International Journal of eISSN: 2475-5559 IPCSE

Petrochemical Science & Engineering
Research Article
Volume 3 Issue 3

Resistivity fractal dimension for characterizing shajara reservoirs of the permo-carboniferous shajara formation Saudi Arabia

Khalid Elyas Mohamed Elameen Alkhidir
Department of Petroleum and Natural Gas Engineering, King Saud University, Saudi Arabia
Received: April 11, 2018 | Published: June 19, 2018

Correspondence: Khalid Elyas Mohamed Elameen Alkhidir, Department of Petroleum and Natural Gas Engineering, King Saud University, Saudi Arabia, Email

Citation: Alkhidir KEME. Resistivity fractal dimension for characterizing shajara reservoirs of the permo-carboniferous shajara formation Saudi Arabia. Int J Petrochem Sci Eng. 2018;3(3):109‒112. DOI: 10.15406/ipcse.2018.03.00084

Abstract

 Sandstone samples were collected from the surface type section of the Permo-Carboniferous Shajara Formation for detailed reservoir characterization. Capillary pressure experiment was performed to contact porosity and permeability was derived from the Data. Resistivity was calculated from the distribution of pores and the fractal dimension was proven from the relationship between water saturation and resistivity. In addition to field observation and obtained results of fractal dimension, the Shajara reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three fractal dimension units. The units from bottom to top are: Lower Shajara Resistivity Fractal dimension Unit, Middle Shajara Resistivity Fractal Dimension Unit, and Upper Shajara Resistivity Fractal Dimension Unit. These units were also proved by geometric relaxation time of induced polarization fractal dimension. It was found that the resistivity fractal dimension is similar to the geometric relaxation time of induced polarization. It was also reported that the obtained fractal dimension speeds with increasing resistivity and relaxation time due to an increase in pore connectivity.

Keywords: shajara reservoirs, shajara formation, resistivity fractal dimension

Introduction

The pore microgeometrical parameters paly an importent role in the physical properties of low-resistivity sandstone reservoir was investigated by Cerepi et al.1 Oil finding in low resistivity reservoir was reported by Pramudhita et al.2 Low resistivity pay zones dislay low resistivity due to the presenece of conductive minerals such as pyrite, sulphides and graphite in the reservoir was reported by Mashaba et al.3 The features of low amplitude structure, high clay content, high irreducible water saturation, and high formation water salinity are attributed to the origin of low resistivity oil layer was described by Feng et al.4 An increase of permeability with an increase of geometric and arithmetic relaxation time of induced polarization and increasing porosity was documented by Moasong et al.5 An increase of bubble pressure fractal dimension and pressure head fractal dimension and decreasing pore size distribution index and fitting parameters m*n due to possibility of having interconnected channels was confirmed by Al-khidir6 An increase of fractal dimension with increasing permeability and relaxation time of induced polarization due to increase in pore connectivity was reported by Alkhidir.7

 

 

Method and materials

Samples were collected from the surface type section of the Shajara reservoirs of the Permo-carboniferous shajara formation at latitude 26° 52′ 17.4″ , longitude 43° 36′ 18″. Porosity was measured and permeability was derived from the measured capillary pressure data

The resistivity can be scaled as

S w = [ Ω Ω max ] [3Df] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHtbWdamaaBaaabaqcLbmapeGaaC4Daaqcfa4daeqaa8qa cqGH9aqpcaaMc8+aamWaa8aabaWdbmaalaaapaqaa8qacaWHPoaapa qaa8qacaWHPoWdamaaBaaabaqcLbmapeGaaCyBaiaahggacaWH4baa juaGpaqabaaaaaWdbiaawUfacaGLDbaadaahaaqabeaajugWaiaacU facaaIZaGaeyOeI0IaamiraiaadAgacaGGDbaaaaaa@4D60@ ………… (1)

Where Sw is the water saturation, Ω = resistivity in ohm meter.

 Ω max = maximum resistivity in ohm meter.

 Df = fractal dimension.

Equation 1 can be proofed from

k=[ 1 120 * 1 F 3 * 1 σ 2 ]  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHRbGaeyypa0JaaGPaVpaadmaapaqaa8qadaWcaaWdaeaa peGaaGymaaWdaeaapeGaaGymaiaaikdacaaIWaaaaiaacQcadaWcaa WdaeaapeGaaGymaaWdaeaapeGaaCOra8aadaahaaqabeaajugWa8qa caaIZaaaaaaajuaGcaGGQaWaaSaaa8aabaWdbiaaigdaa8aabaWdbi aaho8apaWaaWbaaeqabaqcLbmapeGaaGOmaaaaaaaajuaGcaGLBbGa ayzxaaGaaiiOaaaa@4B98@ ……….. (2)

Where k = permeability in millidarcy (md).

