1 Introduction

Carbon capture and storage (CCS) or geologic carbon sequestration in general is an attempt to reduce the amount of greenhouse gases emitted to the atmosphere. \(\hbox {CO}_{2}\) is separated from the flue gas of fossil-fuel power plants or other industries and stored underground. There are three likely underground storage options: injection into depleted oil or gas fields (including enhanced oil recovery), injection into unmineable coal seams, and injection into deep saline aquifers, that is, into highly permeable sandstones that are filled with brine.

Independent of the type of storage, monitoring is crucial. Assuming that after injection of \(\hbox {CO}_{2}\) into the storage site, suddenly, a leak forms in the caprock (a low permeable seal of the reservoir). How can we detect the leak? In this study, we investigate the wave speeds of porous media filled with \(\hbox {CO}_{2}\) and water in order to determine changes in the wave speeds and to evaluate whether it is possible to detect a leak with seismic waves.

We focus on deep storage sites where the \(\hbox {CO}_{2}\) is in a supercritical state, which means that the temperature and pressure have to be above \(31.1\,^{\circ }\hbox {C}\) and \(7.38\,\hbox {MPa}\). Assuming a geothermal gradient of \(30\,\hbox {K}\,\hbox {km}^{-1}\) and a pressure gradient of \(10\,\hbox {MPa}\,\hbox {km}^{-1}\), the minimum depth for a storage site is \(700\,\hbox {m}\) (see Fig. 1). Statistical investigations by Kopp et al. (2009) show that saline aquifers suitable for \(\hbox {CO}_{2}\)-storage have a median depth of \(1524\,\hbox {m}\), which is the depth we will later use for our calculations.

Fig. 1
figure 1

Phase diagram for pure \(\hbox {CO}_{2}\) (data from Span and Wagner 1996) including subsurface conditions (green curve) assuming a geothermal gradient of \(30\,\hbox {K}\,\hbox {km}^{-1}\), a pressure gradient of \(10.5\,\hbox {MPa}\,\hbox {km}^{-1}\) as well as a surface temperature of 25 \(^{\circ }\hbox {C}\) and a surface pressure of \(0.1\,\hbox {MPa}\)

Full size image

In case of a leak, \(\hbox {CO}_{2}\) will rise and start to boil as soon as it reaches temperatures or pressures below the critical point. Therefore, there will be gaseous \(\hbox {CO}_{2}\) in the pores above this depth. The physical properties of gaseous \(\hbox {CO}_{2}\) and supercritical \(\hbox {CO}_{2}\) are very different (especially in terms of density), and thus it is likely that seismic waves propagating through a porous medium filled with gaseous \(\hbox {CO}_{2}\) and water have different speeds than through media filled with supercritical \(\hbox {CO}_{2}\) and water or only with water.

Biot (1955, 1956a, b) developed a theory for wave propagation in porous and elastic media, which enables the computation of compressional and shear wave speeds from the physical properties of the fluid that fills the pores and the material that forms the skeleton. Biot’s theory is limited to saturated porous media. This work is an extension of Biot’s theory to unsaturated porous media. Previously, among others, Albers (2009) and Santos et al. (1990a, b) presented approaches to extend Biot’s theory to porous media saturated by two fluids. Albers (2009) used an approach based on a macroscopic linear model, which accommodates both the Biot model and the Simple Mixture Model of Wilmanski (1998). Santos et al. (1990a) present a Lagrangian derivation using a thermodynamical approach and including the principle of virtual complementary work and energy density functions. The derivation of our equations is loosely based on Coussy (2004).

This paper both presents an approach to extend Biot’s theory and intends to answer the question whether it is possible to directly detect leakage of a caprock of a \(\hbox {CO}_{2}\)-storage site by seismic waves. Therefore, we first derive the equations for wave propagation in unsaturated porous media and afterwards use these equations to calculate wave speeds for a leak in the caprock of a storage site at \(1524\,\hbox {m}\) depth.

2 Theory

In this section, we derive equations that allow us to calculate wave speeds in unsaturated porous media, i.e., for multiple fluids in the pore space. The procedure is as follows: first, we derive the storage equation for multiple fluids, describing the fluid mass balance; next, we set up the balance of linear momentum and linearize the resulting equations; finally, we determine the wave speeds. For a more detailed introduction into seismic wave propagation and poromechanics, the reader is referred to Aki and Richards (2002) and Coussy (2004), respectively.

2.1 Storage Equation for Multiple Fluids

Consider a porous solid with current pore volume d\(\varOmega _t^p\) and total volume d\(\varOmega _t\). The pores are filled with \(\alpha \) immiscible fluids \(f_\alpha \) whose density is denoted by \(\rho _{f_\alpha }\) and whose saturation is denoted by \(S_{f_\alpha }\). The mass balance of the fluids \(f_\alpha \) is then given by

$$\begin{aligned} \frac{\mathrm{d}^{f_\alpha }}{\mathrm{d}t}\left( \rho _{f_\alpha } \varphi S_{f_\alpha } \mathrm{d}\varOmega _t \right) = 0, \end{aligned}$$
(1)

where \(\frac{\mathrm{d}^{f_\alpha }}{\mathrm{d}t}\) is the substantial derivative and \(\varphi \) the Eulerian porosity. For a homogeneous medium, \(\nabla S_{f_\alpha } = \nabla \varphi = \nabla \rho _{f_\alpha } = 0\). Therefore, using the product rule and \(\frac{\mathrm{d}^{f_\alpha }}{\mathrm{d}t}\mathrm{d}\varOmega _t = \varvec{\nabla }\cdot \mathbf{v}^{f_\alpha } \mathrm{d}\varOmega _t\), where \(\mathbf{v}^\beta = \dot{\mathbf{u}}^\beta \) with \(\beta = {f_\alpha }, s\) denotes the velocity of fluid \(f_\alpha \) or the solid \(s\), it follows that

$$\begin{aligned} \rho _{f_\alpha }\frac{\partial }{\partial t}\left( \varphi S_{f_\alpha }\right) +\varphi S_{f_\alpha }\frac{\partial \rho _{f_\alpha }}{\partial t}+\rho _{f_\alpha }\varphi S_{f_\alpha }\varvec{\nabla }\cdot \mathbf{v}^{f_\alpha }=0. \end{aligned}$$
(2)

