# A Numerical Model of Tracer Transport in a Non-isothermal Two-Phase Flow System for \({\text{ CO}}_2\) Geological Storage Characterization

- 1.5k Downloads
- 6 Citations

## Abstract

For the purpose of characterizing geologically stored \(\text{ CO}_{2}\) including its phase partitioning and migration in deep saline formations, different types of tracers are being developed. Such tracers can be injected with \(\text{ CO}_{2}\) or water, and their partitioning and/or reactive transfer from one phase to another can give information on the interactions between the two fluid phases and the development of their interfacial area. Kinetic rock–water interactions and geochemical reactions during two-phase flow of \(\text{ CO}_{2}\) and brine have been incorporated in numerical simulators (e.g., Xu et al., TOUGHREACT User’s Guide: A Simulation Program for Non-isothermal Multiphase Reactive Geochemical Transport in Variably Saturated Geologic Media. LBNL Report 55460, V.1.2., Berkeley, CA, 2004). However, chemical equilibrium between the fluid phases is typically assumed, and multi-component, multiphase, non-isothermal codes for \(\text{ CO}_{2}\)–brine systems that incorporate kinetic mass transfer of tracers between the two fluid phases are not readily available. New models or further developments of existing models are therefore needed to provide the capability for interpreting the signals of novel tracers, including tracers with kinetic/time-dependent interface transfer. This paper presents such new numerical model of tracer transport in a non-isothermal two-phase flow system. The model consists of five different governing equations describing liquid phase (aqueous) flow, gas (\(\text{ CO}_{2})\) flow, heat transport and the movement of the tracers within the two phases, as well as allowing kinetic transport of the tracers between the two phases. A finite element method is adopted for the spatial discretization and a finite difference approach is used for temporal discretization. Some special technologies and solution strategies are adopted for increasing the convergence, ensuring the numerical stability and eliminating non-physical oscillations. The new numerical model is validated against the code TOUGH2/ECO2N as well as some analytical/semi-analytical solutions. Good agreement between the simulated and analytical results indicates that the model has capability to simulate two-phase flow and tracer transport in a non-isothermal two-phase flow system with high confidence. Finally, the capability to model transport and kinetic mass transfer of tracers between the two fluid phases is demonstrated through examples.

## Keywords

Tracer transport Two-phase flow \(\text{ CO}_{2}\) geological storage Numerical model Finite element method## 1 Introduction

Carbon dioxide (\(\text{ CO}_{2})\) is the most important greenhouse gas, being responsible for over 60 % of the increase in the greenhouse effect (Kiehl and Trenberth 1997). It is widely recognized that an excess of carbon dioxide in the atmosphere increases the average temperature on the earth (e.g., Karl and Trenberth 2003). Geological storage of \(\text{ CO}_{2}\) allows injecting large volumes of \(\text{ CO}_{2}\) into the ground at sites suitable for geological storage (e.g., oil and gas reservoirs, unmineable coal seams, and deep saline reservoirs), and thus contributes to the mitigation of an increased greenhouse effect (Hepple and Benson 2004; Orr and Stanford 2004; Baines and Worden 2001). In order to ensure that the geologically stored \(\text{ CO}_{2}\) remains isolated from the atmosphere in the long term effective site characterization and monitoring techniques are needed and are also being developed.

Typically the storage formations are at depths of 800 to 3,000 m below ground surface and access to these formations is limited to a small number of deep boreholes. In addition, significant uncertainty in the geological properties and their spatial variation between the boreholes usually exists. Consequently, accurate monitoring of the spatial distribution of the injected \(\text{ CO}_{2}\) and its migration and fate is highly challenging. At the same time, in-situ information of \(\text{ CO}_{2}\) migration and trapping processes are crucial both in terms of improving our understanding of the fundamental phenomena as well as in commercial \(\text{ CO}_{2}\) storage projects, in which monitoring of the injected \(\text{ CO}_{2}\) is an important requirement. As a part of the EU-FP7 MUSTANG project (Niemi et al. 2012), different tracers are being developed and tested, specifically for the purpose of characterization of geologically stored \(\text{ CO}_{2}\) and its phase partitioning and migration in deep saline formations. These tracers include both more traditional partitioning tracers used in oil reservoir applications (e.g., Tomich et al. 1973) and \(\text{ CO}_{2}\) storage research (e.g., Zhang et al. 2011), and a set of novel reactive, kinetic interface-sensitive (KIS) tracers (Schaffer et al. in press; Licha et al. 2011; Behrens et al. 2006). Due to reactive processes at the \(\text{ CO}_{2}\)–brine interface, these tracers transfer from the \(\text{ CO}_{2}\) phase to the aqueous phase and thereby carry with them information about the \(\text{ CO}_{2}\)–brine active interfacial area, its development during the migration of the \(\text{ CO}_{2}\) plume. Hence they constitute a potentially powerful tool for in-situ \(\text{ CO}_{2}\) monitoring and research.

