Introduction:
An increase of electro kinetic coupling coefficient with increasing permeability that can explain by the effect of variation in surface conductivity was examined by.[1] The reduction of streaming potential coefficient during compaction suggests that the tortuosity of the hydraulic network increases faster than tortuosity of the electric network was reported by.[2] An increase of streaming potential coupling coefficient with increasing permeability was demonstrated by.[3] A new method to calculate total electric conductivity from the ratio of pore area cross-section to length of the effective rock capillaries which derived from streaming potential measurements was investigated by.[4] Capillary pressure follows the scaling law at low water saturation was reported by.[5] Streaming potential coupling coefficient measurements to monitor flow in saline surface environments such as deep saline aquifers and hydrocarbon reservoirs was studied by.[6] Electro kinetic relationship to produce a function that links the mean grain size of a rock to its effective mean pore radius was described by.[7]An increase of volumetric flow rate, which, in turn, increases the penetration distance of the acid before it is being spent due to application of direct current to improve acidizing operations was investigated by.[8]A novel method for estimating permeability with streaming current and electro osmosis pressure was developed by.[9] Good agreement between electro kinetically deduced and gas measured permeability was presented by.[10] An increase of bubble pressure fractal dimension and pressure head fractal dimension with decreasing pore size distribution index and fitting parameters m×n due to possibility of having inter connected channels was proofed by.[11] An increase of fractal dimension with increasing permeability, relaxation time of induced polarization, due to an increase in pore connectivity was confirmed by.[12]
Materials and Methods
Samples were collected from the Surface type section of the Shajara Reservoirs of Permo-Carboniferous Shajara Formation, latitude 26° 52’ 17.4,”longitude 43° 36’ 18” (Figure 1). Porosity was measured and permeably was calculated from the measured capillary pressure data obtained by mercury intrusion technique on powder sandstone samples. Pore radius was determined from distribution of pores which enables the calculation of electro kinetic fractal dimension.
he electro kinetic fractal dimension can be scaled as
Where Sw the water saturation, CEK the electro kinetic coefficient in ampere / (pascal * meter), CEKmax the maximum electro kinetic coefficient in ampere /(pascal * meter), and Df the fractal dimension. Equation 1 can be proofed from
Where V the flow velocity in meter second, CEK the electro kinetic coefficient in ampere / (pascal* meter). E the electric field in volt / meter. The flow velocity can be scaled as
Where V the flow velocity in meter second, Q the flow rate in cubic meter / second, A the area in square meter.
Insert equation 3 into equation 2
The flow rate can be scaled as
Where Q the flow rate in cubic meter second, r the pore radius in meter, Δp the differential pressure in pascal, μ the fluid viscosity, L the capillary length in meter.Insert equation 5 into equation 4
Equation 6 after rearrangement will become
The maximum pore radius can be scaled as
Divide equation 7 by equation 8
Equation 9 after simplification will become
Take the fourth root of equation 10
Equation 11 after simplification will become
Take the logarithm of equation 12
Insert equation 14 into equation 13
Equation 15 after log removal will become
Equation 16 the proof of equation 1 which relates the water saturation, electro kinetic coefficient, maximum electro kinetic coefficient, and the fractal dimension.The capillary pressure can be scaled as
Where Sw the water saturation, Df the fractal dimension, Pc the capillary pressure.
Results and Discussion
Based on field observation the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three units as described in (Figure 1). These units from bottom to top are: Lower, Middle and Upper Shajara Reservoir. Their developed results of electro kinetic and capillary fractal dimensions are presented in (Table 1). The results show equalities between electro kinetic fractal dimension and capillary pressure fractal dimension. A maximum fractal dimension value of about 2.7872 was informed from sample SJ13 as demonstrated in (Table 1). But, a minimum fractal dimension value 2.4379 allocates to sample SJ3 from the Lower Shajara Reservoir as defined in (Table 1). The electro kinetic and capillary pressure fractal dimensions were observed to increase with increasing permeability owing to the possibility of having interconnected channels as verified in (Table 1). Regarding the Lower Shajara Reservoir, it is represented by six sandstone samples as shown in (Figure 1), four of which label as SJ1, SJ2, SJ3, and SJ4 were selected for capillary pressure measurements to determine the fractal dimension. Their positive slopes of the first procedure (log ratio of electro kinetic to maximum elector kinetic versus log water saturation (Sw) and negative slopes of the second procedure (log capillary pressure (Pc) versus log water saturation (Sw) were described in (Figure 2, Figure 3, Figure 4, Figure 5 and Table 1). Their electro kinetic fractal dimension and capillary pressure fractal dimension values are shown in Table 1. As we progress from sample SJ2 to SJ3 a noticeable reduction in permeability from 1955 md to 56 md was observed due to compaction which reflects change in electro kinetic fractal dimension from 2.7748 to 2.4379 as delineated in (Table 1). Such drastic change in permeability can account for heterogeneity which is a key parameter in reservoir quality assessment. Once more, an increase in grain size and permeability was reported from sample SJ4 whose electro kinetic fractal dimension and capillary pressure fractal dimension was found to be 2.6843 as illustrated in (Table 1).