Next, we replace the change in density by the change in pressure using the definition for the bulk modulus \(K_{f_\alpha }\) of fluid \(f_\alpha \), namely

$$\begin{aligned} K_{f_\alpha } = \rho _{f_\alpha } \frac{\partial p_{f_\alpha }}{\partial \rho _{f_\alpha }}, \end{aligned}$$
(3)

where \(p_{f_\alpha }\) is the partial pressure of fluid \(f_\alpha \). Upon insertion of Eq. 3 in Eq. 2, we obtain

$$\begin{aligned} S_{f_\alpha } \frac{\partial \varphi }{\partial t} + \varphi \frac{\partial S_{f_\alpha }}{\partial t} + \frac{\varphi S_{f_\alpha }}{K_{f_\alpha }}\frac{\partial p_{f_\alpha }}{\partial t}+\varphi S_{f_\alpha }\varvec{\nabla }\cdot \mathbf{v}^{f_\alpha } = 0. \end{aligned}$$
(4)

In the case of two fluids in a porous medium, a pressure discontinuity, caused by interfacial interactions, occurs. Neglecting hysteresis effects, this capillary pressure \(p_\mathrm{c} = p_{f_1} - p_{f_2}\) can be described as a unique function under isothermal conditions (Lo et al. 2005). The change in capillary pressure with time can be expressed as

$$\begin{aligned} \frac{\partial p_\mathrm{c}}{\partial t} = \frac{\mathrm{d} p_\mathrm{c}}{\mathrm{d} S_{f_\alpha }} \frac{\partial S_{f_\alpha }}{\partial t} = \frac{\partial p_1}{\partial t} - \frac{\partial p_2}{\partial t}. \end{aligned}$$
(5)

Therefore, the change in saturation of one fluid phase can be expressed as

$$\begin{aligned} \frac{\partial S_{f_\alpha }}{\partial t} = \frac{\mathrm{d} S_{f_\alpha }}{\mathrm{d} p_\mathrm{c}} \left( \frac{\partial p_1}{\partial t} - \frac{\partial p_2}{\partial t} \right) . \end{aligned}$$
(6)

The capillary pressure curve \(\left( \hbox {i.e.}, \frac{\mathrm{d} p_c}{\mathrm{d} S_{f_1}} \left( S_{f_1} \right) \right) \) can be described using the model of van Genuchten (1980):

$$\begin{aligned} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} = \frac{\rho _2 g}{m n \chi } \left[ \left( 1-S_{f_1}\right) ^{-\frac{1}{m}}-1\right] ^{\frac{1}{n}-1} \left( 1-S_{f_1}\right) ^{-\left( \frac{1}{m}+1\right) }, \end{aligned}$$
(7)

where \(n\) and \(\chi \) are fitting parameters and \(m=1-\frac{1}{n}\).

The incremental state equation (Coussy 2004) is given by

$$\begin{aligned} \frac{\mathrm{d}\varphi }{\mathrm{d}t}=\left( b-\varphi \right) \varvec{\nabla }\cdot \mathbf{v}^\mathrm{s} + \frac{1}{N}\frac{\mathrm{d}p_f}{\mathrm{d}t} -3\alpha _\mathrm{s} \left( b-\phi _0\right) \frac{\mathrm{d}T}{\mathrm{d}t}, \end{aligned}$$
(8)

where \(N=\frac{K_\mathrm{s}}{b-\varphi }\), with the bulk modulus of the rock denoted by \(K_\mathrm{s}\). This equation relates the change in porosity to the strain variation, the change in pressure, and the change in temperature. We neglect the term for the volumetric thermal dilation (\( -3\alpha _\mathrm{s} \left( b-\phi _0\right) \frac{\mathrm{d}T}{\mathrm{d}t}\)) because we assume the temperature to be constant and insert this equation, as well as Eq. 6 into Eq. 4. Writing this equation explicitly for two fluids \(\alpha =1,2\) using \(\frac{\partial S_2}{\partial t} = - \frac{\partial S_1}{\partial t}\) and solving for \(\frac{\partial p_{f_1}}{\partial t}\) and \(\frac{\partial p_{f_2}}{\partial t}\) yields

$$\begin{aligned} \frac{\partial p_{f_1}}{\partial t}&= - M \left\{ K_{f_1} N \left( K_{f_2} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_1} S_{f_2} \right) \left( b-\varphi \right) \varvec{\nabla } \cdot \mathbf{v}_\mathrm{s} \right. \nonumber \\&\quad +\, K_{f_1} S_{f_1} \left[ S_{f_2} N \varphi + K_{f_2} \left( S_{f_2}^2 +N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} \right) \right] \varvec{\nabla } \cdot \mathbf{v}_{f_1}\nonumber \\&\quad \left. +\, K_{f_1} K_{f_2} S_{f_2} \left( N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} - S_{f_1} S_{f_2}\right) \varvec{\nabla }\cdot \mathbf{v}_{f_2}\right\} \end{aligned}$$
(9)
$$\begin{aligned} \frac{\partial p_{f_2}}{\partial t}&= - M \left\{ K_{f_2} N \left( K_{f_1} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_1} S_{f_2} \right) \left( b-\varphi \right) \varvec{\nabla } \cdot \mathbf{v}_\mathrm{s} \right. \nonumber \\&\quad +\, K_{f_1} K_{f_2} S_{f_1} \left( N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} - S_{f_1} S_{f_2}\right) \varvec{\nabla }\cdot \mathbf{v}_{f_1}\nonumber \\&\quad \left. +\, K_{f_2} S_{f_2} \left[ S_{f_1} N \varphi + K_{f_1} \left( S_{f_1}^2 +N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} \right) \right] \varvec{\nabla } \cdot \mathbf{v}_{f_2}\right\} , \end{aligned}$$
(10)

with

$$\begin{aligned} \frac{1}{M}&= S_{f_1} S_{f_2} \left( S_{f_1} K_{f_1} + S_{f_2} K_{f_2} + N \varphi \right) \nonumber \\&\,+\left( K_{f_2} S_{f_1} N \varphi + K_{f_1} K_{f_2} + K_{f_1} S_{f_2} N \varphi \right) \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}}. \end{aligned}$$
(11)

