Sonic velocity discontinuity for CO2 brine mixtures in the context of carbon storage in aquifers

Accurate modelling of fluid acoustic parameters and saturation distributions in heterogeneous aquifers is essential for predicting CO₂ plume behaviour. Conventional approaches, such as Batzle & Wang’s (1992) equations, fail to capture the complexities of CO₂–brine interactions under reservoir conditions. In this paper, we apply thermodynamic modelling to calculate sonic velocity and density of CO₂–brine mixtures, addressing the applicability of conventional averaging bounds for mixture bulk modulus. Results show that for low CO₂ mole fractions $(x_{\text{CO}2}\le 0.1-0.2)$, mixture properties remain relatively constant and sonic velocity follows the Gassmann-Hill bound. However, a discontinuity occurs at mole fractions of 0.1–0.2, where sonic velocity shifts to the Gassmann-Wood bound, driven by a sudden density drop as CO₂ patches form. Importantly, values lower than the harmonic bound are observed, consistent with Landau–Lifshitz’s theorem. Based on the results of this study, we demonstrate that the sonic velocity discontinuity coincides with the two-phase boundary between the CO₂ plume and brine. At low CO₂ mole fractions, dissolved gas slightly increases mixture density, but beyond $\sim 0.2$ mole fraction, brine evaporation into CO₂ causes a sharp density decrease. This behaviour implies that monitoring sonic velocity discontinuities provides a practical means to detect CO₂ dispersion within aquifers, thereby improving seismic interpretation accuracy in storage projects.

1 Introduction and background

Energy-related carbon dioxide (CO₂) emissions, resulting from the combustion of fossil fuels for electricity and heat generation, constitute more than 40% of total emissions (IEA, 2015), highlighting the critical need to address global CO₂ reduction efforts (Bui et al., 2021). Geothermal energy has emerged as a promising solution to this challenge, offering a sustainable means to meet heat demands. When combined with carbon capture, utilisation, and storage (CCUS) technologies, geothermal systems have the capacity to sequester significant amounts of CO₂ (10th of MtCO₂) underground for prolonged periods (BEIS, 2022; Busby and Terrington, 2017; Quaranta, 2023). Traditionally, brine has been functioning as the primary working fluid in geothermal systems, with Groundwater Heat Pumps (GWHP) representing common examples, operating on closed or open-loop water cycles. In 2010, Saar et al. introduced the innovative concept of CO₂-plume geothermal (CPG), that can potentially transform geothermal technology (Saar et al., 2010). CPG introduces a hybrid approach that employs CO₂ as the working fluid to extract heat from low-enthalpy geothermal resources. By injecting CO₂, a distinctive funnel-shaped plume forms within the aquifer (from cold injection well to hot production well), giving rise to the name CO₂-plume geothermal (CPG). This novel concept not only enables efficient heat extraction but also offers a viable solution for CO₂ storage.

Using CO₂ as a substitution for water features various advantages. CO₂ has higher mobility, lower solubility of amorphous silica, and higher density sensitivity to temperature compared to brine which makes CPG an attractive option for CCUS (Adams, 2015; Kolditz et al., 2010). After an initial sequestration stage where hot brine is produced, in a second stage of circulation, the geothermally heated CO₂ is produced, expanded in a turbine to generate electricity, cooled, and reinjected through another well back into the reservoir (Norouzi et al., 2023; 2022a). Also when back-produced CO₂ reaches to temperatures that are uneconomic for electricity generation, some of CO₂ will be left to remain stored underground (Fleming et al., 2020). Therefore before and after the circulation stage, some of CO₂ will be sequestered. During the initial sequestration stage and the circulation stage, the density of CO₂ changes substantially between the geothermal reservoir and surface plant, resulting in a thermosiphon that reduces or eliminates pumping requirements compared to brine (Adams et al., 2014). Thus, CPG for low enthalpy systems is recommended for sandstone reservoirs that feature high degrees of heterogeneity of pore space at different spatial length scales.

