PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, February 11-13, 2013 SGP-TR-198 FRACTURE CONNECTIVITY OF FRACTAL FRACTURE NETWORKS ESTIMATED USING ELECTRICAL RESISTIVITY Lilja Magnusdottir and Roland Horne Stanford University, Department of Energy Resources Engineering 367 Panama Street Stanford, CA, 94305-2220, USA e-mail: [email protected] information about the subsurface resistivity, which ABSTRACT can then be used to infer fracture locations. Other geophysical surveys used commonly to find hidden This paper discusses a method of characterizing geothermal resources are self-potential and fracture connectivity in geothermal reservoirs using magnetotelluric surveys. Garg et al. (2007) described conductive fluid injection and electrical resistivity how self-potential, magnetotelluric and direct current measurements. Discrete fractal fracture networks surveys were all used to explore the Beowawe were modeled and a flow simulator was used first to geothermal field in the Basin and Range Province of simulate the flow of a conductive tracer through the the western USA. However, usually these surveys are reservoir. Then, the simulator was applied to solve performed on the surface with very low resolution the electric fields at each time step by utilizing the when exploring deeper portions of the reservoirs, analogy between Ohm‟s law and Darcy‟s law. The making it impossible to characterize fractures that are time history of the electric potential difference small-scaled compared to the size of the reservoir. between the injector and the producer gives Therefore, the possibility of placing the electrodes information about the fracture network because the inside geothermal wells was considered in this study, potential difference drops as conductive fluid fills the in order to measure the resistivity more precisely in fracture paths. Therefore, the fractional connected the deeper parts of the reservoir. area of a fractal fracture network could be estimated with inverse modeling using the time history of the A conductive tracer is injected into the reservoir to electric potential. The fractional connected area was increase the contrast in resistivity between the rock also estimated with an inverse analysis using tracer and fracture zones. Slater et al. (2000), and Singha return curves at the producers but the study showed and Gorelick (2005) have shown a way of using that locations of connected areas were estimated tracer injection with Electrical Resistivity better using the electric potential approach. Tomography (ERT) to observe tracer migration in experimental tanks with cross-borehole electrical INTRODUCTION imaging. In these studies, many electrodes were used Fracture characterization in Enhanced Geothermal to obtain the resistivity distribution for the whole Systems (EGS) is crucial to ensure adequate supply tank at each time step and then this resistivity of geothermal fluids and efficient thermal operation distribution was compared to the distribution without of the wells. The interconnected conductive fractures any tracer to observe resistivity changes in each control mass and heat transport in the system and block visually. Using their approach for a whole inappropriate placing of injection or production wells reservoir would require a massive parameter space can lead to premature thermal breakthrough. Such and likely not be solvable, except at very low premature thermal breakthroughs have occurred in resolution. In the method considered in this study, the numerous geothermal reservoirs, as described by potential difference between wells, which Horne (1982), and observed in The Geysers (Beal et corresponds to changes in apparent resistivity, would al., 1994). be measured and plotted as a function of time while the conductive tracer flows through the fracture This study aimed to estimate the connectivity of network. That response, i.e. potential difference vs. fracture networks using direct current resistivity time, would then be used in an inverse modeling measurements. In these surveys, a direct current is process to characterize the connectivity of the sent into the ground through electrodes and the reservoir. voltage differences between them are recorded. The input current and measured voltage difference give Field studies by Rouleau and Gale (1985) suggest that fracture connectivity is dependent on fracture orientation, spacing and trace length data but GPRS was used not only to simulate the flow of a connectivity has also been quantified by the size of a conductive tracer through the reservoir but also to group of linked fractures, known as a „cluster‟ solve the electric field at each time step while the (Stauffer, 1985). The cluster size is measured by the fluid fills up the fracture paths from the injector to length of the largest connected group of fractures as a the producers. GPRS can be used to solve Ohm‟s proportion of the total fracture length in the network law, describing the flow of an electric current, due to (Odling 1997). Alternatively, the connectivity