The following assumptions define the physical scope of the framework and indicate how they reduce mathematical and computational complexity while highlighting their limitations.

The governing equations for coupled flow–deformation processes in a saturated porous medium originate from two fundamental principles: mass conservation for the pore‐water phase and force equilibrium for the deformable solid skeleton. The former governs the flow of the fluid, while the latter describes the mechanical deformation of the solid matrix. These macroscopic forms are well established for deformable porous media under small‐strain conditions .

All dissolved aqueous elements are transported by the same bulk liquid phase and therefore experience a common flow field. Likewise, the deformation of the porous skeleton responds to the total pore‐fluid pressure rather than to the concentrations of individual solutes. Consequently, the flow and deformation equations remain unchanged regardless of the number or type of chemical elements present; only the solute transport equation varies with chemical composition.

The pore‐water flow equation is expressed as 1Srnβ∂p∂t+Sr∇⋅vs=1ρwg∇⋅K⋅∇p, where porosity n is the volume fraction of the void space and controls fluid storage and transmission, ρw is the density of pore water and g is the gravitational acceleration. The coefficient β represents the effective compressibility of the pore fluid mixture and is written as β = Srkwo+1-Sr+rhSrpa+p0. Here, the bulk modulus of water is taken as kwo = 2000 MPa, the volumetric fraction of dissolved air is rh = 0.02 m, and the atmospheric pressure is p0 = 100 kPa, while pa denotes the apparent capillary pressure.

In Eq. (), vs represents the velocity of the solid skeleton, defined as the time derivative of the displacement vector us = (ux,uy,uz); its components are the displacement rates along the Cartesian axes. The hydraulic conductivity tensor K is taken as diagonal, with principal components Kx, Ky, and Kz aligned with the coordinate axes to represent possible anisotropy of the porous medium.

The deformation equation is formulated using the displacement vector us as the primary variable. 2a∇⋅μ∇us+μ∇usT+λItr∇us-00(1-n)(ρs-Srρw)g∇⋅us=∇p,2bλ=2Gν1-2ν,μ=G.

The body force vector in Eq. () accounts for the self‐weight of the solid skeleton. Because gravity acts vertically, only the z component is non-zero. In Eq. (), λ and μ are the Lamé parameters defining the elastic response of the porous skeleton. The shear modulus G characterises resistance to shear deformation, and the Poisson’s ratio ν describes the lateral contraction under uniaxial loading.

The governing equation for solute migration follows the classical advection–dispersion framework widely used in studies of porous media transport . A separate transport equation is solved for each chemical element of interest. As highlighted in Sect. , the elemental formulation ensures that chemical reactions merely redistribute each element between the aqueous and solid phases without creating or destroying it. Therefore, no additional reaction source or sink terms are required to enforce mass conservation. Species‐level concentrations can be reconstructed by coupling these elemental balances with reaction relations, but the governing transport operator itself remains the same for all elements.

The solute transport equation is written as 3Srn+(1-n)ρsKd∂Y.()∂t=SrnD:∇2Y.()+Srn∇Y.()⋅∇⋅D-Srnvf⋅∇Y.()-(1-n)ρsKdvs⋅∇Y.()+(Srnβ∂p∂t+Srnβvf⋅∇p)Y.(),4vf=-1ρwgSrnK⋅∇p+vs,

In Eq. (), Y.() denotes the dissolved concentrations of aqueous elements, for example Y.C for the total dissolved carbon and Y.Ca for calcium in the liquid phase. Mineral‐phase concentrations, such as Ys.Calcite, are tracked in the geochemical reaction module but do not appear explicitly in the physical governing equations for flow, deformation or transport. The vector vf represents the pore‐fluid velocity that drives advective migration of the dissolved elements (see Eq. ).