 1/120 is a constant

 F= formation electrical resistivity factor in zero dimension

 σ = quadrature conductivity in Siemens / meter

But 1 σ 2 = Ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaaGymaaWdaeaapeGaaC4Wd8aadaahaaqa beaajugWa8qacaaIYaaaaaaajuaGcaaMc8Uaeyypa0JaaGPaVlaayk W7caWHPoWdamaaCaaabeqaaKqzadWdbiaaikdaaaaaaa@44BD@ ……….. (3)

 Insert equation 3 into equation 2

k=[ 1 120 * 1 F 3 * Ω 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHRbGaeyypa0JaaGPaVpaadmaapaqaa8qadaWcaaWdaeaa peGaaGymaaWdaeaapeGaaGymaiaaikdacaaIWaaaaiaacQcadaWcaa WdaeaapeGaaGymaaWdaeaapeGaaCOra8aadaahaaqabeaajugWa8qa caaIZaaaaaaajuaGcaGGQaGaaCyQd8aadaahaaqabeaajugWa8qaca aIYaaaaaqcfaOaay5waiaaw2faaaaa@4951@ ………… (4)

If[ 1 120 * 1 F 3 * Ω 2 ]= r 2 8*F =k  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGjbGaaCOzaiaaykW7daWadaWdaeaapeWaaSaaa8aabaWd biaaigdaa8aabaWdbiaaigdacaaIYaGaaGimaaaacaGGQaWaaSaaa8 aabaWdbiaaigdaa8aabaWdbiaahAeapaWaaWbaaeqabaqcLbmapeGa aG4maaaaaaqcfaOaaiOkaiaahM6apaWaaWbaaeqabaqcLbmapeGaaG OmaaaaaKqbakaawUfacaGLDbaacqGH9aqpcaaMc8UaaGPaVpaalaaa paqaa8qacaWHYbWdamaaCaaabeqaaKqzadWdbiaaikdaaaaajuaGpa qaa8qacaaI4aGaaiOkaiaahAeaaaGaaGPaVlabg2da9iaaykW7caaM c8UaaC4Aaiaacckaaaa@5B2E@ ……. (5)

r in equation 5 is the pore throat radius. Equation 5 after rearrange will become

8*F* Ω 2 =[ 120* F 3 * r 2   ]  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaI4aGaaiOkaiaahAeacaGGQaGaaCyQd8aadaahaaqabeaa jugWa8qacaaIYaaaaKqba+aacaaMc8+dbiabg2da9iaaykW7caaMc8 +aamWaa8aabaWdbiaaigdacaaIYaGaaGimaiaacQcacaWHgbWdamaa CaaabeqaaKqzadWdbiaaiodaaaqcfaOaaiOkaiaahkhapaWaaWbaae qabaqcLbmapeGaaGOmaaaajuaGcaGGGcaacaGLBbGaayzxaaGaaiiO aaaa@5269@ ……….. (6)

Equation 6 after simplification will result in

Ω 2 =[ 15* F 2 * r 2 ]  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHPoWdamaaCaaabeqaaKqzadWdbiaaikdaaaqcfaOaeyyp a0JaaGPaVlaaykW7daWadaWdaeaapeGaaGymaiaaiwdacaGGQaGaaC Ora8aadaahaaqabeaajugWa8qacaaIYaaaaKqbakaacQcacaWHYbWd amaaCaaabeqaaKqzadWdbiaaikdaaaaajuaGcaGLBbGaayzxaaGaai iOaaaa@4BF6@ ………… (7)

Take the square root of both sides of Equation 7

Ω 2 = [ 15* F 2 * r 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGcaaWdaeaapeGaaCyQd8aadaahaaqabeaajugWa8qacaaI YaaaaaqcfayabaGaeyypa0JaaGPaVlaaykW7daGcaaWdaeaapeWaam Waa8aabaWdbiaaigdacaaI1aGaaiOkaiaahAeapaWaaWbaaeqabaqc LbmapeGaaGOmaaaajuaGcaGGQaGaaCOCa8aadaahaaqabeaajugWa8 qacaaIYaaaaaqcfaOaay5waiaaw2faaaqabaaaaa@4B30@ …………… (8)

 Equation 8 after simplification will become

Ω= 15 *F*r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHPoGaeyypa0JaaGPaVlaaykW7daGcaaWdaeaapeGaaGym aiaaiwdaaeqaaiaacQcacaWHgbGaaiOkaiaahkhaaaa@40C5@ ……………. (9)

