Changes

|competition=June 2015

|competition=June 2015

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Fracture is a nonconstanly rock movement that usually seen as a separation in rock feature locally made and has natural sequence. Most of geological formations on top of earth crust crack at some parts of it. The fracture represents mechanical failure of rock stability because of geological pressure such as plate tectonics, lithostatic pressure exchange, thermal pressure, high pressure fluids, drilling activity, and even fluids movement, because fluids has a role as burden in the overburden rock. Even though, petroleum reservoar rocks can be found in most of depth, in deeper depth, the overburden pressure is enough to deform sedimentary rock plastically. It cannot hold on from shear pressure for a long period and flows until balanced state.

[[Fracture]] is a non constant rock movement that usually seen as a separation in rock feature locally made and has natural sequence. Most of geological formations on top of earth [[crust]] crack at some parts of it. The fracture represents mechanical failure of rock stability because of geological pressure such as plate tectonics, lithostatic pressure exchange, thermal pressure, high pressure fluids, drilling activity, and even fluids movement, because fluids has a role as burden in the overburden rock. Even though, petroleum reservoar rocks can be found in most of depth, in deeper depth, the overburden pressure is enough to deform sedimentary rock plastically. It cannot hold on from shear pressure for a long period and flows until balanced state.

Fracture can appear as a long micro rock separation in micrometer or continental fault until thousands of kilometers. It can be limited by a rock formation or a layer of rock, a fracture is a nonconstanly straight or curve line because brittle deformation process in earth crust. The plane is a weak part of rock’s feature from pressure exchange on earth crust because of fracture from one or more different ways, depends on pressure direction and type of rock. A fracture contain 2 uncommon rock’s surface, and contact each other. Volume between surfaces is called by fracture gap.

Fracture can appear as a long micro rock separation in micrometer or continental fault until thousands of kilometers. It can be limited by a rock formation or a layer of rock, a fracture is a nonconstanly straight or curve line because brittle [[deformation]] process in earth crust. The plane is a weak part of rock’s feature from pressure exchange on earth crust because of fracture from one or more different ways, depends on pressure direction and type of rock. A fracture contain 2 uncommon rock’s surface, and contact each other. Volume between surfaces is called by fracture gap.

Naturally, rock is fractured can be categorized into 3 type geologically, based on porosity system :

Naturally, rock is fractured can be categorized into 3 type geologically, based on porosity system :

==Definition of double porosity==

==Definition of double porosity==

On fractured reservoir, total porosity is the sum of primary porosity and secondary porosity

On fractured reservoir, total porosity is the sum of primary porosity and secondary porosity

:<math>Equation</math>

:<math>\phi_t = \phi_1 + \phi_2</math>

the value of porosity is equal with static definition of rock capacity or total void volume of rock.

the value of porosity is equal with static definition of rock capacity or total void volume of rock.

From experiment, expert has found that with various rock types, fractured porosity has a smaller value than matrix porosity. Both porosities are defined with common definition, and relatively to total volume of fracture and matrix.

From experiment, expert has found that with various rock types, fractured porosity has a smaller value than matrix porosity. Both porosities are defined with common definition, and relatively to total volume of fracture and matrix.

φ₁ = void volume of matrix or total bulk volume  

 

φ₂ = void volume of fracture or total bulk volume

φ₁= void volume of matrix or total bulk volume  

φ₂=void volume of fracture or total bulk volume

Correlation between matrix porosity (φm) and fracture porosity (φf), is matrix porosity that refers to total matrix in rock, can be formulated as :

Correlation between matrix porosity (φ_m) and fracture porosity (φ_f), is matrix porosity that refers to total matrix in rock, can be formulated as :

:<math>Equation</math>

:<math>\phi_m = \frac{\text{the volume of empty space of the matrix}}{\text{the overall volume of the rock matrix}}</math>

while fracture porosity is

while fracture porosity is

:<math>Equation</math>

:<math>\phi_2 \approx \phi_f</math>

in this case, primary porosity is functionalized as matrix porosity, with this formula,

in this case, primary porosity is functionalized as matrix porosity, with this formula,