Since the introduction of CPG, numerous studies have explored the performance and thermal potential assessment of CPG sites, as well as investigating various aspects of CPG systems. Recent research has focused on mineral precipitation (Cui et al., 2017; 2018; Norouzi et al., 2022b), optimizing well spacing (Adams et al., 2021; Norouzi et al., 2022a), determining optimal well patterns and reservoir boundary conditions (Babaei, 2019), refining surface power plant cycles (Garapati et al., 2020; Schifflechner et al., 2022; 2020; Norouzi et al., 2023), assessing water and scCO₂ saturation and mass-fraction at production wells (Ezekiel et al., 2022; Fleming et al., 2020), evaluating CPG combined with natural gas recovery (Ezekiel et al., 2020), and examining CPG economic feasibility (Schifflechner et al., 2022). There are also few studies about monitoring CO₂ leakage, techniques for monitoring leakages, and identifying the leakage paths (Yang et al., 2024; Ndlovu et al., 2024). However, despite the well-documented sensitivity of CO₂ plumes to reservoir heterogeneity (Figure 1a), none of the aforementioned studies have addressed the critical need for implementing a comprehensive geophysical monitoring system. Such a system would enable real-time tracking of CO₂ plume propagation within the reservoir and facilitate accurate prediction of system performance. Incorporating advanced geophysical monitoring techniques, such as time-lapse seismic imaging, electromagnetic surveys, and micro-seismic monitoring, could offer valuable insights into the dynamic behaviour of CO₂ plumes, aiding in the optimisation of CPG operations and enhancing overall project efficiency.

Figure 1

Figure 1. (a) The effect of small-scale heterogeneity (50 m by 5 m) in lateral CO₂ plume extent (red arrow) (Jackson and Krevor, 2020). (b) Relative changes in seismic p-wave velocity and density versus CO₂ saturation (Eiken, 2019) for Sleipner. (c) and (e): The discontinuity seen in the calculation of sonic velocity $(c_s)$ between Gassmann-Wood and thermodynamic calculation of Firoozabadi & Pan (Firoozabadi and Pan, 2000; Niknam et al., 2017), (d) Landau-Lifshitz impact of small fractions of bubbles in water on sonic velocity (Kieffer, 1977). (f) Experimental data for bulk modulus $(\kappa )$ from depressurisation experiments to induce the exsolution of carbon dioxide from water in a Berea sandstone sample (Chapman et al., 2021).

For actual heterogeneous aquifers, seismic response modelling (Müller et al., 2010; Marjanović et al., 2019; Wang et al., 2010; Mortazavi et al., 2025) is used to study CO₂ distribution within the aquifer. The compositional interactions between CO₂ and resident brine will affect the elastic properties of fluid-saturated porous media, such that seismic-response characterisation for monitoring of injected CO₂ will be affected. The conventional methods of calculating the sonic/acoustic velocity $(c_s)$ for mixtures are based on Gassmann-Wood (Wood, 1955; Hill, 1963) method, (i.e., through bulk modulus of fluids divided by mixture density $c_s=\sqrt{{\kappa }_f/{\rho }_m}$). Since sound travels through porous media adiabatically, one can write $c_s=\sqrt{1/{\beta }_S{\rho }_m}$, where ${\beta }_S$ is the adiabatic/isentropic compressibility of the fluid.

For hydrocarbon fluid mixtures in reservoirs, Kandil (2021) examined the alterations in the bulk modulus of the hydrocarbon fluids following the injection of scCO₂. The study indicated that conventional models, such as the Gassmann model, which are only applicable to the isothermal regime, have limited predictive capabilities, This is because they often treat fluids by characterising their mechanical properties solely by their densities. However, in different environments, particularly when scCO₂ is introduced into the geological formation, the fluid phase and its mechanical properties can change significantly. Similarly, in an other work, Kandil et al. (2016) stated that some adjustments are required to conventional equations of states to predict properties of CO₂ and reservoir fluid mixtures.