The hydrodynamic‐dispersion tensor D is defined using the local‐scale Scheidegger model, which treats dispersion as the combined effect of molecular diffusion and velocity‐dependent mechanical dispersion under a spatially smooth Darcy‐scale velocity field. 5Dxx=αLvx2v‾+αTHvy2v‾+αTVvz2v‾+Dm,Dyy=αTHvx2v‾+αLvy2v‾+αTVvz2v‾+Dm,6Dzz=αTVvx2v‾+αTVvy2v‾+αLvz2v‾+Dm,Dyx=Dxy=(αL-αTH)vxvyv‾,7Dxz=Dzx=(αL-αTV)vxvzv‾,Dyz=Dzy=(αL-αTV)vyvzv‾. Here, the molecular diffusion coefficient Dm accounts for Brownian‐motion‐driven spreading in the absence of bulk flow. The longitudinal and transverse mechanical‐dispersion coefficients αL, αTH and αTV describe velocity‐dependent spreading caused by microscopic variations in pore‐scale flow paths. The modulus length of the velocity vector v‾ is calculated from 8v‾=vfx-vsx2+(vfy-vsy)2+(vfz-vsz)2=1ρwgSrn-Kx∂p∂x2+-Ky∂p∂y2+-Kz∂p∂z2, where vfx, vfy, vfz and vsx, vsy, vsz are the fluid and solid velocity components in the x, y, and z directions, respectively.

Equation () includes a simple linear adsorption term represented by the distribution coefficient Kd to model instantaneous equilibrium partitioning between the aqueous phase and the solid matrix. If a more detailed description of adsorption–desorption kinetics or competitive sorption among multiple elements is needed, this linear term can be removed and replaced by reaction‐controlled source–sink terms calculated by the geochemical reaction module, which updates both aqueous and solid‐phase concentrations. 9Srn∂Y.()∂t=SrnD:∇2Y.()+Srn∇Y.()⋅∇⋅D-Srnvf⋅∇Y.()+Srnβ∂p∂t+Srnβvf⋅∇pY.().

The overall integration workflow is illustrated in Fig. . At the start of the simulation, PhreeqcRM performs an initialisation stage that reads the configuration parameters, loads the prescribed geochemical conditions, and computes the initial aqueous-element concentrations and mineral-phase abundances. In coordination with OpenFOAM, these initial values are mapped onto the computational mesh so that each control volume is assigned its starting aqueous and mineral concentrations. These fields serve as the baseline for subsequent transport and reaction calculations.

For each global time step thereafter, OpenFOAM first solves the hydro-mechanical (HM) subsystem, pore pressure, mechanical deformation, and elemental advection–dispersion transport. Although the flow and deformation equations are strongly coupled through the pore-pressure–displacement interaction, solving them in a fully monolithic manner is numerically demanding and often computationally expensive. To improve robustness and efficiency, the present framework adopts a segregated iterative strategy, which has been widely used in CFD. The pressure equation is solved first using the previous iteration's displacement field. The resulting pressure distribution is then used to update the solid skeleton deformation, and only explicit terms are refreshed at each iteration. This process repeats until the change between successive iterations falls below a prescribed tolerance, at which point convergence is declared. The convergent fields of pore pressure p and displacement us are then passed in a one-way manner to the transport module to define the pore-fluid velocity vf and solid velocity vs required by the advection–dispersion equation for solute migration.

After the HM and transport updates, the solver checks whether the current simulation time falls within the user-defined reaction-activation window. If the chemical module is inactive, the solver skips the reaction step and proceeds directly to the next global time step. If the chemical module is active, OpenFOAM first assembles the elemental concentration fields from all computational cells into a contiguous vector conforming to the data structure required by PhreeqcRM. This vector is passed to PhreeqcRM, which performs aqueous speciation and kinetic-reaction calculations to update both aqueous concentrations and mineral-phase abundances. Upon completion, PhreeqcRM overwrites the same vector with the updated results, which are then mapped back into the OpenFOAM fields of each cell. The simulation proceeds to the next global HM–transport step using these updated concentrations.

Within the OpenFOAM framework adopted in this study, the governing equations were discretised using the finite volume method. Diffusive and dispersive fluxes were evaluated using the Gauss linear scheme, while the advective term of solute transport was discretised using a Gauss limitedLinear scheme with a limiter coefficient of 1 in order to suppress spurious oscillations. Temporal integration was performed using the implicit Euler method. The pressure, displacement and concentration equations were solved using the GAMG solver with a DILU preconditioner, with absolute and relative tolerances of 10-9 and zero, respectively. The coupled HMC system was advanced using a segregated outer-iteration strategy, and convergence was achieved when the residuals of all primary variables dropped below 10-6. Further implementation details are provided in the companion paper .

