Fractal Characterization of The Geysers Rock

Fractures play an important role in steam production from geothermal reservoirs. It has been a challenge for a long time to characterize materials with a high density of fractures such as The Geysers rock. Experimental data showed that the capillary pressure curves of The Geysers rock are very different from that of Berea sandstone. Methods to characterize the difference between the two have been few. It was also found that the frequently-used Brooks-Corey model could not be used to represent the capillary pressure curves of The Geysers rock samples studied. To this end, a fractal technique was proposed to model the features of the capillary pressure curves and to characterize the difference between The Geysers rock and Berea sandstone. The calculated values of the fractal dimension of all the core samples studied (six from The Geysers and one Berea sandstone) were in the range from 2 to 3. The results demonstrated that The Geysers rock with a high density of fractures had a greater fractal dimension than Berea sandstone that is almost without fractures. This shows that The Geysers rock has greater heterogeneity, as expected. The significance of fractal dimension inferred from capillary pressure data is consistent with the traditional qualitative observation from the frequency graph of pore size distribution. The proposed fractal technique may be extended from core scales to reservoir scales to characterize fracture density in geothermal reservoirs. Introduction Reserves of geothermal reservoirs depend mainly on the porosity of the rock matrix while steam and water production rates primarily depend on fractures. It is relatively easy to measure the values of porosity in the rock matrix. On the other hand, it is difficult to measure or characterize the fracture systems quantitatively as well as the entire rock with both matrix and fractures. Recently attention has been paid to the fractal characterization of the fracture surface (Babadagli and Develi, 2000) and the space distribution of fractures in geothermal reservoirs (Sammis et al., 1992; Tateno et al., 1995; Tsuchiya and Nakatsuka, 1995). Tateno et al. (1995) demonstrated that the distribution of open fracture width in a geothermal reservoir was fractal. Sammis et al. (1992) studied the fracture pattern and the distribution of fractures in two dimensions at The Geysers geothermal field. Sammis et al. (1992) found that the values of the fractal dimension measured at different scales, the outcrop, roadcut, and regional scales, were almost the same. However publications on the characterization of the entire rock with both matrix and fractures have been few. Studies on the features of capillary pressure curves of The Geysers rock and the K. Li and R. N. Horne 2 corresponding pore size distribution have also been few in the literature. In this study, the capillary pressure curves of The Geysers rock were measured using a mercury-injection technique. A fractal approach was proposed to characterize the rock heterogeneity quantitatively using the data from the capillary pressure curves. The values of the fractal dimension inferred from capillary pressure curves were used to represent rock heterogeneity as well as the differences between The Geysers rock and the more uniform, unfractured Berea sandstone. Methodology There have been many methods to characterize heterogeneity in nature. Of all the approaches, fractal geometry has been utilized widely in many areas. Fractal geometry is a branch of mathematics and is used mainly to characterize extremely disordered or heterogeneous systems that appear similar in some way when observed at different scales. This feature is known as selfsimilarity or scale invariance. The Geysers rock is highly heterogeneous and extremely disordered and may be fractal in nature. In this study, we chose fractal geometry as a tool to characterize the heterogeneity of The Geysers rock with both matrix and fractures. According to the basic concept of fractal geometry, the following expression applies to a fractal object: f D r r N − ∝ ) ( (1) where r is the radius (or characteristic length) of a unit chosen to fill the fractal object, N(r) is the number of the units (with a radius of r) required to fill the entire fractal object, and Df is the socalled fractal dimension. The fractal dimension is a representation of the heterogeneity of the fractal object. The greater the fractal dimension, the more heterogeneous the fractal object. Capillary pressure curves measured by a mercury-injection technique are often used to infer the pore size distribution of rock samples. In making this inference, rock with solid skeleton and pores is represented by using a capillary tube model. N(r) can be calculated easily once capillary pressure curves measured using a mercury-injection technique are available. The unit chosen in this study was a cylindrical capillary tube with a radius of r and a length of l. So the volume of the unit is equal to πrl and N(r) at a given radius of r is then calculated easily. Once N(r) is known, the value of fractal dimension, Df, can be determined from the relationship between N(r) and r. The relationship between N(r) and r should be linear on a log-log plot if the pore system of the rock is fractal. Experimental Measurements Six core samples from different wells at The Geysers geothermal field were used in this study. Unfortunately the core samples were irregular and too small to drill a plug for permeability measurements. The measured porosity of the core samples ranged from 0.1 to 4.0%. One core sample of Berea sandstone was also used as a comparison. The porosity of the Berea sample was about 23.0% and the air permeability was about 804 md. K. Li and R. N. Horne 3 The surface tension of air/mercury is 480 mN/m and the contact angle through the mercury phase is 140 (Purcell, 1949). Results Capillary pressure curves of the six core samples from The Geysers geothermal field and one core sample of Berea sandstone were measured using the mercury-injection technique. The results are shown in Fig. 1. There are substantial differences between The Geysers rock and Berea sandstone. The normalized wetting-phase saturation is calculated as follows: wr wr w w S S S S − − = 1 * (2) where * w S is the normalized wetting-phase saturation (the wetting-phase in this study is air), Sw is the wetting-phase saturation, and Swr is the residual saturation of the wetting-phase. 0.01 0.1 1 10 10