This equation can be reduced to the equation for the saturated case by setting \(S_{f_1}=1\) and \(S_{f_1}=0\):

$$\begin{aligned} \frac{\mathrm{d}p_f}{\mathrm{d}t} = -M \left\{ \varvec{\nabla } \cdot \left[ \left( b-\varphi \right) \mathbf{v}^\mathrm{s} \right] + \varvec{\nabla } \left( \varphi \mathbf{v}^f\right) \right\} , \end{aligned}$$
(12)

with

$$\begin{aligned} \frac{1}{M} = \frac{1}{N}+\frac{\varphi }{K_f}. \end{aligned}$$
(13)

2.2 Balance of Momentum

The balance of linear momentum may be written as

$$\begin{aligned} \varvec{\nabla } \cdot \left[ \left( 1-\varphi \right) \varvec{\sigma }^\mathrm{s} \right] + \left( 1-\varphi \right) \rho _\mathrm{s} \left( \mathbf{b} - \mathbf{a}^\mathrm{s} \right) + \mathbf{f}^{\rightarrow \mathrm{s}}&= \mathbf{0} , \end{aligned}$$
(14)
$$\begin{aligned} \varvec{\nabla } \cdot \left( \varphi S_{f_\alpha } \varvec{\sigma }^{f_\alpha } \right) + \varphi S_{f_\alpha } \rho _{f_\alpha } \left( \mathbf{b} - \mathbf{a}^{f_\alpha } \right) + \mathbf{f}^{\rightarrow f_\alpha }&= \mathbf{0} , \end{aligned}$$
(15)

for the solid part, s, and the fluids, \(f_\alpha \), respectively, where \(\varvec{\sigma }^\mathrm{s}\) denotes the partial stress tensor of the solid, \(\varvec{\sigma }^{f_\alpha }\) the partial stress tensor of fluid \(f_\alpha \), \(\mathbf{b}\) the body force per unit mass, \(\mathbf{a}^s\) and \(\mathbf{a}^{f_\alpha }\) the solid and fluid acceleration, respectively, and \(\mathbf{f}^{\rightarrow \mathrm{s}}\) and \(\mathbf{f}^{\rightarrow f_\alpha }\) capture the interaction with the solid and the fluids, respectively. These interactions have to be balanced, so the momentum transfer is given by

$$\begin{aligned} \mathbf{f}^{\rightarrow \mathrm{s}} + \sum \limits _\alpha \mathbf{f}^{\rightarrow f_\alpha } = \mathbf{0}. \end{aligned}$$
(16)

2.2.1 Fluids

The extended Darcy’s law is given by

$$\begin{aligned} \varphi S_{f_\alpha } \left( \mathbf{v}^{f_\alpha } - \mathbf{v}^\mathrm{s}\right) = -\varvec{\lambda }_{f_\alpha } \cdot \left[ \varvec{\nabla } p_{f_\alpha } - \rho _{f_\alpha } \left( \mathbf{b} - \mathbf{a}^{f_\alpha } \right) \right] , \end{aligned}$$
(17)

with

$$\begin{aligned} \varvec{\lambda }_{f_\alpha } = \frac{k_{\mathrm{r},{f_\alpha }} \mathbf {k}}{\mu _{f_\alpha }}, \end{aligned}$$
(18)

where \(\varvec{\lambda }_{f_\alpha }\) and \(\mu _{f_\alpha }\) are the mobility tensor and the viscosity of the fluid \(f_\alpha \), respectively, and \(\mathbf {k}\) is the permeability tensor of the skeleton (compare to Neumann 1977). \(k_{\mathrm{r},{f_\alpha }} = k_{\mathrm{r},{f_\alpha }} \left( S_{f_\alpha } \right) \) is the saturation-dependent relative permeability. The most commonly used hydraulic conductivity models are those by Brooks and Corey (1964) and van Genuchten (1980). In this work, the model by van Genuchten (1980) is used, where the relative permeability is given by

$$\begin{aligned} k_{\mathrm{r},{f_\alpha }} \left( S_{f_\alpha }\right) = \sqrt{S_{f_\alpha }} \left[ 1-\left( 1- {S_{f_\alpha }}^{\frac{1}{m}}\right) ^m\right] ^2 ,\end{aligned}$$
(19)

with the fitting parameters \(m\) and \(n\), related by \(m=1-\frac{1}{n}\).

