**5.5 Convection of CO2-enriched brine in the reservoir**

Convection – here defined as flow at the reservoir scale induced by gradients in density, concentration or heat – can potentially move large quantities of dissolved CO2 through the formation. A resistance threshold has to be overcome for convection to commence, this threshold can be assessed with the Rayleigh number Ra (equation 22) (Riaz et al. 2006).

$$Ra = \frac{K\Delta\rho\text{g}H}{\phi D\mu} \tag{22}$$

where

K = permeability


Dissolution Trapping of Carbon Dioxide in

with

*D*

*D <sup>z</sup> <sup>Z</sup>*

*D ut <sup>t</sup>* φ

*uH Pe* φ

where

<sup>2</sup> ( )

β

*C*

2 2 , *CO*

*<sup>H</sup>* <sup>=</sup> , dimensionless time

*CO saturated*

boundary conditions: *Initial conditions:* 

*Boundary conditions:* 

CD = 0 for tD = 0 and all ZD

at ZD = 1 : 0 *<sup>D</sup>*

at ZD = 0 : CD = 1 for tD > 0

2

0

π

and *erfc erf* () 1 () β= −

*z <sup>t</sup> erf e dt*

*D C Z* <sup>∂</sup> <sup>=</sup> <sup>∂</sup>

The solution for equation (25) is then (Lake 1989)

<sup>−</sup> <sup>=</sup> ∫ is the error function

β

1

CO2. Higher dispersivity generally supports dissolution trapping.

*D*

*C*

Equation 24 can be re-written in dimensionless form

Reservoir Formation Brine – A Carbon Storage Mechanism 253

medium is considered to be isotropic and homogenous. The aquifer thickness is set to 100m, porosity to 20%, temperature to 323 K, pressure to 7.6 MPa and DCO2-H2O to 3 x <sup>9</sup> 10<sup>−</sup> m2/s.

> *C C C Pe Z t Z*

*<sup>D</sup>* <sup>=</sup> , Peclet number = ratio between transport by convection/transport by molecular

Özgür (2006, 2010) solved the diffusion-convection equation (25) for one set of initial and

2 2 2 2

*Zt e Zt C erfc erfc t t*

is the complementary error function.

Özgür (2006, 2010) also conducted numerical modelling studies, and found that convection rate strongly increases with increasing permeability and dissolution trapping is strongly accelerated thereby. In diffusion dominated systems the dissolution rate is very slow, however, and only after <sup>7</sup> 10 years the considered aquifer was completely saturated with

It is moreover interesting to note that the mixing zone length ΔzD (which is defined as the distance between the points CD = 0.1 and CD = 0.9 in the reservoir, Lake 1989) reaches a value of 0.9 in diffusion dominated systems only after <sup>5</sup> 10 years in Özgür's model. Increased porosity slightly increases ΔzD, and ΔzD strongly increases with permeability once

*Z Pe <sup>D</sup> D D D D*

⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ − − <sup>=</sup> <sup>+</sup>

*D D*

(26)

*Pe Pe*

⎝⎠ ⎝⎠

*D D D D*

<sup>∂</sup> ∂ ∂ − = <sup>∂</sup> <sup>∂</sup> <sup>∂</sup> (25)

2 2 2 1 *<sup>D</sup>*

*D*

*C*<sup>=</sup> , normalized concentration of CO2 in brine at CCS conditions

*H*<sup>=</sup> , normalized reservoir height, H is the total thickness of the aquifer

diffusion (Bear 1972)

D = diffusion coefficient

µ = brine viscosity

φ= porosity

Above a critical Rayleigh number Rac convection will occur; Rac is a function of the boundary conditions of the system (Weatherill et al. 2004), e.g. for a homogenous reservoir where the horizontal boundaries are impermeable and perfect heat conductors Rac = 4π<sup>2</sup> (Lindeberg and Wessel-Berg 1997).

One mechanism which can trigger convective flow is dissolution of CO2 into brine, which can increase brine density by 1% under CCS conditions (Ennis-King and Paterson 2005).

Based on an expression suggested by Garcia (2001) Özgür (2006) developed an equation (23) with which the effect of aqueous CO2 concentration on brine density can be estimated.

$$\rho\_{CO\_2,brine} = \frac{\rho\_{brine}}{1 - Y\_{CO\_2} \left(1 - \frac{V\_{m,brine}}{M} \rho\_{brine}\right)}\tag{23}$$

where

ρCO2,brine = density of CO2-enriched brine

ρbrine = original brine density

YCO2 = dissolved CO2 mass fraction

Vm,brine = apparent molar volume of CO2 in brine

M = molecular weight of CO2

#### **6. Reservoir scale dissolution trapping**

In the context of reservoir flow where dissolved CO2 molecules are transported, convection, dispersion, diffusion and maximum CO2 solubilities can all play a significant role. This is an active field of research, and three literature examples are presented where CO2 solute transport was modelled at reservoir scale.

