Simulation of geothermal processes

Proceedings World Geothermal Congress 2010 Bali, Indonesia, 25-29 April 2010 Simulation of geothermal processes Klára Císařová, Jiří Kopal, Jiřina Královcová and Jiří Maryška Faculty of Mechatronics Informatics and Interdisciplinary Studies, Technical University of Liberec, Studentská 2, Liberec, Czech epublic jirina.kralovcova@tul.cz Keywords: Underground flow, heat transport, simulation. ABSTACT This paper describes a mathematical model of geothermal processes in a significantly heterogeneous geological environment. The model is based on mixed-hybrid formulation of finite elements and it has been implemented in our simulation code and prepared for putting through capability testing. The model enables to cover up subdomains of different dimensions. In this way, important fractures and fractured zones coupled with the regional extent of the simulated domains can be involved. esults of some model tasks are presented to demonstrate the behaviour of the model. As examples of real applications The Melechov massif (Central Europe) and Cajamarca (Peru) sites have been selected. Geology and hydrogeology of both localities have been studied. On the basis of the real data, conceptual models and computational meshes have been prepared. According to geological properties physical properties have been specified. The real applications involve underground flow model as well as heat transport model for the case of pumping underground water. 1. INTODUCTION Our work is focused on the simulation of geothermal processes on applications for generating electricity and space heating in particular. For the sake of underground processes simulations our own simulation code named at whole FLOW123D (Královcová at al. 2005) was developed, consisting of: (i) the module FLOW123D, that allows simulation of groundwater flow both in fractured systems and porous media. (ii) The module TAN123D, solves heat transport involving heat diffusion in rock, heat transfer from rock to groundwater and convection of heated water. These models reach steady states after several transient states. Therefore a steady state is used for solving heat transfer from rock to the groundwater. In the paper, we first deal with a mathematical description of both models, the given initial and boundary conditions. Next, model implementation and verification with respect to a possible real configuration are described. Two real simulations are depicted. The first computes heat and water resources in the area of Cajamarca, where a thermal spa might be built in the future. The second model estimates heat conditions for evaluating the possibilities of geothermal heating in a region with properties characteristic for the Czech epublic rock massifs. 2. UNDEGOUND FLOW MODEL The underground flow model is based on Darcy s law and the continuity equation. Let s define the area Ω as a cylinder with an irregular base and a general surface area so that the boundary conditions be properly set and well identified (water-shed, impermeable bedrock, ). Let s 1 denote Ω the boundary of the area with lower indices referring to a particular part. The mathematical description of groundwater flow is described by Darcy s law p u = K + z = ρ K g φ where u denotes groundwater flux, K hydraulic conductivity tensor, p pressure, g gravitational constant and ρ water density. The continuity equation in the conserving mass form is ( ρ ) = ρ q (1) u. Considering incompressible liquid it reduces into conserving volume form u = q, (2) where q denotes density of groundwater resources in the area Ω. For stating the boundary conditions the boundary is divided into several parts: (i) On the bedrock, in most cases considered impermeable, a homogenous Neumann boundary condition is imposed ( u n = 0, where n is outer normal vector). Near faults a Newton boundary condition (general Neumann boundary condition) could be considered as well. (ii) On the surface area a homogenous Neumann boundary condition is imposed near watersheds. On the rest of the surface area a general Neumann boundary condition is given. (iii) On the top a Dirichlet boundary condition is imposed in the parts where the piezometric head is known. In the infiltration area the measured and modified values of the infiltration flux u n = q are used. For the solution of groundwater flow a discretization into