We consider Lauwerier problem settings (Lauwerier, 1955; Stauffer et al., 2013) involving the injection of hot (or cold) fluid into a confined aquifer located between bedrock and caprock, with lateral flow along the coordinate φ. The latter can represent the radial coordinate in an axisymmetric setting or x in Cartesian coordinates; i.e., φ=r or x. Figure 1 illustrates a summary of the problem, while Appendix F provides a summary of the nomenclature.

Downstream, along the flow path away from the injection point, heat is exchanged between the aquifer and the impermeable confining rock layers. Within the confining layers, heat is transported by conduction alone. The heat exchange and thermal variations in the aquifer induce changes in the solubility of the minerals (i.e., saturation concentration, cs(T)), which in turn trigger undersaturation and dissolution reactions or, conversely, supersaturation and precipitation reactions that modify the aquifer porosity, θ. Both removal or accumulation of minerals can occur depending on the injection temperature (colder or warmer than ambient) and the prograde or retrograde nature of the reactive minerals. Our radial setup pertains to injection from a single well or mimics natural localized thermal upwelling in fractured/faulted media of deep origin, discharging into the shallower aquifer (Craw, 2000; Micklethwaite and Cox, 2006; Roded et al., 2013, 2023; Tripp and Vearncombe, 2004). The planar source setup simulates injection wells arranged in a straight line (Lauwerier, 1955).

Here, the THC conceptual model shown in Fig. 1 is mathematically described using conservation equations for heat and reactive transport along with initial and boundary conditions. The thermal Lauwerier solution and the mathematical model involve several simplifying assumptions, the major ones of which are listed below. For a more comprehensive overview, expanded versions of the conservation equations are provided in Appendix A.

The underlying thermal assumptions include negligible basal (background) geothermal heat flow and an initial geothermal gradient compared to the heat input by the injected fluid. The aquifer is located at a significant depth, preventing heat transport to the surface; otherwise, greater heat exchange would occur between the aquifer and the caprock. This assumption regarding the depth also depends on the timescale of interest: the thermal front, which ascends with time, may not reach the surface on a short timescale. However, it may transport heat to the surface after a longer time (which can be estimated using tC).

Heat transport in the layers confining the aquifer is described by conduction and only in the vertical direction (z), neglecting lateral (φ) heat conduction. This assumption limits the applicability of the solution to scenarios involving large injected fluid fluxes. To assess the validity of this assumption, a thermal Péclet number, which compares heat advection in the aquifer to lateral heat conduction, PeT=uAl/αb, is used. PeT involves a length scale, l, at which substantial temperature variation occurs (e.g., larger than 2 % from the total temperature change, ΔT). An analysis using the parameter values in Table 1 and the results in Sect. 3 (i.e., a posteriori inspection) confirm that PeT≫1 at all times. Additionally, beyond the very early moments, the length scale of l should be larger than the vertical dimension of the aquifer, H, at which complete thermal mixing is assumed (l≫H). This assumption may not be applicable if a thick aquifer (i.e., large H) is considered and substantial vertical temperature gradients are expected to develop.

Furthermore, conduction and solute diffusion within the aquifer groundwater are neglected because the respective thermal (PeT) and solute (Pes) Péclet numbers are assumed to be large. Fluid and solid properties, such as density and heat conductivity, are considered constant and independent of temperature. It is noted that for CO₂ applications, the assumption of constant density and incompressibility may not be appropriate for a CO₂-rich phase (supercritical or gas) with moderate temperature changes (e.g., ΔT>40 °C).

Furthermore, the specific reactive surface area, As (L2 to L-3 of the porous medium), is considered constant here and assumed not to change as reaction progresses. In most instances, this assumption does not weaken the applicability of the solution since As may vary widely across different rock lithologies, e.g., from 10⁻¹ m⁻¹ in fractured media (Deng and Spycher, 2019; Pacheco and Van der Weijden, 2014) to above 10⁵ m⁻¹ for porous rocks (Mostaghimi et al., 2013; Noiriel et al., 2012; Seigneur et al., 2019) and can often only be estimated very roughly (e.g., within an order of magnitude accuracy). Furthermore, As can evolve with the reactive flow in a way that is difficult to estimate (Noiriel, 2015; Seigneur et al., 2019). If large porosity changes occur, the inherent assumption of a constant As can limit the applicability of the solutions.

