PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, February 11-13, 2013 SGP-TR-198

FRACTURE CONNECTIVITY OF 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 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 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 Dlim log N(r)/log(1/ r) (2) unstructured control volume finite-difference r0 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 l0 (Archie, 1942), (4) B  (l )a max   a b  (5) 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  Cl 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. In this study, the output parameters of cluster sizes provides a complete measure of the were first the potential differences between wells as a connectivity of fractures within the system. FCA is function of time and then the tracer return curves at defined by the summed area of all clusters within the the producers. The input parameters were the fracture network divided by the total sample area. fractional connected area of the reservoir. The FCA is also a good indicator of the overall fracture objective function measures the difference between density. The area of a cluster is delineated by the the model calculation (the calculated voltage simplest polygon around the extremities of a fracture difference between the wells or tracer return at the cluster, see Figure 1. producers) and the corresponding observed data measured at the reservoir, illustrated in Figure 2. Genetic algorithm (Holland, 1975) was used to find the network with the most similar characteristics by proposing new parameter sets that improve the match iteratively.

RESULTS An inverse analysis was performed to match the time histories of electric potential between different well pairs for a reservoir modeled with fractal dimension D = 1.3, see Figure 3a).

Figure 1: Connected area.

Figure 2: The inverse analysis.

Figure 3: a) Reservoir (connected area in red) b) Electric potential difference between well pairs.

One injection well was modeled and three production better connection towards producer 2 and producer 3 wells. Water was injected at the rate of 10 kg/s and than towards producer 1. The curves do not give tracer was 22 wt% of the water injected. The much information about the area between producer 1 production well was modeled to deliver against a and producer 3, and producer 2 and producer 3 bottom-hole pressure of 106 Pa with productivity because the injected tracer is flowing towards the index of 4×10-12 m3. The initial pressure was set to producers and might not reach these areas. However, 106 Pa and the temperature to 25°C. The porosity of the curves do decrease as the tracer flows towards the fractures was set to 0.9 and the permeability was producer 3 and might add some information about the determined by: reservoir.

w2 The simple tracer return curves at the producers were k  (7) examined as well, as shown in Figure 4. 12 where w is the width of the fractures. The matrix blocks were given a porosity value of 0.1 and the permeability was set as 1×10-10 m2. For the resistivity calculations the pores and fractures were modeled to be filled with water before any tracer was injected into reservoir so the initial tracer mass was set to 0.05 wt%.

As an example, GPRS was used to calculate the electric potential distribution for the reservoir in Figure 3a. An electric current was set equal to 1 A at the injector and as -1 A at Producer 1 and the potential field was calculated based on the resistivity of the field at each time step. Then, the same procedure was repeated for all the other well pairs, Figure 4: Tracer return curves for the reservoir. with results shown in Figure 3b. The electric potential curves calculated between the injector and The tracer return curves also indicate a good producer 1, injector and producer 2, and injector and connection towards producer 2 and producer 3, producer 3, drop once tracer is injected into the reaching producer 2 after about 6 days and producer reservoir due to the lower resistivity of the fluid 3 after about 7 days. The tracer does not reach injected. The fluid reaches the area between the other producer 1 until after about 14 days because of few well pairs later so the corresponding electric potential fractures in the area between the injector and curves decrease slower at the start. Then, some of the producer 1. curves drop relatively quickly and correctly indicate a

Figure 5: a) The fracture network that gave the best match b) Electric potential difference between well pairs.

