Thermal Effects of Carbon Dioxide Sequestration in the Subsurface

Transcription

1 Universität Stuttgart - Institut für Wasserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung Prof. Dr.-Ing. Rainer Helmig Diplomarbeit Thermal Effects of Carbon Dioxide Sequestration in the Subsurface Submitted by Anozie Ebigbo Matrikelnummer Stuttgart, 17th February 2005 Examiners: Prof. Dr.-Ing. Rainer Helmig, Dr.-Ing. Holger Class Supervisor: Dipl.-Geoökol. Andreas Bielinski

2 Contents 1 Introduction The Greenhouse Effect Carbon Dioxide Sequestration Natural Carbon Sequestration Carbon Dioxide Sequestration in Geological Formations Model Concept Fundamental Terms Phases and Components Primary Variables Secondary Variables Mass/Mole Fraction Diffusion Capillary Pressure Relative Permeability Model Mathematical Formulation Conservation of Momentum Conservation of Mass Conservation of Energy Numerical Scheme Box Method Linearisation Time Discretisation MUFTE UG Physical Properties Phase Diagram of Carbon Dioxide Enthalpy Enthalpy of Water Enthalpy of Carbon Dioxide Enthalpy of Brine I

3 CONTENTS II 3.3 Heat Conductivity Other Physical Properties Simulations Definition of the Domain Initial and Boundary Conditions in the Domain Definition of the Conditions in the Well Injection at m Case Case Case Case Injection at other depths Case Case Summary Outlook

4 List of Figures 1.1 Increase in atmospheric CO 2 concentration Capillary pressure-saturation relation Relative permeability-saturation relation Model concept Box method Newton-Raphson method MUFTE UG Phase diagram of CO Specific enthalpy of water, h(p) at constant temperature Specific enthalpy of water, h(t) at constant pressure Specific enthalpy of CO 2, h(p) at constant temperature Specific enthalpy of CO 2, h(t) at constant pressure Specific enthalpy of water and CO Maximum solubility of NACL in water Enthalpy of brine Enthalpy of NaCl and enthalpy of dissolution for saturated solutions Change of enthalpy of brine w.r.t. NaCl concentration Heat conductivity of a fluid-filled porous medium (1) Heat conductivity of a fluid-filled porous medium (2) Definition of the domain Conditions in the well CO 2 saturation for case 1 at different time steps Temperature, pressure and velocity distribution for case Temperature, pressure and velocity distribution for case Velocities for case Temperature, pressure and velocity distribution for case CO 2 phase state in the domain for case CO 2 Saturation for case 4 at different time steps Saturation, temperature, pressure and phase state of CO 2 for case Temperature for case III

5 LIST OF FIGURES IV 4.12 Definition of the domain for case Temperature, pressure and velocity distribution for case Definition of the domain for case Temperature, pressure and velocity distribution for case Pressure difference and CO 2 saturation for case

6 List of Tables 1.1 Concentration of some greenhouse gases in the atmosphere Parameters implemented in the simulation for case Parameters implemented in the simulation for case Parameters implemented in the simulation for case Parameters implemented in the simulation for case Parameters implemented in the simulation for case Parameters implemented in the simulation for case V

7 Nomenclature Symbol Unit Definition k r Relative permeability p P a Pressure p c P a Capillary pressure h m Piezometric head g m/s 2 Acceleration due to gravity z m Vertical coordinate h J/kg Specific enthalpy u J/kg Specific internal energy v m 3 /kg Specific volume c J/(kgK) Specific heat capacity v α m/s Velocity of phase α x Mole fraction m mol/kg Molality S Saturation K m 2 Intrinsic permeability K eff m/s Effective permeability D pm m 2 /s Diffusion coefficient J kg/(m 2 s) Mass flux J H J/(m 2 s) Heat flux T K Temperature X Mass fraction M g/mol Molecular mass U J Internal energy H J Enthalpy V m 3 Volume φ Porosity ϱ kg/m 3 Density µ kg/(ms) Dynamic viscosity λ W/(mK) Heat conductivity

8 Nomenclature VII Symbol Unit Definition ϑ C Temperature Subscript α w CO 2 g f pm ls Definition Phase Water phase CO 2 phase Gas phase Fluid Porous medium Solution Superscript κ w CO 2 Definition Component Water component CO 2 component

9 Chapter 1 Introduction 1.1 The Greenhouse Effect Part of the energy emitted by the sun gets absorbed by the surface of the earth which gets warmer and in turn emits energy into space through the atmosphere. The greenhouse gases absorb part of that energy preventing all of it from leaving the planet. The natural greenhouse effect (so called because the responsible gases and processes already existed before the existence of human beings) keeps the temperature of the atmosphere near to the surface of the earth at an average of 15 C compared to -6 C for a fictive atmosphere completely free of greenhouse gases. The anthropogenic greenhouse effect is the additional effect caused by the greenhouse gases emitted into the atmosphere as a result of human activity such as burning of fossil fuels and deforestation. The gases responsible for the natural greenhouse effect are water vapour and CO 2. Most of the water vapour in the atmosphere gets there through evaporation from the surface of the oceans. Human activity does not significantly affect its concentration in the atmosphere. This is not the case for CO 2. The concentration of CO 2 in the atmosphere has increased considerably (25 %) since the beginning of the industrial revolution (eighteenth century). This increase leads to an increase in the greenhouse effect and hence to global warming. The global temperature has risen by 0.5 C in the same time (Houghton (1994) [11]).