To correctly interpret the tracer signals, a numerical simulator which incorporates two-phase flow of \(\text{ CO}_{2}\) and brine as well as tracer transport and kinetic mass transfer of tracers between the two fluid phases is needed. Kinetic models have been developed for geochemical reactions and interactions between dissolved species in the aqueous phase and the solid phase (e.g., TOUGHREACT, Xu et al. 2004) during two-phase flow. However, commonly used two-phase flow and transport codes (Xu et al. 2004; Pruess et al. 1999; Pruess 2005; Hoteit and Firoozabadi 2008; Geiger et al. 2004; Helming 1997; Bear and Bachmat 1991; Aziz and Settari 1979) do not incorporate kinetic mass transfer between the two fluid phases. In this paper, a new numerical model is presented which incorporates kinetic reactions and kinetic tracer mass transfer from one fluid phase to the other in a non-isothermal two-phase flow system of \(\text{ CO}_2\) and brine. The model can be used to interpret tracer signals obtained from field tests and subsequently aid the characterization of the geologically stored \(\text{ CO}_{2}\) including its phase partitioning and migration in deep saline formations.

The paper is organized as follows: First, we present the derivation of the governing equations for tracer transport in a non-isothermal two-phase flow system. The numerical discretization in space and time, and the solution strategies are introduced next (Sect. 3), followed by a discussion of the main constitutive models and parameters involved in the fourth section. Finally, the model verification against a semi-analytical solution and numerical simulations with the TOUGH2/ECO2N code (Pruess 2005) are presented along with model demonstration for the reactive tracer transfer over the phase interface.

## 2 Governing Equations

### 2.1 Aqueous Phase (Liquid) Flow Equation

**g**is the gravitational acceleration vector, and \(\rho _\mathrm{l}\) is the liquid density. The relative velocity of the liquid phase can be expressed as: \(\mathbf{v}_\mathrm{l}^\mathrm{r} =\phi ^\mathrm{l}\mathbf{v}_\mathrm{l} \), where \(\phi ^\mathrm{l}\) is the liquid content. The velocity of the liquid phase (\(\mathbf{v}_\mathrm{l})\) can then be written as:

### 2.2 Gas (\(\text{ CO}_2\)) Flow Equation

### 2.3 Heat Transport Equation

### 2.4 Tracer Transport Equation

In field experiments on \(\text{ CO}_{2}\) transport and trapping in brine aquifers, tracers can be injected into the aqueous or the gas (supercritical \(\text{ CO}_{2})\) phase and their evolution in the two phases can be used as an indicator of partitioning between the phases.

In this study, we are especially focusing on a case where tracers are injected with the gas phase (supercritical \(\text{ CO}_{2})\), and their concentration in the liquid (water) is measured at later times. Therefore, it is necessary to distinguish between the tracer concentrations in gas and liquid phases. Here two independent variables of tracer concentration are needed (the concentration of tracer in gas (\(C_\mathrm{g})\) and the concentration of tracer in liquid (\(C_\mathrm{l}))\), and the movement of tracers in the gas and the movement of tracers in the liquid need to be described by their respective governing equations.

#### 2.4.1 Transport Equation of Tracer in Gas (\(\text{ CO}_2\))

#### 2.4.2 Transport Equation of Tracer in Liquid (Water)

## 3 Main Constitutive Models and Parameters

### 3.1 Characteristic Curves

### 3.2 Density

The density of carbon dioxide is based on its equation of state as presented by Sterner and Pitzer (1994), applicable for temperatures from 215 to 2,000 K and for pressures from zero to over 10 GPa. For a given temperature and pressure, the equation of state of \(\text{ CO}_2\) can be simplified as a nonlinear function of the density of \(\text{ CO}_2\). Thereby, it can be directly solved in the presented model using the so-called dichotomy method by Allen and Isaacson (1997), which is a method for numerical solution of equations in a single unknown.

### 3.3 Viscosity

For the dynamic viscosity of \(\text{ CO}_{2}\), we use the model utilized in the TOUGH2 simulator (Pruess et al. 1999; Battistelli et al. 1997), where viscosity of \(\text{ CO}_{2}\) is a function of pressure and temperature and calculated using the correlation quoted by Pritchett et al. (1981). This formula is based on data tabulated by Vargaftik (1975).