The inverse analysis compared the time histories of between the well pairs correspond to the locations of the electric potential difference between well pairs the connected areas. for the reservoir in Figure 3 to a „library‟ of up to 180 fracture networks. The fractal dimensions of the These results indicate a good possibility of using networks were ranging from D = 1.0 to 1.8 but other electric potential calculations while injecting a variables such as the size of the network, two conductive tracer into reservoirs to predict fractional principal fracture orientations and relationship used connected area as well as the locations of the for aperture, were the same as for the reservoir. The connected fractures. In addition, the fractal curves for the reservoir were compared to up to 180 dimensions of the two networks were similar, D = 1.3 networks using a genetic algorithm to find the best for the real reservoir and D = 1.4 for the best match. match. The chosen network is seen in Figure 5, with Thus, it would be of interest to also investigate the the curves for the electric potential showing very possibility of using this method to predict the fractal similar behavior to the curves for the true reservoir dimension of fractal fracture networks. described previously in Figure 3. The tracer return curves for the best match, see The Fractional Connected Area (FCA) of the network Figure 6, show somewhat different behavior than the in Figure 5 was compared to the FCA of the true tracer return curves for the reservoir, shown in Figure reservoir to investigate the possibility of predicting 4. FCA using the electric potential difference between well pairs with conductive fluid injection. The results were FCA = 23.3% for the reservoir, and FCA = 23.4% for the best match. Thus, FCA matches very well for these two networks, indicating that FCA can be predicted using this method in this example.

The location of the connected areas is another similarity that can be seen between the best match and the real reservoir. In both cases the connected fractures are located similarly in the upper right corners of the figures and the connected area at the bottom left was also predicted correctly, despite the electric potential in that area giving very limited information. In the upper right corner the fractures form connected paths toward the middle, resulting in Figure 6: Tracer return curve for the best match. a drop in potential difference between the injector and producer 2, as well as between the injector and producer 3. Thus, the drops in electric potential

Figure 7: a) The best match when using tracer return curves b) Electric potential difference between well pairs.

In order to test the advantages of using electrical connected path from the upper right corner towards measurements instead of only using tracer return the center. The fractal dimension is also different, D curves, the inverse analysis was performed again for = 1.1 while for the true reservoir D = 1.3. Advantages the reservoir in Figure 3, but this time the objective of using the electric measurements include having function measured the difference between the model more extensive data and being able to see the changes calculation of just the simple tracer return curves and as the conductive fluid flows through the network the corresponding tracer return curves for the true even before it has reached the production wells. reservoir. The best match when comparing the tracer Future work includes performing more simulations return curves using a genetic algorithm as before, can and a sensitivity analysis to further study the be seen in Figure 7. The time histories of the electric possibilities and advantages of using electrical potential difference between the wells do not match measurements to estimate the connected area. as well as when they were used to find the best match. However, in this case the tracer return curves CONCLUSION match better than before, see Figure 8. In this paper, the Fractional Connected Area (FCA) of a fractured geothermal reservoir was estimated using electrical resistivity measurements with conductive fluid injection. Inverse analysis was used to match the time histories of the electric potential between the wells of the reservoir to the time histories of a library of 180 fracture networks to find the best match. The reservoir and the best match had the same FCA and the connected areas had similar locations. The inverse analysis was also performed matching only the tracer return curves at the producers. The best match gave the same FCA but the locations of the connected areas were a little different from the reservoir.