10 1.1 The Greenhouse Effect 2 Figure 1.1: Increase in atmospheric CO 2 concentration as measured at the Mauna Loa observatory, Hawaii since 1957 (monthly averaged values) (Keeling & Whorf (2004) [12]) Other greenhouse gases whose concentrations in the atmosphere have been increasing as a result of human activity include methane (CH 4 ) and nitrous oxide (N 2 O). Gas Concentration in the atmosphere in parts per million (ppm v ) Carbon dioxide (CO 2 ) 356 Methane (CH 4 ) 1.8 Nitrous oxide (N 2 O) 0.3 Table 1.1: Concentration of some greenhouse gases in the atmosphere (1993)(Houghton (1994) [11]) If nothing is done to reduce the rate of increase of CO 2 emissions into the atmosphere, the average global temperature will probably rise by about 2.5 C within a century. The negative effects of such global warming include: Global sea level rise with all its consequences (flooding of coastal areas, intrusion of salt water into aquifers compromising our water supplies etc.).

11 1.2 Carbon Dioxide Sequestration 3 Increased occurrence of droughts. Extremes in regional climate and increased storms. The changes in climate can affect forests and agricultural productivity and expand deserts. Spread of infectious diseases. 1.2 Carbon Dioxide Sequestration The capture and long-term storage of CO 2 is one way of attempting to reduce the amount of atmospheric CO 2. It must however be accompanied by efforts to reduce CO 2 emissions by for example Reduction of the use of fossil fuels as an energy source and greater efforts to change to renewable energy such as biomass energy, solar, wind and water power. Reduction of deforestation and increased reforestation. Increased energy saving methods Natural Carbon Sequestration The most important natural process involved in the capture of CO 2 is photosynthesis. However most of the carbon captured by photosynthesis gets back into the atmosphere. An example for the natural, long-term storage of carbon can be found in the ocean. The huge plankton population in the oceans captures CO 2 and produces a lot of organic matter. The remains of the plankton sink deep into the ocean. About 1 % reaches the ocean floor and remains there for centuries or even millenia Carbon Dioxide Sequestration in Geological Formations The method of sequestration dealt with in this thesis is the injection of CO 2 into deep geological formations. CO 2 is injected at about 1 km below the ground surface into salty aquifers. At such depths the pressure and temperature are so high that the state of the CO 2 is supercritical. Supercritical CO 2 is denser and more viscous than gaseous CO 2 and therefore also has a lower mobility and buoyancy. The formation should have a layer with low permeability through which almost none of the CO 2 can flow. Since the groundwater flow at such depths is very slow, the CO 2 is trapped. The temperature distribution during the process is important because it has direct consequences for the physical properties of CO 2.

12 Chapter 2 Model Concept 2.1 Fundamental Terms Phases and Components A phase is homogeneous matter with physical properties that are constant in space (Class (2001) [5]). Fluid phases are immiscible fluids separated by a sharp interface, across which discontinuities in fluid properties exist (Helmig (1997) [10]). Different phases may exist as liquids but there can only be one gas phase. A phase consists of different components which influence its physical properties. The composition of a phase can be expressed using the mole or mass fractions of the different components in the phase Primary Variables Primary variables are the parameters needed to describe the properties of a system (Helmig (1997)[10]). The non-isothermal two-phase two-component model consists of three equations with three unknowns (primary variables); see 2.3. These are pressure p, temperature T and saturation S Secondary Variables Secondary variables further describe the properties of the system but can be calculated from the primary variables using equations of state and constitutive relationships. They include density ϱ, viscosity µ, specific enthalpy h, mass fraction X, capillary pressure p c, relative permeability k r, heat conductivity λ.

13 2.1 Fundamental Terms Mass/Mole Fraction The mole fraction x κ α expresses the composition of a phase. The mass fraction X κ α can be calculated from the mole fraction using the following equation: X κ α = xκ αm κ Σ κ x κ α M κ. (2.1) The physical constraint for the sum of the mass fractions of the components κ within a phase α is expressed as follows: Σ κ X κ α = 1. (2.2) The mole fraction of the component water in the gaseous CO 2 phase, x w CO 2 is calculated using Dalton s law of partial pressures which states that the pressure of a gas phase p CO2 is equal to the sum of the partial pressures of all the gases in the phase. p CO2 = Σ κ p κ CO 2, 1 = Σ κ p κ CO 2 p CO2. (2.3) Combining (2.2) and (2.3) gives x κ α = pκ CO 2 p CO2. (2.4) The partial pressure of water in the gaseous CO 2 phase is assumed to be equal to the saturation vapour pressure of water. and therefore p w CO 2 = p w sat. (2.5) p CO 2 CO 2 = p CO2 p w sat. (2.6) The solubility of water in CO 2 is very low at high pressures and hence neglected in the model. The phase is therefore assumed to consist only of pure CO 2. The mass fraction of CO 2 in the water phase is calculated using Henry s law. The presence of salt in brine reduces the solubility of CO 2 (salting-out) and so the value of the Henry s law constant depends on the salt content of the brine (Battistelli et al. (1997) [4]): H CO 2 b (ϑ, Xb NaCl ) = H CO 2 w (ϑ) 10 [m k b(ϑ)] [P a] k b salting-out coefficient, H CO 2 w Henry s law constant for pure water, H CO 2 b Henry s law constant for brine, m salt molality, m = 1000 XNaCl b [ mol M(1 Xb NaCl NaCl ) kg water ], M molecular mass of CO 2 [ g ]. mol (2.7)