We insert Darcy’s law into the balance of linear momentum for the fluids (Eq. 15) to obtain

$$\begin{aligned} \mathbf{f}^{\rightarrow f_\alpha } = S_{f_\alpha } \left[ -\varphi ^2 S_{f_\alpha } \varvec{\lambda }_{f_\alpha }^{-1} \cdot \left( \mathbf{v}^{f_\alpha } - \mathbf{v}^\mathrm{s}\right) + p_{f_\alpha } \varvec{\nabla } \varphi \right] + \varphi p_{f_\alpha } \varvec{\nabla } S_{f_\alpha }. \end{aligned}$$
(20)

Upon insertion of the last equation into Eq. 15, we obtain the balance of linear momentum for the fluids \(f_\alpha \):

$$\begin{aligned} -\varphi S_{f_\alpha } \varvec{\nabla } p_{f_\alpha } + \varphi ^2 S_{f_\alpha }^2 \varvec{\lambda }_{f_\alpha }^{-1} \left( \mathbf{v}^\mathrm{s} - \mathbf{v}^{f_\alpha } \right) + \varphi S_{f_\alpha } \rho _{f_\alpha } \mathbf{b} = \varphi S_{f_\alpha } \rho _{f_\alpha } \mathbf{a}^{f_\alpha }. \end{aligned}$$
(21)

2.2.2 Solid

Writing out the momentum transfer (Eq. 16) and inserting Eqs. 14 and 20 yields

$$\begin{aligned} \left( 1-\varphi \right) \rho _\mathrm{s} \mathbf{a}^\mathrm{s}&= \varvec{\nabla } \cdot \left[ \left( 1-\varphi \right) \varvec{\sigma }^\mathrm{s} \right] -p_f \varvec{\nabla } \varphi + \varphi ^2 \sum \limits _\alpha S_{f_\alpha }^2 \varvec{\lambda }_{f_\alpha }^{-1} \cdot \left( \mathbf{v}^{f_\alpha } - \mathbf{v}^\mathrm{s}\right) \nonumber \\&-\, \varphi \sum \limits _\alpha p_{f_\alpha } \varvec{\nabla } S_{f_\alpha } + \left( 1-\varphi \right) \rho _\mathrm{s} \mathbf{b}. \end{aligned}$$
(22)

According to Biot (1955), the total stress is given by

$$\begin{aligned} \varvec{\sigma } = \varvec{\sigma }^{\prime \mathrm{s}} - bp_f \mathbf{I}, \end{aligned}$$
(23)

where \(\varvec{\sigma }\) denotes the total stress, \(\varvec{\sigma }^{\prime \mathrm{s}}\) the effective stress, \(\mathbf{I}\) the identity tensor, and \(b\) is Biot’s coefficient, which depends on the compressibility of the solid matrix and is defined by

$$\begin{aligned} b = 1 - \frac{K^\mathrm{s}}{K_\mathrm{s}}. \end{aligned}$$
(24)

Here, \(K^\mathrm{s}\) denotes the bulk modulus of the matrix and \(K_\mathrm{s}\) denotes the bulk modulus of the grains. By using this expression and the definition of the partial stress

$$\begin{aligned} \varvec{\sigma } = \left( 1-\varphi \right) \varvec{\sigma }^\mathrm{s} + \varphi \varvec{\sigma }^f = \left( 1-\varphi \right) \varvec{\sigma }^\mathrm{s} - \varphi p_f \mathbf{I}, \end{aligned}$$
(25)

we finally obtain the balance of linear momentum for the solid:

$$\begin{aligned} \begin{aligned} \left( 1-\varphi \right) \rho _\mathrm{s} \mathbf{a}^\mathrm{s} =\,&\varvec{\nabla } \varvec{\sigma }^{\prime \mathrm{s}} - \psi \varvec{\nabla } p_f - \varphi ^2 \sum \limits _\alpha S_{f_\alpha }^2 \varvec{\lambda }_{f_\alpha }^{-1} \cdot \left( \mathbf{v}^\mathrm{s} - \mathbf{v}^{f_\alpha }\right) \\&- \varphi \sum \limits _\alpha p_{f_\alpha } \varvec{\nabla } S_{f_\alpha } + \left( 1-\varphi \right) \rho _\mathrm{s} \mathbf{b}. \end{aligned} \end{aligned}$$
(26)

For convenience, we have defined

$$\begin{aligned} \psi = \psi (\varphi ) \equiv b-\varphi . \end{aligned}$$
(27)

2.3 Linearization of the Balance of Linear Momentum

To linearize the balance of momentum, we need to integrate the storage equation (Eqs. 9 and 10) to obtain an expression for the partial pressure \(p_{f_\alpha }\), \(\alpha =1,2\). We neglect the body forces \(\mathbf{b}\) and assume small fluid and solid motions. Thus, we obtain

$$\begin{aligned} p_{f_1}&= p_{f_1,0} + M \left\{ -K_{f_1} N \left( K_{f_2}\frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_c} + S_{f_1} S_{f_2}\right) \psi \varvec{\nabla } \cdot \mathbf{u}_\mathrm{s} \right. \nonumber \\&+\, K_{f_1} S_{f_1} \left[ S_{f_2}N \varphi + K_{f_2}\left( S_{f_2}^2 +N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_c} \right) \right] \varvec{\nabla } \cdot \mathbf{u}_{f_1} \nonumber \\&\left. +\, K_{f_1} K_{f_2}S_{f_2}\left( N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_c} - S_{f_1} S_{f_2}\right) \varvec{\nabla }\cdot \mathbf{u}_{f_2}\right\} \end{aligned}$$
(28)
$$\begin{aligned} p_{f_2}&= p_{f_2,0} + M \left\{ -K_{f_1}N \left( K_{f_1} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_c} + S_{f_1} S_{f_2}\right) \psi \varvec{\nabla } \cdot \mathbf{u}_\mathrm{s} \right. \nonumber \\&+\, K_{f_1} K_{f_2}S_{f_1}\left( N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_c} - S_{f_1} S_{f_2}\right) \varvec{\nabla }\cdot \mathbf{u}_{f_1} \nonumber \\&\left. +\, K_{f_2}S_{f_2}\left[ S_{f_1} N \varphi + K_{f_1} \left( S_{f_1}^2 +N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_c} \right) \right] \varvec{\nabla } \cdot \mathbf{u}_{f_2}\right\} . \end{aligned}$$
(29)