The pore throat radius r can be scaled as

v r 3Df MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG2bGaaGPaVlabg2Hi1kaaykW7caWHYbWdamaaCaaabeqa aKqzadWdbiaaiodacqGHsislcaWHebGaaCOzaaaaaaa@4204@ ……….. (10)

Where v is the cumulative pore volume. differentiate equation 10 with respect to r

dv dr r 2Df MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaaCizaiaahAhaa8aabaWdbiaahsgacaWH YbaaaiaaykW7cqGHDisTcaaMc8UaaGPaVlaahkhapaWaaWbaaeqaba qcLbmapeGaaGOmaiabgkHiTiaahseacaWHMbaaaaaa@46B7@ …………. (11)

Integrate equation 11

dv=constant* rmin r r 2Df *dr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaubiae qabeqaaiaaygW7aeaaqaaaaaaaaaWdbiabgUIiYdaacaWHKbGaaCOD aiaaykW7cqGH9aqpcaaMc8UaaGPaVlaahogacaWHVbGaaCOBaiaaho hacaWH0bGaaCyyaiaah6gacaWH0bGaaiOkamaawahabeWdaeaajugW a8qacaWHYbGaaCyBaiaahMgacaWHUbaajuaGpaqaaKqzadWdbiaahk haaKqba+aabaWdbiabgUIiYdaacaWHYbWdamaaCaaabeqaaKqzadWd biaaikdacqGHsislcaWHebGaaCOzaaaajuaGcaGGQaGaaGPaVlaahs gacaWHYbaaaa@5FAC@ ………… (12)

v= constant 3Df *[ r 3Df r min 3Df ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWH2bGaaGPaVlabg2da9iaaykW7caaMc8+aaSaaa8aabaWd biaahogacaWHVbGaaCOBaiaahohacaWH0bGaaCyyaiaah6gacaWH0b aapaqaa8qacaaIZaGaeyOeI0IaaCiraiaahAgaaaGaaGPaVlaaykW7 caGGQaGaaGPaVlaaykW7daWadaWdaeaapeGaaCOCa8aadaahaaqabe aajugWa8qacaaIZaGaeyOeI0IaaCiraiaahAgaaaqcfaOaeyOeI0Ia aCOCa8aadaWgaaqaaKqzadWdbiaah2gacaWHPbGaaCOBaaqcfa4dae qaamaaCaaabeqaaKqzadWdbiaaiodacqGHsislcaWHebGaaCOzaaaa aKqbakaawUfacaGLDbaaaaa@6420@ ………… (13)

 The total pre volume can be integrated as follows:

vtotal= rmin rmax r 2Df *dr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaubiae qabeqaaiaaygW7aeaaqaaaaaaaaaWdbiabgUIiYlaaykW7aaGaaCOD aiaahshacaWHVbGaaCiDaiaahggacaWHSbGaaGPaVlabg2da9iaayk W7daGfWbqab8aabaqcLbmapeGaaCOCaiaah2gacaWHPbGaaCOBaaqc fa4daeaajugWa8qacaWHYbGaaCyBaiaahggacaWH4baajuaGpaqaa8 qacqGHRiI8aaGaaCOCa8aadaahaaqabeaajugWa8qacaaIYaGaeyOe I0IaaCiraiaahAgaaaqcfaOaaiOkaiaahsgacaWHYbaaaa@5C86@ ………… (14)

The result of total pore volume integral

v total = constant 3Df *[ r max 3Df r min 3Df ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWH2bWdamaaBaaabaqcLbmapeGaaCiDaiaah+gacaWH0bGa aCyyaiaahYgaaKqba+aabeaapeGaeyypa0JaaGPaVlaaykW7daWcaa WdaeaapeGaaC4yaiaah+gacaWHUbGaaC4CaiaahshacaWHHbGaaCOB aiaahshaa8aabaWdbiaaiodacqGHsislcaWHebGaaCOzaaaacaaMc8 UaaiOkaiaaykW7daWadaWdaeaapeGaaCOCa8aadaWgaaqaaKqzadWd biaah2gacaWHHbGaaCiEaaqcfa4daeqaamaaCaaabeqaaKqzadWdbi aaiodacqGHsislcaWHebGaaCOzaaaajuaGcqGHsislcaWHYbWdamaa BaaabaqcLbmapeGaaCyBaiaahMgacaWHUbaajuaGpaqabaWaaWbaae qabaqcLbmapeGaaG4maiabgkHiTiaahseacaWHMbaaaaqcfaOaay5w aiaaw2faaaaa@6B48@ …………. (15)