:<math>Equation</math>

:<math>\phi_1 = (1 - \phi_2) \phi_m</math>

and effective primary porosity that contain oil, define with formula,

and effective primary porosity that contain oil, define with formula,

:<math>Equation</math>

:<math>\phi_{\text{eff}} = (1 - \phi_2) \phi_m(1 - S_{\text{wi}})</math>

porosity is illustrated schematically in image above, whereas unit of total rock is scaled on top and total of matrix in rock is scaled in bottom. With matrix porosity (φ_m), water saturated part, oil saturated part, and each parts are expressed as percentage of all matrix units in rock.

porosity is illustrated schematically in image above, whereas unit of total rock is scaled on top and total of matrix in rock is scaled in bottom. With matrix porosity (φ_m), water saturated part, oil saturated part, and each parts are expressed as percentage of all matrix units in rock.

==Fractured reservoir permeability==

==Fractured reservoir permeability==

The existence of two systems (matrix and fracture), permeability can be defined as matrix permeability, fracture permeability and system permeability between fracture and matrix.

The existence of two systems (matrix and [[fracture]]), permeability can be defined as matrix permeability, fracture permeability and system permeability between fracture and matrix.

Primary definition of permeability is fracture permeability can be interpretated as single fracture permeability or fracture system permeability or sometime fracture permeability of total fracture volume of rock. So permeability will be defined more detail.

Primary definition of permeability is fracture permeability can be interpretated as single fracture permeability or fracture system permeability or sometime fracture permeability of total fracture volume of rock. So permeability will be defined more detail.

Evaluation of relative permeability has generate formula.

Evaluation of relative permeability has generate formula.

Capillary pressure, desaturation curve is expected with formula,

[[Capillary pressure]], desaturation curve is expected with formula,

:<math>Equation</math>

:<math>\frac{1}{P_c^2} = \text{B S*oe}</math>

Where B is constanta and S*oe is effective saturation in oil (percentage of volume in effective porous for run off the fluids). Relative permeability is expressed to function of saturation,

Where B is constant and S*oe is effective saturation in oil (percentage of volume in effective porous for run off the fluids). Relative permeability is expressed to function of saturation,

:<math>Equation</math>

:<math>K_{ro} = \text{S*}_o^4</math>

:<math>Equation</math>

:<math>K_{rg} = (1 - \text{S*}_o)^2 [1 - (\text{S*}_o)^2]</math>

:<math>Equation</math>

:<math>\text{S*}_o = \frac{\text{S*}_o - \text{S*}_{or}}{1 - \text{S*}_{or}}</math>

In carbonaceous rock whereas secondary porosity vuggy tipe is developed as a extra of intergranular porosity, conventional calculation is asked if the secondary porosity is significant and distributed.

In carbonaceous rock whereas secondary porosity vuggy tipe is developed as a extra of intergranular porosity, conventional calculation is asked if the secondary porosity is significant and distributed.

Velocity, in pressure with low reduction ΔP/Δt very important for replace relative permeability curve in porous limestone with Kg/Ko value specifically.  

Velocity, in pressure with low reduction ΔP/Δt very important for replace relative permeability curve in porous limestone with Kg/Ko value specifically.  

Relative permeability from fracture and porous generate secondary porosity of limestone that has been tested with simplified model which has been developed by Erlich, 1971. It contains isolated porous in matrix, and connected porous with one or two direction cracks. Porous and fracture that connected is expressed with probability f, that indicates degree between porous connection with fracture. Value variation f from 0 – 1 is ilustrated in image below. The example of relation between fracture and porous, shows comparative size in matrix with probability fracture factor f = 0,5. Residual saturation of oil has been evaluated by Erlich model with dimension of 12 x 12 with random process based on probability realtion between porous and fracture. In this case, there are 312 fracture location possibilities. If some of the fracture filled with oil and water, transmigration process can be simulated. For water and oil transmigration, residual saturation of oil is used as parameter function f. Product of this simulation has been compard with core data and adjusment, shows no execption in core with huge secondary porosity, where transmigration cannot be mixed and value of residual saturation is producted separately. This model on fracture-matrix part known as pseudo-function so it is possible to make a better closure to transmigration evaluation.