To strengthen the numerical credibility of the proposed HMC framework, a systematic sensitivity analysis was conducted with respect to spatial discretisation, global time-step size and chemical sub-stepping in the ChemWindow controller. A one-at-a-time strategy was adopted, whereby only one numerical factor was varied while all others were kept identical.

The numerical deviations were quantified using the mean relative error (MRE) and maximum relative error (MaxRE), defined as 11MRE=1N∑i=1Nyi-yirefyiref,MaxRE=max⁡1≤i≤Nyi-yirefyiref, where yi denotes the computed value at the ith comparison point and yiref is the corresponding value obtained from the finest reference solution. The MRE reflects the overall deviation level, whereas the MaxRE captures the largest local discrepancy associated with potential non-linear or synchronisation effects.

The reference configuration corresponds to the finest grid (120 × 1 × 100), the smallest global time step (Δt = 1.58 ×105 s), and the smallest chemical sub-step (Δtc = 3000 s). The medium discretisation employs a grid of 60 × 1 × 50 with Δt=3.15×105s and Δtc = 6000 s, while the coarse setting uses 30 × 1 × 25, Δt = 6.31 × 10⁵ s, and Δtc = 12 000 s.

Table  together with Figs.  and summarises the grid-, time-step- and chemical-substep-independence results for pore pressure p, solid displacement us and solute concentration Y.C. For the hydro-mechanical variables p and us, both the MRE and MaxRE decrease rapidly with mesh refinement and global time-step reduction. In all medium–fine comparisons, the maximum relative errors remain below 0.5 %, indicating excellent numerical convergence of the hydro-mechanical part of the solver.

The solute concentration Y.C exhibits a higher sensitivity to spatial resolution, as expected for advection–dispersion–reaction dominated processes. Nevertheless, the MaxRE decreases from 11.5 % in the coarse–fine comparison to 3.8 % in the medium–fine comparison, while the MRE reduces from 5.5 % to 2.1 %, demonstrating satisfactory convergence of the transport component. In contrast, the sensitivity to the global time-step size is negligible, with maximum discrepancies below 0.2 %. More importantly, decreasing the chemical sub-step size from 12 000 to 3000 s results in less than 1 % variation in solute concentration, with the MaxRE reducing from 1.1 % to 0.6 %. This confirms that the proposed ChemWindow synchronisation scheme is numerically stable and does not introduce artificial splitting errors.

Based on these results, the medium discretisation settings (60 × 1 × 50, Δt = 3.15 × 10⁵ s, Δtc = 6000 s) are adopted in the remainder of this study as a balanced compromise between numerical accuracy and computational efficiency.

Two benchmark problems were used to verify different components of the present model and to examine the numerical stability of the solvers. Laboratory data is not available in the literature for direct comparison. However, benchmarking against published numerical results is a well‐established practice in the field of reactive solute transport, and provides a practical means to confirm model implementation before applying it to new scenarios.

In OpenFOAM, a quasi-2-D domain (Fig. ) was built as a thin 3-D slab of length Lx = 20 m and height Lz = 1 m, with a negligible out-of-plane thickness to emulate two-dimensional behaviour. Along the top boundary, three contaminant inlets (depicted by blue arrows) are prescribed as boundary molar fluxes J1–J3 with units of mol (m² s)⁻¹. Each inlet is applied uniformly over a top-boundary patch of width 4 m (i.e., an inlet area Ai = 4 m × 0.02 m, where 0.02 m is the out-of-plane thickness of the quasi-2-D OpenFOAM model). The intervening top-boundary segments are assigned zero solute flux. The entire top surface is subjected to a uniform mechanical load. The bottom boundary comprises five localised drainage outlets (red arrows) with a prescribed hydraulic head, with the rest rendered impermeable; both lateral boundaries are also impermeable and mechanically fixed to prevent rigid body motion.

The coordinate system is defined such that x is positive to the right and z is positive upward. The geochemical initial condition is spatially uniform: the entire computational domain is assigned Solution 1 from the PHREEQC input file, thus providing the same initial aqueous composition throughout the liner and the surrounding geomaterial. Each OpenFOAM cell is directly mapped to a corresponding PHREEQC cell, ensuring a one-to-one correspondence between the flow and geochemical grids. If spatially variable initial compositions are required (i.e., different Solution definitions), the OpenFOAM mesh must be preconfigured to assign the appropriate PHREEQC solution number to each computational cell.