Interesting conclusions extracted from these computations are that dissolution trapping is favourably done in a high permeability reservoir. Moreover CO2 dissolution can significantly reduce reservoir pressure (the pressure is increased by CO2 injection, but only a maximum reservoir pressure is tolerable, the fracture pressure), improving injectivity, i.e. CO2 can be injected at a faster rate, and more CO2 can be stored in total – provided that CO2 dissolves at an adequate rate or CO2 injection is slow enough.

#### **6.1 Özgür model (2006)**

Özgür (2006, 2010) modelled diffusion and convection in an aquifer with the diffusion-2

convection equation (24). 2 *<sup>c</sup> <sup>D</sup> z* ∂ ∂ is the diffusion term and *u c* φ *z* ∂ ∂ is the convection term.

$$D\frac{\partial^2 c}{\partial z^2} - \frac{\mu}{\phi} \frac{\partial c}{\partial z} = \frac{\partial c}{\partial t} \tag{24}$$

This model considers one-dimensional vertical flow in an aquifer, temperature and CO2 gas cap pressure are assumed to be constant; chemical reactions are ignored and the porous medium is considered to be isotropic and homogenous. The aquifer thickness is set to 100m, porosity to 20%, temperature to 323 K, pressure to 7.6 MPa and DCO2-H2O to 3 x <sup>9</sup> 10<sup>−</sup> m2/s. Equation 24 can be re-written in dimensionless form

$$\frac{1}{\text{Pe}} \frac{\partial^2 \mathbf{C}\_{\text{D}}^2}{\partial \mathbf{Z}\_{\text{D}}^2} - \frac{\partial \mathbf{C}\_{\text{D}}}{\partial \mathbf{Z}\_{\text{D}}} = \frac{\partial \mathbf{C}\_{\text{D}}}{\partial \mathbf{t}\_{\text{D}}} \tag{25}$$

with

252 Mass Transfer - Advanced Aspects

Above a critical Rayleigh number Rac convection will occur; Rac is a function of the boundary conditions of the system (Weatherill et al. 2004), e.g. for a homogenous reservoir where the horizontal boundaries are impermeable and perfect heat conductors Rac = 4π<sup>2</sup>

One mechanism which can trigger convective flow is dissolution of CO2 into brine, which can increase brine density by 1% under CCS conditions (Ennis-King and Paterson 2005). Based on an expression suggested by Garcia (2001) Özgür (2006) developed an equation (23) with which the effect of aqueous CO2 concentration on brine density can be estimated.

2

*Y*

, 1 1

In the context of reservoir flow where dissolved CO2 molecules are transported, convection, dispersion, diffusion and maximum CO2 solubilities can all play a significant role. This is an active field of research, and three literature examples are presented where CO2 solute

Interesting conclusions extracted from these computations are that dissolution trapping is favourably done in a high permeability reservoir. Moreover CO2 dissolution can significantly reduce reservoir pressure (the pressure is increased by CO2 injection, but only a maximum reservoir pressure is tolerable, the fracture pressure), improving injectivity, i.e. CO2 can be injected at a faster rate, and more CO2 can be stored in total – provided that CO2

Özgür (2006, 2010) modelled diffusion and convection in an aquifer with the diffusion-

2 2 *c uc c <sup>D</sup>*

*z* φ*z t*

is the diffusion term and *u c*

This model considers one-dimensional vertical flow in an aquifer, temperature and CO2 gas cap pressure are assumed to be constant; chemical reactions are ignored and the porous

φ *z* ∂ ∂

<sup>∂</sup> ∂ ∂ − = <sup>∂</sup> <sup>∂</sup> <sup>∂</sup> (24)

is the convection term.

*brine*

ρ

<sup>=</sup> ⎛ ⎞ − − ⎜ ⎟

*m brine CO brine V*

⎝ ⎠

ρ

(23)

*M*

2

dissolves at an adequate rate or CO2 injection is slow enough.

2 2 *<sup>c</sup> <sup>D</sup> z* ∂ ∂

ρ

,

*CO brine*

D = diffusion coefficient µ = brine viscosity

(Lindeberg and Wessel-Berg 1997).