Relative permeability from [[fracture]] and porous generate secondary porosity of limestone that has been tested with simplified model which has been developed by Erlich, 1971. It contains isolated porous in matrix, and connected porous with one or two direction cracks. Porous and fracture that connected is expressed with probability f, that indicates degree between porous connection with fracture. Value variation f from 0 – 1 is ilustrated in image below. The example of relation between fracture and porous, shows comparative size in matrix with probability fracture factor f = 0,5. Residual saturation of oil has been evaluated by Erlich model with dimension of 12 x 12 with random process based on probability realtion between porous and fracture. In this case, there are 312 fracture location possibilities. If some of the fracture filled with oil and water, transmigration process can be simulated. For water and oil transmigration, residual saturation of oil is used as parameter function f. Product of this simulation has been compard with core data and adjusment, shows no execption in core with huge secondary porosity, where transmigration cannot be mixed and value of residual saturation is producted separately. This model on fracture-matrix part known as pseudo-function so it is possible to make a better closure to transmigration evaluation.

Relative permeability in fracture-matrix system or fracture-crack-matrix system can be tested with core in laboratory with conventional method. This procedure more successfu as primary direction determinant from visible fracture. This closure can be done with using unconformity rock method so can make pseudo- relative permeability curve.

Relative permeability in fracture-matrix system or fracture-crack-matrix system can be tested with core in laboratory with conventional method. This procedure more successfu as primary direction determinant from visible fracture. This closure can be done with using unconformity rock method so can make pseudo- relative permeability curve.

* Shape of 2 relative permeability curve K_ro, K_rw can be various depends on water fracture saturation function (S_w2).

* Shape of 2 relative permeability curve K_ro, K_rw can be various depends on water fracture saturation function (S_w2).

* Connection between two fracture:

* Connection between two fracture:

:<math>Equation</math>

:<math>K_{ro} = \left [\frac{K_2}{K} + \left (1 - \frac{K_2}{K} \right ) (1 - S_{w1}^2) (1 - S_{w1})^2 \right] (1 - S_{w2})^2 (1 - S_{w2}^2)</math>

:<math>Equation</math>

:<math>K_{rw} = \left [ \frac{K_2}{K} + \left ( 1 - \frac{K_2}{K} \right ) S_{w2}^4 \right ] S_{w2}^4</math>

Which is similar to Corey’s calculation.

Which is similar to Corey’s calculation.

Flow condition is illustrated in image below, where trickle flow (first image) may the bazaar phase from bubble to follow one wall; bubble flow (second image) where bubble flow both fractured walls so there is bouyancy force; string flow (third image) fluids that appear constantly and wet phase follows fractured wall.

Flow condition is illustrated in image below, where trickle flow (first image) may the bazaar phase from bubble to follow one wall; bubble flow (second image) where bubble flow both fractured walls so there is bouyancy force; string flow (third image) fluids that appear constantly and wet phase follows fractured wall.

Physical parameter of fluids and rock that effects flow in the fractured media are density, viscosity, surface tension, bazaar properties, fracture length, fractured wall roughness, and tilt degree. First thing that must be known is 2 phases flow in relative permeability which can be seen in simplified model Romm, 1966., that produce 10 until 20 paralel fracture. Saturation is counted with electric resistance value and evaluated permeability of velocity flow of fluids, water, and kerogen. The result will show linear tred of diagram Kf vs saturation. It also determine the limit value of 2 phase from unreducted fracture flow become a result of a fracture. Relation between fractures can be modified each flow characteristics perfectly.

Physical parameter of fluids and rock that effects flow in the fractured media are density, viscosity, surface tension, bazaar properties, fracture length, fractured wall roughness, and tilt degree. First thing that must be known is 2 phases flow in relative permeability which can be seen in simplified model Romm, 1966., that produce 10 until 20 paralel fracture. Saturation is counted with electric resistance value and evaluated permeability of velocity flow of fluids, water, and [[kerogen]]. The result will show linear tred of diagram Kf vs saturation. It also determine the limit value of 2 phase from unreducted fracture flow become a result of a fracture. Relation between fractures can be modified each flow characteristics perfectly.

