From Canadian Geotechnical Journal, ©2000 National Research Council Canada. Used with permission.

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Equivalent plane strain solution

The theory of consolidation of vertical drains under axisymmetric conditions has gained wide acceptance in geotechnical engineering because of its simplicity. According to Hansbo (1981), for axisymmetric flow the average degree of consolidation U ¯ h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGabmyvayaaraWaaSbaaSqaaiaajI gaaeqaaaaa@33E1@  on a horizontal plane at a depth z and at time t may be predicted from

 

U ¯ h =1exp( 8 T h μ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGabmyvayaaraWaaSbaaSqaaiaajI gaaeqaaOGaeyypa0JaaGymaiabgkHiTiGacwgacaGG4bGaaiiCamaa bmaabaGaeyOeI0YaaSaaaeaacaaI4aGaamivamaaBaaaleaacaqIOb aabeaaaOqaaiabeY7aTbaaaiaawIcacaGLPaaaaaa@4073@

[1]

where T h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamivamaaBaaaleaacaqIObaabe aaaaa@33C8@  is the time factor, and the effect of smear and well resistance (μ) is given by

 

μ = ln ( n s ) + ( k h k h ) ln ( s ) 0.75 + π ( 2 l z z 2 ) k h q w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaeqiVd0Maeyypa0JaciiBaiaac6 gadaqadaqaamaalaaabaGaamOBaaqaaiaadohaaaaacaGLOaGaayzk aaGaey4kaSYaaeWaaeaadaWcaaqaaiaadUgadaWgaaWcbaGaaKiAaa qabaaakeaaceWGRbGbauaadaWgaaWcbaGaaKiAaaqabaaaaaGccaGL OaGaayzkaaGaciiBaiaac6gadaqadaqaaiaadohaaiaawIcacaGLPa aacqGHsislcaaIWaGaaiOlaiaaiEdacaaI1aGaey4kaSIaeqiWda3a aeWaaeaacaaIYaGaamiBaiaadQhacqGHsislcaWG6bWaaWbaaSqabe aacaaIYaaaaaGccaGLOaGaayzkaaWaaSaaaeaacaWGRbWaaSbaaSqa aiaajIgaaeqaaaGcbaGaamyCamaaBaaaleaacaqI3baabeaaaaaaaa@56EA@

[2]

In the above, n=R/ r w , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamOBaiabg2da9maalyaabaGaam OuaaqaaiaadkhadaWgaaWcbaGaaK4DaaqabaGccaGGSaGaaGzaVlaa ygW7aaaaaa@3AA9@  where R is the radius of the influence zone of the drain and r w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamOCamaaBaaaleaacaqI3baabe aaaaa@33F5@  is the radius of the drain; s= r s / r w , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaam4Caiabg2da9maalyaabaGaam OCamaaBaaaleaacaqIZbaabeaaaOqaaiaadkhadaWgaaWcbaGaaK4D aaqabaGccaGGSaaaaaaa@38EE@  where r s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamOCamaaBaaaleaacaqIZbaabe aaaaa@33F1@  is the radius of the smear; l is the length of the drain having one-way drainage or half this value for two-way drainage; z is the depth of the drain under consideration; q w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamyCamaaBaaaleaacaqI3baabe aaaaa@33F4@  is the discharge capacity of the drain; and k h  and  k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaam4AamaaBaaaleaacaqIObaabe aakiaabccacaqGHbGaaeOBaiaabsgacaqGGaGabm4AayaafaWaaSba aSqaaiaajIgaaeqaaaaa@3A06@  are the coefficients of horizontal permeability outside and inside the smeared zone, respectively.

In equation [2], the final term represents the well resistance, which is a function of the drain length l and depth z for l>z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamiBaiabg6da+iaadQhaaaa@34C8@  and is inversely proportional to the drain discharge capacity ( q w ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaWaaeWaaeaacaWGXbWaaSbaaSqaai aajEhaaeqaaaGccaGLOaGaayzkaaaaaa@3587@ .