can be the analogy between Darcy‟s law and Ohm‟s law defined by the fraction of the total area that is (Muskat, 1932). The potential distribution in steady- connected by clusters, as described by Ghosh and state flow through a porous medium is exactly the Mitra (2009). In this project, the connectivity is same as the potential distribution in an electrically characterized by the Fractional Connected Area conducting medium. Therefore, the efficiency can be (FCA) because it provides a good indicator of the increased by using GPRS for both the fluid flow overall fracture density instead of only relating the simulations as well as to simulate the electric current. cluster size to the connectivity within the largest That way, the same grid can be used when cluster. calculating the distribution of a conductive tracer in the reservoir as well as when solving the electric METHODOLOGY difference between the wells at each time step. The method of using GPRS to solve electric flow was A series of simulations were conducted on discrete demonstrated earlier by Magnusdottir and Horne fractal fracture networks using General Purpose (2012). Reservoir Simulator (GPRS) developed at Stanford University (Cao, 2002). The objective was to Fractal Fracture Networks investigate the influence of connectivity on the time history of electric potential between well pairs to Several field studies performed on fault systems at study the possibility of using changes in electric different length scales have demonstrated that potential with conductive tracer injection to fracture populations can follow a power-law length characterize fracture connectivity. distribution (Shaw and Gartner, 1986, and Main et al., 1990). Therefore, the relationship between the Simulation using GPRS fractal dimension D within an L × L square domain and N(r), the number of boxes of size r that include General Purpose Reservoir Simulator (GPRS) was the center point of fractures, can be represented by a used to simulate the flow of a conductive tracer fractal equation using the box-counting approach through discrete fracture networks (DFN). A DFN [Barton and Larsen, 1985], approach introduced by Karimi-Fard et al. (2003) was used to create realistic fracture networks where Dlim log N(r)/log(1/ r) (2) unstructured control volume finite-difference r0 formulation was used with element connections assigned using a connectivity list. The computational where r = L/k (k=1,2,3,…). The length distribution of grid was formed using Triangle, a triangular mesh the fractures is also fractal and can be described by generator developed by Shewchuk (1996). The the fractal equations (Nakaya et al., 2003), conductive tracer was assumed to be a NaCl solution and the resistivity of the solution calculated using a N(l) Bl a three-dimensional regression formula established by (3) Ucok et al. (1980). Then, the resistivity of the water a limlog N(l)/log(1/l) saturated rock, ρ, was calculated using Archie‟s law l0 (Archie, 1942), (4) a B (lmax) b (5) a w (1) where a is the fractal dimension of fracture length distribution, lmax is the maximum fracture length and where is the porosity of the rock and a and b are N(l) is the number of fractures with lengths larger empirical constants. Archie (1942) concluded that for than l, so l=lmax when N(l)=1. Discrete-fracture typical sandstones of oil reservoirs the coefficient a is networks with fractal dimensions ranging from D = approximately 1 and b is approximately 2 but Keller 1.0 to 1.8 with 0.1 increments were created using a and Frischknecht (1996) showed that this power law method described by Nakaya et al. (2003). The is valid with varying coefficients based on the rock fracture locations were determined randomly and the type. In this case, a was set as 0.62 and b as 1.95, angles normal to the fractures were chosen randomly which corresponds to well cemented sedimentary to be either 45° or 135° with a standard deviation of rocks with porosity 5-25% (Frischknecht, 1996). 5°. The maximum fracture length was set as lmax=600 m and the aperture was defined by, w Cl e (6) In fractured reservoirs, the connected area has high max influence on the heat and mass transport in the system. The aim of this project was to characterize where wmax is the aperture and C is a constant. Olson the size and location of connected areas in the (2003) describes how this power law equation was reservoirs to ensure efficient operation of the wells. used to fit various fracture datasets of different sizes, usually with e = 0.4. Here, e was set as 0.4, C as Inverse Analysis 0.002 m3/5 and the size of the reservoir was set as 1000 × 1000 m2. An inverse analysis is used with the time history of the electric potential to estimate the connectivity of Fractional Connected Area the fracture network. In inverse modeling the results of actual observations are used to infer the values of Ghosh and Mitra (2009) concluded that the Fractional the parameters characterizing the system under Connected Area (FCA) combined with a distribution investigation.