### 3.4 Dissolution of Carbon Dioxide

*m*is salt molality. \(K_\mathrm{h}\) is Henry’s constant for pure water (in units: Pa), and can be calculated as follows by a polynomial fit: \(K_\mathrm{h} =\sum _{i=0}^5 {B(i)T^{i}}\) (Cramer 1982). Here,

*T*is temperature, and the coefficients \(B(i)\) have the following values: \(B(0)=78.3666, B(1)=1.96025, B(2)=8.20574 \times 10^{-2}, B(3)=-7.40974 \times 10^{-4}, B(4)=2.1838 \times 10^{-6},B(5)=-2.20999 \times 10^{-9}\).The salting-out coefficient is expressed as: \({K}_\mathrm{b} =\sum _{i=0}^4 {C(i)T^{i}} \), where the coefficients \(C(i)\) have the following values: \(C(0)=0.119784,C(1)=-7.17823 \times 10^{-4},C(2)=4.93854 \times 10^{-6}, C(3)=-1.03826 \times 10^{-8}, C(4)=1.08233 \times 10^{-11}\).

### 3.5 Tracer Transfer Between Liquid and Gas Phases

*R*is the universal gas constant, and

*T*is temperature at the gas/liquid interface in \({^\circ }\text{ C}\). This specific approach extends common tracer methods (based on simple partitioning) by including the rate of a chemical reaction at the fluid–fluid interface and thus incorporates true kinetically controlled mass transfer from one phase to the other. This new model allows a preliminary assessment of the potential of such kinetic interface-sensitive tracers which constitute a novel investigation tool in multiphase flow applications (Schaffer et al. in press).

## 4 Numerical Discretization and Solution Strategy

### 4.1 Numerical Discretization

#### 4.1.1 Spatial Discretization

The governing equations (5), (8), (9), (13), and (14) are nonlinear differential equations. To solve them, they must be appropriately discretized in space and time. A Galerkin finite element solution approach (Zienkiewicz and Taylor 2000) is used for the spatial discretization of Eqs. (5), (8), and (9). But for equations (13) and (14), the standard Galerkin method, in which interpolation functions themselves serve as weighting functions, cannot be directly used because this will generate numerical solutions with artificial and non-physical oscillations (Younes and Ackerer 2005). In the present numerical model, interpolation functions are linear function of space. In order to ensure local mass conservation, for a given Gauss integration point, the weighting functions of a node are set to zero when the gas (or water) pressure of the node is higher than the gas (or water) pressure of the Gauss integration point, and the sum of other nodes is equal to 1. Simulations show that this method performs well for eliminating any non-physical oscillations, thereby obeying local mass conservation and providing sufficiently accurate results.

When simulating \(\text{ CO}_{2}\) injection (migration of \(\text{ CO}_{2}\) in a brine aquifer), the first order derivative of water saturation with respect to the spatial coordinates is not continuous in the non-wetting phase (\(\text{ CO}_{2})\) front if the initial degree of water saturation is 1.0. Therefore, some numerical oscillations will occur if the interpolation functions themselves serve as weighting functions for spatial discretization (Helming 1997). One of methods adopted in this numerical model to eliminate the numerical oscillations is to set the weighting functions as constants. The numerical aspects of this model, developed for best computational performance, are further discussed in Tong et al. (2012).

#### 4.1.2 Temporal Discretization

*S*) is assumed to be a linear function of time, and can be expressed as:

### 4.2 Solution Strategies

To solve these nonlinear governing equations, convergence, numerical stability and computational efficiency are always the three key components in the numerical solution. When solving the above Eqs. (5), (8), (9), (13), and (14), the Eqs. (5) and (8) are strongly coupled and need to be solved iteratively, while the heat transport and tracer transport equations are relatively independent, at least in this preliminary application, and are therefore not solved simultaneously. The general route for iterative solution is to calculate the two-phase flow equations first, after which the temperature is obtained by solving heat transport equation, followed by solving the tracer transport equations.

Besides the route of solving Eqs. (8) and (24) together, another available iterative route is to first calculate gas pressure by solving Eq. (24) and then calculate the water saturation by solving Eq. (8). This circulating iteration will be kept until the values of gas pressure and water saturation do not change in a given time step. Because the water saturation is mainly related to gas pressure in Eq. (8), and capillary pressure does not appear in Eq. (8), the numerical stability of solving Eq. (8) is better than that of solving Eq. (5). In the examples presented in this paper, the number of iterations is limited to 5 and the convergence criterion is a relative error for gas saturation of less than 0.0001 in every time step. Calculations show that the above described iterative method has a good convergence behavior and numerical stability.