We substitute these equations into Eq. 21 to obtain the linearized balance of linear momentum of the fluids

$$\begin{aligned}&M N K_{f_1} S_{f_1} \left( K_{f_2} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_1} S_{f_2} \right) \varphi \psi \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_\mathrm{s}\right) \nonumber \\&\quad +\, M K_{f_1} S_{f_1}^2 \left[ K_{f_2} S_{f_2}^2 + N \varphi \left( K_{f_2} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_2} \right) \right] \varphi \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_{f_1}\right) \nonumber \nonumber \\&\quad +\, M K_{f_1} K_{f_2} S_{f_1} S_{f_2} \left( N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} - S_{f_1} S_{f_2}\right) \varphi \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_{f_2}\right) \nonumber \\&\quad +\, \varphi ^2 S_{f_1}^2\varvec{\lambda }_{f_1}^{-1} \frac{\partial \mathbf{u}_s}{\partial t} - \varphi ^2 S_{f_1}^2\varvec{\lambda }_{f_1}^{-1} \frac{\partial \mathbf{u}_{f_1}}{\partial t} - \varphi S_{f_1} \rho _{f_1} \frac{\partial ^2 \mathbf{u}_{f_1}}{\partial t^2} = 0 \end{aligned}$$
(30)
$$\begin{aligned}&M N K_{f_2} S_{f_2} \left( K_{f_1} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_1} S_{f_2} \right) \varphi \psi \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_s\right) \nonumber \\&\quad +\, M K_{f_1} K_{f_2} S_{f_1} S_{f_2} \left( N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} - S_{f_1} S_{f_2}\right) \varphi \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_{f_1}\right) \nonumber \\&\quad + M K_{f_2} S_{f_2}^2 \left[ K_{f_1} S_{f_1}^2 + N \varphi \left( K_{f_1} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} +S_{f_1} \right) \right] \varphi \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_{f_2}\right) \nonumber \\&\quad +\, \varphi ^2 S_{f_2}^2\varvec{\lambda }_{f_2}^{-1} \frac{\partial \mathbf{u}_s}{\partial t} - \varphi ^2 S_{f_2}^2\varvec{\lambda }_{f_2}^{-1} \frac{\partial \mathbf{u}_{f_2}}{\partial t} - \varphi S_{f_2} \rho _{f_2} \frac{\partial ^2 \mathbf{u}_{f_2}}{\partial t^2} = 0 . \end{aligned}$$
(31)

We also substitute the linearized storage equation (Eqs. 28 and 29) into the equation that describes the balance of linear momentum of the solid (Eq. 26). In addition, we use Hooke’s law and the Lamé moduli \(\lambda ^s\) and \(\mu ^\mathrm{s}\) for isotropic media to replace the stress tensor \(\varvec{\sigma }^{\prime \mathrm{s}}\) by the displacement gradients \(\nabla \mathbf{u}^\mathrm{s}\) according to

$$\begin{aligned} \varvec{\sigma }^{\prime \mathrm{s}}&= {\mathbf{E}}^\mathrm{s} : {\varvec{\varepsilon }}^\mathrm{s} = 2\mu ^\mathrm{s}{\varvec{\varepsilon }}^\mathrm{s} + \lambda ^\mathrm{s} \mathrm{tr}\left( {\varvec{\varepsilon }}^\mathrm{s}\right) \mathbf{I} \end{aligned}$$
(32)
$$\begin{aligned} \varvec{\nabla } \varvec{\sigma }^{\prime \mathrm{s}}&= \left( \lambda ^\mathrm{s}+2\mu ^\mathrm{s}\right) \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}^\mathrm{s}\right) - \mu ^\mathrm{s} \varvec{\nabla }\times \left( \varvec{\nabla }\times \mathbf{u}^\mathrm{s}\right) , \end{aligned}$$
(33)

where \({\mathbf{E}}^\mathrm{s}\) denotes the elastic tensor and \({\varvec{\varepsilon }}^\mathrm{s}\) the linear strain tensor (compare to Jeffreys and Jeffreys 1950). Thus, we finally obtain the linearized balance of linear momentum equation for the solid:

$$\begin{aligned}&\left\{ \lambda ^\mathrm{s}+2\mu ^\mathrm{s} + M N \left[ K_{f_1} K_{f_2} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_1} S_{f_2} \left( K_{f_1} S_{f_1} + K_{f_2} S_{f_2} \right) \right] \psi ^2 \right\} \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_s\right) \nonumber \\&\quad + M N K_{f_1} S_{f_1} \left( S_{f_1} S_{f_2} + K_{f_2} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} \right) \varphi \psi \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_{f_1}\right) \nonumber \\&\quad + M N K_{f_2} S_{f_2} \left( S_{f_1} S_{f_2} + K_{f_1} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} \right) \varphi \psi \varvec{\nabla }\left( \varvec{\nabla }\cdot \mathbf{u}_{f_2}\right) \nonumber \\&\quad - \varphi ^2 \left( S_{f_1}\varvec{\lambda }_{f_1}^{-1} + S_{f_2}\varvec{\lambda }_{f_2}^{-1}\right) \cdot \frac{\partial \mathbf{u}_\mathrm{s}}{\partial t} +\varphi ^2 S_{f_1}\varvec{\lambda }_{f_1}^{-1} \cdot \frac{\partial \mathbf{u}_{f_1}}{\partial t} + \varphi ^2 S_{f_2}\varvec{\lambda }_{f_2}^{-1} \cdot \frac{\partial \mathbf{u}_{f_2}}{\partial t} \nonumber \\&\quad - \left( 1-\varphi \right) \rho _\mathrm{s} \frac{\partial ^2 \mathbf{u}_\mathrm{s}}{\partial t^2} - \mu ^\mathrm{s} \varvec{\nabla }\times \left( \varvec{\nabla }\times \mathbf{u}_\mathrm{s}\right) = 0. \end{aligned}$$
(34)

2.4 Propagation of Harmonic Waves

2.4.1 Longitudinal Waves

Let us consider the propagation of harmonic longitudinal waves in the \(\mathbf{e}_x\) direction:

$$\begin{aligned} \mathbf{u}^\mathrm{s}&= u^\mathrm{s}_0 {\mathrm{Re}}\left\{ \exp \left[ i\left( sx+\omega t\right) \right] \right\} \mathbf{e}_x, \end{aligned}$$
(35)
$$\begin{aligned} \mathbf{u}^{f_\alpha }&= u^{f_\alpha }_0 {\mathrm{Re}}\left\{ \exp \left[ i\left( sx+\omega t\right) \right] \right\} \mathbf{e}_x, \end{aligned}$$
(36)

where \(s\) denotes the wave number and \(\omega \) denotes the angular frequency.