For CO₂ and water, Nichita et al. (2010) used thermodynamics calculation of ${\beta }_S$ (adiabatic/isentropic compressibility of the fluid) through flash calculation and showed a discontinuity in ${\beta }_S$ versus pressure profile for 2% CO₂-98% water mixtures. The discontinuity occurs around the bubble point pressure. However, such a discontinuity is not captured by volume averaging $({\beta }_S^{*}=S_{{\text{CO}}_2}{\beta }_{{\text{CO}}_2}+S_{\text{water}}{\beta }_{\text{water}})$. A quick calculation of fluid mixture bulk modulus can be carried out using a MATLAB implementation of the Gassmann fluid-substitution formulation (Kumar, 2006), adapted for CO₂–brine mixtures. Using the coefficients of isothermal and isentropic compressibility factors calculated by Nichita et al. (2010) for the same 2% CO₂–98% water mixture at $T$ = 335.15 K, reveals that for the thermodynamic and Woods mixing approaches, the calculated bulk-modulus values are 1.1 GPa and 2.5 GPa, respectively. These values highlight discrepancy of the fluid mixture bulk modulus at the discontinuity. From the thermodynamic low fluid-mixture bulk modulus that is typical of gas-bearing mixtures, where gas has an extremely low bulk modulus, a strong decrease in P-wave velocity $V_p$ or $c_s$ and impedance will be observed, and fluid front should be better detectable associated with the CO₂ plume edge. A quantitative example of $V_p$ calculation is provided by Broseta et al. (2009), where a similar but not identical thermodynamic method to the proposed methodology here was used to calculate $V_p$. The authors refer to their method as Gassmann-Landau & Lifshitz method.

Such discontinuities have been reported for other subsurface fluid compositions under two-phase conditions as well (Firoozabadi and Pan, 2000; Niknam et al., 2017). The difference is shown in Figures 1c,e. Clearly, the sonic velocity (or bulk modulus of fluid) is lower for the thermodynamic approach than for the averaging method.

Above can be explained by Landau-Lifshitz regime of relaxed phases through mass transfer (Landau and Lifshitz, 1959), making the mixture fluid more compressible. The averaging method is based on unrelaxed fluid phases where there is no mass transfer between phases and hence fluid pressures are unequilibrated. This condition occurs when the liquid and gas phases are “frozen” or “patchy”, i.e., they do not have enough time to exchange matter at the passage of the pressure wave (Toms et al., 2006; Lebedev et al., 2009; Khalid et al., 2014). The patchy fluid distribution will also result in a higher mixture fluid bulk modulus and higher bound of sonic velocity $(c_s)$ (Figures 1b,d). The sensitivity of ${\kappa }_f$ is exacerbated for low gas saturations (e.g., as shown by the study at the Sleipner CO₂ storage site (Rubino et al., 2011)) or near bubble point conditions that can lead to over-optimistic sonic velocity and therefore no detection of low gas saturations. Finally, recent depressurisation experiments of Chapman et al. (2021) Figure 1f proves this point. In summary: $K_{\text{wood}}\gg K_{\text{experiment}}$, and $K_{\text{wood}}\gg K_{\text{thermodynamics}}$, therefore: $u_{\text{wood}}\gg u_{\text{experiment}}$, and $u_{\text{wood}}\gg u_{\text{thermodynamics}}$.

1.1 This study

Sonic velocity will be governed by the dynamic interplay of pressure and temperature, thermodynamics of CO₂-water interaction and macroscopic two-phase flow in CO₂ storage in the aquifers. For the typical seismic frequency bands of 1 – ${10}^2$ Hz, the seismic velocity calculated using averaging methods underestimates low CO₂ saturation under two-phase conditions. Due to the thermodynamics of CO₂-water interaction at the boundary of CO₂ plume, the bulk modulus and sonic velocity calculated by the thermodynamic model will be lower than Gassmann-Wood values (unrelaxed state of phase pockets).

The overall aim of this research is to improve performance prediction and monitoring of CO₂ storage. The objectives of this are as follows. I: Improve the accuracy of seismic response modelling in CO₂-brine distribution. II: Utilise the strong sensitivity of seismic velocity to low saturations of free CO₂ for monitoring purposes. To this end, the sonic velocity of CO₂-brine mixtures is calculated using thermodynamic modelling. It is coupled with the CO₂ storage simulator results to provide a realistic sonic velocity distribution in an aquifer. Results of the present study provide useful guidance for accurate study of CO₂ plume distribution inside an aquifer and prevent under/over-estimations occurring through conventional averaging methods. Although the present study focuses on CPG systems, the thermodynamic modelling approach and resulting insights into CO₂–brine acoustic behaviour are equally applicable to conventional CCS aquifers, thereby ensuring the generalisability of the findings.