Other experiments conducted by du Prey, 1973, which resulted in two models where fluid is injected from one and more holes. Relative permeability derived from the value of Q_o and Q_w through the equation:

Other experiments conducted by du Prey, 1973, which resulted in two models where fluid is injected from one and more holes. Relative permeability derived from the value of Q_o and Q_w through the equation:

:<math>Equation</math>

:<math>K_{ro} = \frac{12 Q_o \mu_o L_f}{a b^3 \Delta P_o}</math>

:<math>Equation</math>

:<math>K_{rw} = \frac{12 Q_w \mu_w L_f}{a b^3 \Delta P_w}</math>

Because of the pressure ΔP_o and ΔP_w can’t be measured if the fluid is moving, so that the problem should be avoided by maintaining a phase that doesn’t move and forget about the possibility of them, moving as the effects of segregation fluid. In this case, instead of a decrease in pressure, gravity  

Because of the pressure ΔP_o and ΔP_w can’t be measured if the fluid is moving, so that the problem should be avoided by maintaining a phase that doesn’t move and forget about the possibility of them, moving as the effects of segregation fluid. In this case, instead of a decrease in pressure, [[gravity ]]

<math>Equation</math>  

:<math>\Delta \gamma \times g \times \sin{\alpha}</math>  

plays a role. Saturation is given in varying conditions of flow by counting the length of time or the time of trial.

plays a role. Saturation is given in varying conditions of flow by counting the length of time or the time of trial.

Permeability is more influenced by the dimension of fracture than the matrix porosity or total porosity. The curve which shows equations to block up and down as follows :

Permeability is more influenced by the dimension of fracture than the matrix porosity or total porosity. The curve which shows equations to block up and down as follows :

# Block up

# Block up

#*<math>Equation</math>

#*<math>dLU = \frac{1}{4.2 \times 10^{-2} FII^2 - 2.43 \times 10^{-5}} </math>

# Block down

# Block down

#*<math>Equation</math>

#*<math>dLD = \frac{1}{9.44 \times 10^{-3}exp(FII) - 9.3 \times 10^{-3}}</math>

Where:

Where:

:dLU = lateral distance to the fault plane for block up, feet  

:dLU = [[lateral]] distance to the fault plane for block up, feet  

:dLD = lateral distance to the fault plane for block down, feet  

:dLD = lateral distance to the fault plane for block down, feet  

:FII = fracture intensity index, fraction

:FII = fracture intensity index, fraction

In-situ value of fracture intensity index estimated from:

In-situ value of fracture intensity index estimated from:

:<math>Equation</math>

:<math>FII = \frac{\frac{1}{R_{xo}} - \frac{1}{R_t}}{\frac{1}{R_{mf}} - \frac{1}{R_w}}</math>

The equation for blocks up can be used for distances of 250 to 5000 feet and FII value range is 7 to 25%, and equations for blocks down can be used for distances of 250 to 1250 feet and FII value range is 7 to 25%. That correlations were developed from the well data which obtained in Pirson near to Luling-Mexia fault in Austin. The main function of both these correlations is at the time of exploration and discovered the presence of fault of seismic data where the data obtained is only the distance from the fault. It is important to know the influence of FFI of several factors, including the number and geometry of fractures and not all of fractures are the result of fault.

The equation for blocks up can be used for distances of 250 to 5000 feet and FII value range is 7 to 25%, and equations for blocks down can be used for distances of 250 to 1250 feet and FII value range is 7 to 25%. That correlations were developed from the well data which obtained in Pirson near to Luling-Mexia fault in Austin. The main function of both these correlations is at the time of exploration and discovered the presence of fault of seismic data where the data obtained is only the distance from the fault. It is important to know the influence of FFI of several factors, including the number and geometry of fractures and not all of fractures are the result of fault.