## 5 Model Verification and Demonstration

The verification of the developed numerical model includes three parts. The first part concerns the validation of the heat transport, which was presented by Tong et al. (2009, 2010), and is not repeated here. The second part is the verification of two-phase flow model, which is presented through comparisons against a semi-analytical solution and against numerical simulation results obtained with the TOUGH2/ECO2N model (Pruess 2005). The third part is a comparison between simulated and analytical results for the validation of tracer transport.

### 5.1 Conceptual and Numerical Model

Material properties of the reservoir layer

Permeability | \(1.0\times 10^{-13}\) |

Porosity | 0.15 |

Capillary pressure | Eq. (15) with \(p_{0}=0.133\) bar, \(\lambda =0.4118\) |

Relative permeability of liquid | Eq. (16) with \(s_\mathrm{lr}=0.3,\lambda =0.4118\) |

Relative permeability of gas | Eq. (17) with \(s_\mathrm{lr}=0.3, s_\mathrm{lr}=0.05\) |

Thermal conductivity | \(1.28-0.71/[1+\text{ exp}(10S-6.5)]\) |

The kinetic transfer of the tracers between CO2 and liquid water | Eqs. (21) and (22) with \(k_\mathrm{l}= 6.8 \times 10^{-7}, a_\mathrm{na}= 154.8 \times (1-S),\; A=2.02\times 10^{13}, E_\mathrm{a}=90.0, R=0.008314\) |

Initial conditions and boundary conditions

Initial conditions | \(\text{ CO}_{2}\) pressure | 15.6 MPa |

Temperature | \(50^\circ \text{ C}\) | |

Water saturation | 1.0 | |

Tracer in \(\text{ CO}_{2}\) | 0.52 mol/kg | |

Tracer in liquid | 0.0 | |

Boundary conditions | Gas flow equation, Eq. (8) | Left side: A given \(\text{ CO}_{2}\) injection rate (\(1.0\times 10^{-5}\) m/s or \(2.71 \times 10^{-6}\) m/s) |

Total mass equation, Eq. (24) | Right side: \(P_\mathrm{g}=15.6\) MPa Above and below side: Impermeable | |

Above and below side: Impermeable | ||

Heat flow equation, Eq. (9) | \(T= 50\,{^\circ }\text{ C}\) |

Quadrilateral elements are used to discretize the \(100\;\text{ m} \times 10 \;\text{ m}\) model. After a grid convergence test with different element sizes, the final calculation mesh consists of a total of 2,000 elements and 2,112 nodes.

### 5.2 Verification of the Two-Phase Flow Model

#### 5.2.1 Verification Against a Semi-analytical Solution

For the verification, we first need an analytical or semi-analytical solution. Although the Buckley and Leverett (1942) solution is a well-known analytical solution, it cannot be adopted in this verification because it suppresses the capillary drive term and is only suitable for use when externally applied driving forces are large in relation to the gradient of capillary pressure. In common cases, capillary pressure can not be neglected because it has a significant effect on two-phase flow (Hoteit and Firoozabadi 2008). Until now, although there are still problems in purely analytical approaches, the capillary drive has been considered through numerical solution (Yortsos and Fokas 1983; Chen 1988; Van Duijn and De Neef 1998; McWhorter and Sunada 1990). Based on Fučīk et al. (2007), a semi-analytical solution was obtained for the validation of the new model. Note that, for this verification case, the gravity term is not included. This essentially renders the comparison to be one-dimensional, even though we still simulate the two-phase flow problem on the two-dimensional rectangular domain using the new model. Temperature is constant (\(50\,^\circ \text{ C}\)). The compression of gas was neglected and the flow rate of gas was set to \(1.0 \times 10^{-5}\) m/s on left injection boundary.

#### 5.2.2 Verification Against TOUGH2/ECO2N

Figures 3 and 4 show the comparison between the spatial distributions of \(\text{ CO}_{2}\) saturation simulated with the new model and with TOUGH2/ECO2N at 12 and 20 days after the start of the injection, respectively. The results show a good agreement in most of the model domain, with only differences occurring in the region of the \(\text{ CO}_{2}\) front. The main reason for this difference is that different interpolation functions are adopted in the new model and in TOUGH2/ECO2N for the spatial discretization of the governing differential equations. In comparison to the finite volume method (FVM) used in TOUGH2/ECO2N, the finite element method (FEM) adopted in the new numerical model has a higher precision in the regions of large gradient of \(\text{ CO}_{2}\) saturation. Thus the gradient of \(\text{ CO}_{2}\) saturation simulated by the new model is bigger than that simulated by TOUGH2/ECO2N, in the area of \(\text{ CO}_{2}\) front. This indicates that the simulated \(\text{ CO}_{2}\) front (interface) is sharper in the new model than in TOUGH2/ECO2N. One the other hand, TOUGH2/ECO2N has a better numerical stability, and the distribution of \(\text{ CO}_{2}\) saturation simulated by TOUGH2/ECO2N at the front is smoother than that of the new model.