 Divide equation 13 by equation 15

v v total = [ constant 3Df *[ r 3Df r min 3Df ] ] [ constant 3Df *[ r max 3Df r min 3Df ] ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaaCODaaWdaeaapeGaaCODa8aadaWgaaqa aKqzadWdbiaahshacaWHVbGaaCiDaiaahggacaWHSbaajuaGpaqaba aaa8qacaaMc8UaaGPaVlabg2da9iaaykW7caaMc8+aaSaaa8aabaWd bmaadmaapaqaa8qadaWcaaWdaeaapeGaaC4yaiaah+gacaWHUbGaaC 4CaiaahshacaWHHbGaaCOBaiaahshaa8aabaWdbiaaiodacqGHsisl caWHebGaaCOzaaaacaGGQaWaamWaa8aabaWdbiaahkhapaWaaWbaae qabaqcLbmapeGaaG4maiabgkHiTiaahseacaWHMbaaaKqbakabgkHi TiaahkhapaWaaSbaaeaajugWa8qacaWHTbGaaCyAaiaah6gaaKqba+ aabeaadaahaaqabeaajugWa8qacaaIZaGaeyOeI0IaaCiraiaahAga aaaajuaGcaGLBbGaayzxaaaacaGLBbGaayzxaaaapaqaa8qadaWada WdaeaapeWaaSaaa8aabaWdbiaahogacaWHVbGaaCOBaiaahohacaWH 0bGaaCyyaiaah6gacaWH0baapaqaa8qacaaIZaGaeyOeI0IaaCirai aahAgaaaGaaiOkamaadmaapaqaa8qacaWHYbWdamaaBaaabaqcLbma peGaaCyBaiaahggacaWH4baajuaGpaqabaWaaWbaaeqabaWdbiaaio dacqGHsislcaWHebGaaCOzaaaacqGHsislcaWHYbWdamaaBaaabaqc LbmapeGaaCyBaiaahMgacaWHUbaajuaGpaqabaWaaWbaaeqabaqcLb mapeGaaG4maiabgkHiTiaahseacaWHMbaaaaqcfaOaay5waiaaw2fa aaGaay5waiaaw2faaaaaaaa@8FF2@ ……………. (16)

Equation 16 after simplification will become

Sw= [ r r max ] [ 3Df ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGtbGaaC4DaiaaykW7cqGH9aqpcaaMc8UaaGPaVpaadmaa paqaa8qadaWcaaWdaeaapeGaaCOCaaWdaeaapeGaaCOCa8aadaWgaa qaaKqzadWdbiaah2gacaWHHbGaaCiEaaqcfa4daeqaaaaaa8qacaGL BbGaayzxaaWdamaaCaaabeqaa8qadaWadaWdaeaajugWa8qacaaIZa GaeyOeI0IaaCiraiaahAgaaKqbakaawUfacaGLDbaaaaaaaa@4EE7@ ……………….. (17)

Insert equation 9 into equation 17

Sw= [ ( Ω 15 *F ] [ ( Ω max 15 *F ] ] ] [ 3Df ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHtbGaaC4Daiabg2da9iaaykW7caaMc8+aamWaa8aabaWd bmaalaaapaqaa8qadaqcWaWdaeaapeWaaSaaa8aabaWdbiaahM6aa8 aabaWdbmaakaaapaqaa8qacaaIXaGaaGynaaqabaGaaiOkaiaahAea aaaacaGLOaGaayzxaaaapaqaa8qadaWadaWdaeaapeWaaKama8aaba Wdbmaalaaapaqaa8qacaWHPoWdamaaBaaabaqcLbmapeGaaCyBaiaa hggacaWH4baajuaGpaqabaaabaWdbmaakaaapaqaa8qacaaIXaGaaG ynaaqabaGaaiOkaiaahAeaaaaacaGLOaGaayzxaaaacaGLBbGaayzx aaaaaaGaay5waiaaw2faa8aadaahaaqabeaapeWaamWaa8aabaqcLb mapeGaaG4maiabgkHiTiaahseacaWHMbaajuaGcaGLBbGaayzxaaaa aaaa@5AD2@ ………….. (18)

Equation 18 after simplification will become

Sw= [ Ω Ω max ] [ 3Df ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHtbGaaC4DaiaaykW7cqGH9aqpcaaMc8+aamWaa8aabaWd bmaalaaapaqaa8qacaWHPoaapaqaa8qacaWHPoWdamaaBaaabaqcLb mapeGaaCyBaiaahggacaWH4baajuaGpaqabaaaaaWdbiaawUfacaGL DbaapaWaaWbaaeqabaWcpeWaamWaaKqba+aabaqcLbmapeGaaG4mai abgkHiTiaahseacaWHMbaajuaGcaGLBbGaayzxaaaaaaaa@4E6F@ ……………… (19)