ρCO2,brine = density of CO2-enriched brine

Vm,brine = apparent molar volume of CO2 in brine

**6. Reservoir scale dissolution trapping** 

transport was modelled at reservoir scale.

**6.1 Özgür model (2006)** 

convection equation (24).

ρbrine = original brine density YCO2 = dissolved CO2 mass fraction

M = molecular weight of CO2

= porosity

φ

where

2 2 , *CO D CO saturated C C C*<sup>=</sup> , normalized concentration of CO2 in brine at CCS conditions

$$\begin{aligned} Z\_D &= \frac{z}{H}, \text{ normalized reservoir height, H is the total thickness of the aquifier} \\ t\_D &= \frac{\mu t}{\phi H}, \text{dimensionless time} \\ Pe &= \frac{\mu H}{\phi D}, \text{ Peeler number} = \text{ ratio between transport by convection/transport by molecular} \\ \text{diffusion (Bear 1972)} \end{aligned}$$

Özgür (2006, 2010) solved the diffusion-convection equation (25) for one set of initial and boundary conditions:

*Initial conditions:* 

CD = 0 for tD = 0 and all ZD

*Boundary conditions:* 

$$\begin{array}{ll} \text{at } \mathbf{Z\_D = 0}: & \mathbf{C\_D = 1} \text{ for } \mathbf{t\_D > 0} \\ \text{at } \mathbf{Z\_D = 1}: & \frac{\partial \mathbf{C\_D}}{\partial \mathbf{Z\_D}} = \mathbf{0} \end{array}$$

The solution for equation (25) is then (Lake 1989)

$$\mathbf{C}\_{D} = \frac{1}{2} \text{erfc} \left( \frac{\mathbf{Z}\_{D} - \mathbf{t}\_{D}}{2 \sqrt{\frac{\mathbf{t}\_{D}}{Pe}}} \right) + \frac{e^{\mathbf{Z}\_{D}Pe}}{2} \text{erfc} \left( \frac{\mathbf{Z}\_{D} - \mathbf{t}\_{D}}{2 \sqrt{\frac{\mathbf{t}\_{D}}{Pe}}} \right) \tag{26}$$

where

2 0 <sup>2</sup> ( ) *z <sup>t</sup> erf e dt* β π <sup>−</sup> <sup>=</sup> ∫ is the error function

and *erfc erf* () 1 () β = − βis the complementary error function.

Özgür (2006, 2010) also conducted numerical modelling studies, and found that convection rate strongly increases with increasing permeability and dissolution trapping is strongly accelerated thereby. In diffusion dominated systems the dissolution rate is very slow, however, and only after <sup>7</sup> 10 years the considered aquifer was completely saturated with CO2. Higher dispersivity generally supports dissolution trapping.

It is moreover interesting to note that the mixing zone length ΔzD (which is defined as the distance between the points CD = 0.1 and CD = 0.9 in the reservoir, Lake 1989) reaches a value of 0.9 in diffusion dominated systems only after <sup>5</sup> 10 years in Özgür's model. Increased porosity slightly increases ΔzD, and ΔzD strongly increases with permeability once

Dissolution Trapping of Carbon Dioxide in

**6.4 Summary of reservoir models** 

seismic imaging (Iglauer 2011).

current research; this is an active area of research.

carbonate reservoirs) enhance convection or slow it down.

**7. Multiphase flow in the reservoir – flow of the scCO2 phase** 

results.

Reservoir Formation Brine – A Carbon Storage Mechanism 255

tc influences the penetration depth of the diffusive boundary layer δ(t), which again influences Rac = Ra(δ(t)). tc can vary over several magnitudes, mainly because permeability can span several magnitudes in a geological formation. For a permeability increase from 1 mD to 3 Darcy Riaz et al. (2006) calculated a tc decrease from 2000 years to below 10 days,

The model also demonstrates that Ra has a strong influence on the finger-like flow in the reservoir, including finger thickness and shape. In terms of the numerical model they found that grid size plays an important role and a fine grid is required to resolve disturbances at small times. Correct identification of such early disturbances is necessary to obtain reliable

One important conclusion they make is that dissolution trapping is strongly enhanced in high permeability reservoirs. They estimate that the onset of gravitational instabilities essentially induced by molecular diffusion mass transfer processes – occurs after several hundreds of years for typical aquifers with average permeability. It should be noted that

There are other reservoir models described in the literature, e.g. Ennis-King and Paterson (2005) conclude that anisotropy of the reservoir has a strong effect on dissolution trapping, but this is beyond the scope of this book chapter and the reader is encouraged to check

