Electro Kinetic Fractal Dimension for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia

1.       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]. Toledo et al 1994 in Li and Williams 2007 [5] reported that capillary pressure follows the scaling law at low water saturation. 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].

2.       Experimental Procedure

Two procedures for obtaining the fractal dimension have been utilized. The first procedure was completed by plotting the logarithm of the ratio between streaming potential to electro osmosis coupling coefficient divided by the logarithm of maximum value describing the ratio between streaming potential and electro osmosis coupling coefficient versus logarithm water saturation. The slope of the first procedure is positive = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was determined by plotting the logarithm of capillary pressure versus the logarithm of water saturation. The slope of the second procedure is negative = Df -3.

3.       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.  Effective pore radius was determined from distribution of pores which enables the calculation of electro kinetic fractal dimension.

The electro kinetic fractal dimension can be scaled as

                   1          

 Sw= water saturation.

 Cs= streaming potential coefficient in volt / pascal

 CE =electro osmosis coefficient in pascal / volt

 Df the fractal dimension.

Equation 1 can be proofed from

                 2

 r_{eff =} effective pore radius in meter

 η=fluid viscosity in pascal. Second

σ=fluid electric conductivity in Siemens per meter.

Take the square root of equation 2

                 3

Equation 3 after simplification will become

                 4

The effective pore radius can be scaled as

                         5

 A= the pore radius

 a=packing constant

Insert equation 5 into equation 4

               6

The maximum pore radius can be scaled as

          7

Divide equation 6 by equation 7

                 8

Equation 8 after simplification

                                 9

Take the logarithm of equation 9

                 10

Insert equation 11 into equation 10

                        12

Remove the logarithm from equation 12

                            13

Equation 13 the proof of equation 1 which relates the water saturation, the ratio of streaming potential coefficient to electro osmosis coefficient and the fractal dimension.

The capillary pressure fractal dimension was determined from

                             14

Where Sw the water saturation Pc the capillary pressure and Df the fractal dimension

4.       Results and Discussion

Based on field observation the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three units as explained in (Figure 1). These units from bottom to top are: Lower, Middle and Upper Shajara Reservoir. Their acquired 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 reported from sample SJ13 as illustrated in (Table 1). However, a minimum fractal dimension value 2.4379 allocates to sample SJ3 from the Lower Shajara Reservoir as delineated in (Table 1). The electro kinetic and capillary pressure fractal dimensions were noticed to increase with increasing permeability due 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 streaming potential to electro osmosis coupling coefficient divided by maximum streaming potential to electro osmosis coefficient ratio versus log water saturation (Sw) and negative slopes of the second procedure (log capillary pressure versus log water saturation are delineated 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 remarkable 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 demonstrated in (Table 1). Such drastic change in permeability can account for heterogeneity which is a key parameter in reservoir quality assessment. Again, 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 demonstrated in (Table 1).

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 chosen for fractal dimension determination as specified in (Table 1). Their positive and negative slopes of the first and second procedures were presented in (Figure 6), (Figure 7), and (Figure 8) and (Table 1) Additionally, their electro kinetic and capillary pressure fractal dimension exhibit equal values as displayed 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 illustrated 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) and (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 clarified 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.

5.       Conclusions

1.       The sandstones of the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three units based on electro kinetic fractal.

2.       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.

3.       These units were also confirmed by capillary pressure fractal dimension

4.       Variation in fractal dimensions can account for heterogeneity which is a key parameter in reservoir quality assessment.

5.       The heterogeneity increases with increasing permeability, increasing fractal dimension, decreasing compaction owing to possibility of having interconnected channels.

6.       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.

Figure 1: Surface type section of the Shajara Reservoirs of Permo-Carboniferous Shajara Formation. Latitude 26^° 52' 17.4,"longitude 43^° 36' 18".

Table 1: Petro physical model showing the three Shajara Reservoirs with their corresponding values of electro kinetic and capillary pressure fractal dimension.