The following equation can be used to estimate the fracture area and fracture permeability in the reservoir which spreader:

The following equation can be used to estimate the fracture area and fracture permeability in the reservoir which spreader:

:<math>Equation</math>

:<math>w_f = \frac{0.064}{\phi_t} [(1 - S_{iw})\text{FII}]^{1.315}</math>

:<math>Equation</math>

:<math>k_f = 1.5 \times 10^7 \phi_t [(1 - S_{wi})\text{FII}]^{2.63}</math>

Where porosity, FII and water saturation were not reduced be expressed as fractions and fracture width and permeability in cm and mD, respectively. Fracture porosity can be directly estimated using the formula:

Where porosity, FII and water saturation were not reduced be expressed as fractions and fracture width and permeability in cm and mD, respectively. Fracture porosity can be directly estimated using the formula:

:<math>Equation</math>

:<math>\phi_f = \left [ R_{mf} \left ( \frac{1}{R_{LLS}} - \frac{1}{R_{LLD}} \right ) \right ]^2</math>

Where the range of coefficient C_T between 2/3 and 3/4. R_mf, R_LLS and R_LLD is mudfiltrate, laterolog shallow and laterolog deep resistivity in ohm-m, R_LLS dan R_LLD equivalent to R_xodan R_t,

Where the range of coefficient C_T between 2/3 and 3/4. R_mf, R_LLS and R_LLD is mudfiltrate, laterolog shallow and laterolog deep resistivity in ohm-m, R_LLS dan R_LLD equivalent to R_xodan R_t,

===Solution A===

===Solution A===

By using the existing formula, fracture intensity index is:

By using the existing formula, [[fracture]] intensity index is:

:<math>Equation</math>

:<math>\text{FII} = \frac{ \frac{1}{R_{xo}} - \frac{1}{R_t}}{ \frac{1}{R_{mf}} - \frac{1}{R_w}} = \frac{ \frac{1}{9.7} - \frac{1}{95}}{ \frac{1}{0.17} - \frac{1}{0.19}} = 0.15</math>

The distance to the closest fault estimated from the correlation of the blocks up and can be estimated using the equation:

The distance to the closest fault estimated from the correlation of the blocks up and can be estimated using the equation:

:<math>Equation</math>

:<math>\text{d}_{\text{LU}} = \frac{1}{4.2 \times 10^{-2} \text{FII}^2 - 2.43 \times 10^{-5}}</math>

:<math>Equation</math>

:<math>\text{d}_{\text{LU}} = \frac{1}{4.2 \times 10^{-2} (0.15)^2 - 2.43 \times 10^{-5}} \approx 1,100 \text{ ft}</math>

The distance to the fault can be estimated using the formula above, so that the value of FII is 15%, and its distance estimated 1100 feet.

The distance to the fault can be estimated using the formula above, so that the value of FII is 15%, and its distance estimated 1100 feet.

# Known R_LLS = R_xo = 12 and R_LLD= R_t = 85,  

# Known R_LLS = R_xo = 12 and R_LLD= R_t = 85,  

#* fracture intensity index using the formula:

#* fracture intensity index using the formula:

#** <math>Equation</math>

#** <math>\text{FII} = \frac{ \frac{1}{R_{xo}} - \frac{1}{R_t}}{ \frac{1}{R_{mf}} - \frac{1}{R_w}} = \frac{ \frac{1}{12} - \frac{1}{85}}{ \frac {1}{0.165} - \frac{1}{0.18}} = 0.1417</math>

#* Saturation of mud filtrate in the washing zone are:

#* Saturation of mud filtrate in the washing zone are:

#** <math>Equation</math>

#** <math>\text{S}_{\text{xo}} = \text{S}_\text{w}^{\text{Cx}} = 0.24^{0.25} = 0.70</math>

#* Porosity coefficient using the formula :

#* Porosity coefficient using the formula:

#** <math>Equation</math>

#** <math>v = \frac{R_w}{\phi_t (S_w - S_{xo})} \left ( \frac{1}{R_t} - \frac{1}{R_{xo}} \right ) = \frac{0.18}{0.17 (0.24 - 0.70)} \left ( \frac{1}{85} - \frac{1}{12} \right ) = 0.165</math>

#** <math>Equation</math>

# Using the Formula:

# Using the Formula:

#* <math>Equation</math>

#* <math>w_f = \frac{0.064}{\phi_t} [(1 - S_{wi}) \text{FII}]^{1.315} =  \frac{0.064}{0.17}[(1 - 0.24)(0.1417)]^{1.315} = 0.02 \text{ cm}</math>