2.4.1.1 Two Fluids

Substituting this into Eqs. 30, 31, and 34 yields

$$\begin{aligned} \left( -\mathrm{s}^2\mathsf{S}+ i \omega \mathsf{K}+ \omega ^2\mathsf{M}\right) \cdot \left( \begin{matrix} u^\mathrm{s}_0 \\ u^{f_1}_0 \\ u^{f_2}_0 \end{matrix} \right) = 0, \end{aligned}$$
(37)

with the stiffness matrix \(\mathsf{S}\), the damping matrix \(\mathsf{K}\), and the mass matrix \(\mathsf{M}\) defined as

$$\begin{aligned} \mathsf{S}&= \left( \begin{array}{lll} \mathsf{s}_{11} &{} \mathsf{s}_{12} &{} \mathsf{s}_{13}\\ \mathsf{s}_{21} &{} \mathsf{s}_{22} &{} \mathsf{s}_{23}\\ \mathsf{s}_{31} &{} \mathsf{s}_{32} &{} \mathsf{s}_{33}\\ \end{array} \right) ,\end{aligned}$$
(38)
$$\begin{aligned} \mathsf{K}&= \varphi ^2 \left( \begin{matrix} - \left( \frac{S_{f_1}^2}{\lambda _{f_1}} + \frac{S_{f_2}^2}{\lambda _{f_2}} \right) &{} &{} \frac{S_{f_1}^2}{\lambda _{f_1}} &{} &{} \frac{S_{f_2}^2}{\lambda _{f_2}} \\ \frac{S_{f_1}^2}{\lambda _{f_1}} &{} &{} -\frac{S_{f_1}^2}{\lambda _{f_1}} &{} &{} 0 \\ \frac{S_{f_2}^2}{\lambda _{f_2}} &{} &{} 0 &{} &{} -\frac{S_{f_2}^2}{\lambda _{f_2}} \end{matrix} \right) ,\end{aligned}$$
(39)
$$\begin{aligned} \mathsf{M}&= \left( \begin{matrix} \left( 1-\varphi \right) \rho _\mathrm{s} &{} &{} 0 &{} &{} 0 \\ 0 &{} &{} \varphi S_{f_1} \rho _{f_1} &{} &{} 0 \\ 0 &{} &{} 0 &{} &{} \varphi S_{f_2} \rho _{f_2} \end{matrix} \right) . \end{aligned}$$
(40)
$$\begin{aligned} \mathsf{s}_{11}&= \lambda ^\mathrm{s}+2\mu ^\mathrm{s} + M N \left[ K_{f_1} K_{f_2} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_1} S_{f_2} \left( K_{f_1} S_{f_1} + K_{f_2} S_{f_2} \right) \right] \psi ^2\end{aligned}$$
(41)
$$\begin{aligned} \mathsf{s}_{22}&= M K_{f_1} S_{f_1}^2 \left[ K_{f_2} S_{f_2}^2 + N \varphi \left( K_{f_2} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_2} \right) \right] \varphi \end{aligned}$$
(42)
$$\begin{aligned} \mathsf{s}_{33}&= M K_{f_2} S_{f_2}^2 \left[ K_{f_1} S_{f_1}^2 + N \varphi \left( K_{f_1} \frac{d S_{f_1}}{d p_\mathrm{c}} +S_{f_1} \right) \right] \varphi \end{aligned}$$
(43)
$$\begin{aligned} \mathsf{s}_{12}&= \mathsf{s}_{21} = M N K_{f_1} S_{f_1} \left( K_{f_2} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} + S_{f_1} S_{f_2} \right) \varphi \psi \end{aligned}$$
(44)
$$\begin{aligned} \mathsf{s}_{13}&= \mathsf{s}_{31} = M N K_{f_2} S_{f_2} \left( K_{f_1} \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} +S_{f_1} S_{f_2} \right) \varphi \psi \end{aligned}$$
(45)
$$\begin{aligned} \mathsf{s}_{23}&= \mathsf{s}_{32} = M K_{f_1} K_{f_2} S_{f_1} S_{f_2} \left( N \varphi \frac{\mathrm{d} S_{f_1}}{\mathrm{d} p_\mathrm{c}} - S_{f_1} S_{f_2}\right) \varphi \end{aligned}$$
(46)

To obtain the dispersion relation for the harmonic longitudinal waves, we have to solve the following equation for wave number \(s\) as a function of frequency \(\omega \):

$$\begin{aligned} \mathrm{det}\left( -s^2\mathsf{S}+ i \omega \mathsf{K}+ \omega ^2\mathsf{M}\right) = 0. \end{aligned}$$
(47)

The resulting \(s\) reveals both the speed for the longitudinal waves \(c=\frac{\omega }{{\mathrm{Re}}\left( s\right) }\) and the corresponding attenuation coefficient \(\alpha =\mathrm{{Im}}\left( s\right) \).