14 2.1 Fundamental Terms 6 H CO 2 w and k b are calculated using correlations given by Battistelli et al. (1997) [4]. The mole fraction of CO 2 in the water phase is then x CO 2 w = pco2 CO 2 H CO 2 b. (2.8) Diffusion Molecular diffusion is mainly dependent on the concentration gradients. Mechanical dispersion depends on the velocity fluctuation of a fluid in the porous medium and since the velocity is very low, will be neglected. The diffusive flux is calculated using Fick s law, J κ α = ϱ α D κ α X κ α (2.9) and the following condition for the total flux in each phase. Σ κ J κ α = 0 κ {CO 2, w}. (2.10) D κ α is the diffusion coefficient of the component κ within the phase α. The diffusive mass flux is usually accompanied by a diffusive heat flux: Capillary Pressure J H α = Σ κj κ α hκ α. (2.11) At the interface of a multiphase system, there is a difference between the pressure of the wetting and non-wetting phases. This difference which is dependent on the pore space geometry and on the saturation is the capillary pressure. It can be calculated with the method by van Genuchten (Class (2001) [5]). p c = 1 α S e = Sw Swr 1 S wr m = 1 1. n ( S 1 m e 1 ) 1 n. (2.12) S e is the effective water saturation, S w is the water saturation and S wr is the residual water saturation, m, n and α are curve-fitting parameters based on experimental data. Figure 2.1 shows the capillary pressure with respect to water saturation using the following values (which are the values used in the simulation): parameter value S wr 0.2 S gr 0.05 α n 1.67

15 2.2 Model 7 1.0E E E E p c [Pa] 6.0E E E+05 k r [-] k r w k r n 3.0E E E E S w [-] S w [-] Figure 2.1: Capillary pressure-saturation relation Figure 2.2: Relative permeabilitysaturation relation Relative Permeability The effective permeability for multiphase flow K eff [m/s] is defined as follows K eff = Kk rα ϱ α g µ α. (2.13) The corresponding velocity is then calculated with Darcy s law as follows v α = ( ) p K f α ϱ αg v α = K krα µ α ( p α ϱ α g). (2.14) The intrinsic permeability, K [m 2 ] is only dependent on the properties of the porous medium whereas the relative permeability, k r [-] (which scales the intrinsic permeability to give the effective permeability) is a function of pore space geometry, tortuosity and saturation. The relative permeability-saturation relation by van Genuchten for the same parameters as used above is shown in figure 2.2 (Class (2001) [5]). 2.2 Model k rw = S e [1 k rn = (1 S e ) 1 3 ( 1 S 1 m e ) m ] 2. [ 1 S 1 m e ] 2m. (2.15) For the CO 2 -water system in the subsurface, we assume a non-isothermal two-phase two-component model. The phases are water (w) and CO 2 (CO 2 ) and the components

16 2.3 Mathematical Formulation 8 are water (w) and CO 2 (CO 2 ). The solubility of CO 2 in the water phase is dependent on temperature and pressure as well as on salt concentration. The presence of salt reduces the CO 2 solubility in the water phase (salting-out). For a gaseous CO 2 phase and a liquid water phase, the processes involved in mass transfer from one phase to the other are shown in figure 2.3. CO phase 2 waterphase CO 2 water evaporation condensation water CO 2 dissolution degassing Figure 2.3: Model concept 2.3 Mathematical Formulation In order to describe the two-phase flow and transport processes in porous media, one needs the conservation laws of mass, momentum and energy Conservation of Momentum For the balance of momentum, we use a generalised form of Darcy s law (Helmig (1997) [10]) Conservation of Mass v α = K k rα µ α ( p α ϱ α g), α {w, CO 2 }. (2.16) The balance of mass is taken for each component in both phases and therefore any transfer of mass from one phase to the other can be accounted for (Class (2001) [5]).

17 2.3 Mathematical Formulation 9 We have two equations per phase. φ (ΣαϱαXκ α Sα) t { k rα accumulation term } Σ α div µ α ϱ α Xα κk ( p α ϱ α g) convection term Σ α div { Dpm κ ϱ } α Xα κ diffusion term q κ source/sink term = 0, κ {w, CO 2 }, α {w, CO 2 }. (2.17) In the above equation (2.17), equation 2.14 (Darcy s law) has been inserted for the velocity Conservation of Energy Just as it is with mass, the balance of energy is done in order to describe the heat transfer in the multiphase system (Class (2001) [5]). It states that the change of internal energy U in a system equals the sum of the thermal energy flux Q (heat conduction) across the system boundaries, the work done by change in volume W V and the dissipation work W D (in our context the dissipation work can be neglected since the velocity gradients are very small). φ (ΣαϱαuαSα) t + (1 φ) ϱscst accumulation term t div(λ pm { T ) } conduction term k Σ α div rα µ α ϱ α h α K ( p α ϱ α g) convection term Σ α Σ κ div { } Dpmϱ κ α h κ α Xα κ diffusion term q h source/sink term = 0, κ {w, CO 2 }, α {w, CO 2 }. (2.18) c s is the specific heat capacity of the soil grains, λ pm is the average heat conductivity of the porous medium. In (2.18), we consider the thermal energy balance for all fluid phases plus the balance for the solid phase. We can assume local thermal equilibrium because of low flow velocities and relatively small grain sizes (T matrix = T fluids = T ). This allows us to formulate a single balance of thermal energy for the whole fluid-filled porous medium. We now have a system of three coupled non-linear partial differential equations with the following supplementary constraints: Σ α S α = 1, p w = p CO2 p c. (2.19) p c is the capillary pressure between the two phases.