Equation 19 is the proof of equation 1

The geometric relaxation time of induced polarization can be scaled as

Sw= [  IPT g 1.57 1 2   IPTgma x 1.57 1 2 ] [ 3Df ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHtbGaaC4Daiabg2da9iaaykW7daWadaWdaeaapeWaaSaa a8aabaWdbiaacckacaWHjbGaaCiuaiaahsfacaWHNbWdamaaCaaabe qaaKqzadWdbiaaigdacaGGUaGaaGynaiaaiEdal8aadaahaaqcfaya beaal8qadaWcaaqcfa4daeaajugWa8qacaaIXaaajuaGpaqaaKqzad WdbiaaikdaaaaaaaaajuaGcaGGGcaapaqaa8qacaWHjbGaaCiuaiaa hsfacaWHNbGaaCyBaiaahggacaWH4bWdamaaCaaabeqaaKqzadWdbi aaigdacaGGUaGaaGynaiaaiEdal8aadaahaaqcfayabeaal8qadaWc aaqcfa4daeaajugWa8qacaaIXaaajuaGpaqaaKqzadWdbiaaikdaaa aaaaaaaaaajuaGcaGLBbGaayzxaaWdamaaCaaabeqaa8qadaWadaWd aeaajugWa8qacaaIZaGaeyOeI0IaaCiraiaahAgaaKqbakaawUfaca GLDbaaaaaaaa@6715@ …………… (20)

Where Sw = water saturation

 IPTg = geometric relaxation time of induced polarization in milliseconds.

IPTgmax = maximum geometric relaxation time of induced polarization in milliseconds.

Df = fractal dimension.

Equation 20 can be proofed from

k=9.6* ( IPTg* Φ 4 ) 1.57   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHRbGaeyypa0JaaGPaVlaaykW7caaI5aGaaiOlaiaaiAda caGGQaGaaGPaVlaaykW7daqadaWdaeaapeGaaCysaiaahcfacaWHub GaaC4zaiaaykW7caGGQaGaaGPaVlaahA6apaWaaWbaaeqabaqcLbma peGaaGinaaaaaKqbakaawIcacaGLPaaapaWaaWbaaeqabaqcLbmape GaaGymaiaac6cacaaI1aGaaG4naaaajuaGcaGGGcaaaa@548D@ ……………. (21)

 k= permeability in millidarcy

9.6 = constant.

Φ = porosity.

1.57 = constant.

The maximum permeability can be scaled as

kmax=9.6* ( IPTgmax* Φ 4 ) 1.57 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHRbGaaCyBaiaahggacaWH4bGaaGPaVlabg2da9iaaykW7 caaMc8UaaGyoaiaac6cacaaMc8UaaGOnaiaacQcadaqadaWdaeaape GaaCysaiaahcfacaWHubGaaC4zaiaah2gacaWHHbGaaCiEaiaaykW7 caGGQaGaaCOPd8aadaahaaqabeaajugWa8qacaaI0aaaaaqcfaOaay jkaiaawMcaa8aadaahaaqabeaajugWa8qacaaIXaGaaiOlaiaaiwda caaI3aaaaaaa@5712@ …………….. (22)

Divide equation 21 by equation 22

[ k kmax ]= [ 9.6* [ IPTg* Φ 4 ] 1.57 ] [ 9.6* [ IPTgmax* Φ 4 ] 1.57 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWadaWdaeaapeWaaSaaa8aabaWdbiaahUgaa8aabaWdbiaa hUgacaWHTbGaaCyyaiaahIhaaaaacaGLBbGaayzxaaGaaGPaVlaayk W7cqGH9aqpcaaMc8UaaGPaVpaalaaapaqaa8qadaWadaWdaeaapeGa aGyoaiaac6cacaaI2aGaaiOkamaadmaapaqaa8qacaWHjbGaaCiuai aahsfacaWHNbGaaiOkaiaahA6apaWaaWbaaeqabaqcLbmapeGaaGin aaaaaKqbakaawUfacaGLDbaapaWaaWbaaeqabaqcLbmapeGaaGymai aac6cacaaI1aGaaG4naaaaaKqbakaawUfacaGLDbaaa8aabaWdbmaa dmaapaqaa8qacaaI5aGaaiOlaiaaiAdacaGGQaWaamWaa8aabaWdbi aahMeacaWHqbGaaCivaiaahEgacaWHTbGaaCyyaiaahIhacaGGQaGa aCOPd8aadaahaaqabeaajugWa8qacaaI0aaaaaqcfaOaay5waiaaw2 faa8aadaahaaqabeaajugWa8qacaaIXaGaaiOlaiaaiwdacaaI3aaa aaqcfaOaay5waiaaw2faaaaaaaa@7032@ ……………… (23)