Figure 8: Tracer return curves for the best match The study showed a successful attempt to use time when using tracer return curves. histories of electric potential with conductive fluid injection to estimate the connectivity of a reservoir. Here, estimated FCA was the same as for the More cases need to be studied and more simulations previous case, FCA = 23.4%, and the fracture performed to test the reliability of this method. A network (Figure 7a) does have a connected area sensitivity analysis will also be performed to between the injector and producer 2 as well as demonstrate the advantages of using electrical between producer 1 and producer 3. However, the resistivity measurements instead of only using the connected area between the injector and producer 2 is tracer return curves at the producers. relatively smaller than for the true reservoir we are trying to match (Figure 3a) and it does not form a distributions on rock deformation in the brittle ACKNOWLEDGEMENT field, in Deformation Mechanisms, Rheology and Tectonics, edited by Knipe, R.J. and Rutter," This research was supported by the US Department E. H.,” Geol. Spec. Publ., London, 54, 71-79. of Energy, under Contract DE-EE0005516. The Stanford Geothermal Program is grateful for this Muskat, M., (1932), “Potential Distributions in Large support. Cylindrical Disks with Partially Penetrating Electrodes”, , 2, 329-364. REFERENCES Nakaya, S., Yoshida, T. and Shioiri, N., (2003), Archie, G.E. (1942), “The Electrical Resistivity Log “Perlocation Conditions in Binary Fractal as an Aid in Determining some Reservoir Fracture Networks: Applications to Rock Characteristics,” Transaction of the American Fractures and Active and Seismogenic Faults,” J. Institute of Mining, Metallurgical and Petroleum Geophys. Res., 108(B7), 2348, doi: Engineers, 146, 54-62. 10.1029/2002JB002117. Beal, J.J., Adams, M.C. and Hrtz, P.N., (1994), “R- Odling, N. E., (1997), “Scaling and connectivity of 13 Tracing of Injection in The Geysers,” systems in sandstones from western Geothermal Resources Council, 18, 151-159. Norway,” Journal of Structural Geology, 19, 1251–1271. Cao, H., (2002), “Development of Techniques for General Purpose Simulators,” PhD Thesis, Olson, J.E., (2003), “Sublinear scaling of fracture Stanford University. aperture versus length: an exception or the rule?” Journal of Geophysical Research, 108, 2413, Garg, S.K., Pritchett, J.W., Wannamaker, P.E. and doi:10.1029/2001JB000419. Combs, J., (2007), “Characterization of Geothermal Reservoirs with Electrical Surveys: Rouleau, A., and J. E. Gale, (1985), “Statistical Beowawe geothermal field”, Geothermics, 36, characterization of the fracture system in the 487-517. Stripa granite.” International Journal of Rock Mechanics and Mining Sciences and Ghosh, K. and Mitra, S., (2009), “Two-Dimensional Geomechanics Abstracts, Sweden, 22, 353–367, Simulation of Controls of Fracture Parameters on doi:10.1016/0148-9062 (85)90001-4. Fracture Connectivity,” AAPG Bulletin, 93, 1517-1533, doi: 10.1306/07270909041. Shaw, H. R. and Gartner, A. E., (1986), “On the graphical interpretation of paleoseismic data.” Holland, J.H., (1975), “Adaptation in Natural and U.S. geol. Surv. Open-File Rep., 86-394. Artificial Systems,” University of Michigan Press, Ann Arbor, 54. Shewchuk J.R., (1996), “Triangle: Engineering a 2D Quality ;Mesh Generator and Delaunay Horne, R.N., (1982), “Effects of Water Injection into Triangulator,” Applied Computational Fractured Geothermal Reservoirs, a Summary of Geometry: Towards Geometric Engineering, Experience Worldwide,” Geothermal Resources 1148, 203-222. Council, Davis, CA, 12, 47–63. Singha, K. and Gorelick, S.M., (2005), “Saline Karimi-Fard, M., Durlofsky, L.J. and Aziz, K., Tracer Visualized with Three-dimensional (2003), “An Efficient Discrete Fracture Model Electrical Resistivity Tomography: Field-scale Applicable for General Purpose Reservoir Spatial Moment Analysis”, Water Resources Simulators”, SPE 79699, SPE Reservoir Research, 41, W05023. Simulation Symposium, Houston, TX. Slater, L., Binley, A.M., Daily, W. and Johnson, R., Keller, G.V. and Frischknecht, F.C., (1996), (2000), “Cross-hole Electrical Imaging of a “Electrical Methods in Geophysical Controlled Saline Tracer Injection”, Journal of Prospecting”, Pergamon, London. Applied Geophysics, 44, 85-102. Magnusdottir, L. and Horne, R. N., (2012), Stauffer, D., (1985), “Introduction to percolation “TOUGH2 flow simulator used to simulate an theory,” Taylor and Francis, London, 190. electric field of a reservoir with a conductive tracer for Fracture characterization,” Geothermal Ucok, H., Ershaghi, I. and Olhoeft, G.R., (1980), Resources Council, 36th Annual Meeting and “Electrical Resistivity of Geothermal Brines,” GEA Trade Show, Reno, Nevada. Journal of Petroleum Technology, 32, 717-727. Main, I.G., Meredith, P.G., Sammonds, P.R, and Jones, C., (1990), “Influence of fractal flaw