A mechanical load is applied at the top boundary and expressed through the local force-balance condition. 12-n⋅σ′=Q(t)-n⋅pI, where n is the outward unit normal, σ′ denotes the effective stress tensor, and I is the identity tensor. The effective stress is defined following , and the corresponding displacement gradient on the loaded surface is given by 13n⋅∇us=-Q(t)-n⋅μ∇usT+λItr∇us-(μ+λ)∇us+np2μ+λ.

The bottom boundary is constrained to emulate a stiff foundation: both vertical and lateral components of displacement are fixed at zero, suppressing downward settlement and horizontal sliding along the base. The lateral boundaries are laterally fixed to prevent rigid-body motion of the entire domain, but their vertical displacement remains free unless otherwise specified.

For pore pressure, the top surface is assumed to be in contact with an overlying ponded leachate at atmospheric pressure and is therefore assigned a zero normal gradient condition, 14∂p(x,y,0,t)∂z=0, which is consistent with a flat free surface in static equilibrium and negligible vertical seepage. Under near-saturated conditions, the liquid phase remains continuous up to the surface while the gas phase is disconnected, so no capillary head or hydrostatic jump develops across the interface. If a sustained head difference develops between the ponded leachate and the liner interior, a non-zero gradient or a Dirichlet-type pressure specification would be required to represent advective inflow.

The lateral boundaries are treated as impermeable for flow, enforcing no-drainage (zero-normal-flux) conditions. At the bottom boundary, drainage is permitted only at the designated outlet segments where hydraulic head is prescribed, while the remaining sections of the bottom share the same no-drainage condition as the lateral boundaries.

The boundary of solute concentration at the top is specified as 15∂Y.(x,y,0,t)∂z=-J(i)SrnDm(i), where J(i) the imposed vertical solute-mass flux of elements i, and Dm(i) the molecular diffusion coefficient. This Robin-type, diffusion-controlled boundary is appropriate when the overlying leachate pond has negligible vertical seepage yet maintains a concentration gradient that drives diffusive mass transfer into the liner. If appreciable advective inflow occurs, a prescribed advective flux or a Dirichlet-type concentration boundary would be more suitable. The lateral boundaries and the impermeable portions of the bottom boundary are assigned zero normal gradient (Neumann) conditions for concentration. The boundary conditions are summarised in Table .

It is emphasised that the imposed concentration boundary conditions are adopted as controlled demonstration settings to explore the sensitivity of coupled hydro-mechanical–chemical responses to source strength and drainage configuration. The leakage cases considered correspond to localised defects or preferential pathways rather than spatially uniform barrier failure, which is not regarded as realistic for engineered containment systems. In the absence of site-specific investigation data, the present simulations should therefore be interpreted as scenario-based analyses aimed at mechanistic understanding, rather than predictive landfill exposure assessments.

The simulated liner is modelled as a low permeability clayey soil under a small-strain linear elastic framework (Table ). Hydraulic and mechanical properties—including porosity n, degree of saturation Sr, hydraulic conductivity K, shear modulus G, Poisson's ratio ν, and compressibility coefficients—were taken from representative ranges for compacted clay liners . These values are consistent with the small-strain assumption, allowing the use of linear elasticity and the Eulerian formulation without large-deformation corrections. Geochemical reactions are implemented via the PhreeqcRM interface, in which standard PHREEQC input blocks are specified using OpenFOAM chemistry dictionaries. Details of the corresponding configuration options are provided in Appendix .

For reactive-transport simulations, the linear distribution coefficient Kd, a simplified equilibrium sorption term, was set to zero for all elements so that changes in breakthrough and spatial distribution arise solely from geochemical reactions such as mineral dissolution and precipitation. The PHREEQC reaction module is, in principle, capable of representing more complex sorption mechanisms (e.g. nonlinear Freundlich or Langmuir isotherms and competitive or kinetic adsorption–desorption), but these functionalities are reserved for future extensions and are not used in this work.