4.2.1.2 One Fluid

We can also obtain the well-known equation for fluid-saturated porous media (e.g., Coussy 2004). For the saturated case, we only consider one fluid and the corresponding saturation is \(S_f = 1\). Therefore, we find

$$\begin{aligned} \left( -s^2\mathsf{S}+ i \omega \mathsf{K}+ \omega ^2\mathsf{M}\right) \cdot \left( \begin{matrix} u^s_0 \\ u^f_0 \end{matrix} \right) = 0. \end{aligned}$$
(48)

In this case, the stiffness matrix \(\mathsf{S}\), damping matrix \(\mathsf{K}\), and mass matrix \(\mathsf{M}\) are defined by

$$\begin{aligned} \mathsf{S}&= \left( \begin{matrix} \lambda ^\mathrm{s}+2\mu ^\mathrm{s} + \psi ^2 M &{} &{} \varphi \psi M \\ \varphi \psi M &{} &{} \varphi ^2 M \end{matrix} \right) , \end{aligned}$$
(49)
$$\begin{aligned} \mathsf{K}&= \frac{\varphi ^2}{\lambda _f} \left( \begin{matrix} - 1 &{} &{} 1 \\ 1 &{} &{} - 1 \end{matrix} \right) , \end{aligned}$$
(50)
$$\begin{aligned} \mathsf{M}&= \left( \begin{matrix} \left( 1-\varphi \right) \rho _s &{} &{} 0\\ 0 &{} &{} \varphi \rho _f \end{matrix} \right) . \end{aligned}$$
(51)

2.4.2 Transverse Waves

Let us now consider the propagation of harmonic transverse waves traveling in the \(\mathbf{e}_y\) direction and polarized in the \(\mathbf{e}_x\) direction:

$$\begin{aligned} \mathbf{u}^s&= u^s_0 {\mathrm{Re}}\left\{ \exp \left[ i\left( sy+\omega t\right) \right] \right\} \mathbf{e}_x \end{aligned}$$
(52)
$$\begin{aligned} \mathbf{u}^{f_\alpha }&= u^{f_\alpha }_0 {\mathrm{Re}}\left\{ \exp \left[ i\left( sy+\omega t\right) \right] \right\} \mathbf{e}_x. \end{aligned}$$
(53)

4.2.2.1 Two Fluids

Substituting this into Eqs. 30, 31, and 34 yields

$$\begin{aligned} \left( -s^2 \mathsf{T} + i \omega \mathsf{K}+ \omega ^2\mathsf{M}\right) \cdot \left( \begin{matrix} u^s_0 \\ u^{f_1}_0 \\ u^{f_2}_0 \end{matrix} \right) = 0 \end{aligned}$$
(54)

with the damping and mass matrices defined according to Eqs. 39 and 40 and the matrix \(\mathsf{T}\) defined by

$$\begin{aligned} \mathsf{T} = \left( \begin{matrix} \mu ^\mathrm{s} &{} &{} 0 &{} &{} 0 \\ 0 &{} &{} 0 &{} &{} 0 \\ 0 &{} &{} 0 &{} &{} 0 \end{matrix} \right) . \end{aligned}$$
(55)

To obtain the dispersion relation for the harmonic transverse waves, we have to solve the following equation for the wave number \(s\) as a function of the frequency \(\omega \):

$$\begin{aligned} {\mathrm{det}}\left( -s^2 \mathsf{T} + i \omega \mathsf{K}+ \omega ^2\mathsf{M}\right) = 0. \end{aligned}$$
(56)

The resulting \(s\) reveals both the speed for the transverse wave \(c=\frac{\omega }{{\mathrm{Re}}\left( s\right) }\) and the corresponding attenuation coefficient \(\alpha =\mathrm{{Im}}\left( s\right) \).

2.4.2.2 One Fluid

For the case of one fluid we have

$$\begin{aligned} \left( -s^2 \mathsf{T} + i \omega \mathsf{K}+ \omega ^2\mathsf{M}\right) \cdot \left( \begin{matrix} u^\mathrm{s}_0 \\ u^f_0 \end{matrix} \right) = 0 \end{aligned}$$
(57)

with the damping and mass matrices defined according to Eqs. 50 and 51 and with the matrix \(\mathsf{T}\) defined by

$$\begin{aligned} \mathsf{T} = \left( \begin{matrix} \mu ^\mathrm{s} &{} &{} 0 \\ 0 &{} &{} 0 \end{matrix} \right) . \end{aligned}$$
(58)

3 Application

Having obtained the necessary equations for calculating wave speeds in unsaturated porous media, we apply our theory for a representative storage site. Kopp et al. (2009) determined characteristic parameters for typical reservoirs using statistical calculations based on a database of more than 1,200 reservoirs. They found that an average reservoir has a depth of 1,524 m with a reservoir pressure of \(15.47\,\hbox {MPa}\) at a temperature of 55.13 \(^{\circ }\hbox {C}\). The density of the water therein is \(1025.5\,\hbox {kg}\,\hbox {m}^{-3}\). With these values, we calculated the density and saturation of \(\hbox {CO}_{2}\) as well as the saturation of water for various depths, starting at 1,524 m, using a flash calculation (Gor and Prévost 2014; Prévost 1981). The capillary pressure curve of a \(\hbox {CO}_{2}\)/water system was measured by e.g., Plug and Bruining (2007) and Pini et al. (2012). We calibrated the capillary pressure curve model by van Genuchten (1980), which was given in Eq. 7, with the data presented by Pini et al. (2012) and obtained \(n=2.28\) and \(\chi =0.69\,\hbox {m}^{-1}\). These and all other values used are listed in Tables 1 and 2. We used our equations for wave speeds and attenuation of longitudinal (Eq. 47) and transverse waves (Eq. 56) to determine profiles as a function of depth \(h\) and frequency \(\omega \).