The simulation results are very important for CCS assessments and project planning, but it must be emphasized that more experimental research should be conducted, in the laboratory and especially at field scale to evaluate the quality of the model predictions. In addition, it is important to stress the approximative character of these models, real field situations are much more complex, e.g. it is not clear whether Fick's law can describe diffusion in the field or whether very heterogeneous pore structures (for instance in

The flow of the scCO2 phase affects the dissolution process as it determines interfacial areas and overall position of the CO2 in the reservoir. Reservoir models predict that the injected CO2 phase rises upwards and is stopped by the caprock (Qi et al. 2009, Juanes 2006, Hesse 2008). This behaviour has been confirmed experimentally in the Sleipner formation by

Small residual CO2 clusters at the trailing edge of the rising CO2 plume - trapped by capillary forces (Iglauer et al. 2010, Juanes et al. 2006) - strongly increase CO2-brine interfacial areas. Hence CO2 dissolution speed is predicted to be accelerated, especially if combined with convective flow of saturated/undersaturated brine. However experimental reservoir monitoring data is needed to confirm these predictions. Optimal conditions would be to bring undersaturated brine continuously into contact with residual micrometer-sized CO2 bubbles while removing saturated or highly CO2-enriched brine simultaneously. Engineering this dissolution phenomenon can be a promising topic for future research. Moreover, and most likely even more significant in the short term - thereby strongly affecting the economics of CCS schemes are the fluid dynamics associated with CO2 injection. CO2 injectivity and CO2-wellbore effects can strongly impact CCS schemes. For

and associated with that δ(t) changed from 55 m (for 1 mD permeability) to 0.07 m.

these estimates are quite rough because of the assumptions made.

a threshold (when convection sets in) is reached. In convection dominated systems, porosity decreases ΔzD, while dispersivity slightly increases ΔzD.

#### **6.2 The Lindeberg/Wessel-Berg model (1997)**

Lindeberg and Wessel-Berg (1997) modelled the onset of convection in aquifers into which CO2 has been injected. The water column in such aquifers can be unstable because of the density gradient introduced by molecular CO2 diffusion into the brine.

In their simulation they solved the Darcy equation, heat conduction equation, equation of continuity and energy equation (details are described very thoroughly by Bear (1972)). They also included the equation of diffusion (27) so that diffusive mass transfer was considered.

$$\frac{\partial \rho}{\partial t} + \frac{j}{\phi} \nabla \rho = \nabla \cdot \left(\frac{D}{\phi} \nabla \rho\right) \tag{27}$$

where j is the volume flux.

They calculated the Rayleigh numbers for an array of model reservoirs, spanning a temperature range from 303-363 K, a pressure range from 10-30 MPa, a permeability range from 100-2000 mD, while the porosity was a constant 30%. Variations in pressure and temperature resulted in brine density variations between 1013.5-1036 kg/m3 and molecular diffusion coefficients between 2.2-6.3 x <sup>9</sup> 10<sup>−</sup> m2/s. The brine density difference Δρ was 14.42 kg/m3 due to difference in dissolved CO2 concentration, while Δρ was only 2.847-2.910 kg/m3 due to differences in temperature.

In Lindeberg and Wessel-Berg's model the water column is stable if only thermal gradients are considered, Ra lies then in the range 3.53-29.3 and no convection occurs (Rac = 39.5 in this case). However, if molecular CO2 diffusion is considered (resulting in a significantly higher Δρ), convection is predicted to occur. Lindeberg and Wessel-Berg define a stability criterion S which is the sum of the temperature and concentration effect on convective stability. S is analogous to Ra, and for infinite CO2 dilution or an infinite molecular diffusion coefficient S becomes equal to Ra. The computed S values range from 1046-24204, and they are much higher than Rac. This means that convection will occur in aquifers under CCS conditions, which strongly enhances dissolution trapping and storage security. This convection is caused by the concentration gradient, not the temperature gradient. Lindeberg and Wessel-Berg (1997) suggest improvements for their model, especially a more sophisticated description of the concentration gradient should be implemented (they used a linearized concentration gradient). Moreover the Soret effect should be considered.