2 Methods

To calculate sonic velocity in a CPG aquifer with varying temperature and pressure, first, thermodynamic properties of the CO₂-brine mixture are calculated and then, these results are used and coupled with the results of the aquifer model simulation. The thermodynamics and the aquifer model are described in the following sections.

2.1 Thermodynamic model and validation

Conventionally, the sonic velocity in a mixture is calculated using pure phase properties weighted by saturation. Gassmann-Wood (Wood, 1955) and Gassmann-Hill (Hill, 1963) are the lower and upper bounds of mixing, which are uniform and patchy, respectively. Using these methods, sonic velocity is calculated using Equations 1a–3:

$\frac{1}{K_{\text{f}}}=\frac{S_{\text{brine}}}{K_{\text{brine}}}+\frac{S_{{\text{CO}}_2}}{K_{{\text{CO}}_2}}(\mathrm{1a})$

$K_{\text{f}}=S_{\text{brine}}\times K_{\text{brine}}+S_{{\text{CO}}_2}\times K_{{\text{CO}}_2}(\mathrm{1b})$

$K_{\text{f}}=-V\frac{dP}{dV}=\rho \frac{dP}{d\rho }=\frac{1}{{\beta }_{\text{S}}}(2)$

$c_s=\sqrt{\frac{1}{\rho {\beta }_{\text{S}}}}(3)$

However, based on Introduction, for a two-phase ${\text{CO}}_2$–water system, the effective isentropic compressibility of the mixture cannot be expressed as a simple linear saturation-weighted average of the individual fluid compressibilities (Equation 4):

$K_{\text{f}}\ne S_{\text{brine}}\times K_{\text{brine}}+S_{{\text{CO}}_2}\times K_{{\text{CO}}_2},(4)$

When ${\text{CO}}_2$ appears or disappears as a separate phase, the system undergoes a thermodynamic phase transition. The onset of a second fluid phase introduces additional degrees of freedom related to mass transfer between phases. As a result, the pressure response of the mixture volume becomes non-smooth, leading to a kink or discontinuity in the isentropic compressibility as a function of pressure, temperature, or ${\text{CO}}_2$ saturation. This discontinuity is a fundamental thermodynamic effect associated with phase equilibrium constraints and is not a numerical artifact. It is particularly pronounced near phase boundaries and close to the critical point of ${\text{CO}}_2$, where small changes in saturation or pressure can produce large changes in compressibility.

Here we use SLB Symmetry to calculate $c_s$ based on the thermodynamics. Interaction between the vapour and liquid phases are extracted using thermodynamic equations. The multiphase flash is to solve the thermodynamic equilibrium calculations. SLB Symmetry software package implements Peng Robinson equation of state to calculate the CO₂-brine mixture properties. Once the flash calculation is done, the output isothermal compressibility and the heat capacity ratio are known, so the sonic velocity can be calculated as

$c_s=\sqrt{{\left(\frac{\text{d}P}{\text{d}\rho }\right)}_S}=\sqrt{\frac{C_{\text{P,Bulk}}}{C_{\text{V,Bulk}}}\left(\frac{V_{\text{Bulk}}^2}{M{W}_{\text{Bulk}}}\right){\left(\frac{\text{d}P}{\text{d}V}\right)}_T}(5)$

where, heat capacity ratio $(\gamma =C_{\text{P,Bulk}}/C_{\text{V,Bulk}})$, ${(\text{d}P/\text{d}V)}_T$, bulk volume $(V_{\text{Bulk}})$, bulk molar weight $(M{W}_{\text{Bulk}})$ are known or calculated from flash calculation results. Temperature ranges between 35 and 135 $°$C, and pressure ranges between 20 and 40 MPa, and CO₂ mole fraction ranges between 0-1. All the parameters in Equation 5 are plotted versus pressure, temperature, and mole fraction in Section 3. The results are compared with two experimental studies to validate Symmetry for calculating properties of CO₂-brine mixtures. Figure 2a shows the sonic velocity compared with those reported by Tahani (2011) and the results for isentropic compressibility compared with the results provided by Firoozabadi et al. (1988). Based on the validation results, it is observed that for high pressures (above 25 MPa), Symmetry provides acceptable and reasonable results compared to the experimental results. Therefore, Symmetry is used to calculate the thermodynamic properties of CO₂-brine mixtures and $c_s$. In this study, mole fraction $(x_{\mathit{CO2}})$ is used in thermodynamic calculations (equations of state and flash analysis).