### 5.3 Verification of the Tracer Transport

### 5.4 Simulation of Tracer Transport in Two-Phase Flow Including Kinetic Transfer of the Tracer

As a whole, it is necessary to present a simulation of tracer transport in two-phase flow, including the tracer transfer from one phase to another. There are, however, no analytical or numerical models that can be used as basis for comparison, so the simulation at this point only provides a demonstration example. Eventually, our objective is to obtain experimental data that will form the basis of a validation. In the present simulation, the injection flow rate of \(\text{ CO}_{2}\) is \(2.71 \times 10^{-6}\) m/s. The concentration of tracer in \(\text{ CO}_{2}\) on the left hand side injection boundary is 0.52 mol/kg. The initial concentration of tracer in the water is zero. The other computational conditions were defined previously in Sect. 5.1 Based on Grant and Gerhard (2007b); Brusseau et al. (2006), and Miller et al. (1990), the main parameters related to the kinetic mass transfer of tracer between the \(\text{ CO}_{2}\) and and aqueous phases are set as follows. The average mass transfer coefficient is \(k_\mathrm{l}= 6.8 \times 10^{-7}\), the total specific interfacial area is \(a_\mathrm{na} =154.8 \times (1-S_\mathrm{l})\), the pre-exponential factor \(A = 2.02 \times 10 ^{13}\), the activation energy \(E_\mathrm{a} = 90.0\) and the universal gas constant R = 0.008314. In this simulation, we focus on advective transport and kinetic transfer of the tracer, thus we have neglected the dispersion effect.

## 6 Concluding Remarks

We present a new numerical model of tracer transport in a non-isothermal two-phase flow system of \(\text{ CO}_{2}\) and brine, adding capability to interpret tracer signals including tracers with kinetic/time-dependent interface transfer. The model consists of five different governing equations, describing liquid phase (aqueous) flow, gas (\(\text{ CO}_{2})\) flow, heat transport and the movement of the tracers within the two fluid phases, as well as allowing kinetic transport of tracers between the phases. Geochemical reactions and interactions between the dissolved species in the aqueous phase and the solid rock have been incorporated in some two-phase flow codes (e.g., Xu et al. 2004). However, kinetic reactions at the interface between the two fluid phases and kinetic mass transfer over the fluid–fluid interface are features which typically are not available in codes used for \(\text{ CO}_{2}\) storage simulation, and therefore the motivation for the presented model development.

Validation against existing analytical, semi-analytical, and numerical solutions shows good agreement, indicating that the model can simulate non-isothermal two-phase flow and tracer transport with confidence. The numerical model could have been also compared with similarity solutions of gravity-driven flow under various approximations (e.g., Huppert and Woods 1995; Nordbotten and Celia 2006; Hesse et al. 2007). However, given that (i) our numerical solution is shown to be in close agreement with those obtained by the widely used and accepted TOUGH2/ECO2N code and (ii) the TOUGH2/ECO2N code has been verified against similarity solutions (see Pruess 2005), it may not be necessary to perform the verification step with similarity solution for gravity currents.

Although the motivation to develop the model was characterization of geologically stored \(\text{ CO}_{2}\) and its phase partitioning and migration in deep saline formations, the model can easily be adapted for other two-phase flow applications, such as NAPL flow and transport.

The solution methodology for the two-phase flow is first to calculate the gas pressure using the equation for conservation of total mass, and then to calculate the water saturation using the gas flow equation. Simulations show that this iterative route is efficient for improving convergence and numerical stability of the two-phase flow calculations, while yet providing accurate results. The Galerkin FEM approach cannot directly be used to solve the tracer transport equations, as artificial and non-physical oscillations will be generated. To overcome this difficulty the weighting functions of a node are set to zero when the gas (or water) pressure of the node is bigger than the gas (or water) pressure of the Gauss integration point. The numerical mesh and the length of the time step also influence the numerical stability and the calculation precision. The numerical results can be further improved if the size of elements can adapt well to the gradient of the primary variables, particularly in the area of wetting front.

