdbiabeE7aOjaadAgajuaGpa WaaSbaaSqaaKqzGeWdbiaad+gaaSWdaeqaaaaaaOWdbiaawIcacaGL PaaajuaGdaGfWbGcbeWcpaqaaKqzGeWdbiaaigdaaSWdaeaajugib8 qacqaH3oaAa0Wdaeaajugib8qacqGHRiI8aaGaeqySdeMaeq4TdGMa amOzaKqba+aadaWgaaWcbaqcLbmapeGaam4BaaWcpaqabaqcLbsape GaamizaiabeE7aOjaacckacaGGGcGaaiiOaiaacckacaGGGcGaamOz aSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaqcfa4dbmaabmaak8 aabaqcLbsapeGaaGymaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcaaI WaGaaiiOaiaacckaaOqaaKqzGeGaamOzaSWdamaaDaaabaqcLbmape GaaGOmaaWcpaqaaKqzadWdbiaacEcaaaqcLbsacqGH9aqpjuaGdaWc aaGcpaqaaKqzGeWdbiaaigdaaOWdaeaajugib8qacqaH3oaAcaWGMb WcpaWaaSbaaeaajugWa8qacaWGVbaal8aabeaaaaqcfa4dbmaawaha keqal8aabaqcLbsapeGaaGymaaWcpaqaaKqzGeWdbiabeE7aObqdpa qaaKqzGeWdbiabgUIiYdaacqaHXoqycqaH3oaAcaWGMbqcfa4damaa Baaaleaajugib8qacaaIXaaal8aabeaajugib8qacaWGKbGaeq4TdG MaeyOeI0scfa4aaSaaaOWdaeaajugib8qacaWGMbWcpaWaaSbaaeaa jugWa8qacaaIXaaal8aabeaaaOqaaKqzGeWdbiabeE7aOjaadAgal8 aadaqhaaqaaKqzadWdbiaad+gaaSWdaeaajugWa8qacaaIYaaaaaaa juaGdaGfWbGcbeWcpaqaaKqzGeWdbiaaigdaaSWdaeaajugib8qacq aH3oaAa0Wdaeaajugib8qacqGHRiI8aaGaeqySdeMaeq4TdGMaamOz aKqba+aadaWgaaWcbaqcLbsapeGaam4BaaWcpaqabaqcLbsapeGaam izaiabeE7aOjaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa dAgajuaGpaWaaSbaaSqaaKqzadWdbiaaikdaaSWdaeqaaKqba+qada qadaGcpaqaaKqzGeWdbiaaigdaaOGaayjkaiaawMcaaKqzGeGaeyyp a0JaaGimaiaacckacaGGGcGaaiiOaaaak8aacaGL7baaaaa@1D1A@ (21)

By implementing simple mathematical operations, results of the Equation (21) could be defined as three following statements:

f o ( η )= 1 8 ( 1+α )( η 2 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbWcpaWaaSbaaeaajugWa8qacaWGVbaal8aabeaajuaG peWaaeWaaOWdaeaajugib8qacqaH3oaAaOGaayjkaiaawMcaaKqzGe Gaeyypa0JaeyOeI0scfa4aaSaaaOWdaeaajugib8qacaaIXaaak8aa baqcLbsapeGaaGioaaaajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacq GHRaWkcqaHXoqyaOGaayjkaiaawMcaaKqbaoaabmaak8aabaqcLbsa peGaeq4TdGwcfa4damaaCaaaleqabaqcLbsapeGaaGOmaaaacqGHsi slcaaIXaaakiaawIcacaGLPaaaaaa@5315@

f 1 ( η )= 1 8 α( η 2 1 ) α 4 logx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbWcpaWaaSbaaeaajugWa8qacaaIXaaal8aabeaajuaG peWaaeWaaOWdaeaajugib8qacqaH3oaAaOGaayjkaiaawMcaaKqzGe Gaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaaIXaaak8aabaqcLbsa peGaaGioaaaacqaHXoqyjuaGdaqadaGcpaqaaKqzGeWdbiabeE7aOL qba+aadaahaaWcbeqaaKqzadWdbiaaikdaaaqcLbsacqGHsislcaaI XaaakiaawIcacaGLPaaajugibiabgkHiTKqbaoaalaaak8aabaqcLb sapeGaeqySdegak8aabaqcLbsapeGaaGinaaaaciGGSbGaai4Baiaa cEgacaWG4baaaa@585C@ (22)