Figure 2

Figure 2. (a) Validating the Symmetry results with experimental data for sonic velocity (m/s) from (Tahani, 2011) and isothermal compressibility (1/Pa) from (Firoozabadi et al., 1988) and (b) schematic of the aquifer and the mesh used for aquifer modelling.

2.2 Aquifer modelling

To calculate the sonic velocity in a CO₂-brine system with varying temperature and pressure, a 2D rectangular Cartesian CPG aquifer with a domain of 2000 m$\times$ 100 m at a depth of 3000 m is considered and is discretized into $200\times 100$ grid blocks. The sensitivity of results to grid resolution was previously studied by the authors in the previous works (Norouzi et al., 2022a). The initial pressure of the aquifer (30 MPa) equals the hydrostatic pressure at the aquifer depth. The initial temperature of the aquifer (120 $°$C) is a result of the temperature gradient (35 $°$C/km) multiplied by the aquifer depth, added by the surface temperature (15 $°$C). Vertical boundaries of the aquifer are closed to flow, however, for the left and the right boundaries pore volume modification is used to represent an aquifer with side boundaries open to flow. Additionally, conductive heat transfer occurs at all boundaries.

The aquifer is initially filled with brine with a salinity of 20% by weight at 120 $°$C. The topic of aquifer is at 2000 m depth with an initial pressure of 30 MPa, the aquifer is 100 m thick, porosity is 0.17, horizontal permeability is 225.5 mD, and vertical to horizontal permeability ratio is 0.1. Rock density is 2650 kg/${\text{m}}^3$, rock heat capacity is 1987.5 kJ/${\text{m}}^3$/K, and Rock thermal conductivity is 2.1 W/m/K. CO₂ is injected at the constant rate of 20 kg/s for the duration of 5 years. Assuming that saturated liquid CO₂ at the temperature of 22 $°$C and pressure of 6 MPa is injected into the aquifer and that CO₂ isentropically compresses to a supercritical phase at the aquifer depth, the CO₂ injection temperature will be about 50 $°$C. The aquifer is modelled using two-phase flow simulations of CO₂ and brine with mutual solubility and is discretised into $2000\times 1000\times 100$ grid-blocks. The aquifer simulator uses the Peng-Robinson Equation of State (Peng and Robinson, 1976) to calculate the thermodynamic properties of each phase. A schematic of the aquifer is presented in Figure 2b. The mutual solubilities of CO₂ and ${\text{H}}_2$O are calculated to match experimental data for typical CO₂ storage conditions: typically 12-250 $°$C and up to 600 bars. They are calculated following the procedure given by Spycher and Pruess (2005), Spycher and Pruess (2010), based on fugacity equilibration between water and a CO₂ phase. Water fugacity is obtained by Henry’s law, while CO₂ fugacity is calculated using a modified Redlich-Kwong equation of state.

First, the thermodynamic properties of CO₂-brine mixture are calculated through the thermodynamic model by SLB Symmetry software. For these calculations, a temperature range of 35–125 $°$C, a pressure range of 20–40 MPa, and CO₂ mole fraction of 0–1 are considered. In total, the thermodynamic properties are calculated for 12100 different CO₂-brine mixtures with various temperatures, pressure, and mole fraction. Using these points, a 4D matrix of data that relates these parameters to the mixture properties is generated and is coupled to the results obtained from the aquifer simulator. Second, a CPG aquifer is modelled using SLB E CLIPSE E300 simulator for the injection time of 5 years. Then, taking temperature, pressure, and components mole fraction for each grid block from the simulation results and using interpolation methods, the sonic velocity for each grid block is calculated form the matrix of data generated in the first step. The calculated thermodynamic properties for CO₂-brine mixtures, as well as the results of the CPG aquifer modelling, are provided in the following sections.