Equation 23 after simplification will become

[ k kmax ]= [ IPTg ] 1.57 [ IPTgmax ] 1.57 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWadaWdaeaapeWaaSaaa8aabaWdbiaahUgaa8aabaWdbiaa hUgacaWHTbGaaCyyaiaahIhaaaaacaGLBbGaayzxaaGaaGPaVlaayk W7cqGH9aqpcaaMc8UaaGPaVpaalaaapaqaa8qadaWadaWdaeaapeGa aCysaiaahcfacaWHubGaaC4zaaGaay5waiaaw2faa8aadaahaaqabe aajugWa8qacaaIXaGaaiOlaiaaiwdacaaI3aaaaaqcfa4daeaapeWa amWaa8aabaWdbiaahMeacaWHqbGaaCivaiaahEgacaWHTbGaaCyyai aahIhaaiaawUfacaGLDbaapaWaaWbaaeqabaqcLbmapeGaaGymaiaa c6cacaaI1aGaaG4naaaaaaaaaa@5C86@ …………. (24)

Take the square root of equation 24

[ k kmax ] = [ IPTg ] 1.57 [ IPTgmax ] 1.57 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGcaaWdaeaapeWaamWaa8aabaWdbmaalaaapaqaa8qacaWH Rbaapaqaa8qacaWHRbGaaCyBaiaahggacaWH4baaaaGaay5waiaaw2 faaaqabaGaeyypa0JaaGPaVlaaykW7daGcaaWdaeaapeWaaSaaa8aa baWdbmaadmaapaqaa8qacaWHjbGaaCiuaiaahsfacaWHNbaacaGLBb GaayzxaaWdamaaCaaabeqaaKqzadWdbiaaigdacaGGUaGaaGynaiaa iEdaaaaajuaGpaqaa8qadaWadaWdaeaapeGaaCysaiaahcfacaWHub GaaC4zaiaah2gacaWHHbGaaCiEaaGaay5waiaaw2faa8aadaahaaqa beaajugWa8qacaaIXaGaaiOlaiaaiwdacaaI3aaaaaaaaKqbagqaaa aa@5A5C@ …………….. (25)

Equation 25 can also be written as

[ k 1 2 kma x 1 2 ]=[ IPT g 1.57 1 2 IPTgma x 1.57 1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWadaWdaeaapeWaaSaaa8aabaWdbiaahUgapaWaaWbaaeqa baWcpeWaaSaaaKqba+aabaqcLbmapeGaaGymaaqcfa4daeaajugWa8 qacaaIYaaaaaaaaKqba+aabaWdbiaahUgacaWHTbGaaCyyaiaahIha paWaaWbaaeqabaWcpeWaaSaaaKqba+aabaqcLbmapeGaaGymaaqcfa 4daeaajugWa8qacaaIYaaaaaaaaaaajuaGcaGLBbGaayzxaaGaaGPa VlaaykW7cqGH9aqpcaaMc8UaaGPaVpaadmaapaqaa8qadaWcaaWdae aapeGaaCysaiaahcfacaWHubGaaC4za8aadaahaaqabeaajugWa8qa caaIXaGaaiOlaiaaiwdacaaI3aWcpaWaaWbaaKqbagqabaWcpeWaaS aaaKqba+aabaqcLbmapeGaaGymaaqcfa4daeaajugWa8qacaaIYaaa aaaaaaaajuaGpaqaa8qacaWHjbGaaCiuaiaahsfacaWHNbGaaCyBai aahggacaWH4bWdamaaCaaabeqaaKqzadWdbiaaigdacaGGUaGaaGyn aiaaiEdal8aadaahaaqcfayabeaal8qadaWcaaqcfa4daeaajugWa8 qacaaIXaaajuaGpaqaaKqzadWdbiaaikdaaaaaaaaaaaaajuaGcaGL BbGaayzxaaaaaa@7375@ …………… (26)