Two types of reactive‐transport scenarios were designed to demonstrate and evaluate the capabilities of the proposed HMC framework. The first case considers a single reactive mineral that undergoes multicomponent aqueous reactions, enabling a clear examination of the fundamental coupling between hydro-mechanical processes and geochemical reaction fronts. This case is also used to explore the sensitivity of the system response to external mechanical loading and to variations in the initial degree of saturation, highlighting the interaction between pore pressure dissipation, deformation, and reactive solute migration.

The second case extends the system to a multi‐mineral, multi‐component reaction network to showcase the stability of the framework to capture simultaneous dissolution–precipitation and competitive aqueous reactions among several solid phases. This progression from a simplified to a more complex geochemical setting highlights the versatility of the solver and its ability to handle general reactive transport problems.

This subsection investigates how external loading and aqueous reactions jointly influence solute migration and calcite dissolution by varying the magnitude of loading, deformation behaviour, and dissolution-reaction pathways.

Under bottom drainage conditions, all three loading cases demonstrate that a higher load produces a larger initial pore-water pressure, slower dissipation, and greater ultimate settlement (Fig. a, b). A larger initial excess pressure requires a longer time to dissipate, thus extending the deformation period despite a greater absolute pressure drop. At mid-depth (z = 0.5 m), which is closer to the drainage outlet, the pore pressure dissipates more rapidly and the displacement stabilises earlier. Overall, Case D exhibits the highest pore pressure response and settlement, followed by Case A and then Case E, confirming that increasing load amplifies both pressure evolution and deformation.

Figure c–h illustrate the temporal evolution of aqueous concentrations within the calcite-dissolution system under different mechanistic scenarios. In Fig. c (elemental carbon), Case B – which excludes aqueous reactions – yields markedly lower concentrations than the other cases because chemical reactions are suppressed. Case C, where matrix deformation is neglected, exhibits a crossover behaviour: its concentration is lower than that of the fully coupled Case A at early times but becomes higher at later stages. This reversal arises because deformation during consolidation reduces the pore volume and flow ability, thereby restricting early advective release of dissolved species. In contrast, the absence of deformation preserves pore connectivity and facilitates solute transport, resulting in higher late-time concentrations.

The influence of mechanical compression is further highlighted under varying loading magnitudes (Fig. d). Initially, higher loads accelerate the release of the solute through increased flow and reduction of porosity, but as compression progresses, the decrease in pore volume and flow ability restrict later stage transport. Consequently, long-term concentrations under heavy loads become lower than those observed under lighter loads.

Unlike Y.C, Y.Ca has no external solute source, the initial calcium concentration is negligible and increases only by calcite dissolution. Accordingly, Fig. e shows a gradual increase in aqueous Ca²⁺ over time, while Case B remains essentially zero because no calcium derived from dissolution is supplied. Figure e–f further demonstrate that higher loads or the inclusion of deformation compress the soil skeleton, reducing porosity and flow capacity. These mechanical changes alter the transport pathways and influence the rate at which calcium accumulates, underscoring the coupled role of stress, deformation, and chemical reactions in governing long-term redistribution of solutes.

Figures  and present 2-D contour maps at t = 20 years for dissolved carbon (Y.C) originating from calcite dissolution (Fig. ) and calcium (Y.Ca) released by the same process (Fig. ). Three external source zones are clearly visible, with strengths increasing from left to right. This gradient produces markedly different peak concentrations across the sources – for instance, under Case A, the rightmost source exhibits approximately two to three times higher concentrations than the leftmost one.

In Fig. , the carbon concentration field follows a similar spatial pattern, reaffirming that neglecting deformation or applying a smaller load accelerates dissolution and enhances solute accumulation. The three distinct source-controlled peaks along the horizontal direction are clearly delineated. Initially, the migration of the solute occurs predominantly in the vertical direction following the main drainage pathway, and with time, it spreads laterally. These maps confirm that the mechanisms identified in the temporal profiles, specifically the effects of deformation, loading magnitude, and reaction coupling, also govern horizontal solute dispersion.