following elements was used: 3D elements representing porous or micro-fractured media, 2D elements representing significant fissures, fissure systems and permeable faults and 1D elements representing linear features such as wells and tunnels. The approximation of the problem follows Mixed-Hybrid FEM. This method forces balanced flux on inner edges of computational mesh and enables a transparent description including interconnection of 3D, 2D and 1D elements. An indefinite system matrix coupled with and higher number of variables are disadvantages of this method. The Mixed-Hybrid FEM and its numerical implementation were described in (Maryška at al., 2005). 3. HEAT TANSPOT MODEL Heat transport model respects heat convection and diffusion. The heat transport in rock is defined by following equations: N

ϑ τ = λ ϑ (1 n) ρ c α ( ϑ ϑ (1 n) ρ c W ) (3) ϑ A = A ϑ A, (9) t ϑ A ( tn+ 1, = AΔtϑA ( tn, (10) ϑ τ W α = u ϑw ) + ( ϑ ϑ ), (4) nρ c ( W W W whereϑ denotes rock temperature, ϑ W λ heat conductivity, n rock porosity, water temperature, ρ rock density, c water heat c rock heat capacity, ρ W water density, W capacity, u vector of groundwater flow velocity and α heat transmissivity coefficient between water and rock. In the equation (3), the first term is heat convection in the rock and the second term represents the heat exchange between rock and water. The second equation (4) characterizes the heat transport by groundwater flow. Here the first term is heat convection and the latter heat exchange between rock and water. Let s denote ϑ ϑ =, ϑw then the equations (3), (4) can be rephrased as In the second step heat diffusion in rock matrix is computed. Let s define the initial temperature distribution as ϑ t ϑ ( t ), thus this step can be written as: D ( n, = A n+ 1, x ϑ D = D ϑ D, (11) t ϑ D( tn+ 1, = DΔtϑD ( tn,. (12) To solve this step we use MH-FEM on the same mesh as in the previous examples. A detailed description is introduced in (Severýn at al., 2006). In the third step heat exchange in element volumes is computed. For the purpose we suppose the solution for α, which represents equilibrium state of the heat exchange. Let s denote the initial state before exchange as ϑ t ϑ ( t ), thus the final state can be X ( n, = D n+ 1, x rephrased as ϑ τ = A ( ϑ) + D( ϑ) + X( ϑ), (5) ϑ X = X ϑ X, (13) t where A denotes convection operator, D diffusion operator and X rock-water heat exchange operator. For more detail the operators are defined as follows: 0 A ( ϑ) =, (6) ( uϑ W ) λ ϑ D ( ϑ) = (1 n) ρ c, (7) 0 α ( ϑ ϑw ) (1 n) ρ ( ) = c X ϑ (8) α ( ϑ ϑw ) nρw cw Dirichlet boundary conditions are set according to the depth and the assumed or measured temperature gradient (increasing with depth from 3 to 7 centigrade per 100 m). We solve the problem by splitting the operator into particular physical tasks, which are after that computed by suitable numerical methods. The time period is divided into N equidistant time subintervals. Then we integrate the contribution of the partial physical processes on the intervals by the superposition method. At first we solve advection by the upwind scheme of FEM. Due to the balance on the element edges we solve the task on the same mesh as in the case of flow task. We denote ϑ t = ϑ( t ) and then the advection step can be A ( n, n, x written down as ϑ X ( tn+ 1, = XΔtϑX( tn,. (14) The final state after the computation of the heat exchange between rock and groundwater is used as the final state of the temperature distribution at the end of particular time step and it is used as the initial condition for the solution of solute transport in the next time step. Thus ϑ ( tn+ 1, = ϑx( tn,. (15) 4. IMPLEMENTATION The module FLOW123D of our simulation code solves groundwater flow, solute transport and water-rock interactions. As to the water-rock interactions nonequilibrium mobile-immobile pore exchange and non-linear sorption with arbitrary number of variables in fractures or continuum are considered. FLOW123D allows solving steady state flow, unsteady state flow and variable density flow. Solute transport is solved by operator splitting, convection by FVM, non-linear adsorption by Newton s method. In the case of the non-equilibrium mobile-immobile pore exchange an analytical solution is used. The module TANS123D of our simulation code enables the solution of composite heat transport between rock and groundwater. The operator splitting method is employed. The solution of heat convection in rock is based on Mixed- Hybrid FEM, whilst heat transport by the groundwater flux is solved by FVM. Heat exchange between rock and water is solved analytically. TAN123D employs the vector of filtration velocities, generated by the FLOW123D module. 2