f 2 ( η )= π 2 α 2 24( 1+α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbWcpaWaaSbaaeaajugWa8qacaaIYaaal8aabeaajuaG peWaaeWaaOWdaeaajugib8qacqaH3oaAaOGaayjkaiaawMcaaKqzGe Gaeyypa0tcfa4aaSaaaOWdaeaajugib8qacqaHapaCl8aadaahaaqa beaajugWa8qacaaIYaaaaKqzGeGaeqySde2cpaWaaWbaaeqabaqcLb mapeGaaGOmaaaaaOWdaeaajugib8qacaaIYaGaaGinaKqbaoaabmaa k8aabaqcLbsapeGaaGymaiabgUcaRiabeg7aHbGccaGLOaGaayzkaa aaaaaa@51F3@ + α 2 [ ( logx 1 2 log ( x ) 2 +logxlog( 1+x )polylog( 2,2x )+polylog( 2,x ) ) ]/( 2+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHRaWkcqaHXoqyl8aadaahaaqabeaajugWa8qacaaIYaaa aKqbaoaadmaak8aabaqcfa4dbmaabmaak8aabaqcLbsapeGaeyOeI0 IaciiBaiaac+gacaGGNbGaamiEaiabgkHiTKqbaoaalaaak8aabaqc LbsapeGaaGymaaGcpaqaaKqzGeWdbiaaikdaaaGaciiBaiaac+gaca GGNbqcfa4aaeWaaOWdaeaajugib8qacaWG4baakiaawIcacaGLPaaa juaGpaWaaWbaaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaci iBaiaac+gacaGGNbGaamiEaiGacYgacaGGVbGaai4zaKqbaoaabmaa k8aabaqcLbsapeGaaGymaiabgUcaRiaadIhaaOGaayjkaiaawMcaaK qzGeGaeyOeI0IaamiCaiaad+gacaWGSbGaamyEaiGacYgacaGGVbGa ai4zaKqbaoaabmaak8aabaqcLbsapeGaaGOmaiaacYcacaaIYaGaam iEaaGccaGLOaGaayzkaaqcLbsacqGHRaWkcaWGWbGaam4BaiaadYga caWG5bGaciiBaiaac+gacaGGNbqcfa4aaeWaaOWdaeaajugib8qaca aIYaGaaiilaiabgkHiTiaadIhaaOGaayjkaiaawMcaaaGaayjkaiaa wMcaaaGaay5waiaaw2faaKqzGeGaai4laKqbaoaabmaak8aabaqcLb sapeGaaGOmaiabgUcaRiaaikdacqaHXoqyaOGaayjkaiaawMcaaaaa @855F@

Regarding the Equation (15) the amount of flux accumulation could be written as follows:

0 t αdt=C t α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaybCaOqabSWdaeaajugWa8qacaaIWaaal8aabaqcLbmapeGa amiDaaqdpaqaaKqzGeWdbiabgUIiYdaacqaHXoqycaWGKbGaamiDai abg2da9iaadoeacaWG0bWcpaWaaWbaaeqabaqcLbmapeGaeqySdega aaaa@477C@ (23)

Where C is the constant standing for flux accumulation and equals:

C=2π N 4 0 1 ηfdη MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGdbGaeyypa0JaaGOmaiabec8aWjaad6eajuaGpaWaaWba aSqabeaajugWa8qacaaI0aaaaKqbaoaawahakeqal8aabaqcLbmape GaaGimaaWcpaqaaKqzadWdbiaaigdaa0Wdaeaajugib8qacqGHRiI8 aaGaeq4TdGMaamOzaiaadsgacqaH3oaAaaa@4B77@ (24)

Thus, characteristic radius r o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadkhal8aadaWgaaqaaKqzadWdbiaad+gaaSWdaeqa aaaa@3A43@ and interface between the two fluids could be indicated with respect to C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadoeaaaa@378D@ as follows:

r o 4 = C t α+1 2π 0 1 ηfdη MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGYbWcpaWaa0baaeaajugWa8qacaWGVbaal8aabaqcLbma peGaaGinaaaajugibiabg2da9Kqbaoaalaaak8aabaqcLbsapeGaam 4qaiaadshajuaGpaWaaWbaaSqabeaajugWa8qacqaHXoqycqGHRaWk caaIXaaaaaGcpaqaaKqzGeWdbiaaikdacqaHapaCjuaGdaqfWaGcbe WcpaqaaKqzadWdbiaaicdaaSWdaeaajugWa8qacaaIXaaan8aabaqc LbsapeGaey4kIipaaiabeE7aOjaadAgacaWGKbGaeq4TdGgaaaaa@55F3@ (25)

h=( C 2π 0 1 ηdη ) t α f( η ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObGaeyypa0tcfa4aaeWaaOWdaeaajuaGpeWaaSaaaOWd aeaajugib8qacaWGdbaak8aabaqcLbsapeGaaGOmaiabec8aWLqbao aavadakeqal8aabaqcLbmapeGaaGimaaWcpaqaaKqzadWdbiaaigda a0Wdaeaajugib8qacqGHRiI8aaGaeq4TdGMaamizaiabeE7aObaaaO GaayjkaiaawMcaaKqzGeGaamiDaSWdamaaCaaabeqaaKqzadWdbiab eg7aHbaajugibiaadAgajuaGdaqadaGcpaqaaKqzGeWdbiabeE7aOb GccaGLOaGaayzkaaaaaa@5698@ (26)