Neglecting heat conduction in the radial direction, r, the heat conduction equation in the rock confining the aquifer above and below is given by 1∂T∂t=αb∂2T∂z2z≤-H2,z≥H2, where T represents temperature; t denotes time; z is the vertical coordinate, with its origin at the center of the injection well; and H is the aquifer thickness (see Fig. 1). The quantity αb=Kb/Cpb is the thermal diffusivity [L² T⁻¹], where the subscript b indicates bulk rock, K is the thermal conductivity, and Cp is the volumetric heat capacity (Chen and Reddell, 1983; Stauffer et al., 2013).

Assuming that heat transport in the fluid along the aquifer is governed by advection and that complete mixing occurs in the aquifer transverse direction (z), a depth-averaged heat-transport equation can then be formulated for the aquifer region: 2CpbH∂T∂t=-CpfH1r∂(ruT)∂r-Θ(r,t)for-H2≤z≤H2, where subscript f denotes fluid and u(r) is the radial superficial velocity (or Darcy flux), which can be determined from the total volumetric flow rate, Q, using u=Q/(H2πr) (assuming u to be uniform along the z direction of the aquifer; Andre and Rajaram, 2005; Lauwerier, 1955). Function Θ accounts for the heat exchange between the aquifer and the confining rock located above and below, calculated using Fourier's law, with continuous temperature assumed at the interfaces: 3Θ=-2Kb∂T∂zz=H2. The factor of 2 accounts for the rock both above and below the horizon (Stauffer et al., 2013).

The solute transport advection–reaction equation in the aquifer is 40=-u∂c∂r-Ω(r,t)for-H2≤z≤H2. Here, c is the solute concentration [M L⁻³] and Ω is the reaction term (Chaudhuri et al., 2013; Szymczak and Ladd, 2012). Equation (4) is derived by neglecting transient and diffusive terms in the advection–diffusion–reaction equation (Eq. A3). The justification for the quasi-static approximation used when deriving Eq. (4) lies in the separation of timescales between heat conduction (tC) in the confining rocks and mineral alteration (tM) and the relaxation of solute concentration (tA) (for in-depth analysis and discussion, see Appendix B and, e.g., Detwiler and Rajaram, 2007; Ladd and Szymczak, 2017; Lichtner, 1991; Roded et al., 2020; Sanford and Konikow, 1989).

Here, we assume a surface-controlled reaction and first-order kinetics: 5Ω=AsλΛ, where As is the specific reactive surface area and λ is the kinetic reaction rate coefficient [L T⁻¹], here assumed to be constant (Dreybrodt et al., 2005; Seigneur et al., 2019). Λ is denoted as the solute disequilibrium and is defined as the difference between the concentration of dissolved ions and saturation (equilibrium) concentrations, cs, 6Λ=c-cs(T). Thus, the solute disequilibrium, Λ, is negative for undersaturation and positive for supersaturation. cs is calculated as follows: 7cs(T)=csT0+βT-T0. Here, T0 represents the initial temperature in the aquifer and the parameter β=∂cs/∂T. Equation (7) assumes a linear relationship between cs and T, with a constant proportionality factor, β, which is positive for minerals of prograde solubility and negative for minerals of retrograde solubility (Al-Sulaimi, 2015; Corson and Pritchard, 2017; Woods, 2015).

Given the reaction rate (Eq. 5), the change in porosity, θ, can be calculated as follows: 8∂θ∂t=-Ωνcsolfor-H2≤z≤H2, where csol is the concentration of the soluble solid mineral and ν accounts for the stoichiometry of the reaction. In the case of planar flow and Cartesian coordinates, r can be replaced by x in the equations above, while Eq. (2) takes the following form: 9Cpb∂(HT)∂t=-uCpfH∂T∂x-Θ(x,t)for-H2≤z≤H2, where u here is the spatially uniform fluid velocity in the x direction.

The initial conditions involve a uniform value of temperature T0 throughout the medium. The boundary conditions at the injection well (φ=0) include a constant rate of fluid injection at temperature Tin and initially zero solute disequilibrium, Λ=0 (Eq. 6). The caprock and bedrock thickness and aquifer extent are assumed to be infinite.