The mentioned modules use the same computational mesh i.e. composite meshes with compatible or incompatible connection of elements of different dimensions are supported. The computational meshes as well as outputs are mainly generated by the GMSH programme (http://geuz.org/gmsh/). To illustrate the basic capabilities of the modules FLOW123D and TANS123D a simple task is presented below. A model problem is shown in the following three images. Let s consider a block of rock of 800 800 800m. The piezometric head on the block boundary is set to zero. The hydraulic conductivity of the whole rock is K=3.17 10-6 m/year. We simulate groundwater pumping (650 m deep, discharge of 80 L/s). The vectors of groundwater flux are in the Fig. 1. Figure 1: Groundwater flux vectors during the pumping. At a depth of 800 m the temperature is set to 80 C. Towards surface the temperature decreases linearly down to 20 C. The heat conductivity of rock is λ = 3.0 W/m/K, porosity n =10 %, rock density ρ =3000 kg/m 3, heat capacity of rock c = 1.0 J/g/K. The temperature field without and during pumping is shown in Fig. 2 and Fig. 3. 5. APPLICATIONS 5.1 Cajamarca Model Cajamarca is a region in Peru, where the development of a possible spa centre is being considered by the Peruvian side. Pumping of thermal groundwater may significantly affect the area. Over-pumping may cause decrease in temperature and due to the infiltration of surface waters pollutants from a near industrial area could badly affect the thermal waters. Therefore, a mathematical hydrogeological model was designed to investigate the groundwater flow and, afterwards, the heat flux in the area. Geological and hydrogeological survey has been performed by several institutions (Peruvian geological survey INGEMET, Czech Geological Survey) in the region. Figure 2: Cross-section of the temperature field without pumping. Figure 3: Cross-section of the temperature field during the pumping (steady-state). Figure 4: Delimitation of the simulated region (white rectangle) on the map, regional tectonic faults (black lines) and thermal springs (blue circles). The simulated region (white rectangle in the Fig. 4) was determined with respect to the natural conditions and to the physical processes expected under the ground. The area is approximately 11 14.6 km. It covers regional tectonic faults and the infiltration area (pink colour in the south on the Fig. 4). There are two significant thermal springs Perolitos and Tragadero there. 3

Precipitation and spring discharge data were provided by the Peruvians. The precipitation was taken from nearby meteorological stations. According to the provided data the precipitation is lower than 800 mm per year. The springs, together with nearby discharges, yield up to 130 L/s. Table 1: Velocity of the infiltration in accordance to precipitation per year and percentage of the infiltration. [L/s] 20 % 30 % 35 % 40 % 600 mm 45.8 68.7 80.2 91.6 650 mm 49.6 74.4 86.8 99.2 700 mm 53.4 80.1 93.4 106.8 750 mm 57.2 85.8 100.1 114.4 800 mm 60.1 91.5 106.8 122 The groundwater aquifer consists of a well-permeable folded layer of quartzite and quartzite sandstones Chimu pink layer on the profile in the Fig. 5. This layer is present within the whole area and outcrops in the core of the anticline Yumagual, in the southern part of the area. The Yumagual anticline is supposed to be the main infiltration area. The intersection of regional tectonic faults and the Banos de Inca anticline forms the thermal springs. In the central part, the layer Chimu is connected to another permeable layer of the Farrat sandstones. In the rest of the area the aquifer is covered by impermeable upper layers comprising claystones and siltstones (formations Santa and Carhuaz). The simulated domain was constrained to cover the hydrologically substantial structures and the area of infiltration. The model was constructed just for the main aquifer. The used computation mesh (Fig. 6) consists of 32512 elements (656 triangles, others are