Take the Logarithm of equation 26

log[ k 1 2 kma x 1 2 ]=log[ IPT g 1.57 1 2 IPTgma x 1.57 1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaciGGSbGaai4BaiaacEgacaaMc8+aamWaa8aabaWdbmaalaaa paqaa8qacaWHRbWdamaaCaaabeqaaSWdbmaalaaajuaGpaqaaKqzad WdbiaaigdaaKqba+aabaqcLbmapeGaaGOmaaaaaaaajuaGpaqaa8qa caWHRbGaaCyBaiaahggacaWH4bWdamaaCaaabeqaaSWdbmaalaaaju aGpaqaaKqzadWdbiaaigdaaKqba+aabaqcLbmapeGaaGOmaaaaaaaa aaqcfaOaay5waiaaw2faaiabg2da9iaaykW7ciGGSbGaai4BaiaacE gacaaMc8+aamWaa8aabaWdbmaalaaapaqaa8qacaWHjbGaaCiuaiaa hsfacaWHNbWdamaaCaaabeqaaKqzadWdbiaaigdacaGGUaGaaGynai aaiEdal8aadaahaaqcfayabeaal8qadaWcaaqcfa4daeaajugWa8qa caaIXaaajuaGpaqaaKqzadWdbiaaikdaaaaaaaaaaKqba+aabaWdbi aahMeacaWHqbGaaCivaiaahEgacaWHTbGaaCyyaiaahIhapaWaaWba aeqabaqcLbmapeGaaGymaiaac6cacaaI1aGaaG4naSWdamaaCaaaju aGbeqaaSWdbmaalaaajuaGpaqaaKqzadWdbiaaigdaaKqba+aabaqc LbmapeGaaGOmaaaaaaaaaaaaaKqbakaawUfacaGLDbaaaaa@778A@ ……………. (27)

But log[ k 1 2 kma x 1 2 ]=log Sw [ 3Df ]     MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaciGGSbGaai4BaiaacEgacaaMc8+aamWaa8aabaWdbmaalaaa paqaa8qacaWHRbWdamaaCaaabeqaaSWdbmaalaaajuaGpaqaaKqzad WdbiaaigdaaKqba+aabaqcLbmapeGaaGOmaaaaaaaajuaGpaqaa8qa caWHRbGaaCyBaiaahggacaWH4bWdamaaCaaabeqaaSWdbmaalaaaju aGpaqaaKqzadWdbiaaigdaaKqba+aabaqcLbmapeGaaGOmaaaaaaaa aaqcfaOaay5waiaaw2faaiabg2da9iaaykW7ciGGSbGaai4BaiaacE gacaaMc8+aaSaaa8aabaWdbiaahofacaWH3baapaqaa8qadaWadaWd aeaapeGaaG4maiabgkHiTiaahseacaWHMbaacaGLBbGaayzxaaaaai aacckacaGGGcGaaiiOaaaa@604E@ ………………. (28)

Insert equation 28 into equation 27

log Sw 3Df =log[ IPT g 1.57 1 2 IPTgma x 1.57 1 2 ]   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaciGGSbGaai4BaiaacEgapaqbaeqabiqaaaqaa8qadaWcaaWd aeaapeGaaC4uaiaahEhaa8aabaqcLbmapeGaaG4maiabgkHiTiaahs eacaWHMbaaaKqbakabg2da9iaaykW7caaMc8UaciiBaiaac+gacaGG NbGaaGPaVpaadmaapaqaa8qadaWcaaWdaeaapeGaaCysaiaahcfaca WHubGaaC4za8aadaahaaqabeaajugWa8qacaaIXaGaaiOlaiaaiwda caaI3aWcpaWaaWbaaKqbagqabaWcpeWaaSaaaKqba+aabaqcLbmape GaaGymaaqcfa4daeaajugWa8qacaaIYaaaaaaaaaaajuaGpaqaa8qa caWHjbGaaCiuaiaahsfacaWHNbGaaCyBaiaahggacaWH4bWdamaaCa aabeqaaKqzadWdbiaaigdacaGGUaGaaGynaiaaiEdal8aadaahaaqc fayabeaal8qadaWcaaqcfa4daeaajugWa8qacaaIXaaajuaGpaqaaK qzadWdbiaaikdaaaaaaaaaaaaajuaGcaGLBbGaayzxaaaapaqaa8qa caGGGcaaaaaa@6CEF@ ………. (29)

If we remove the Log from equation 29

Sw= [ IPT g 1.57 1 2 IPTgma x 1.57 1 2 ] [ 3Df ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWHtbGaaC4DaiaaykW7cqGH9aqpcaaMc8UaaGPaVpaadmaa paqaa8qadaWcaaWdaeaapeGaaCysaiaahcfacaWHubGaaC4za8aada ahaaqabeaajugWa8qacaaIXaGaaiOlaiaaiwdacaaI3aWcpaWaaWba aKqbagqabaWcpeWaaSaaaKqba+aabaqcLbmapeGaaGymaaqcfa4dae aajugWa8qacaaIYaaaaaaaaaaajuaGpaqaa8qacaWHjbGaaCiuaiaa hsfacaWHNbGaaCyBaiaahggacaWH4bWdamaaCaaabeqaaKqzadWdbi aaigdacaGGUaGaaGynaiaaiEdal8aadaahaaqcfayabeaal8qadaWc aaqcfa4daeaajugWa8qacaaIXaaajuaGpaqaaKqzadWdbiaaikdaaa aaaaaaaaaajuaGcaGLBbGaayzxaaWdamaaCaaabeqaa8qadaWadaWd aeaajugWa8qacaaIZaGaeyOeI0IaaCiraiaahAgaaKqbakaawUfaca GLDbaaaaaaaa@67E4@ ……………. (30)