#### **6.3 The Riaz model (2006)**

Riaz et al. (2006) conducted a linear analysis and numerical simulations of the stability of the diffusive boundary layer (i.e. the brine layer adjacent to the scCO2 phase into which the CO2 diffuses) in a semi-infinite domain. Their calculations are based on Boussinesq-flow in a horizontal porous layer. The model neglects dispersion and geochemical reactions and assumes a homogenous and isotropic porous medium. Riaz et al. (2006) describe a critical time tc (equation 28) which is a criterion for the onset of gravitational instability. For times larger than tc convection will occur.

$$t\_c = 146 \frac{\phi \mu^2 D}{\left(K \Lambda \rho \chi\right)^2} \tag{28}$$

a threshold (when convection sets in) is reached. In convection dominated systems, porosity

Lindeberg and Wessel-Berg (1997) modelled the onset of convection in aquifers into which CO2 has been injected. The water column in such aquifers can be unstable because of the

In their simulation they solved the Darcy equation, heat conduction equation, equation of continuity and energy equation (details are described very thoroughly by Bear (1972)). They also included the equation of diffusion (27) so that diffusive mass transfer was considered.

*j D*

<sup>∂</sup> ⎛ ⎞ + ∇ =∇⋅ ∇ ⎜ ⎟ <sup>∂</sup> ⎝ ⎠

They calculated the Rayleigh numbers for an array of model reservoirs, spanning a temperature range from 303-363 K, a pressure range from 10-30 MPa, a permeability range from 100-2000 mD, while the porosity was a constant 30%. Variations in pressure and temperature resulted in brine density variations between 1013.5-1036 kg/m3 and molecular diffusion coefficients between 2.2-6.3 x <sup>9</sup> 10<sup>−</sup> m2/s. The brine density difference Δρ was 14.42 kg/m3 due to difference in dissolved CO2 concentration, while Δρ was only 2.847-2.910

In Lindeberg and Wessel-Berg's model the water column is stable if only thermal gradients are considered, Ra lies then in the range 3.53-29.3 and no convection occurs (Rac = 39.5 in this case). However, if molecular CO2 diffusion is considered (resulting in a significantly higher Δρ), convection is predicted to occur. Lindeberg and Wessel-Berg define a stability criterion S which is the sum of the temperature and concentration effect on convective stability. S is analogous to Ra, and for infinite CO2 dilution or an infinite molecular diffusion coefficient S becomes equal to Ra. The computed S values range from 1046-24204, and they are much higher than Rac. This means that convection will occur in aquifers under CCS conditions, which strongly enhances dissolution trapping and storage security. This convection is caused by the concentration gradient, not the temperature gradient. Lindeberg and Wessel-Berg (1997) suggest improvements for their model, especially a more sophisticated description of the concentration gradient should be implemented (they used a

linearized concentration gradient). Moreover the Soret effect should be considered.

Riaz et al. (2006) conducted a linear analysis and numerical simulations of the stability of the diffusive boundary layer (i.e. the brine layer adjacent to the scCO2 phase into which the CO2 diffuses) in a semi-infinite domain. Their calculations are based on Boussinesq-flow in a horizontal porous layer. The model neglects dispersion and geochemical reactions and assumes a homogenous and isotropic porous medium. Riaz et al. (2006) describe a critical time tc (equation 28) which is a criterion for the onset of gravitational instability. For times

( )

ρ

*K g* φμ

<sup>2</sup> 146 *<sup>c</sup> <sup>D</sup> <sup>t</sup>*

<sup>=</sup> <sup>Δ</sup>

2

ρ (27)

(28)

 φ

ρ

φ

decreases ΔzD, while dispersivity slightly increases ΔzD.

density gradient introduced by molecular CO2 diffusion into the brine.

*t* ρ

**6.2 The Lindeberg/Wessel-Berg model (1997)** 

where j is the volume flux.

**6.3 The Riaz model (2006)** 

larger than tc convection will occur.

kg/m3 due to differences in temperature.

tc influences the penetration depth of the diffusive boundary layer δ(t), which again influences Rac = Ra(δ(t)). tc can vary over several magnitudes, mainly because permeability can span several magnitudes in a geological formation. For a permeability increase from 1 mD to 3 Darcy Riaz et al. (2006) calculated a tc decrease from 2000 years to below 10 days, and associated with that δ(t) changed from 55 m (for 1 mD permeability) to 0.07 m.

The model also demonstrates that Ra has a strong influence on the finger-like flow in the reservoir, including finger thickness and shape. In terms of the numerical model they found that grid size plays an important role and a fine grid is required to resolve disturbances at small times. Correct identification of such early disturbances is necessary to obtain reliable results.

One important conclusion they make is that dissolution trapping is strongly enhanced in high permeability reservoirs. They estimate that the onset of gravitational instabilities essentially induced by molecular diffusion mass transfer processes – occurs after several hundreds of years for typical aquifers with average permeability. It should be noted that these estimates are quite rough because of the assumptions made.
