Chapter 4 – Saturation 4.12 4.2 Resistivity The presence of hydrocarbons is identified by the electrical resistance of the formation. These electrical properties of rocks depend on the pore geometry and fluid distribution. That is, the size, type and interconnection of the pore space and the type and distribution of the fluids in the pore space. Rock matrix act as a perfect insulator conducting no electrical current; therefore all conduction is through the fluids in the pore space. This conductivity is known as ionic or electrolytic conduction produced by the movement of ions in the formation water. The ions are typically Na+ and Cl- residing in the water phase. Subsequently, the greater the salinity of the water the higher the conductivity since more ions are present to transmit the current. Oil and gas behave as insulators, therefore a hydrocarbon-bearing formation will exhibit lower conductivity than a water-bearing formation of the same porosity. The electrical properties of rocks and the pore geometry are combined into Archie’s Law which describes the volume fraction of fluids within the pore space. Understanding this equation is the cornerstone to interpreting well logs. Fundamentals To begin our understanding of this process, we must first establish fundamental concepts and clarify terms. To illustrate this concept consider an open-top cubic tank one meter in all dimensions. It is electrically nonconducting except for two opposing metal electrodes (Fig. 4.8). First the tank is filled with water containing sodium chloride to simulate average formation water. A voltage (v) is applied and the resulting current (i) is measured. Figure 4.8 Definition of water resistivity i v A L waterChapter 4 – Saturation 4.13 Using Ohm's Law the resistance (rw) of the water can be calculated, (4.4) The resistivity is defined as the resistance scaled by the aspect ratio of area to length, (4.5) where Rw is in units of ohm-meter. This resistivity is inversely proportional to water salinity and temperature; i.e., as either property is increased the resultant resistivity will decrease. Next, sand is added to the water-filled tank, displacing some of the water. The result is a porous media 100% saturated with water. Since the rock grains act as an insulator, the current will be reduced and therefore the resistance will increase. This resistance with respect to the porous rock and associated water content is known as ro, and the resistivity as Ro. i v Water+sand A L La Figure 4.9 Definition of a 100% water saturated sand The increased resistance is due to the tortuous path the electrical current must take to circumvent the sand grains. This tortuous path length is defined as La and the corresponding cross-sectional area as Aa. The resistivity, Ro, is proportional to Rw since only the water conducts electricity, thus RwRo = F (4.6) ivwr LAwrwR Chapter 4 – Saturation 4.14 where the proportionality constant F is known as the formation resistivity factor. A plot of formation factor vs. water conductivity is a horizontal line as shown in Figure 4.10; thus F is independent of the formation water resistivity. FCwClean sandShaly sands Figure 4.10 Relationship between F and water conductivity The ratio of resistivities can also be expressed in terms of the aspect ratios; (4.7) where ro rw for the parallel circuit developed in the water- sand mixture. If we define tortuosity () as: (4.8) and porosity () as: (4.9) then the formation factor can be expressed in terms of variables which represent the pore geometry. (4.10) LaLaAAwRoR2LaLAaAFChapter 4 – Saturation 4.15 Example 4.3 Consider a synthetic rock sample made of an insulator material and shaped as a cube of length L as shown below. There is a square tube of dimension L/2 through the cube. Assume the inner square tube is filled with brine of resistivity, Rw and that the current will flow perpendicular to the faces. Calculate F, the relationship between F and porosity, and porosity. L/2L Solution 1. The expression for formation factor is, In this simple geometry the tortuous path length is equal to the cube length (La = L), therefore the only difference is the cross-sectional area. 2. Since the lengths are equal, then  = 1 and the formation-factor – porosity relationship is: The result is that porosity is 25%. 3. To verify this result, calculate porosity in terms of the pore volume/bulk volume ratio.  4222LLLLFaLaAwrLAorwRoRF1FChapter 4 – Saturation 4.16 This is in agreement with the previous results. The previous derivation applies to theoretical models of simple geometry. This approach develops an awareness of the fundamentals in petrophysics; however is not practical in application. The theoretical models do not capture the variations in the porous media found in the real world. An alternative method is to directly measure the formation factor and porosity of rock samples in a laboratory. Figure 4.11 is an example from a sandstone formation. Figure 4.11 Formation factor – porosity relationship from core samples, [Helander,1983] Lab-measured values provide excellent results for a particular rock type. However, they are limited to specific formations where core has been retrieved and analyzed in the lab. Frequently, core is not available for the formation of interest; therefore we rely on empirical  41322LLLbVpVChapter 4 – Saturation 4.17 correlations developed in the past. Archie developed a general relationship between porosity and formation factor; where a is defined as the cementation factor and m as the cementation exponent. The term cementation was proposed by Archie because he noticed a dependency of the exponent (m) on the degree of cementation (Fig. 4.12). Figure 4.12 F- relationship for various degrees of cementation, [Archie,1942] Note that for an increase in cementation from unconsolidated to highly cemented rocks the exponent increases from 1.3 to 2.2. For a formation factor of 20 this variation translates into a change in porosity of 10 to 26%, respectively. Subsequent work has shown that the empirical constants are dependent on numerous variables such as cementation, tortuosity, and wettability. Many empirical