3 Results and discussions

Results are presented in two sections. First, the thermodynamic properties of CO₂-brine mixtures with various temperatures, pressure, and mole fraction are reported. In this section, all the contributing parameters in Equation 5 are reported to identify the dominant parameter.

3.1 CO₂-brine mixture thermodynamic properties calculation

Figure 3a shows the bulk density of CO₂-brine mixtures with different $T$, $P$, and CO₂ mole fraction $(x_{{\text{CO}}_2})$. The density of water-CO₂ solutions may primarily be influenced by significant volume changes occurring upon CO₂ dissolution or release. Additionally, the exchange of water between liquid and vapour phases, occurring alongside CO₂ movement, further influences fluid volume alterations and subsequent changes in density (Straus and Schubert, 1979). The main parameter that affects the mixture bulk density is the CO₂ mole fraction. For low mole fractions $(x_{{\text{CO}}_2}<0.1-0.2)$, it is observed that adding CO₂ to the mixture slightly increases the bulk density. This is more noticeable as the mixture temperature decreases to 80 $°$C, in which the bulk density is increased from 1050 kg/${\text{m}}^3$ for $x_{{\text{CO}}_2}=0$ to about 1070 kg/${\text{m}}^3$ for $x_{{\text{CO}}_2}=0.2$ for the 40 MPa case. This is because at this stage, CO₂ is dissolving into brine and no CO₂ patches are formed yet. However, as the CO₂ mole fraction increases, the situation becomes vice versa, CO₂ patches form and it seems that brine is dissolving into CO₂, and therefore, a discontinuity happens in density versus mole fraction at $x_{{\text{CO}}_2}\approx 0.1-0.2$. The mole fraction at which CO₂ patches begin to form, and the mixture density exhibits a sharp change, can be considered the critical mole fraction. This value provides a useful indicator for identifying and locating the CO₂ plume edge.

Figure 3

Figure 3. (a) Density and critical mole fraction and (b) sonic velocity of CO₂-brine mixtures versus pressure (25–40 MPa), temperature (80, 100, 120 $°$C), and CO₂ mole fraction (dots: SLB Symmetry, and dashed lines: conventional averaging bounds).

From Figure 3a, it is also observed that, as the CO₂ mole fraction increases and CO₂ patches form, the mixture moves from an incompressible to a compressible fluid and the effects of pressure on bulk density increase. Also, as the CO₂ density is more sensitive to temperature compared to brine, with an increase in CO₂ mole fraction the effect of temperature on bulk density amplifies. For instance, for $x_{{\text{CO}}_2}=0$, an increase in temperature from 80 $°$C to 120 $°$C results in the bulk density dropping from 1050 to 1010 kg/${\text{m}}^3$, while for $x_{{\text{CO}}_2}=0.8$, the bulk density drops from 920 kg/${\text{m}}^3$ to 785 kg/${\text{m}}^3$ for the 40 MPa case.

Mixing the vapour and liquid phases is commonly done by arithmetic or harmonic averaging within the rock physics models. The harmonic averaging has been regularly employed to calculate the fluid bulk modulus. However, the results from (Behzadi et al., 2011) showed that harmonic averaging may not be valid, and the sonic velocity may even be below the harmonic average value. Therefore, we require a new fluid mixing approach that includes the interaction between phases. In a two-fluid mixture, the measured sonic velocity can be one order of magnitude smaller than that of its constituents. For example, for water and air in normal conditions, the sonic velocity in the mixture can be about 23 m/s while it is 1500 m/s in water and 330 m/s in air (this water–air case is presented only as a conceptual illustration of the extreme reduction in sonic velocity that can occur in two-phase mixtures, and is not intended as a direct analogue for supercritical CO₂–brine systems).