Here, we discuss conditions for thermally induced reactivity in carbonates and sandstone aquifers and the parameter values assigned in the simulations (Table 1). Regarding the description of the kinetics of these systems, calcite dissolution can often be complex, involving various chemical species and reactions of varying orders (Dreybrodt, 1988; Plummer et al., 1978). However, for a wide range of pH values, it can be simplified and described by assuming a linear dependence on undersaturation or acid concentration. Specifically, first-order kinetics are commonly employed to study natural karst formations (pH ∼ 6; Dreybrodt et al., 2005; Palmer, 1991), and dissolution under the acidic conditions common in engineering applications (pH ∼ 3; Hoefner and Fogler, 1988; Peng et al., 2015) or in geothermal systems of high CO₂ partial pressure, PCO₂ (pH∼ 5; Coudrain-Ribstein et al., 1998; Lu et al., 2020; Roded et al., 2023). Silica precipitation can be well described by first-order kinetics (Carroll et al., 1998; Ji et al., 2023; Pandey et al., 2015; Rimstidt and Barnes, 1980).

We also exploit approximately linear temperature–solubility dependence over the temperature range studied here (between T0=20 °C and Tin=60 °C) and assign a constant value to β (Eq. 7; Andre and Rajaram, 2005; Glassley, 2014; Rimstidt and Barnes, 1980; Roded et al., 2023). Additionally, it should be noted that in carbonates, the temperature–solubility relation strongly depends on PCO₂: higher PCO₂ values result in larger increases in cs as the water cools (i.e., the magnitude of β is larger; see Fig. 2b in Roded et al., 2023, and Andre and Rajaram, 2005; Palmer, 1991). Here, in accordance with typical conditions in geothermal systems, we consider injection of water with PCO₂ = 0.03 MPa (Coudrain-Ribstein et al., 1998; Lu et al., 2020).

In the simulations, we assign characteristic porosity (θ) and reactive surface area (As) for the different aquifer types. In accordance with common field observations, we consider a carbonate aquifer in which flow and dissolution are focused in the permeable fracture network and a porous sandstone aquifer characterized by high intergranular permeability (Bear and Cheng, 2010; Jamtveit and Yardley, 1996). The different aquifer characteristics are reflected in significant differences in θ and As for the different aquifer types. Specifically, carbonates are often characterized by permeability contrasts spanning orders of magnitudes between the fractures and the rock matrix (Dreybrodt et al., 2005; Lucia, 2007). Consequently, transport in the matrix occurs mostly by slow diffusion, and the reaction within the matrix can be neglected. Hence, only the reactive surface area, As, of the fractures effectively participates in the reaction (Deng and Spycher, 2019; Maher et al., 2006; Pacheco and Alencoão, 2006; Seigneur et al., 2019). In this case, θ can be minimal (Lucia, 2007) and As is orders of magnitude smaller compared to its value in porous sandstones (Hussaini and Dvorkin, 2021; Lai et al., 2015; Pacheco and Alencoão, 2006; Pacheco and Van der Weijden, 2014; Seigneur et al., 2019). This disparity can lead to substantial differences in characteristic alteration rates and Damköhler numbers in these systems (Ladd and Szymczak, 2021; Lucia, 2007; Seigneur et al., 2019).

Specifically, in the case of fractured rocks as described above, we calculate the reactive surface area using As=2⋅κ⋅RF, where κ is the fracture density (defined as the number of fractures per unit volume), the factor of 2 accounts for the presence of two surfaces, and RF is the roughness factor (Deng et al., 2018). Assuming κ=1/33 m⁻³ and RF = 1.35 results in As=0.1 m⁻¹. Typical values of κ and fracture spacing can span a substantial range and may be higher or lower (Narr and Suppe, 1991; Scholz, 2019). Here, it is further assumed that the fracture density is high and the network is of high connectivity, allowing it to be treated as a continuum (Anderson et al., 2015; Sahimi, 2011). We consider here an injection flow rate of Q=500 m³ d⁻¹, which falls within the typical range of flow rates observed in relevant applications, such as geothermal systems (Glassley, 2014) or groundwater storage and recovery (Maliva, 2019). The injection temperature is set to Tin=60 °C, and aquifer ambient temperature is set to T0=20 °C (ΔT=40 °C). To obtain the results, in this section, the solutions were implemented in MATLAB code (MATLAB, 2022). Appendix D details the use of the approximated Eq. (D2) in calculating the results in Figs. 2 and 3.