Indraratna and Redana (1997) showed that the degree of consolidation at a depth z in plane strain condition can be represented by

 

U ¯ hp = 1 u ¯ u ¯ 0 = 1 exp ( 8 T hp μ p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGabmyvayaaraWaaSbaaSqaaiaajI gacaqIWbaabeaakiabg2da9iaaigdacqGHsisldaWcaaqaaiqadwha gaqeaaqaaiqadwhagaqeamaaBaaaleaacaaIWaaabeaaaaGccqGH9a qpcaaIXaGaeyOeI0IaciyzaiaacIhacaGGWbWaaeWaaeaadaWcaaqa amaaCaaaleqabaGaeyOeI0caaOGaaGioaiaadsfadaWgaaWcbaGaaK iAaiaajchaaeqaaaGcbaGaeqiVd02aaSbaaSqaaiaajchaaeqaaaaa aOGaayjkaiaawMcaaaaa@49A3@

[3]

where u ¯ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8rrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaaceWG1bGbaebadaWgaaWcba GaaGimaaqabaaaaa@352C@ is the initial pore pressure, u ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGabmyDayaaraaaaa@32E2@  is the pore pressure at time t (average values), T hp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamivamaaBaaaleaacaqIObGaaK iCaaqabaaaaa@34C3@  is time factor in plane strain, and

 

μ p =[ α+β k hp k hp +θ( 2lz z 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaeqiVd02aaSbaaSqaaiaajchaae qaaOGaeyypa0ZaamWaaeaacqaHXoqycqGHRaWkcqaHYoGydaWcaaqa aiaadUgadaWgaaWcbaGaaKiAaiaajchaaeqaaaGcbaGabm4Aayaafa WaaSbaaSqaaiaajIgacaqIWbaabeaaaaGccqGHRaWkcqaH4oqCdaqa daqaaiaaikdacaWGSbGaamOEaiabgkHiTiaadQhadaahaaWcbeqaai aaikdaaaaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@4BC0@

[4]

where k hp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaam4AamaaBaaaleaacaqIObGaaK iCaaqabaaaaa@34DA@  and k hp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGabm4AayaafaWaaSbaaSqaaiaajI gacaqIWbaabeaaaaa@34E6@  are the undisturbed horizontal and corresponding smear zone permeabilities, respectively. Ignoring higher order terms, the geometric parameters α and β and the flow term θ are given by

 

α = 2 3 2 b s B ( 1 b s B + b s 2 3 B 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaeqySdeMaeyypa0ZaaSaaaeaaca aIYaaabaGaaG4maaaacqGHsisldaWcaaqaaiaaikdacaWGIbWaaSba aSqaaiaajohaaeqaaaGcbaGaamOqaaaadaqadaqaaiaaigdacqGHsi sldaWcaaqaaiaadkgadaWgaaWcbaGaaK4CaaqabaaakeaacaWGcbaa aiabgUcaRmaalaaabaGaamOyamaaDaaaleaacaqIZbaabaGaaKOmaa aaaOqaaiaaiodacaWGcbWaaWbaaSqabeaacaaIYaaaaaaaaOGaayjk aiaawMcaaaaa@46FD@

[5]

 

β = 1 B 2 ( b s b w ) 2 + b s 3 B 3 ( 3 b w 2 b s 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaeqOSdiMaeyypa0ZaaSaaaeaaca aIXaaabaGaamOqamaaCaaaleqabaGaaGOmaaaaaaGcdaqadaqaaiaa dkgadaWgaaWcbaGaaK4CaaqabaGccqGHsislcaWGIbWaaSbaaSqaai aajEhaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSYaaSaaaeaacaWGIbWaaSbaaSqaaiaajohaaeqaaaGcbaGaaG 4maiaadkeadaahaaWcbeqaaiaaiodaaaaaaOWaaeWaaeaacaaIZaGa amOyamaaDaaaleaacaqI3baabaGaaKOmaaaakiabgkHiTiaadkgada qhaaWcbaGaaK4CaaqaaiaajkdaaaaakiaawIcacaGLPaaaaaa@4D0C@