Figure 3a shows the calculated $c_s$ based on the SLB Symmetry thermodynamic model (dots) and based on the conventional averaging bounds (dashed lines) for the sonic velocity in CO₂-brine mixtures. It shows how sonic velocity changes at CO₂ mole fraction of about 20% at a range of pressures and temperatures. This discontinuity is important as it reduces sonic velocity from about 1880 m/s in the single-phase liquid to about 400 m/s in the two-phase region. This discontinuity is not captured by the conventional averaging methods (dashed lines in Figure 3b. Also, it is observed that the harmonic bound is not always a lower bound for sonic velocity and values lower than this bound happens for CO₂-brine mixtures, especially at lower pressures. Additionally, from the contour diagrams in Figure 3b, it is seen that at lower pressures (25 MPa), the discontinuity is more sudden, i.e., it happens at a smaller CO₂ mole fraction window, while for higher pressures (40 MPa) it has a less slope and it happens more gradually, rather sudden.

Figure 4A shows variations of mixture heat capacity ratio for different CO₂ mole fractions, pressures, and temperatures. Based on Equation 5, this parameter is directly effective in sonic velocity in the mixture. Similar behaviour as density is also observed in these figures. Single-phase brine is an incompressible fluid. However, when CO₂ is dissolved into the brine, it is expected that the mixture becomes compressible, but from the figures, it is observed that up to CO₂ mole fractions of 0.1–0.2, the mixture still shows an incompressible behaviour. This is because with such low CO₂ mole fractions no CO₂ patches are formed in the mixture and CO₂ is dissolved into brine. As the CO₂ mole fraction increases, although CO₂ is in the supercritical phase, it has a compressible behaviour and therefore, the mixture becomes compressible. Although both $\gamma$ and $\rho$ show a discontinuity around $x_{{\text{CO}}_2}\approx 0.1-0.2$, comparing the order of magnitude of the variations in these two parameters it is seen that $\gamma$ varies by an order of magnitude of 0.1, while $\rho$ changes by order of magnitude of $10^2$, and therefore, it can be concluded that density is dominant compared to compressibility ratio.

Figure 4

Figure 4. (a) Heat capacity ratio and (b) d$P$/d$V$ ratio in CO₂-brine versus pressure (25–40 MPa), temperature (80, 100, 120 $°$C), and CO₂ mole fraction.

Another parameter that affects sonic velocity in the mixture, is the ratio of pressure change over volume change (d$P$/d$V$) (Equation 5). This parameter is illustrated for different mixtures in Figure 4b. Based on the results calculated through the SLB Symmetry thermodynamic modelling, firstly, it is observed that this parameter is almost independent of the mixture pressure. Secondly, we can observe that as the temperature increases, d$P$/d$V$ decreases by about 33%. Additionally, there is no discontinuity in the trend of this parameter versus CO₂ mole fraction. Therefore, it is concluded that d$P$/d$V$ is not the predominant parameter in determining sonic velocity in the mixture.

Based on the results in this section, it is concluded that the main parameter that affects sonic velocity within the CO₂-brine mixture is the mixture bulk density, which shows the highest sensitivity to variations in CO₂ mole fraction. The discontinuity in bulk density versus CO₂ mole fraction results in a discontinuity in sonic velocity by decreasing the sonic velocity from the arithmetic bound to the harmonic bound. This discontinuity happens at the boundary of the CO₂ plume which is in contact with brine. As a result, by monitoring sonic velocity and looking for discontinuities it will be possible to study CO₂ plume distribution in the aquifer.

3.2 Sonic velocity calculations for a CPG aquifer

Figures 5a–d show the CO₂ saturation, temperature, CO₂ mole fraction, and pressure distribution within the aquifer that are used as inputs of the thermodynamic model. Using these parameters, and implementing linear interpolation on the thermodynamic data, fluid bulk density, compressibility ratio, and sonic velocity are calculated and illustrated in Figures 5e–g. Due to the CO₂ density-driven flow, the CO₂ plume tends to move upward to the top of the aquifer. This results in a high saturation area on top of the aquifer and a transition area with $x_{{\text{CO}}_2}\approx 0.2$ in the middle of the CO₂ plume column. Besides this location, the boundary of the CO₂ plume also has a $x_{{\text{CO}}_2}\approx 0.2$. Using the thermodynamic model to calculate sonic velocity, it is observed that at the locations with $x_{{\text{CO}}_2}\approx 0.2$ the discontinuity in sonic velocity occurs (Figure 5g). Therefore, it is concluded that the areas with sonic velocity discontinuity represent CO₂ mole fraction of about 0.1–0.2 which represent the CO₂ plum leading boundary and the mid-section of the CO₂ plume column. Although Figure 5 presents fluid-only velocity behaviour, the discontinuities identified at CO₂ mole fractions of 0.1–0.2 will manifest as seismic velocity contrasts with the rock matrix, thereby enhancing detectability of plume boundaries in monitoring applications.