In Fig. 2, the results of CO₂-rich hot water injection into a carbonate aquifer at successive times since the beginning of the injection are shown (Eqs. 10, 16, and D2 are solved for t=0.2, 10, and 100 kyr). During the radial flow within the aquifer, the hot fluid cools by transferring heat into the confining layers, which heat up with time, resulting in the gradual advancement of the thermal front downstream (Fig. 2a). The cooling induces solute disequilibrium (Λ) associated with undersaturation (note that Λ is negative for undersaturation and positive for supersaturation; see Eq. 6). The magnitude of Λ in the aquifer is small compared to the absolute solubility change in the system, Δcs=|cs(Tin)-cs(T0)|, i.e., between cs(Tin) at the injection point and cs(T0) at ambient conditions (|Λ|/Δcs≪1 %; see Fig. 2b). The small magnitude of disequilibrium is associated with relatively high PCO₂ considered here (0.03 MPa) and rapid kinetics under these conditions. The quasi-equilibrium conditions may allow for the simplification and calculation of the local reaction rate from transport processes alone, regardless of kinetics, referred to as the so-called “equilibrium model” (Andre and Rajaram, 2005; Bekri et al., 1995; Golfier et al., 2002; Lichtner, 1991), which will be the subject of a future research.

Although the magnitude of disequilibrium, Λ, is small, it controls the alteration of the aquifer and the evolution of its properties. Significantly, because the water at the inlet is hot and saturated with calcite, c=cs(Tin), disequilibrium, and reaction rate are zero at the inlet, leading to no change in the porosity there (see Figs. 2b and 3c and their magnifications). Disequilibrium (undersaturation) sharply develops downstream of the injection site, first forming a small minimum (at r≈20 m) and gradually increasing to zero at greater distances. Undersaturation and dissolution along the flow path are controlled by the interplay of three processes: (I) dissolution reducing undersaturation (i.e., Λ becomes closer to zero), (II) progressive cooling increasing undersaturation, and (III) advection–transport–reaction products (i.e., calcium ions) radially outward from the well, helping maintain undersaturation. Here, the effect of fluid velocity and advection decays with a distance of 1/r.

High advection and cooling rates near the inlet result in the abrupt formation of undersaturation (i.e., negative Λ). Further downstream, undersaturation diminishes due to dissolution reactions. As the thermal front advances downstream over time and the temperature gradients diminish along the aquifer, the Λ curve flattens and becomes more elongated (see curves for t=10 and 100 kyr in Fig. 2b). Due to the disequilibrium, porosity grows with time. The porosity sharply increases near the inlet and then gradually decreases downstream (Fig. 2c). The porosity changes are extensive and take place over an aquifer area of ∼30 km² within a relatively short geological timescale of 100 kyr, resulting in the addition of significant void space of thousands of cubic meters (∼5×103 m³).

An essential assumption underlying the solutions in Sect. 2 and the results depicted in Fig. 2 is the assumption of spatial uniformity and symmetry of reactive flow. In practical scenarios, however, dissolution instabilities can emerge at the reaction front. These instabilities, owing to the positive feedback between reaction and transport, may evolve into dissolution channels, often referred to as wormholes (Aharonov et al., 1997; Budek and Szymczak, 2012; Chadam et al., 1986; Ortoleva et al., 1987; Roded et al., 2018, 2021). The wormholes concentrate reactive flow, resulting in heterogeneous flow fields that cannot be accurately represented by assuming symmetry and uniformity. In such a case, the results of Fig. 2 can only be regarded as an average solution, which is not accurate locally.

Isothermal dissolution, driven by undersaturation of the incoming solution is known to be unstable in the radial geometry for a large-enough solute Péclet number, Pes, and intermediate Damköhler numbers. The Damköhler number here is given by Da=AsλlA/uA and represents the ratio between advective and reactive timescales (Daccord, 1987; Grodzki and Szymczak, 2019; Kalia and Balakotaiah, 2007; Xu et al., 2020). However, in our case, the cooling of the solution renews its aggressiveness, hence extending the penetration length in the system which may influence the stability of the reactive front (Xu et al., 2020). The effect of renewed aggressiveness by considering solubility gradients was studied for planar reactive flow in Aharonov et al. (1997) and Spiegelman et al. (2001) but requires further investigation for radial flow and taking into account coupling with heat transfer.