tetrahedrons). At first we calibrated the current underground flow to fulfill the overall hydrological balance (assumed infiltration and spring flu and conceptual model. The calibrated hydraulic conductivity of porous media varies from the 0.01 up to 10 10-5 m/s, hydraulic conductivity of tectonic faults varies from 100 to 1000 10-5 m/s, the fracture under the spring is considered to be well permeable with hydraulic conductivity 0.1 m/s. On the boundaries of the simulated layer boundary condition for groundwater flow simulation were set up in accordance to the conceptual model. After the calibration of current state simulations of possible water pumping were carried out to get a presumption of future spa extent. According to the results (see Tab. 2) we suppose a significant decrease of spring flux in the case of pumping more than 50 L/s. Fig. 7 provides a look on the groundwater flux in the simulated domain. On the basis of the calibrated groundwater flow model a simulation of heat transport was performed. For the first calculation we used uniform heat parameters for the whole domain the heat conductivity of rock λ = 2.0 W/m/K, porosity of the porous media n =1 %, porosity of fractures n =10 %, rock density ρ =2500 kg/m 3 and heat capacity of rock c = 0.85 J/g/K. The boundary conditions for heat transport were calculated according with the depth of the particular part of the boundary. A growth of temperature of 70 C per kilometer of depth was considered. Then simulations for the current situation and for the case of pumping up to 100 L/s were performed. Tab. 2 shows the temperature of spring water and temperature obtained as results of simulations for four considered cases. 4 Table 2: Some results of the performed simulations. Water pumping [L/s] 20 50 100 Spring flux[l/s] 97 83 62 29 Spring temperature [ C] 66.7 66.3 65.5 62.2 Temperature of pumped water [ C] 60.0 60.2 60.8 Figure 5: Lengthwise cross-section. Figure 6: The computational mesh. Figure 7: Piezometric head in the main aquifer layer. Figure 8: Groundwater flux in the simulated domain, the position of the spring and water pumping are highlighted by a circle.

5.2 Melechov Model The second real application of our simulation tool is related to the prospecting for localities suitable for geothermal exploitation in the Czech epublic. In this model simultaneous injection and pumping groundwater from the depth of about 2 km were simulated. The model was designed for a fictive region, inspired by geological and hydrogeological configuration of the Melechov massif, central Bohemia. The model was developed to attempt an evaluation of the stability of possible future geothermal resources and their capacity. The studied area and its fundamental geological structures are shown in Fig. 9. There (i) the green line is the boundary of the area, (ii) black lines represent proven regional subvertical tectonic faults, (iii) orange lines show tectonic structures as extended to the rest of the area. At about 150 m below the surface a subhorizontal tectonic fault is considered through the whole region. The Melechov massif consists of four basic rocks Melechov granite, Lipnice granite, Kouty granite and gneiss. These types are distinguished in the Fig. 9. Figure 9: Simulated area with boudaries and main geological structures. A mesh was designed with the respect to all geological structures. It covers an area of approximately 15 km 2, reaching down to 2 000 2 300 m. The computational mesh comprises 47 000 2D and 3D simplex elements, with the characteristic length of lines equal to 200 m. In the first step a calibration of the present hydrogeological conditions was performed, according to the basic regional hydrogeological characteristics: The deep groundwater is recharged by approximately 5 % of the average precipitation amount. Hydrogeological research revealed that the groundwater level mostly lies up to 10 m deep, even at higher parts of the region (hills). Thus, regarding the model resolution, it can be assumed that the profile of piezometric level corresponds to the surface. Figure 10: Vertical W-E cross-section of the actual distribution of piezometric head (values from 360 (blue) to 680 m(red)), W-E cross. The hydraulic conductivity of rock depends on the rock type and the depth. It varies from 10-7 do 10-12 m/s. In tectonic faults the hydraulic conductivity