Equation 30 the proof of equation 20 which relates the water saturation, the geometric relaxation time of induced polarization, the maximum geometric relaxation time of induced polarization, and the fractal dimension

Results and discussions

Petrophysical data characterizing Shajara reservoirs of the permo-Carboniferous Shajara Formation were presented in Table 1. These sandstone reservoirs were divided into three bodies, from bottom to top are: lower, middle, and upper shajara reservoir. Concerning the lower shajara reservoir, it is domenstrated by four sandstone samples named as SJ1, SJ2, SJ3, and SJ4. Their results of resistivity and geometric relaxation time fractal dimensions were displayed in Table 1. Sample SJ1 with a porosity value of about 29% and permeability equal to 1680 millidarcy, whose resistivity and geometric relaxation time fractal dimensions was found to be 2.7859 as revealed in table 1. Sample SJ2 is defined by 35% porosity and permeability around 1955 millidarcy. Its resistivity and geometric relaxation time fractal dimensions of induced polarization equal 2.7748. As we progress from sample SJ2 to sample SJ3 an extreme reduction in permeability was encountered from 1955 millidarcy to 56 millidary which accounts for decrease in fractal dimension from 2.7848 to 2.4379 as explained in Table 1. Such drastic chane in permeability and fractal dimension can account for heterogeneity which is an importment parameter in reservoir qulaity assessment. Again an increase in permeability from 56 millidary to 176 millidarcy was reported as we proceed from sample SJ3 to SJ4 as delineated in Table 1. Such increase in permeability gives rise to fractal dimension from 2.4379 to 2.6843 as presented in Table 1.

However the middle Shajara Reservoir is designated by three sample, so called SJ7, SJ8, and SJ9 as shown in Table 1. Their poroperm data were presented in table 1. Their resistivity and geometric relaxation time fractal dimensions were higher than samples SJ3 and SJ4 due to an increase in their permeabilities as displayed in table 1.

The upper shajara reservoir is illustrated by three samples labeled as SJ11, SJ12, and SJ13 as described in Table 1. Their resistivity and geometric relaxation time fractal dimension values are also higher than samples SJ3 and SJ4 owing to an increase in their flow capacity (permeability) as demonstrated in Table 1. Overall aplot of resistivity fractal dimensions versus geometric relaxation time fractal dimension fractal dimensions of induced polarization (Figure 1) delineates three zones of varing petrophysical characteristics. Such discrepancy in fractal dimension can account for heterogeneity whic is a key parameter in reservoir quality assessment.

Figure 1 Resistivity fractal dimension versus geometric relaxation time fractal dimension of induced polarization.

Formation

Reservoirs

Samples

porosity

Permeability

Resistivity fractal dimension

Geometric relaxation time fractal dimension

Permo-Carboniferous
Shajara Formation

Upper Shajara Reservoir

SJ13

25

973

2.7872

2.7872

SJ12

28

1440

2.7859

2.7859

SJ11

36

1197

2.7586

2.7586

Middle Shajara Reservoir

SJ9

31

1394

2.7786

2.7786

SJ8

32

1344

2.7752

2.7752

SJ7

35

1472

2.7683

2.7683

Lower Shajara Reservoir

SJ4

30

176

2.6843

2.6843

SJ3

34

56

2.4379

2.4379

SJ2

35

1955

2.7748

2.7748

SJ1

29

1680

2.7859

2.7859

Table 1 Petrophysical model showing the the thee Shajara Reservoirs of the Permo-Carboniferous shajara Formation with their corresponding values of resistivity and geometric relaxation time fractal dimensions of induced polarization

Conclusion

  1. The sandstones of the Shajara reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three bodies based on resistivity fractal dimension.
  2. Theses reservoir bodies were also confirmed by geometric relaxation time fractal dimension of induced polarization.
  3. The heterogeneity of the Shajara reservoirs increases with an increase of fractal dimension, permeability, giving rise to an increase in pore size distribution.

Acknowledgements

The author would like to thank college of Engineering, King Saud University, Department of Petroleum and Natural Gas Engineering, Department of Chemical Engineeing, Research Centre at college of Engineeing, King Abdullal institute for research and Consulting Studies for their supports

Conflict of interest

The author declares that there is no conflict of interest.

References

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