The main objective of this work has been to provide a basis for kinetic mass transfer of a tracer between the two fluid phases (the \(\text{ CO}_{2}\) and aqueous phases). The presented model further allows incorporation of a chemical reaction involving or producing a tracer at the fluid–fluid interface, thus allowing reaction-controlled kinetic mass transfer from one fluid phase to the other. In a demonstration example, we show that the tracer concentration in the aqueous phase is sensitive to the size of the interfacial area (\(\text{ a}_\mathrm{an})\) and monotonously increases for increasing \(\text{ a}_\mathrm{an}\). Consequently, a well-designed tracer can provide information about the evolution of the interfacial area over time. Different reactions and kinetic mass transfer models can be chosen and used with the flow and transport model. The mass transfer model presented here is preliminary in nature, as the work continues to find suitable KIS tracers whose specific mass transfer behavior can later be incorporated in our model.

It should be pointed out that the numerical model is not suitable for regimes with viscous instability as discussed in, e.g., Riaz and Tchelepi (2006) and Riaz et al. (2007) or flow with invalidity of volume-averaging (continuum description) as discussed in e.g., Xu et al. (1998) and Yortsos et al. (2001). Finally, non-equilibrium formulations for capillary pressure–interfacial area–saturation (e.g., Joekar-Niasar et al. 2010) have been proposed in the recent literature. Here, although we have adopted the conventional constitutive relationships (e.g., the van Genuchten model) in the numerical exercises, the numerical model is not limited by these constitutive relationships and it is possible to incorporate the non-equilibrium formulations in our numerical model. However, this is beyond the scope of the current study and motivates further research on this topic. Further work is also underway for continued improvements of the solution technique and additional code verification against laboratory- and field-scale experiments.

## Notes

### Acknowledgments

The research leading to these results has received funding from the European Community’s 7th Framework Programme FP7/2007-2013, under Grant Agreement No. 227286, and from the Swedish Research Council for Environment, Agricultural Sciences and Spatial Planning (FORMAS), Project No. 214-2008-1032. The authors are grateful for fruitful discussions with our MUSTANG project partners.