18 2.4 Numerical Scheme Numerical Scheme Box Method The balance equations (2.17, 2.18) can hardly be solved analytically and are therefore solved numerically. To this end, MUFTE UG (which is the simulation program used in this thesis, see 2.5) uses a subdomain collocation finite volume discretisation method (box method). The discretisation describes the primary and secondary variables exactly only at the nodes of the elements. In-between the nodes, the values are approximated using linear ansatz functions, N j. The geometry is also approximated using the same ansatz functions (iso-parametric concept). For example, the approximated temperature function: T = Σ n nodes j=1 T j N j, n nodes = number of nodes. (2.20) Insertion of the approximated values into the weak form (which is obtained by integrating the balance equations over a control volume, G) of the discretisation method produces an error, ε in the equations. This error can be weighted in such a way that after integrating over the whole domain, G the error term becomes zero on the average (weighted residual method). G W iεdg! = 0, i = 1, 2,, n nodes. W i weighting function. For the box method the weighting functions are defined as follows (2.21) W i = { 1 inside the box, Bi. 0 otherwise. (2.22) Using the nodal values, the gradient of a variable (in the balance equation) is then approximated as follows Σ j T j N j = Σ j i (T j T i ) N j. (2.23) The mobility λ α = krα µ α of the convection term is described using the fully-upwind method, which assumes the value of the upstream node.

19 2.4 Numerical Scheme 11 control volume subdomain, B i i center of gravity of element, e finite element mesh j η element e Figure 2.4: Box method η is the set of neighbouring nodes Linearisation The system of differential equations to be solved is non-linear (mainly because of the non-linear capillary pressure-saturation and relative permeability-saturation relations). In order to solve it, a linearisation is necessary. In this case, the Newton-Raphson method (Class (2001) [5]) is used. For the nonlinear differential function, R of the vector of primary variables, u at the current node, the Newton-Raphson method reads R(u) = 0 u n+1 = u n ( ) R 1 u n R(un ). R = Jacobi matrix, calculated by numerical differentiation. u n current iteration step. Figure 2.5 shows this method for a one-dimensional case. (2.24)

20 2.5 MUFTE UG 12 R(u) R u R(u n ) R(u n+1 ) exact solution, u u n+1 u n u Figure 2.5: Newton-Raphson method Time Discretisation A finite difference scheme is used for the time discretisation. Consider the following ordinary differential equation of the form A finite difference scheme yields u t = f(u). (2.25) u t+ t u t t = f(u t+ t ). (2.26) f is evaluated at the next time step t + t which means that the scheme is implicit (implicit Euler) (Helmig (1997) [10]). 2.5 MUFTE UG As mentioned earlier, MUFTE UG is the software program used to implement the numerical scheme in this thesis. MUFTE (Multiphase Flow, Transport and Energy Model) describes the physical laws of the system and implements the discretisation

21 2.5 MUFTE UG 13 methods. See Helmig et al (1994) [9]. UG (Unstructured Grids) is a toolbox for solving the differential equations (including the handling of geometry data, mesh generation, implementation of linear and nonlinear solvers). See Bastian et al (1997) [3]. (Helmig et. al 1997, 1998) (Bastian et. al 1997, 1998) (S. Lang, K. Birken, K. Johannsen et. al 1997) Institute for Hydraulic Engineering (IWS) - problem description - constitutive relationships - physical-mathematical models - discretization methods - numerical schemes - refinement criteria - physical interpretation Interdisciplinary Center for Scientific Computing (IWR) - multigrid data structures - local grid refinement - solvers (multigrid, etc) - r,h,p-adaptive methods - parallelization - user interface - graphic representation MUFTE (Helmig) UG (Wittum, Bastian) Figure 2.6: MUFTE UG

22 Chapter 3 Physical Properties 3.1 Phase Diagram of Carbon Dioxide As can be seen in figure 3.1, the triple point of CO 2 is at about C and 5.1 bar. The critical point is the point at which the differentiation between gas and liquid is no more possible. For CO 2 it is at 31.1 C and 72.8 bar. p [bar] 72.8 solid liquid critical point triple point gas (T sub) (T 3 ) (T crit ) T [K] Figure 3.1: Phase diagram of CO 2

23 3.2 Enthalpy Enthalpy Enthalpy is the amount of energy in a system capable of doing mechanical work. It is defined as the internal energy of the system plus the product of its volume and pressure. The internal energy is the total energy of the molecules of a system. H = U + pv [J] pv volume-changing work. Dividing by the mass of the system gives the specific enthalpy, h. H mass = h = u + pv = u + p ϱ (3.1) [J/kg]. (3.2) The specific enthalpy of compressible gases is strongly dependent on the volumechanging work. For liquids however, the volume-changing work is of little importance and is often neglected Enthalpy of Water The enthalpy of brine strongly depends on the enthalpy of its main constituent which is water. The specific enthalpy of pure water is calculated using equations obtained from the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam (IAPWS-IF97) [2] h [kj/kg] C 40 C E E+06 p [Pa] Figure 3.2: Specific enthalpy of water, h(p) at constant temperature

24 3.2 Enthalpy bar 80 bar h [kj/kg] T [K] Figure 3.3: Specific enthalpy of water, h(t) at constant pressure The enthalpy of pure water is proportional to both temperature and pressure, however, as mentioned earlier, the effect of temperature is much greater than that of pressure. This is as a result of the low compressibility of water.