Figure 1:Surface type section of the Shajara reservoirs of the Shajara Formation, at latitude 26° 52 ′ 17.4 ″ longitude 43° 36 ′ 18 ″
Figure 2:Log (CEK1/4 /CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ1
Figure 3:Log (CEK1/4/ CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ2
Figure 4: Log (CEK1/4 /CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ3
Figure 5: Log (CEK1/4 /CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ4
Table 1: Petrophysical model showing the three Shajara Reservoir Units with their corresponding values of electro kinetic fractal dimension and capillary pressure fractal dimension
| Formation | Reservoir | Sample | Porosity % | k(md) | Positive slope of the first procedure
Slope=3-Df | Negative slope of the second procedure Slope=Df-3 | Electro kinetic fractal dimension | Capillary pressure fractal dimension |
| Permo-Carboniferous Shajara Formation | Upper Shajara Reservoir | SJ13 | 25 | 973 | 0.2128 | -0.2128 | 2.7872 | 2.7872 |
| SJ12 | 28 | 1440 | 0.2141 | -0.2141 | 2.7859 | 2.7859 |
| SJ11 | 36 | 1197 |
0.2414 |
-0.2414 | 2.7586 | 2.7586 |
| Middle Shajara Reservoir |
SJ9 | 31 | 1394 | 0.2214 | -0.2214 | 2.7786 | 2.7786 |
| SJ8 | 32 | 1344 | 0.2248 | -0.2248 | 2.7752 | 2.7752 |
| SJ7 | 35 | 1472 | 0.2317 | -0.2317 | 2.7683 | 2.7683 |
| Lower Shajara Reservoir | SJ14 | 30 | 176 | 0.3157 | -0.3157 | 2.6843 | 2.6843 |
| SJ3 | 34 | 56 | 0.5621 | -0.5621 | 2.4379 | 2.4379 |
| SJ2 | 35 | 1955 | 0.2252 | -0.2252 | 2.7748 | 2.7748 |
| SJ1 | 29 | 1680 | 0.2141 | -0.2141 | 2.7859 | 2.7859 |
Concerning the Middle Shajara Reservoir, it is separated from Lower Shajara Reservoir by an unconformity surface as shown in (Figure 1). It was represented by four samples, three of which named as SJ7, SJ8, and SJ9 were selected for fractal dimension determination as verified in (Table 1). Their positive and negative slopes of the first and second procedures were described in (Figure 6, Figure 7, Figure 8 and (Table 1) Additionally, their electro kinetic and capillary pressure fractal dimension displays equal values as shown in (Table 1) Moreover, their fractal dimension values are higher than those of sample SJ3 and SJ4 from the Lower Shajara Reservoir due to an increase in their permeability as specified in (Table 1). On the other hand, the Upper Shajara reservoir is separated from the Middle Shajara reservoir by yellow green mudstone as revealed in (Figure 1). It is defined by three samples so called SJ11, SJ12, SJ13 as explained in (Table 1). Their positive slopes of the first procedure and negative slopes of the second procedure are displayed in (Figure 9, Figure 10, Figure 11 and Table 1). Moreover, their electro kinetic fractal dimension and capillary pressure fractal dimension are also higher than those of sample SJ3 and SJ4 from the Lower Shajara Reservoir due to an increase in their permeability as testified in (Table 1). Overall a plot of positive slope of the first procedure versus negative slope of the second procedure delineates three reservoir zones of varying petrophysical characteristics as shown in (Figure 12). These zones were also confirmed by plotting electro kinetic fractal dimension versus capillary pressure fractal dimension as shown in (Figure 13). Such variation in fractal dimensions can be used to explain heterogeneity which is a key parameter in reservoir quality assessment.
Figure 6:Log (CEK1/4 /CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ7
Figure 7:Log (CEK1/4 /CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ8
Figure 8:Log (CEK1/4 /CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ9
Figure 9:Log (CEK1/4/CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ11
Figure 10:Log (CEK1/4/CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ12
Figure 11:Log (CEK1/4/CEK1/4max) versus log Sw & log Pc versus log Sw for sample SJ13
Figure 12:Positive slope of the first procedure versus negative slope of the second procedure
Figure 13:Electro kinetic fractal dimension versus capillary pressure fractal dimension
Conclusions
The sandstones of the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three units based on electro kinetic fractal. The Units from bottom to top are Lower Shajara electro kinetic fractal dimension unit, Middle Shajara electro kinetic fractal dimension unit, and Upper Shajara electro kinetic fractal dimension unit. These units were also confirmed by capillary pressure fractal dimension. The heterogeneity increases with increasing permeability, increasing fractal dimension, decreasing compaction owing to possibility of having interconnected channels.
Acknowledgements
The author would like to thank college of Engineering, King Saud University, Department of Petroleum and Natural Gas Engineering, Department of Chemical Engineering, Research Centre at college of Engineering and King Abdulla Institute for Research and Consulting Studies for their supports.