[6]

and

 

θ = 2 k hp 2 k hp B q z ( 1 b w B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaeqiUdeNaeyypa0ZaaSaaaeaaca aIYaGaam4AamaaDaaaleaacaqIObGaaKiCaaqaaiaajkdaaaaakeaa ceWGRbGbauaadaWgaaWcbaGaaKiAaiaajchaaeqaaOGaamOqaiaadg hadaWgaaWcbaGaamOEaaqabaaaaOWaaeWaaeaacaaIXaGaeyOeI0Ya aSaaaeaacaWGIbWaaSbaaSqaaiaajEhaaeqaaaGcbaGaamOqaaaaai aawIcacaGLPaaaaaa@4562@

[7]

where q z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamyCamaaBaaaleaacaqI6baabe aaaaa@33F7@  is the equivalent plane strain discharge capacity of the drain, and dimensions B, b s , and  b w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaamOqaiaacYcacaaMc8UaamOyam aaBaaaleaacaqIZbaabeaakiaacYcacaqGGaGaaeyyaiaab6gacaqG KbGaaeiiaiaadkgadaWgaaWcbaGaaK4Daaqabaaaaa@3DB4@  are defined in Figure 1.

At a given stress level, to maintain the same degree of consolidation at each time step, the average degree of consolidation for both axisymmetric ( U ¯ h ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaWaaeWaaeaaceWGvbGbaebadaWgaa WcbaGaaKiAaaqabaaakiaawIcacaGLPaaaaaa@3574@  and equivalent plain strain ( U ¯ hp ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaWaaeWaaeaaceWGvbGbaebadaWgaa WcbaGaaKiAaiaajchaaeqaaaGccaGLOaGaayzkaaaaaa@366F@  conditions are made equal, hence

 

U ¯ h = U ¯ hp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGabmyvayaaraWaaSbaaSqaaiaajI gaaeqaaOGaeyypa0JabmyvayaaraWaaSbaaSqaaiaajIgacaqIWbaa beaaaaa@37FC@

[8]

Combining equations [3] and [8] with the original Hansbo (1981) theory (equation [1]) defines the time factor ratio by the following equation:

 

T hp T h = k hp k h R 2 B 2 = μ p μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaWaaSaaaeaacaWGubWaaSbaaSqaai aajIgacaqIWbaabeaaaOqaaiaadsfadaWgaaWcbaGaaKiAaaqabaaa aOGaeyypa0ZaaSaaaeaacaWGRbWaaSbaaSqaaiaajIgacaqIWbaabe aaaOqaaiaadUgadaWgaaWcbaGaaKiAaaqabaaaaOWaaSaaaeaacaWG sbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOqamaaCaaaleqabaGaaG OmaaaaaaGccqGH9aqpdaWcaaqaaiabeY7aTnaaBaaaleaacaqIWbaa beaaaOqaaiabeY7aTbaaaaa@4668@

[9]

Indraratna and Redana (1997) showed that if the radius of the axisymmetric influence zone of a single drain (R) were taken to be the same as the half-width (B) in plain strain, then the relationship between k hp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaam4AamaaBaaaleaacaqIObGaaK iCaaqabaaaaa@34DA@  and k hp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGabm4AayaafaWaaSbaaSqaaiaajI gacaqIWbaabeaaaaa@34E6@  is given by