25 3.2 Enthalpy Enthalpy of Carbon Dioxide The specific enthalpy of CO 2 is obtained from Span & Wagner (1996) [17] h [kj/kg] C 40 C 0 0 5E+06 p [Pa] 1E E Figure 3.4: Specific enthalpy of CO 2, h(p) at constant temperature The relatively large changes in specific enthalpy of CO 2 (figure 3.4), compared to that of water, with respect to change in pressure (at constant temperature) result from its change of state. When the pressure achieves the value on the two-phase surface, gaseous CO 2 changes state and becomes liquid. Hence there is a strong and discontinuous reduction in volume which causes the abrupt drop in the specific enthalpy-pressure curve. After the change in state, the specific enthalpy changes only a bit since the compressibility of liquid CO 2 is relatively small. The transition from supercritical to gaseous CO 2 however, is continuous. As can be seen in figure 3.5, the specific enthalpy of CO 2 as a gas changes slowly with temperature, as a result of its relatively low specific heat capacity in that state. The second curve in figure 3.5 (80 bar) shows the change from the liquid state (with very low specific enthalpy as a result of the low specific volume) to the supercritical state accompanied by an increase in volume.

26 3.2 Enthalpy h [kj/kg] bar 80 bar T [K] -100 Figure 3.5: Specific enthalpy of CO 2, h(t) at constant pressure h [kj/kg] h CO2 h water z [m] Figure 3.6: Specific enthalpy of water and CO 2

27 3.2 Enthalpy 19 In figure 3.6, the specific enthalpy of water and CO 2 are plotted with respect to the elevation. The pressure increases hydrostatically with depth. The temperature also increases with depth by 0.03 K/m (geothermal gradient). The effects of temperature and pressure changes are therefore combined. The strong change in specific enthalpy that occurs during the transition from gaseous to supercritical CO 2 is also shown in the diagram Enthalpy of Brine The specific enthalpy of brine obviously is dependent on the specific enthalpy of the dissolved salts and the amount of salt dissolved. Although geothermal fluids are solutions of chloride, sulfate, and carbonate salts in water, only NaCl (which is the most abundant salt) is taken into account when determining the thermophysical properties of brine. To compensate for that, the effects of all the salts are summed up in a socalled equivalent NaCl content (Michaelides (1981) [13]). The mass fraction of NaCl in halite saturated water as a function of temperature is given by Palliser & McKibbin (1997) [14] and Potter (1975) [7]. The two correlations have been plotted and compared in figure 3.7. The differences between the two are negligible X lsat [-] PALLISER & MCKIBBIN POTTER T [K] Figure 3.7: Maximum solubility of NACL in water The specific enthalpy of the salt solution is computed as follows

28 3.2 Enthalpy 20 (Michaelides (1981) [13]): h Solution (X NaCl, ϑ) = (1 X NaCl ) h W ater (ϑ) + X NaCl h NaCl (ϑ) + X NaCl L h(x NaCl, ϑ) kj [ kgsolution ϑ temperature [ C], kgnacl X NaCl mass fraction of NaCl in brine [ kgsolution kj L h heat of dissolution of NaCl in water [ kgnacl (3.3) The specific enthalpy of NaCl is calculated from its specific heat capacity c with 0 C as reference temperature. h = u = c(t )dt. (3.4) The correlations for the calculation of the specific heat capacity of NaCl are given by Daubert & Danner (1989) [6]. The heat of dissolution of NaCl in water is calculated using Michaelides [13] correlations. T h [kj/kg] h lsmichaelides h water T [K] -100 Figure 3.8: Enthalpy of brine Other approaches add an empirically determined value to the specific enthalpy of water. h Solution (X NaCl, ϑ) = h W ater (ϑ) + h(x NaCl, ϑ). (3.5) In further calculations, however the Michaelides approach will be used.

29 3.2 Enthalpy h NaCl enthalpy of dissolution of NaCl h lsmichaelides 150 h [kj/kg] T [K] -100 Figure 3.9: Enthalpy of NaCl and enthalpy of dissolution for saturated solutions h [kj/kg] h brine h water X NaCl [-] Figure 3.10: Change of enthalpy of brine w.r.t. NaCl concentration

30 3.3 Heat Conductivity 22 The process of dissolution of NaCl in water is endothermic. Hence dissolution causes a reduction of the specific enthalpy of the solution as depicted in figure 3.10 (for T = 10 C and p = 1 bar). The lower specific enthalpy of NaCl compared to that of water also causes a reduction in the specific enthalpy Dissolution of Carbon dioxide The dissolution of CO 2 in brine also has an effect on its enthalpy. As a result of the low solubility of CO 2 in water (which is further reduced by the salt in brine), the effect of CO 2 -dissolution on the enthalpy of brine is neglected. 3.3 Heat Conductivity The heat conductivity of a material is the factor of linear proportionality between the heat flux and the temperature gradient. For fluid-filled porous media, we generally define an average or effective value for the whole medium. The assumption of local thermal equilibrium is a pre-condition for the validity of such an average value. J H = λ pm T. (3.6) Two approaches generally used to describe the heat conductivity of fluid-filled porous media are: λ pm = λ s (1 φ) + λ w φs w + λ CO2 φs CO2 (3.7) the equivalent heat conductivity (Class (2001) [5]) and λ pm = λ Sw=0 pm + S w (λ Sw=1 pm λ Sw=0 pm ) (3.8) the approach by Somerton (Class (2001) [5]). In this case, one needs to obtain the values of the effective heat conductivity of the porous medium when fully saturated with CO 2, λ Sw=0 pm and when fully saturated with water, λsw=1 pm. Obtaining such values means averaging the heat conductivities of the solid particles, λ s and the fluid, λ f (which is water for λ Sw=1 pm and CO 2 for λpm Sw=0 ). Possible methods are: (a) λ pm = λ s (1 φ) + λ f φ (3.9) the weighted arithmetic mean, (b) ( (1 φ) λ pm = + φ ) 1 (3.10) λ s λ f the harmonic mean,