#* <math>Equation</math>

# Using the Formula:

# Using the Formula:

#* <math>Equation</math>

#* <math>k_f = 1.5 \times 10^7 \phi_t [(1 - S_{wi}) \text{FII}]^{2.63} = 1.5 \times 10^7 (0.17) [(1 - 0.24) 0.1417]^{2.63} = 7.265 \text{ mD}</math>

#* <math>Equation</math>

#*# If C_T = 3/4, then fracture porosity is:

#*# If C_T = 3/4, then fracture porosity is:

#*#* <math>Equation</math>

#*#* <math>\phi_f = \left [ R_{mf} \left ( \frac{1}{R_{ \text{LLS}}} - \frac{1}{R_{ \text{LLD}}} \right ) \right ]^{C_T} = \left [0.165 \left ( \frac{1}{12} - \frac{1}{85} \right ) \right ]^{ \frac{3}{4}} = 0.0358</math>

#*# If C_T = 2.3, fracture porosity is 0,052, so the value of φf is between 0,036 and 0,052.

#*# If C_T = 2.3, fracture porosity is 0.052, so the value of φf is between 0.036 and 0.052.

#*#* Matrix Porosity is:

#*#* Matrix Porosity is:

#*#* <math>Equation</math>

#*#* <math>\phi_m = \phi_t(1 - \nu) = 0.17(1 - 0.165) = 0.142</math>

Note that total of φ_f (for C_T=3/4) and φ_m is 0.177 so the estimated is equal to total porosity obtained from well logs. So the fracture porosity of this reservoir is 3.6%.

Note that total of φ_f (for C_T=3/4) and φ_m is 0.177 so the estimated is equal to total porosity obtained from well logs. So the fracture porosity of this reservoir is 3.6%.

==Relationship Between Porosity and Permeability in Dual Porosity System==

==Relationship Between Porosity and Permeability in Dual Porosity System==

[[File:UGM_Porosity_Fig_9.png|thumb|500px|{{figure number|9}}]]

[[File:UGM_Porosity_Fig_9.png|thumb|500px|{{figure number|9}}Model of the two dominant unit that has a pore radius. The system shows the different petrophysical properties such as porosity and permeability. (Donaldson, 2004)]]

Petroleum reservoir can be divided into three main classes based on porosity systems:

Petroleum reservoir can be divided into three main classes based on porosity systems:

The total flow through the system are among the total of value stream passing of each system, the system has a different petrophysical properties such as porosity and permeability.

The total flow through the system are among the total of value stream passing of each system, the system has a different petrophysical properties such as porosity and permeability.

:<math>Equation</math>

:<math>Q_t = q_1 + q_2</math>

Using Darcy's law (for q_t) and Poiseuille (for q₁ and q₂), obtained :

Using Darcy's law (for q_t) and Poiseuille (for q₁ and q₂), obtained :

:<math>Equation</math>

:<math>kA_t \frac{\Delta P}{\mu L} = \left [ \frac{n_1 \pi r_{c1}^4}{8} + \frac{n_2 \pi r_{c2}^4}{8} \right ] \frac{\Delta P}{\mu L}</math>

The total area for the system are:

The total area for the system are:

:<math>Equation</math>

:<math>A_t = \frac{n_1 \pi r_{c1}^2}{\phi_1} + \frac{n_2 \pi r_{c2}^2}{\phi_2}</math>

And known also:

And known also:

:<math>Equation</math>

:<math>r_c = \frac{2}{S_{vp}}</math>

By using the substitution equation, obtained:

By using the substitution equation, obtained:

:<math>Equation</math>

:<math>k = \frac{1}{2} \frac{ \left [ \frac{1}{S_{vp1}^4} + \frac{1}{S_{vp2}^4} \right ] }{ \left [ \frac{1}{\phi_1 S_{vp1}^2} + \frac{1}{\phi_2 S_{2p2}^4} \right ] }</math>

The general equation of the formula is :

The general equation of the formula is :