In Fig. , the calcium concentration fields exhibit trends consistent with the earlier time series results, now extending into a full spatial perspective. The transition from vertical to horizontal migration highlights preferential transport pathways shaped by loading and deformation. As expected, Case B remains near zero because there is no dissolution-reaction source, while the other cases display distinct high concentration regions aligned with the three source zones. Enhanced mechanical compression hinders fluid migration and solute dispersion, leading to narrower plumes and lower concentration peaks than in lighter-load or un-deformed cases such as Cases C and E.

Figure  shows the two-dimensional contour maps of pH at t = 20 years. The pH distribution closely mirrors the dissolved carbon and calcium fields, being primarily governed by local ionic equilibria within the aqueous phase. Zones enriched in dissolved calcium and carbonate species exhibit lower pH (more acidic), while areas with limited accumulation of solutes remain near the background weakly acidic level (pH ≈ 6.5–7.0). Accordingly, cases or source zones that generate higher solute concentrations develop broader and deeper low pH regions, while non-reactive regions maintain near-neutral conditions.

Figure  presents two-dimensional contour maps of calcite dissolution at different times. During the early stage (up to approximately 10 years), dissolution remains concentrated beneath the strongest sources of contaminants. In Case A, the upper layer below the rightmost source, where the inflow is strongest, loses nearly 15 % of its initial calcite mass by year 10, whereas the depletion beneath the weakest leftmost source remains below 5 %.

The results reveal that external loading exerts a pronounced influence on the morphology and evolution of the reactive front. With increasing load, the transition between the reacted and unreacted zones becomes progressively smoother and more diffuse, while in the non-deforming case (Case C) it remains narrow and sharply defined. For example, at t = 20 years, the front thickness measured as the horizontal distance of 40 % calcite-depletion isolines—widens from approximately 1.4 m in Case C to 2 m under high load (Case A). Consistently, the 2-D maps of aqueous concentration confirm this behaviour: the concentration front is distinctly sharper in the non-deforming case (Case C), whereas deformation produces a smoother and more diffuse transition across the reactive zone.

This “mechanical smoothing effect” arises from differences in solute distribution and local chemical disequilibrium. Under higher external load, pore pressure redistribution alters the concentration field, redirecting solute transport predominantly along the horizontal direction rather than vertically. As a result, the dissolved products (e.g. Ca²⁺, HCO3-) spread laterally and accumulate near the reaction zone, moderating vertical concentration gradients and producing a more diffuse dissolution front. This accumulation increases the local saturation index (SI_calcite) and reduces the thermodynamic driving force term (1-10nSI), thus slowing and broadening the dissolution process spatially. Consequently, concentration gradients across the front are gradually smoothed, yielding a wider, more diffuse reactive zone. In contrast, the non-deforming case (Case C) maintains stronger undersaturation and steeper concentration gradients, resulting in a sharper, advection-controlled interface that propagates more abruptly through the matrix.

Together, these results demonstrate that dissolution, flow-driven transport, and load-induced pore-space evolution jointly control the timing and magnitude of solute concentrations. The 2-D contour maps corroborate the line-based profiles, confirming that these trends persist in the horizontal direction and revealing peak differences and lateral plume structures, such as spreading and source-zone interactions, not captured by 1-D analyses.

This subsection examines how saturation conditions influence reactive solute transport in deformable porous media. Figure a and b compare excess pore pressure p and vertical displacement uz between three representative cases: the near-saturated baseline Case A (Sr = 0.88), the fully saturated Case F (Sr = 1.0), and the less-saturated Case G (Sr = 0.80). The mechanical response strongly depends on the saturation. The fully saturated Case F exhibits the highest excess pore pressure – approximately 400 kPa at the top, nearly three times that of Case G – because its pores are filled with nearly incompressible water that resists volumetric change under load.

Although all cases reach similar final surface settlements (≈ 0.065 m), Case F requires substantially longer to stabilise, as excess pressure dissipates only through slow water drainage in the absence of compressible gas. Lower saturation reduces both the pore water content and the initial pressure peak, facilitating faster dissipation and earlier stabilisation; Case G, for instance, shows roughly 45 % lower peak p and settles much earlier than Case F.

Figure a–e depict the spatial and temporal evolution of dissolved calcium (Y.Ca) and carbon (Y.C) at different saturation levels. Both solutes exhibit similar behaviour: lower saturation generally enhances aqueous concentrations. Comparison in the three case groups (Fig. c–e) clarifies the distinct and combined functions of deformation, saturation, and geochemical reactions.