Table 1 Depth-independent physical properties used for the calculation of wave speeds
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Table 2 Depth-dependent properties used in the calculation
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4 Results

We found that there is a third mode pressure wave in porous media saturated by two immiscible, whereas there are only two modes of pressure waves in porous media saturated by only one fluid. This was also observed in previous studies (e.g., Santos et al. 1990a, b; Lo et al. 2005; Albers 2009). We have a first mode pressure wave (\(\hbox {P}_1\)), where the solid and fluids are in phase; a second mode pressure wave (\(\hbox {P}_{2}\)), where the solid and the fluids move out of phase; and a third mode pressure wave (\(\hbox {P}_3\)), where the fluids move in phase but not in phase with the solid. The additional wave, compared to the case of one fluid is \(\hbox {P}_{2}\). For every additional fluid there would be an additional pressure wave phase, because the dimension of the matrices is \(\left( \alpha +1\right) \times \left( \alpha +1\right) \) (see Eqs. 47 and 56) and the number of solutions is \(\left( \alpha +1\right) \).

To compare the differences between the saturated case, where all pores are filled with water, and the unsaturated case, where there is \(\hbox {CO}_{2}\) and water in the pore space, we compared these two cases at \(600\,\hbox {m}\) depth (see Fig. 2). This leads us to several interesting observations: \(\hbox {P}_{1}\) is about 15% slower in the case where \(\hbox {CO}_{2}\) is present compared to the case where there is only water in the pores, whereas \(\hbox {P}_{2}\) is about 95% slower in the case where \(\hbox {CO}_{2}\) is present. The additional \(\hbox {P}_3\) wave is extraordinary slow and highly attenuated. The transverse S wave is not affected by the content of the pores, and in both cases \(\hbox {P}_{1}\) and S are only slightly attenuated, whereas \(\hbox {P}_{2}\) and \(\hbox {P}_{3}\) are highly attenuated.

Fig. 2
figure 2

Wave speeds (left) and attenuation (right) of the seismic waves at a depth of \(600\,\hbox {m}\). The upper plots are calculated using the equations for the saturated case and the lower plots are calculated using the equations for the unsaturated case

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For every mode, there is a 3D-plot for both wavespeed and attenuation. These plots are shown in Figs. 3 (\(\hbox {P}_{1}\)), 4 (\(\hbox {P}_{2}\)), 5 (\(\hbox {P}_{3}\)), and 6 (S). From these figures, we conclude that \(\hbox {P}_{1}\), \(\hbox {P}_{2}\), and S suddenly increase in speed by a few percent as soon as the \(\hbox {CO}_{2}\) boils, which is at \(695\,\hbox {m}\), whereas the attenuation behaves conversely only for \(\hbox {P}_{1}\) and S. For \(\hbox {P}_{2}\) the attenuation also increases as soon as the \(\hbox {CO}_{2}\) boils. \(\hbox {P}_{2}\) first decreases in speed as soon as the \(\hbox {CO}_{2}\) boils and then continues to decrease in speed. The corresponding attenuation of \(\hbox {P}_{3}\) behaves in the same way, like it is the case for \(\hbox {P}_{2}\).

Fig. 3
figure 3

First mode pressure wave speed (left) and corresponding attenuation (right) as a function of depth and frequency

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Fig. 4
figure 4

Second mode pressure wave speed (left) and corresponding attenuation (right) as a function of depth and frequency

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Fig. 5
figure 5

Third mode pressure wave speed (left) and corresponding attenuation (right) as a function of depth and frequency

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Fig. 6
figure 6

Shear wave speed (left) and corresponding attenuation (right) as a function of depth and frequency

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5 Discussion

We determined that the wave speed of \(\hbox {P}_{1}\) decreases by about 15% and that of \(\hbox {P}_{2}\) decreases by about 95% if \(\hbox {CO}_{2}\) is present. When analyzing real data, this change in wave speed of \(\hbox {P}_{1}\) is not necessarily related to a pore-content change, because this could also be explained, for example, by a change in the elastic moduli of the rock. The existance of a third mode could provide evidence of an additional phase in the pores, but \(\hbox {P}_{2}\) and \(\hbox {P}_{3}\) are very slow and highly attenuated. This makes it difficult to detect these phases directly.

Previously, among others, Albers (2009) and Santos et al. (1990a, b) dealt with similar problems, although they used different approaches and calculated wave speeds and attenuation for different scenarios. Interestingly, Albers (2009) calculated very similar values for a sandstone filled with an air-water mixture (Albers 2009, Fig. 4) using an approach based on a macroscopic linear model, which accommodates both the Biot model and the Simple Mixture Model of Wilmanski (1998). For the Lagrangian derivation of Santos et al. (1990a), a thermodynamical approach including the principle of virtual complementary work and energy density functions was used. They computed wave speeds for oil-water mixtures in two different sandstone formations (Santos et al. 1990b). The derivation of our equations is loosely based on Coussy (2004). The advantage of our work is that we only use the balances of mass (Eq. 1) and linear momentum (Eqs. 14 and 15), as well as an equation which is based on the equations of state (Eq. 8) and Darcy’s law (Eq. 17). This makes our derivation clear and simple. Nevertheless, the similarity of our numerical results with the results from Albers (2009) and Santos et al. (1990a, b) indicates that our approach is consistent with their work.

6 Conclusion

We have presented an extension of Biot’s theory valid for unsaturated porous media with two different pore-space phases. We found that for each additional fluid phase there is an additional pressure wave (i.e., the number of different kinds of pressure waves is \(\alpha +1\)).

We tested this extension for a representative \(\hbox {CO}_{2}\)-storage scenario to assess whether it is possible to detect leakage of a \(\hbox {CO}_{2}\)-storage site. Our results suggest that it will be difficult to detect the \(\hbox {P}_{3}\)-wave because it is very slow and highly attenuated. More promising attempts involve detection of changes in \(\hbox {P}_{1}\) and \(\hbox {P}_{2}\) wave speeds induced by a leak. This requires a baseline seismic survey before injection and comparison of subsequent surveys to the baseline, i.e., time-lapse seismic migration (Carcione et al. 2006). Another approach involves cross-well monitoring (Morency et al. 2011).