:<math>Equation</math>

:<math>k = \frac{1}{2} \frac{\displaystyle \sum_{i=1}^n \left ( \frac{1}{S_{vpi}^4} \right )}{\displaystyle \sum_{i=1}^n \left ( \frac{1}{\phi_i S_{vpi}^2} \right ) }</math>

For single porosity systems, obtained equation has been reduced :

For single porosity systems, obtained equation has been reduced :

:<math>Equation</math>

:<math>k = \frac{1}{2}\frac{ \left ( \frac{1}{S_{vp}^4} \right ) }{ \left ( \frac{1}{\phi S_{vp}^2} \right ) } = \frac{\phi}{2S_{vp}^2}</math>

Constanta above associated with the formation of capillaries can be replaced with K_T:

Constanta above associated with the formation of capillaries can be replaced with K_T:

:<math>Equation</math>

:<math>k = \frac{\displaystyle \sum_{i=1}^n \left ( \frac{1}{S_{vpi}^4} \right )}{\displaystyle \sum_{i=1}^n \left ( \frac{K_{Ti}}{\phi_i S_{vpi}^2} \right )}</math>

Where

Where

:<math>Equation</math>

:<math>K_T = 6f_{sp} \tau </math>

In the case of formations that contain little number of holes per unit pore volume, as in a reservoir with high storage capacity in the rock matrix, and the storage capacity is very low in the hole, n1>>n2, the equation can be stated as follows:

In the case of formations that contain little number of holes per unit pore volume, as in a reservoir with high storage capacity in the rock matrix, and the storage capacity is very low in the hole, n₁>>n₂, the equation can be stated as follows:

:<math>Equation</math>

:<math>k = \frac{\phi_1}{2S_{pv1}^2} = \frac{\phi_1 r_{c1}^2}{8}</math>

Where subscript 1 for the primary pore space that accommodates almost all fluids. In the case of n2 >> n1, for example, on rocks with fluid accommodated in secondary porosity such as cracks and holes. The above equation would be:

Where subscript 1 for the primary pore space that accommodates almost all fluids. In the case of n2 >> n1, for example, on rocks with fluid accommodated in secondary porosity such as cracks and holes. The above equation would be:

:<math>Equation</math>

:<math>k = \frac{\phi_2}{2S_{pv2}^2} = \frac{\phi2 r_{c2}^2}{8}</math>

Where subscript 2 for secondary pore space. So in the case of n1>>n2 and n2>>n1, dual porosity system can be estimated from a single pore system. n1 approximately equal to n2, because it can not determine the value of n1 and n2, so it can be approached to take the geometric mean of the two capillary system, for example:

Where subscript 2 for secondary pore space. So in the case of n1>>n2 and n2>>n1, dual porosity system can be estimated from a single pore system. n1 approximately equal to n2, because it can not determine the value of n1 and n2, so it can be approached to take the geometric mean of the two capillary system, for example:

:<math>Equation</math>

:<math>k = \sqrt{\left ( \frac{\phi_1 r_{c1}^2}{8} \right ) \left ( \frac{\phi_2 r_{c2}^2}{8} \right )} = \frac{r_{c1} r_{c2}}{8} \sqrt{\phi_1 \phi_2}</math>

Using the average value of rc1 and rc2, and the average value of φ1 and φ2, the equation similar to the Kozeny equation.

Using the average value of rc1 and rc2, and the average value of φ1 and φ2, the equation similar to the Kozeny equation.

==Relationship Between Porosity and Permeability in Fractured Basement Reservoir==

==Relationship Between Porosity and Permeability in Fractured Basement Reservoir==

<gallery mode=packed heights=300px widths=300px>

<gallery mode=packed heights=300px widths=300px>

File:UGM_Porosity_Fig_10.png|{{figure number|10}}

File:UGM_Porosity_Fig_10.png|{{figure number|10}}Units model used to calculate fracture permeability in reservoir fracture. (Donaldson, 2004)

File:UGM_Porosity_Fig_11.png|{{figure number|11}}

File:UGM_Porosity_Fig_11.png|{{figure number|11}}Fracture permeability as a function of the width and spacing of fractures. (Nelson, 2001)