Here, we consider the injection of hot silica-rich water that cools, becoming supersaturated and leading to silica precipitation, consequently reducing void space and permeability. While the previous case involved dissolution, this one involves precipitation; however, the thermal and reactive transport processes are similar in both cases (with approximately mirror-image Λ and θ profiles; see Figs. 2b, c and 3a, b).

Similarly to the previous section, the low magnitude of Λ suggests that the reaction rate (Eq. 5) is relatively high compared to transport processes, effectively reducing disequilibrium, Λ. It is noted that the reaction rates are high in both systems despite the orders-of-magnitude differences in the kinetic rate coefficient (λ=10-6 m s⁻¹ for calcite dissolution compared to 5×10-10 m s⁻¹ for silica precipitation). However, this difference is largely compensated by the contrast between the reactive surface area of the porous sandstone and fractured carbonate aquifers (As=104 m⁻¹ compared to 10 m⁻¹, respectively). It should also be noted that while precipitation of crystalline and non-crystalline (amorphous) silica is characterized by relatively high rates, dissolution of quartz and silica polymorphs is typically slower by several orders of magnitude (Rimstidt and Barnes, 1980).

While the reaction rates are high in both systems, differences exist in the absolute magnitude of porosity change resulting from the injection. For example, the maximal porosity change in the aquifer due to silica precipitation is approximately Δθmax≈0.03, whereas for the carbonate case it is around Δθmax≈0.08 (where Δθmax=|θmax(t=100kyr)-θ0| and θmax denote the maximal porosity change along the profile). The predicted lower porosity change in silica arises mostly due to its lower total solubility change, Δcs, and the reduced dependence of mineral solubility on temperature, expressed here by the β parameter (see Table 1). This conclusion is further supported by the fact that no disequilibriated fluid exits the system: the fluid flows out from the system at r=3000 m at a temperature that is close to the ambient temperature, T0 (Fig. 2a), and chemically equilibrated (Λ=0; Figs. 2b and 3a).

The porosity changes affect the aquifer hydraulics. Here, we calculate the effective aquifer permeability, keff, within a distance, R, around the well. keff is calculated based on the relationship between the local porosity and permeability, utilizing the power-law relation k(r)/k0=(θ(r)/θ0)n, where k0 and θ0 are the initial permeability and porosity (the steps for the calculation of keff are presented in Appendix E). The exponent n depends on various factors, such as medium microstructural details and the nature of the alteration processes (Seigneur et al., 2019; Steefel et al., 2015; Vafaie et al., 2023). The limited predictive capabilities of k–θ relations, including instances where counter trends of porosity and permeability changes occur (Garing et al., 2015), have been previously noted (e.g., Sabo and Beckingham, 2021). Here, it is applied to evaluate general trends, which, with the exception of unique cases, remain valid regardless of the porosity–permeability relation used.

The wide range of heterogeneous microstructures in rocks and sediments and their response to different reactive-flow regimes lead to a large variability in the exponent n values. For example, for relatively uniform spatial dissolution, n can range from ∼3 to a few dozen for the early stages of flow or when wormholes develop (Hao et al., 2013; Roded et al., 2020; Vafaie et al., 2023). For precipitation, n typically ranges from ∼2 up to above 10 (Aharonov et al., 1998; Hommel et al., 2018; Seigneur et al., 2019).

Figure 4 shows keff evolution over time for representative exponent values within a distance of R=3 km. The rapid increase in carbonate aquifer permeability indicates (in agreement with previous works, Agar and Geiger, 2015; Andre and Rajaram, 2005; Dreybrodt et al., 2005) that keff can be substantially altered within relatively short geological timescales. Specifically, the results suggest that keff can even increase by several tens of percents within tens to hundreds of years. Conversely, significant keff alterations due to silica precipitations (10 %–50 % reduction) involve typical timescales of tens of thousands of years. These findings are consistent with previous observations of dissolution and precipitation driven by a solubility gradient (e.g., Aharonov et al., 1997), emphasizing differences between these processes, as embodied in the exponent n. Moreover, under constant pressure (instead of constant flux) boundary conditions, this effect will be enhanced due to a positive (negative) feedback during dissolution (precipitation) (Aharonov et al., 1997).