31 3.3 Heat Conductivity 23 (c) the geometric mean. λ pm = λ (1 φ) s λ φ f (3.11) The values of the arithmetic mean are always very high and those of the harmonic mean very low. These are extreme cases and only useful at obtaining upper and lower boundaries for the effective heat capacity. The geometric mean is empirical and often used in geothermal studies. The Hashin-Shtrikman bounds (Hartmann et al. (2004) [8]) are derived by assuming the solid consists of spheres dispersed in the fluid (for the lower bound HS-) or that the fluid is confined in spherical inclusions in the solid phase (for the upper bound HS+). λ HS = λ f + 1 λ s λ f 1 φ + φ 3λ f, (3.12) φ λ HS+ = λ s +. (3.13) 1 λ f λ s + φ 3λ s A method of obtaining the values for the two extreme conditions (S w = 0 and S CO2 = 0) is by calculating the arithmetic or harmonic mean of the two bounds. Figure 3.11 shows the two HS bounds and the harmonic mean inserted into the Somerton equation (3.8) for a sample soil with λ s = 4.5 W/(mK) (Smoltczyk (2001) [16]) and for values of λ water and λ CO2 at 80 bar and 34 C. Figure 3.12 shows the differences between the equivalent heat conductivity as calculated in equation 3.7 and the effective heat conductivity by Somerton (3.8) for the same conditions as in figure The Somerton approach is more realistic. At low water saturation, a small rise in water saturation leads to a strong increase in the effective heat conductivity. This phenomenon is not included in the computation of the equivalent heat conductivity.

32 3.3 Heat Conductivity heat conductivity [W/(mK)] λ HS- HS water saturation [-] Figure 3.11: Heat conductivity of a fluid-filled porous medium (1) heat conductivity [W/(mK)] SOMERTON equivalent heat conductivity water saturation [-] Figure 3.12: Heat conductivity of a fluid-filled porous medium (2)

33 3.4 Other Physical Properties Other Physical Properties Other properties of CO 2 density and viscosity. Water: and water important for mass and heat transport include CO 2 : The density of water is calculated as given by the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam (IAPWS-IF97) [2]. The viscosity of water is obtained from [15] (Reid et al, 1987). The density of CO 2 is given by Span & Wagner (1996) [17]. The viscosity of CO 2 is calculated as given by Vargaftik (1975) [18].

34 Chapter 4 Simulations The thermal effects of CO 2 sequestration in the subsurface can be depicted by a simple case study consisting of a well with an injection point and a homogeneous domain in the water-saturated zone of the subsurface. The set-up, as described in the following sections is based on the In-situ R&D laboratory for geological storage of CO 2 - CO2SINK project [1]. 4.1 Definition of the Domain The homogeneous domain is assumed to be two-dimensional. The CO 2 is injected on the west boundary of the domain Initial and Boundary Conditions in the Domain The pressure is hydrostatic according to the density of the brine. The domain is assumed to be initially completely saturated with water. The temperature increases with depth according to the geothermal gradient of the formation (which is assumed to be 0.03 K/m). The ground surface temperature is taken to be 10 C. The salinity is assumed to be constant for the whole domain at 0.20 kg salt per kg brine. The boundary conditions are Dirichlet for all three variables and for all the boundaries except the west boundary which is assumed to be a no-flow boundary. At the injection point the boundary conditions are defined by the conditions in the well. See figure 4.1

35 4.2 Definition of the Conditions in the Well Definition of the Conditions in the Well The conditions in the well depend on the condition of CO 2 at the surface and the temperature of the groundwater surrounding the well. The pressure in the well depends on the density of CO 2 and the pressure at the top. The temperature in the well depends on a number of factors including the temperature of the surrounding groundwater, heat flux into or out of the well and most importantly on the average velocity of CO 2 while moving through the well. The two extreme cases no change in temperature of CO 2 while in the well and CO 2 having the same temperature as the groundwater will be used for simulation. The mass flux of CO 2 in the well and into the domain is taken to be 1 kg/s (for cases 3 and 4 it is 0.05 kg/s). In order to keep the mass flux constant, it needs to be regulated by a throttle. This can, for example be done at the top of the well (ground surface) which would be easy to carry out because of the accessibility. Another option is to regulate the mass flux at the bottom of the well in which case one can have separate conditions in the well and in the formation. In the following simulations, only the second option (mass flux regulation at the bottom of the well) will be investigated. The Neumann boundary condition at the point of injection into the domain is given as the mass flux in the well. The specific enthalpy of the CO 2 at the point of injection can be calculated from the temperature and pressure of the CO 2 at that point. The Neumann boundary condition for heat flux can then be obtained by multiplying the specific enthalpy by the mass flux.