File:UGM_Porosity_Fig_12.png|{{figure number|12}}

File:UGM_Porosity_Fig_12.png|{{figure number|12}}fracture porosity as a function of the width and spacing of fractures. (Nelson, 2001)

File:UGM_Porosity_Fig_13.png|{{figure number|13}}

File:UGM_Porosity_Fig_13.png|{{figure number|13}}Porosity versus permeability shown relative position of matrix curve and fracture. (Nelson, 2001)

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Fracture provides a capacity to store and permeability and fluid flow is controlled by the properties of the fracture. The equation for velocity of volumetric flow, combined with Darcy's law, creating basic approaches to estimate the fracture permeability.

Fracture provides a capacity to store and permeability and fluid flow is controlled by the properties of the fracture. The equation for velocity of volumetric flow, combined with Darcy's law, creating basic approaches to estimate the fracture permeability.

Determined a block of rock that cracking naturally with fractures, as shown in [[:File:UGM_Porosity_Fig_10.png|Figure 10]].

Determined a block of rock that [[cracking]] naturally with fractures, as shown in [[:File:UGM_Porosity_Fig_10.png|Figure 10]].

Assumed fracture is a box, smooth and doesn’t contain any mineral, Hagen- Poiseiulle equation is:

Assumed fracture is a box, smooth and doesn’t contain any mineral, Hagen- Poiseiulle equation is:

:<math>Equation</math>

:<math>q = \frac{nh_f w_f^3}{12} \frac{\Delta P}{\mu L}</math>

Darcy's law is:

Darcy's law is:

:<math>Equation</math>

:<math>q = kA \frac{\Delta P}{\mu L}</math>

The equation obtained two calculations and troublesolving of permeability results:

The equation obtained two calculations and troublesolving of permeability results:

:<math>Equation</math>

:<math>k = \frac{nh_f w_f^3}{12A}</math>

Difficulties were obtained using the above equation is the number of fractures, fracture height, and width of fracture should be sought. Thus defined as:

Difficulties were obtained using the above equation is the number of fractures, fracture height, and width of fracture should be sought. Thus defined as:

:<math>Equation</math>

:<math>\phi = \frac{V_p}{V_b} = \frac{nh_f w_f L}{AL}</math>

And

And

:<math>Equation</math>

:<math>A = \frac{nh_f w_f}{\phi}</math>

So it can be substituted:

So it can be substituted:

:<math>Equation</math>

:<math>k = \frac{\phi w_f^2}{12}</math>

Previous equation k = φr2/8, where the capillary radius r and constant 8 exchanged withfracturewidthwf and12.Theequationisusuallyusedtocalculatefracture permeability. Fracture porosity in percent and fracture width in micrometers (m), so the equation becomes:

Previous equation  

:<math>Equation</math>

:<math>k = \frac{\phi r^2}{8} </math>

where the capillary radius r and constant 8 exchanged with fracture width wf and 12. The equation is usually used to calculate fracture permeability. Fracture porosity in percent and fracture width in micrometers (m), so the equation becomes:

:<math>k_f = 8.33 x 10^{-4} w_f \phi_f</math>

Where kf in darcy.

Where kf in darcy.

Previous equation can be used to calculate wf if porosity and permeability are known from well log or well testing :

Previous equation can be used to calculate wf if porosity and permeability are known from well log or well testing :

:<math>Equation</math>

:<math>w_f = \sqrt{12 \frac{k}{\phi}}</math>

So that, when viewed the relationship between the dimensions of fractures and fracture permeability, will be shown in  [[:File:UGM_Porosity_Fig_11png|Figure 11]] (Nelson, 2001). So that, when viewed the relationship between the dimensions of fractures and fracture porosity, will be shown in [[:File:UGM_Porosity_Fig_12.png|Figure 12]] (Nelson, 2001).

So that, when viewed the relationship between the dimensions of fractures and fracture permeability, will be shown in  [[:File:UGM_Porosity_Fig_11png|Figure 11]] (Nelson, 2001). So that, when viewed the relationship between the dimensions of fractures and fracture porosity, will be shown in [[:File:UGM_Porosity_Fig_12.png|Figure 12]] (Nelson, 2001).