## References

- Allen, M., Isaacson, E.: Numerical Analysis for Applied Science. Wiley, New York (1997)CrossRefGoogle Scholar
- Aziz, K., Settari, A.: Petroleum Reservoir Simulation, Environmental Engineering. Elsevier, London (1979)Google Scholar
- Bachmann, J., van der Plog, R.: A review on recent developments in soil water retention theory: interfacial tension and temperature effects. J. Plant Nutr. Soil Sci.
**165**, 468–478 (2002)CrossRefGoogle Scholar - Baines S.J., Worden R.H.: Geological $\text{ CO}_2$ disposal: understanding the long term fate of $\text{ CO}_2$ in naturally occurring accumulations. In: Fifth International Conference on Greenhouse Gas Control Technologies, August 13–16, 2000, Cairns, Australia, CSIRO, pp. 311–316 (2001)Google Scholar
- Battistelli, A., Calore, C., Pruess, K.: The simulator TOUGH2/EWASF for modeling geothermal reservoirs with brines and non-condensible gas. Geothermics
**26**, 437–464 (1997)CrossRefGoogle Scholar - Bear, J., Bachmat, y: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer, Dordrecht (1991)CrossRefGoogle Scholar
- Behrens, H., Ghergut, I., Licha, T., Orzol, J., Sauter, M.: Reactive behaviour of uranine (fluorescein) in a deep geothermal-reservoir tracer test. Geophys. Res. Abstr.
**8**, 10448 (2006)Google Scholar - Brooks R., Corey, A.: Hydraulic properties of porous media. Hydrology Paper No. 3. Colorado State University, Fort Collins, CO (1964)Google Scholar
- Brusseau, M.I., Peng, S., Schnaar, G., Costanza-Robinson, M.S.: Relationships among air–water interfacial area, capillary pressure, and water saturation for a sandy porous medium. Water Resour. Res.
**42**, W03501 (2006)Google Scholar - Buckley, R.L., Leverett, M.C.: Mechanism of fluid displacement in sands. Am. Inst. Min. Metall. Pet. Eng.
**146**, 107–116 (1942)Google Scholar - Chen, Z.X.: Some invariant solutions to two-phase fluid displacement problems including capillary effect. Soc. Pet. Eng. Reser. Eng.
**3**(2), 691–700 (1988)Google Scholar - Corey, A.T.: The interrelation between gas and oil relative permeabilities. Prod. Mon.
**19**, 38–41 (1954)Google Scholar - Cramer, S.D.: The solubility of methane, carbon dioxide and oxygen in brines from $0^\circ $ to $300^\circ {\text{ C}}$. US Bureau of Mines, Report No. 8706, USA, 16pp (1982)Google Scholar
- Fučīk, R., Mikyška, J., Beneš, M., Illangasekare, T.H.: An improved semi-analytical solution for verification of numerical models of two-phase flow in porous media. Vadose Zone J.
**6**, 93–104 (2007)CrossRefGoogle Scholar - Geiger, S., Roberts, S., Matth, S., Zoppou, C., Burri, A.: Combining finite element and finite volume methods for efficient multiphase flow simulations in highly heterogeneous and structurally complex geologic media. Geofluids
**4**(4), 284–299 (2004)CrossRefGoogle Scholar - Grant, G.P., Gerhard, J.I.: Simulating of dissolution of a complex dense nonaquesous phase liquid source zone: 1. Model to predict interfacial area. Water Resour. Res.
**43**, W12410 (2007a)Google Scholar - Grant, G.P., Gerhard, J.I.: Simulating of dissolution of a complex dense nonaquesous phase liquid source zone: 2. Experimental validation of an interfacial area-based mass transfer model. Water Resour. Res.
**43**, W12409 (2007b)Google Scholar - Guvanasen, V., Chan, T.: A three-dimensional numerical model for thermohydromechanical deformation with hysteresis in a fractured rock mass. Int. J. Rock Mech. Min. Sci.
**37**, 89–106 (2000)CrossRefGoogle Scholar - Helming, R.: Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems. Springer, Berlin (1997)CrossRefGoogle Scholar
- Hepple, R.P., Benson, S.M.: Geologic storage of carbon dioxide as a climate change mitigation strategy: performance requirements and the implications of surface seepage. Environ. Geol.
**47**(4), 576–585 (2004)CrossRefGoogle Scholar - Hesse, M.A., Orr, F.M., Tchelepi, H.A.: Gravity currents in horizontal porous layers: transition from early to late self-similarity. J. Fluid Mech.
**577**, 363–383 (2007)CrossRefGoogle Scholar - Hoteit, H., Firoozabadi, A.: Numerical modeling of two-phase flow in heterogeneous permeable with different capillarity pressures. Adv. Water Resour.
**31**, 56–73 (2008)CrossRefGoogle Scholar - Huppert, H., Woods, A.: Gravity-driven flows in porous layers. J. Fluid Mech.
**292**, 55–69 (1995)CrossRefGoogle Scholar - International Association for the Properties of Water and Steam: IAPWS Release on Surface Tension of Ordinary Water Substance, September (1994)Google Scholar
- Joekar-Niasar, V., Hassanizadeh, S.M., Dahle, H.K.: Non-equilibrium effects in capillarity and interfacial area in two-phase flow: dynamic pore-network modelling. J. Fluid Mech.
**655**, 38–71 (2010)CrossRefGoogle Scholar - Jury, W.A., Cardner, W.R., Gardner, W.H.: Soil Physics, 5th edn. Wiley, New York (1991)Google Scholar
- Karl, T.R., Trenberth, K.E.: Modern global climate change. Science
**302**(5651), 1719–1723 (2003)CrossRefGoogle Scholar - Kiehl, J.T., Trenberth, K.E.: Earth’s annual global mean energy budget. Bull. Am. Meteorol. Soc.
**78**(2), 197–208 (1997)CrossRefGoogle Scholar - Li, W.M., Rubin, D., Krempl, E.: Introduction to Continuum Mechanics. Pergamon Press, Oxford (1999)Google Scholar
- Licha, T., Sauter, M., Schaffer, M., Maier, F., Nottebohm, M., Fagerlund, F.: Report on batch experiment results for interfacial processes. D041/WP04/MUSTANG (2011)Google Scholar