36 4.3 Injection at m Injection at m Domain BC: dirichlet: p 60 bar dirichlet: T = 28 C dirichlet: S w = 1 z = m BC: neumann: no flow Initial values: p 60 bar + ϱg(z-m) dirichlet: T = 28 C (z-m) dirichlet: S w = 1 BC: dirichlet: p 60 bar + ϱg(z-m) dirichlet: T = 28 C (z-m) dirichlet: S w = 1 injection point BC: dirichlet: p 90 bar dirichlet: T = 37 C dirichlet: S w = 1 z = 900 m Figure 4.1: Definition of the domain The boundary and initial conditions shown above are implemented in cases 1-4. The top of the domain is m below the ground surface. The CO 2 is injected m below the ground surface on the west boundary of the domain. The dimensions of the domain are the same for all the simulations (cases 1-6): 500 m x 300 m (length x height).

37 4.3 Injection at m 29 Well At the ground surface: Temperature: T top Pressure: p top 1 kg/s Injection point m p bottom T bottom h bottom p domain < p bottom h inject = h bottom ṁ The injection rate is kept constant at 1kg/s. There is a pressure drop from the well into the domain. The enthalpy of the CO 2 at the bottom of the well equals its enthalpy just after injection. At the bottom of the well: Temperature: T bottom Pressure. p bottom ṁ = 1kg/s, J = h(p bottom, T bottom ) ṁ Figure 4.2: Conditions in the well Case 1 The temperature of the CO 2 at the top of the well is taken to be 25 C and the pressure 64 bar. These are values obtained through personal communication with partners in the CO2SINK project [1]. As mentioned in 4.2, the temperature of the CO 2 at the bottom of the well depends on a number of factors. The assumption made in case 1 is that the temperature of the CO 2 is constant throughout the well. Using this assumption, the pressure and density of in the CO 2 in the well with respect to depth can be calculated. p bottom = p top + z=m z=0m ϱ (p(z), T ) gdz. (4.1)

38 4.3 Injection at m 30 Equation 4.1 is implicit since the density is dependent on the pressure. It has been solved iteratively. Table 4.1 shows the temperature and pressure at both ends of the well and the relevant properties of the formation - porosity and absolute permeability used in case 1. The pressure and the temperature distribution in the domain are shown in figure 4.1. Properties of the formation Conditions in the well T [ C] p [bar] Porosity, φ [-] Permeability, K [m 2 ] Top Bottom Table 4.1: Parameters implemented in the simulation for case 1 Results The saturation distributions in figure 4.3 show transport of CO 2 in all directions near the injection point as a result of the pressure gradient at that point. As it gets farther away from the injection point, the pressure gradient reduces and buoyancy takes the upper hand driving the CO 2 upwards. Across the throttle at the injection point, the CO 2 experiences a pressure drop (in this case from 128 bar at the bottom of the well to about 80 bar in the formation) and an increase in volume called Joule-Thompson expansion (based on the assumption of adiabatic conditions at the throttle). The expansion is accompanied by a decrease in temperature of about 3 K (for case 1). The vertical velocity is an indicator for the predominant forces and also for the state of the CO 2 in the domain. The high velocities around m are as a result of the expansion during the transition from the liquid/supercritical to the gaseous state. After the change of phase, the velocities reduce again but remain higher than beforehand since the buoyancy of the gaseous CO 2 is higher than that of the CO 2 in the liquid/supercritical state.

39 4.3 Injection at m 31 Sg Sg day 2 days Sg Sg days 5 days Figure 4.3: CO 2 saturation for case 1 at different time steps

40 4.3 Injection at m 32 Sg pg 9.0E E E E E E E E CO 2 Saturation Pressure [Pa] Te vgy 1.2E E E E E E E E Temperature [K] Velocity (vertical component) [m/s] Figure 4.4: Temperature, pressure and velocity distribution for case 1 at time, t 5 days

41 4.3 Injection at m Case 2 In case 2, the temperature of the CO 2 in the well is idealised as having exactly the same temperature as the surrounding groundwater. This an extreme condition that would be valid if the velocity of the CO 2 in the well is very small and if insulation by the casing of the well can be neglected. Table 4.2 shows the resulting pressure at the bottom of the well. The pressure and the temperature distribution in the domain are shown in figure 4.1. Properties of the formation Conditions in the well T [ C] p [bar] Porosity, φ [-] Permeability, K [m 2 ] Top Bottom Table 4.2: Parameters implemented in the simulation for case 2 Results The cooling effect at the injection point for case 2 is about 6 K (which is greater than that of case 1). The temperature distribution shows an additional cooling effect between m and m depth which, judging from the vertical velocities should be the region around which the CO 2 becomes gaseous. It draws the needed energy for evaporation from its surroundings which as a result cool down. Figure 4.6 shows the velocity of the CO 2 (arrows) as well as its vertical component (colour scheme) for part of the domain.

42 4.3 Injection at m 34 Sg pg 9.0E E E E E E E E CO 2 Saturation Pressure [Pa] Te vgy 1.3E E E E E E E E Temperature [K] Velocity (vertical component) [m/s] Figure 4.5: Temperature, pressure and velocity distribution for case 2 at time, t 5 days

43 4.3 Injection at m vgy 1.3E E E E E E E E Figure 4.6: Velocities for case 2 at time, t 5 days Case 3 The properties (porosity, permeability) of the formation used in the previous cases are quite high. In this case, more realistic (lower) values are used in the simulation. The mass flux into the domain as used in the other cases would, if implemented in case 3, give very high pressures. In order to obtain a similar pressure distribution over time, the injection rate is reduced to only 0.05 kg/s. Results As a result of the lower permeability, the velocity of the CO 2 in the formation is much less than in case 1. It takes 9 weeks for it to reach the top of the domain compared to 5 days in case 1. The cooling effect at the injection point, however is about the same for both cases (though the time taken to reach the mentioned effect differs).