ranges from 10-5 do 10-7, depending on the depth of the structure. Flow simulation provided the piezometric head distribution and velocity vectors of the actual state (Fig. 10 and 11). Consequently simultaneous injection and pumping (100 L/s) were introduced. Points of injection and pumping were situated in the central part of the massif, with spacing of 100 m and about 2 km below the surface. The distance between points was chosen with respect to the scale and the resolution of the model. In the surroundings of the injection and pumping points the groundwater flow is influenced as illustrated in Fig. 12. Figure 11: Velocity field along the vertical W-E crosssection. Next, heat flux and temperature distribution in the massif were computed. Boundary conditions were set according to the average temperature gradient of 3 O C per 100 m. The average surface temperature and the water temperature of injected water were considered to be 10 C for both. The heat conductivity of rock is λ = 3.0 W/m/K, porosity of the porous media n =1%, porosity of fractures n =10%, rock density ρ =2800 kg/m 3, heat capacity of rock c = 0.85 J/g/K. The boundary conditions for heat transport were calculated according to the depth of the particular part of the boundary 5 Figure 12: Velocity field with injection and pumping in a vertical W-E cross-section (in the plane of the simulated wells)

From the results heat flux and temperature of pumped water were acquired. According to the performed simulation, the average heat flux in the simulated domain varies from 20 to 100 mw/m 2 and depends on the rock type and the particular structure. It was revealed that between the points of injection and pumping the heat flux significantly increases up to 3 W/m 2. For the given characteristics of the model the computed temperature of pumped groundwater is 66 C. In Fig. 13 14 some results of the simulations are presented. Figure 13: Distribution of temperature in a vertical W-E cross-section (in the plane of the simulation wells) during injection and pumping. Figure 14: Heat flux in a vertical W-E cross-section (in the plane of the simulation wells) during injection and pumping. The local influence of the wells is apparent. This simulation is one of the first models performed after the new module of heat transport had been implemented. The simulation was performed on a fictive area having geological features of a real locality. This model should be viewed as simplified, predominantly aiming to verify the mathematical model on real data. In our next work, the model will be refined and used to study the possibilities of geothermal exploitation in the Czech epublic. 6. CONCLUSION The paper deals with a heat transport model implemented in our simulation code FLOW123D. The important feature of the software is the ability to simulate underground flow, solute transport and heat transport in the saturated parts of a fractured porous rock environment. The computational mesh of the model could involve subdomains of different dimensions (with 3D, 2D and 1D elements). The first realized tests and real models presented in this paper provide reliable results. The applications describe in the paper will be further refined to provide more details. Further, the software will be employed to support research of geothermal energy exploitation. EFEENCES Maryška J., Severýn O., Vohralík M.: Numeric simulation of the Fracture Flow with a Mixed-hybrid FEM Stochastic Discrete Fracture Network Model. Computational Geosciences 8, (2005), 217-234. Královcová J., Maryška J., Severýn O., Šembera J.: Formulation of mixed-hybrid FE model of flow in fractured porous medium. Proceedings of ENUMATH, Numerical Mathematics and Advanced Application, Santiago de Compostela, Spain (2005). Severýn O.,Maryška J., Královcová J.,Hokr M.: Modeling of groundwater flow and contaminant transport in hard rock using multidimensional FEM/FVM. Proceedings of GeoProc 2006, Nanjing (2006). Severýn O., Maryška J., Královcová J., Hokr M.: Numerical simulation of borehole tests in fractured rock at Potůčky site. Proceedings of 4th Workshop on hard rock hydrogeology of the Bohemian Massif, Jugowice (2006b). Acknowledgements: This work has been supported by the Czech Science Foundation (GAČ), project number 205/09/1879, and the Ministry of Education of the Czech epublic (MŠMT), project number 1M0554. 6