- McWhorter, D.B., Sunada, D.K.: Exact integral solutions for two-phase flow. Water Resour. Res.
**26**, 399–413 (1990)CrossRefGoogle Scholar - Miller, C.T., Pirier-McNeill, M.M., Mayer, A.S.: Dissolution of trapped nonaqueous phase liquids: mass transfer characteristics. Water Resour. Res.
**26**, 2783–2796 (1990)CrossRefGoogle Scholar - Mualem, Y.: A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res.
**12**, 513–522 (1976)CrossRefGoogle Scholar - Niemi, A., et al.: Small-scale $\text{ CO}_2$ injection into a deep geological formation at Heletz, Israel. Energy Proc.
**23**, 504–511 (2012)CrossRefGoogle Scholar - Nordbotten, J.M., Celia, M.A.: Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech.
**561**, 307–327 (2006)CrossRefGoogle Scholar - Pruess, K.: ECO2N user’s guide: a TOUGH2 fluid property module for mixtures of water, NaCl, and $\text{ CO}_2$. Lawrence Berkeley Laboratory Report LBNL-57952, Berkeley, CA (2005)Google Scholar
- Orr Jr, F.M., Stanford, U.: Storage of carbon dioxide in geologic formations. J. Pet. Technol.
**56**(9), 90–97 (2004)Google Scholar - Pruess, K., Oldenburg, C., Moridis, G.: Tough2 user’s guide version. Lawrence Berkeley Laboratory Report LBNL-57952, Berkeley, CA (1999)Google Scholar
- Pritchett, J.W., Rice, M.H., Riney, T.D.: Equation of state for water–carbon dioxide mixtures: implications for Baca reservoir. Report DOE/ET/27163-8, UC-66a, La Jolla, CA (1981)Google Scholar
- Schaffer, M., Maier, F., Licha, T., Sauter, M.: A new generation of tracers for the characterization of interfacial areas during supercritical carbon dioxide injections into deep saline aquifers: concept for kinetic interface-sensitive tracers (KIS tracer). Int. J. Greenh. Gas Control. doi: 10.1016/j.ijggc.2013.01.020
- Riaz, A., Tang, G.-Q., Tchelepi, H.A., Kovscek, A.R.: Forced imbibition in natural porous media: Comparison between experiments and continuum models. Phys. Rev. E
**75**, 011131 (2007)CrossRefGoogle Scholar - Riaz, A., Tchelepi, H.A.: Influence of relative permeability on the stability characteristics of immiscible flow in porous media. Trans. Porous Media
**64**, 315–338 (2006)CrossRefGoogle Scholar - Sterner, S.M., Pitzer, K.S.: An equation of state for carbon dioxide valid from zero to extreme pressures. Contrib. Min. Pet.
**117**, 362–374 (1994)CrossRefGoogle Scholar - Tomich, J.F., Dalton, R.L., Deans, H.A., Shallenberger, L.K.: Single-well tracer method to measure residual oil saturation. J. Pet. Technol.
**25**, 211–218 (1973)Google Scholar - Tong, F.G., Jing, L., Zimmerman, R.W.: A fully coupled thermo-hydro-mechanical model for simulating multiphase flow, deformation and heat transfer in buffer material and rock masses. Int. J. Rock Mech. Min. Sci.
**47**, 205–217 (2010)Google Scholar - Tong, F.G., Jing, L., Zimmerman, R.W.: A fully-coupled finite element code for modeling thermo-hydro-mechanical processes in porous geological media. In: Proceedings 43rd US Rock Mechanics Symposium, Asheville, NC, USA, 28–30 June 2009Google Scholar
- Tong, F.G., Niem,i A., Yang, Z., Fagerlund, F.: A solution scheme for simulating tracer transport in a non-isothermal two-phase flow system. Geophysical Research Abstract, vol. 14, EGU2012-6743-1, EGU General Assembly 2012, Vienna (2012)Google Scholar
- Van Duijn, C., De Neef, M.: Similarity solution for capillary redistribution of two phases in a porous medium with a single discontinuity. Adv. Water Resour.
**21**(6), 451–461 (1998)CrossRefGoogle Scholar - Van Genuchten, M.Th: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**44**, 892–898 (1980)CrossRefGoogle Scholar - Vargaftik, N.B.: Tables on thermophysical properties of liquid and gases, 2nd edn. Wiley, New York (1975)Google Scholar
- Xu, B., Yortsos, Y.C., Salin, D.: Invasion percolation with viscous forces. Phys. Rev. E
**57**(1), 739–751 (1998)CrossRefGoogle Scholar - Xu, T., Sonnenthal, E., Spycher, N., Pruess, K.: TOUGHREACT user’s guide: a simulation program for nonisothermal multiphase reactive geochemical transport in variably saturated geologic media. LBNL Report 55460, V. 1.2., Berkeley, CA, USA (2004)Google Scholar
- Yortsos, Y.C., Xu, B.M., Salin, D.: Delineation of microscale regimes of fully-developed drainage and implications for continuum models. Comput. Geosci.
**5**(3), 257–278 (2001)CrossRefGoogle Scholar - Yortsos, Y.C., Fokas, A.S.: An analytical solution for linear waterflood including the effects of capillary pressure. Soc. Pet. Eng. J.
**23**(1), 115–124 (1983)Google Scholar - Younes, A., Ackerer, P.: Solving the advection–diffusion equation with the Eulerian-Lagrangian localized adjoint method on unstructured meshes and non uniform time stepping. J. Comput. Phys.
**208**, 384–402 (2005)CrossRefGoogle Scholar - Zhang, Y., Freifeld, B., Finsterle, S., Leahy, M., Ennis-King, J., Paterson, L., Dance, T.: Single-well experimental design for studying residual trapping of supercritical carbon dioxide. Int. J. Greenh. Gas Control
**5**, 88–98 (2011)CrossRefGoogle Scholar - Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, vol. 1, Basic Formulations and Linear Problems, 5th edn. Butterworth-Heinemann, London (2000)Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.