Considering the importance of the amount of 0 1 ηfdη MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcfaieaa aaaaaaa8qadaGfWbGcbeWcpaqaaKqzadWdbiaaicdaaSWdaeaajugW a8qacaaIXaaan8aabaqcLbsapeGaey4kIipaaiabeE7aOjaadAgaca WGKbGaeq4TdGgaaa@4333@ in assessing variables of the problem and relations obtained from Equation (21), this integral could be solved with respect to α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiabeg7aHbaa@3864@ .

0 1 ηfdη= 1 32 +α 1+( 4 π 2 /3 ) 16( 1+α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaybCaOqabSWdaeaajugWa8qacaaIWaaal8aabaqcLbmapeGa aGymaaqdpaqaaKqzGeWdbiabgUIiYdaacqaH3oaAcaWGMbGaamizai abeE7aOjabg2da9Kqbaoaalaaak8aabaqcLbsapeGaaGymaaGcpaqa aKqzGeWdbiaaiodacaaIYaaaaiabgUcaRiabeg7aHLqbaoaalaaak8 aabaqcLbsapeGaaGymaiabgUcaRKqbaoaabmaak8aabaqcLbsapeGa aGinaiabgkHiTiabec8aWTWdamaaCaaabeqaaKqzadWdbiaaikdaaa qcLbsacaGGVaGaaG4maaGccaGLOaGaayzkaaaapaqaaKqzGeWdbiaa igdacaaI2aqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaey4kaSIaeq ySdegakiaawIcacaGLPaaaaaaaaa@6079@ (27)

Results and discussion

This study generally represents the function f( η )/( 1+α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadAgajuaGdaqadaGcpaqaaKqzGeWdbiabeE7aObGc caGLOaGaayzkaaqcLbsacaGGVaqcfa4aaeWaaOWdaeaajugib8qaca aIXaGaey4kaSIaeqySdegakiaawIcacaGLPaaaaaa@438C@ as the solution of Equation (14). Figure 3 compares the obtained results with both the numerical results and the results obtained by Li et al19 in α=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiabeg7aHjabg2da9iaaigdaaaa@3A25@ . Also, in Figure 4, the f( η )/( 1+α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiaadAgajuaGdaqadaGcpaqaaKqzGeWdbiabeE7aObGc caGLOaGaayzkaaqcLbsacaGGVaqcfa4aaeWaaOWdaeaajugib8qaca aIXaGaey4kaSIaeqySdegakiaawIcacaGLPaaaaaa@438C@ is presented with regards to the various values of η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiabeE7aObaa@3871@ and α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcLbsaqa aaaaaaaaWdbiabeg7aHbaa@3864@ . In view of the significance of amount fd η 2 /1+α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaGCqcfa4aau biaOqabSqabeaajugibiaaygW7a0qaaKqzGeaeaaaaaaaaa8qacqGH RiI8aaGaamOzaiaadsgacqaH3oaAl8aadaahaaqabeaajugWa8qaca aIYaaaaKqzGeGaai4laiaaigdacqGHRaWkcqaHXoqyaaa@45CB@ in evaluating the position of the interface, Figure 5 compares assessed amount of this variable in this study with results obtained from numerical solutions.

Figure 3 Compares the obtained results with numerical results and the results.
By19 in  α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiiOaKqzGeGaeqySdeMaeyypa0JaaGymaaaa@3B1E@ .

Figure 4 The values of f( η )/( 1+α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbqcfa4aaeWaaOWdaeaajugib8qacqaH3oaAaOGaayjk aiaawMcaaKqzGeGaai4laKqbaoaabmaak8aabaqcLbsapeGaaGymai abgUcaRiabeg7aHbGccaGLOaGaayzkaaaaaa@4362@ respect with to the various values of η and α.

Figure 5 Comparing of the obtained values of ( 0 1 fd η 2   )/( 1+α )  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaOWdaeaajuaGpeWaaybCaOqabSWdaeaajugWa8qacaaI Waaal8aabaqcLbmapeGaaGymaaqdpaqaaKqzGeWdbiabgUIiYdaaca WGMbGaamizaiabeE7aOLqba+aadaahaaWcbeqaaKqzadWdbiaaikda aaqcLbsacaGGGcaakiaawIcacaGLPaaajugibiaac+cajuaGdaqada GcpaqaaKqzGeWdbiaaigdacqGHRaWkcaqGXoaakiaawIcacaGLPaaa jugibiaabckaaaa@50BF@ with results obtained from numerical solutions.