44 4.3 Injection at m 36 Properties of the formation Conditions in the well T [ C] p [bar] Porosity, φ [-] Permeability, K [m 2 ] Top Bottom Table 4.3: Parameters implemented in the simulation for case 3 Figure 4.8 shows the state of the CO 2 in the domain using the following simple definition. The CO 2 is defined as supercritical if either the pressure or the temperature exceeds the value at the critical point. liquid if both pressure and temperature are below the critical values and the pressure is greater than the saturation vapour pressure at the given temperature. gaseous if both pressure and temperature are below the critical values and the pressure is less than the saturation vapour pressure at the given temperature. A comparison of figure 4.8 with the vertical velocities in figure 4.7 proves that the higher vertical velocities on the upper part of the domain are as a result of phase change.

45 4.3 Injection at m 37 Sg pg 9.0E E E E E E E E CO 2 Saturation Pressure [Pa] Te vgy 5.5E E E E E E E E Temperature [K] Velocity (vertical component) [m/s] Figure 4.7: Temperature, pressure and velocity distribution for case 3 at time, t 9 weeks

46 4.3 Injection at m 38 gaseous liquid supercritical Figure 4.8: CO 2 phase state in the domain for case 3 at time, t 9 weeks Case 4 The formation should possess a layer with very low permeability in order to prevent the CO 2 from rising to the ground surface. In case 4, there is such a layer between 678 m and 687 m below the ground surface, with the following properties: porosity, φ = , permeability, K = m 2. The other properties of the domain, the conditions in the well and domain and the boundary and initial conditions are identical to those of case 3 (mass flux in the well and into the formation, ṁ = 0.05 kg/s).

47 4.3 Injection at m 39 Properties of the formation Conditions in the well T [ C] p [bar] Porosity, φ [-] Permeability, K [m 2 ] Top Bottom Table 4.4: Parameters implemented in the simulation for case 4 Results Sg Sg months 4 months Sg Sg months months Figure 4.9: CO 2 Saturation for case 4 at different time steps

48 4.3 Injection at m 40 The temperature near the injection point is about the same as in case 1 and 3. That means the cooling effect does not keep on increasing with time. This can also be seen in figure 4.11 which shows the temperature of a node close to the injection point. The temperature can clearly be seen to stabilise at about 22 C for that node. Sg pg 9.0E E E E E E E E CO 2 Saturation Pressure [Pa] Te gaseous liquid supercritical Temperature [K] Phase state Figure 4.10: Saturation, temperature, pressure and phase state of CO 2 for case 4 at time, t months

49 4.3 Injection at m 41 temperature [ C] time [months] Figure 4.11: Temperature for case 4 at a node close to the injection point

50 4.4 Injection at other depths Injection at other depths In the following two cases, the injection is at depths at which the conditions are nearer to the critical point of CO 2 (i.e. at shallower depths) than in the previous cases Case 5 The injection depth in this case is m below the ground surface. At this depth, the conditions in the domain are below the critical temperature and pressure of CO 2. Figure 4.12 shows the temperature and pressure in the domain. The properties of the formation and the conditions in the well are the same as in case 1 (table 4.5), except that the pressure at the bottom of the well in this case is lower as a result of the shallow depth. BC: dirichlet: p 50 bar dirichlet: T = 25 C dirichlet: S w = 1 z = 500 m BC: neumann: no flow Initial values: p 50 bar + ϱg(z-500m) dirichlet: T = 25 C (z-500m) dirichlet: S w = 1 BC: dirichlet: p 50 bar + ϱg(z-500m) dirichlet: T = 25 C (z-500m) dirichlet: S w = 1 injection point BC: dirichlet: p 80 bar dirichlet: T = 34 C dirichlet: S w = 1 z = m Figure 4.12: Definition of the domain for case 5 Properties of the formation Conditions in the well T [ C] p [bar] Porosity, φ [-] Permeability, K [m 2 ] Top Bottom Table 4.5: Parameters implemented in the simulation for case 5

51 4.4 Injection at other depths 43 Results The injected CO 2 reaches the top of the domain sooner than in all other cases. This is because the CO 2 becomes gaseous earlier in this case than in the other cases and therefore rises quickly Sg pg 8.0E E E E E E E E CO 2 Saturation Pressure [Pa] Te vgy 3.9E E E E E E E E Temperature [K] Velocity (vertical component) [m/s] Figure 4.13: Temperature, pressure and velocity distribution for case 5 at time, t days

52 4.4 Injection at other depths Case 6 In this case, the CO 2 is injected m below the ground surface. BC: dirichlet: p 55 bar dirichlet: T = 26.5 C dirichlet: S w = 1 z = 550 m BC: neumann: no flow Initial values: p 55 bar + ϱg(z-550m) dirichlet: T = 26.5 C (z-550m) dirichlet: S w = 1 BC: dirichlet: p 55 bar + ϱg(z-550m) dirichlet: T = 26.5 C (z-550m) dirichlet: S w = 1 injection point BC: dirichlet: p 85 bar dirichlet: T = 35.5 C dirichlet: S w = 1 z = m Figure 4.14: Definition of the domain for case 6 Properties of the formation Conditions in the well T [ C] p [bar] Porosity, φ [-] Permeability, K [m 2 ] Top Bottom Table 4.6: Parameters implemented in the simulation for case 6 Results Case 6 lies between case 1 and 5. The transition to the gaseous state occurs earlier than in case 1 and later than in case 5.