Figure 5

Figure 5. Results obtained from coupled ECLIPSE E300 and Symmetry for an aquifer in CPG system. (a) CO₂ saturation, (b) temperature, (c) CO₂ mole-fraction, (d) pressure, (e) aquifer fluid density, (f) compressibility factor, and (g) sonic velocity distribution.

4 Conclusions and remarks

Thermodynamic calculations of the sonic velocity and density of fluid mixture in the complex fluid system are conducted to address the applicability of the conventional harmonic and arithmetic averaging bounds for mixture bulk modulus. Solving for an Equation of State and multi-phase flash enables reliable estimation of fluid acoustic properties. The model is validated with experimental results for pressures above 25 MPa. Finally, using this thermodynamic model, sonic velocity and bulk density are calculated for a CPG system with varying temperature, pressure, and fluid composition. Based on the provided results, the key findings can be summarized as below:

• For low CO₂ mole fractions $(x_{{\text{CO}}_2}\le 0.1-0.2)$, the mixture’s thermodynamic properties are almost constant and sonic velocity follows the Gassmann-Hill bound, but at a mole fraction of about 0.1–0.2 a discontinuity happens that shifts the sonic velocity to the Gassmann-Wood bound in a small mole fraction window.

• The main parameter causing the discontinuity is found to be the density of the mixture. At low CO₂-mole fractions, adding gas slightly increases the mixture density, meaning that CO₂ is dissolving in brine (solvent) and there are no CO₂ patches formed. However, as the mole fraction increases, the situation reverses and CO₂ patches (solvent) are formed and brine evaporates in CO₂, resulting in a sudden drop in density.

• The conventionally accepted arithmetic and harmonic bounds for sonic velocity calculation do not fully agree with the calculations based on the thermodynamic model, and values lower than the harmonic bound are observed for $x_{{\text{CO}}_2}>0.1-0.2$. In other words, Landau-Lifshitz’s theorem shifts down the lower boundary and increases the envelop volume between two boundaries.

• The sonic velocity discontinuity happens at the two-phase boundary (transition zone) between the CO₂ plume and brine. As a result, by monitoring sonic velocity and looking for discontinuities, CO₂ dispersion within the aquifer can be modelled.

Future works will extend the analysis to estimate seismic amplitude, impedance contrast, and time-shift responses for realistic plume boundaries, thereby quantifying the monitoring implications of the observed discontinuities. In parallel, the sensitivity of sonic velocity discontinuity thresholds to variations in brine salinity will be investigated, allowing assessment of how salinity influences the density-driven transition between dissolved CO₂ in brine and CO₂ patch formation. Combined, these efforts will refine the applicability of the model to a wider range of aquifer conditions and strengthen its relevance for practical monitoring applications.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

AMN: Writing – original draft, Formal Analysis, Visualization, Methodology, Validation, Software, Conceptualization, Writing – review and editing. MP: Writing – review and editing, Software, Resources. MI: Methodology, Writing – review and editing. MB: Writing – review and editing, Methodology, Formal Analysis, Supervision, Conceptualization.

Funding

The author(s) declared that financial support was received for this work and/or its publication. Amir Mohammad Norouzi has been funded by PDS award from the University of Manchester.

Acknowledgements

The authors acknowledge the University of Manchester President’s Doctoral Scholar (PDS) award to Amir Mohammad Norouzi that made this research possible. Additionally, the authors would like to thank SLB for using their E CLIPSE and Symmetry software in this study, as well as the technical support from the SLB GeoSolutions team.

Conflict of interest

Author MP was employed by SLB.

The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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