Conclusion

General model for interface between two immiscible fluids in porous media especially for case that elastic perturbation are propagating very faster than gravity perturbation have been investigated. Stability analysis of the Boussinesq equation as governing equation in this case has been done. The result showed that gravity is unstable if the lower fluid is lighter. Also the Boussinesq equation for dynamic movement of interface between two immiscible fluids has been solved by the K-Method. With a Power-law flux at the origin and dimensional analysis, the drilling mud invasion into a permeable aquifer problems becomes a nonlinear boundary value problem that have been investigated in this paper. The obtained results show that the K-Method has a good ability for investigation of B-L problems such as this solved problem.

Acknowledgements

None.

Conflict of interest

The author declares that there is no conflict of interest.

References

  1. Calugaru CI, Calugaru DG, Crolet JM, et al. Evolution of fluid-fluid interface in porous media as the model of gas-oil fields. Electronic Journal of Differential Equations. 2003;2003(72):1–13.
  2. Whitham. Linear and Nonlinear Waves. New York: John Wiley& Sons; 1974. 638 p.
  3. Dagan G. second-order theory of shallow free surface flow in porous media. QJ Mech Appl Math. 1967;20(4):517–526.
  4. Bear J. Dynamics of Fluid in Porous Media. American Elsevier New York: American Elsevier; 1972. 764 p.
  5. Barenblatt GI, Entov VM, Ryzhik VM. Theory of Fluid Flows through Natural Rocks. Dordrecht: Kluwer Academic Publishers; 1990. 395 p.
  6. Childs E. Drainage of groundwater resting on a sloping bed. Water Res journal. 1971;7(5):1256–1263.
  7. Aravin VI, numerov SN. Theory of fluid flow in under formable porous media. Isr Program for Science Translation Jerusalem; 1965. 511 p.
  8. Verhoest Niko EC, Troch PA. Some analytical solutions of the linearized Boussinesq equation with recharge for a sloping aquifer. Water Resources research. 2000;36(3):793–800.
  9. Pistiner A. Radial Oil–Water Transport toward an Extraction Well, Transp. Porous Med. 2018;124(2):479–493.
  10. Teo HT, Jeng DS, Seymour BR, Barry DA. A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches. Advances In Water Resources. 2003;26(12):1239–1247.
  11. Gomes AFC, Marinho JLG, Oliveira LMTM. Numeric study of a drilling fluid leak in a rock formation, permeability aspects. Brazilian Journal of Petroleum and Gas. 2016;10(4):221–232.
  12. Dussan E, Auzerias FM. Buoyancy-induced flow in porous media generated near a drilled oil well. The accumulation of filtrate at a horizontal impermeable boundary. J Fluid Mech. 1993;254:283–311.
  13. Doll HG. Filtrate invasion in highly permeable sands. USA: Petrol Eng; 1955. 115 p.
  14. Tang Y, Jiang Q, Zhou C. Approximate analytical solution to the Boussinesq equation with a sloping water-land boundary. Water Resour. Res. 2016;52(4):2529–2550.
  15. Bartlett MS, Porporato A. A Class of Exact Solutions of the Boussinesq Equation for Horizontal and Sloping Aquifers. Water Resources Research. 2018;54(2):767–778.
  16. Lu D, Shen J, Cheng Y. The approximate solutions of nonlinear Boussinesq equation. Journal of Physics: Conference Series. 2016;11(2):1–9.
  17. Bansal RK. Approximate Analytical Solution of Boussinesq Equation in Homogeneous Medium with Leaky Base. Applications and Applied Mathematics: An International Journal. 2016;11(1):184–191.
  18. Telyakovskiy AS, Kurita S, Allen MB. Polynomial-based approximate solutions to the Boussinesq equation near a well. Advances in Water Resources. 2016;96:68–73.
  19. Li L, Lockington DA, Parlange MB, et al. Similarity solution of axisymmetric flow in porous media. Advances in water resources. 2005;28(10):1076–1082.
  20. Mortensen J, Sayaka O, Jean-Yves P, et al. Approximate similarity solution to a nonlinear diffusion equation with spherical symmetry. International Journal of Numerical Analysis & Modeling. 2012;9(1):105–114.
  21. Shahnazari MR. A novel homotopy perturbation method: Koroush method for a thermal boundary layer in a saturated porous medium. International Journal of Engineering (IJE). 2012;25(1):59–64.
Creative Commons Attribution License

©2018 Shahnazari, et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.