 

k hp = k h [ α + β k hp k hp + θ ( 2 l z z 2 ) ] [ ln ( n s ) + ( k h k h ) ln ( s ) 0.75 + π ( 2 l z z 2 ) k h q w ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaam4AamaaBaaaleaacaqIObGaaK iCaaqabaGccqGH9aqpdaWcaaqaaiaadUgadaWgaaWcbaGaaKiAaaqa baGcdaWadaqaaiabeg7aHjabgUcaRiabek7aInaalaaabaGaam4Aam aaBaaaleaacaqIObGaaKiCaaqabaaakeaaceWGRbGbauaadaWgaaWc baGaaKiAaiaajchaaeqaaaaakiabgUcaRiabeI7aXnaabmaabaGaaG OmaiaadYgacaWG6bGaeyOeI0IaamOEamaaCaaaleqabaGaaGOmaaaa aOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaamaadmaabaGaciiBai aac6gadaqadaqaamaalaaabaGaamOBaaqaaiaadohaaaaacaGLOaGa ayzkaaGaey4kaSYaaeWaaeaadaWcaaqaaiaadUgadaWgaaWcbaGaaK iAaaqabaaakeaaceWGRbGbauaadaWgaaWcbaGaaKiAaaqabaaaaaGc caGLOaGaayzkaaGaciiBaiaac6gadaqadaqaaiaadohaaiaawIcaca GLPaaacqGHsislcaaIWaGaaiOlaiaaiEdacaaI1aGaey4kaSIaeqiW da3aaeWaaeaacaaIYaGaamiBaiaadQhacqGHsislcaWG6bWaaWbaaS qabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaSaaaeaacaWGRbWaaSba aSqaaiaajIgaaeqaaaGcbaGaamyCamaaBaaaleaacaqI3baabeaaaa aakiaawUfacaGLDbaaaaaaaa@7271@

[10]

If both the smear and well resistance are ignored, then the simplified ratio of plane strain to axisymmetric horizontal permeability k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaGaam4AamaaBaaaleaacaqIObaabe aaaaa@33DF@  is represented by

 

k hp k h = 0.67 [ ln ( n ) 0.75 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaWaaSaaaeaacaWGRbWaaSbaaSqaai aajIgacaqIWbaabeaaaOqaaiaadUgadaWgaaWcbaGaaKiAaaqabaaa aOGaeyypa0ZaaSaaaeaacaaIWaGaaiOlaiaaiAdacaaI3aaabaWaam WaaeaaciGGSbGaaiOBamaabmaabaGaamOBaaGaayjkaiaawMcaaiab gkHiTiaaicdacaGGUaGaaG4naiaaiwdaaiaawUfacaGLDbaaaaaaaa@453A@

[11]

If the effect of well resistance is ignored, the permeability in the smear zone can be isolated by neglecting the final terms of the denominator and numerator in equation [10] to give

 

k hp k hp = β k hp k h [ ln ( n s ) + ( k h k h ) ln ( s ) 0.75 ] α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefmuySLMyYLgarqqtubsr4rNCHbGeaGqiVu0J e9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGa ciGaaiaabeqaaeaadaabaaGcbaWaaSaaaeaaceWGRbGbauaadaWgaa WcbaGaaKiAaiaajchaaeqaaaGcbaGaam4AamaaBaaaleaacaqIObGa aKiCaaqabaaaaOGaeyypa0ZaaSaaaeaacqaHYoGyaeaadaWcaaqaai aadUgadaWgaaWcbaGaaKiAaiaajchaaeqaaaGcbaGaam4AamaaBaaa leaacaqIObaabeaaaaGcdaWadaqaaiGacYgacaGGUbWaaeWaaeaada Wcaaqaaiaad6gaaeaacaWGZbaaaaGaayjkaiaawMcaaiabgUcaRmaa bmaabaWaaSaaaeaacaWGRbWaaSbaaSqaaiaajIgaaeqaaaGcbaGabm 4AayaafaWaaSbaaSqaaiaajIgaaeqaaaaaaOGaayjkaiaawMcaaiGa cYgacaGGUbWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGaeyOeI0IaaG imaiaac6cacaaI3aGaaGynaaGaay5waiaaw2faaiabgkHiTiabeg7a Hbaaaaa@58E4@

[12]