The Case B, J, and K group disables geochemical reactions to isolate the influence of mechanical deformation. Although total compression is nearly identical in all cases, the drainage rate varies markedly with saturation. In fully saturated condition, excess pore pressure slowly dissipates and the resulting limited fluid mobility restricts advective transport, leading to lower solute concentrations. In contrast, partially saturated conditions allow faster drainage and stronger advection, which promote vertical redistribution. Thus, the observed concentration difference primarily reflects drainage-controlled advection capacity rather than differences in overall deformation magnitude.

In contrast, the Case C, H, I group suppresses deformation but retains geochemical reactions, thus isolating the reaction-driven response to varying saturation. Here, decreasing Sr elevates the concentrations mainly because the available water volume (θw = nSr) diminishes; thus, the same mass of the dissolution-derived solute is distributed into a smaller liquid volume, producing uniformly higher aqueous concentrations. This 1/θw scaling amplifies the contrast between saturation levels.

Finally, the Case A, F, G group activates both deformation and geochemical reactions, and the observed patterns reflect their combined influence. Geochemical processes increase the levels of solutes through ongoing dissolution, while lower saturation further magnifies these increases by reducing the aqueous volume. Meanwhile, mechanical compression partially counteracts this enhancement by reducing the level of pore connectivity and limiting the degree of advective redistribution. In general, geochemical reactions amplify saturation-driven contrasts, whereas mechanical deformation provides a modest damping effect on solute concentrations under any given saturation.

Figures a–i present the two-dimensional distributions of dissolved carbon (Y.C) at t = 20 years for all case groups. Compared with earlier vertical profiles, these maps emphasise the pronounced horizontal variability shaped by the three contaminant sources. Peak concentrations consistently occur near the rightmost source where the inflow is strongest, reaching Y.C ≈ 3.0 mol L⁻¹, approximately two to three times higher than those near the weakest leftmost source. Horizontal plumes are progressively broadened from right to left, revealing distinct source–plume separation and limited mixing between adjacent zones.

Lower saturation elevates aqueous concentrations throughout the domain, particularly around the strong right-hand source, where the high-solute cores expand laterally by about 15 %–20 % in Case G relative to Case F. Similar conclusions can be drawn from Cases J and K. In contrast, mechanical deformation exerts an opposing influence: at the same saturation level, the peak Y.C values in deformable cases (e.g., Case G versus Case I) are about 20 % lower, and the corresponding high-concentration plumes appear more confined and compact.

The pH distributions (Fig. j–o) reflect the coupled influence of dissolved calcium, carbonate species, and other ions on local aqueous equilibria. Regions with higher solute concentrations develop broader and more acidic zones, while areas with limited solute accumulation remain close to neutral. Consequently, the extent of acidification closely mirrors both the intensity and lateral extent of the solute plumes.

Overall, the results indicate that saturation variations exert a stronger control on chemical concentrations and plume geometry than on the final mechanical deformation. Although saturation influences response magnitudes, the governing mechanisms and comparative trends remain consistent across Sr 0.8–1.0. Lower saturation amplifies both the magnitude and the horizontal reach of high-concentration plumes, while mechanical deformation slightly suppresses the concentration peaks and contracts the plume footprints. Together, these trends highlight the competing effects of saturation and deformation on the long-term spread of reactive solutes in deformable porous media.

The purpose of this subsection is to demonstrate the capability of the developed solver to handle multi-component and multi-mineral reactions, thus establishing a foundational framework for future model extensions.

Figures  and  illustrate the temporal evolution of aqueous chemistry and mineral reactions driven by three externally supplied contaminants, carbon (Y.C), iron (Y.Fe), and aluminium (Y.Al), with source intensities decreasing from left to right. Among the three, the external influx of iron is the strongest, followed by aluminium, while carbon has the weakest source. However, the retained aqueous concentrations show the opposite order (C > Fe > Al) due to their distinct reaction pathways. Dissolved carbon maintains the highest residual level, with typical peaks around 6.6 mol L⁻¹, while dissolved iron and aluminium reach lower quasi-steady concentrations of approximately 0.2 mol L⁻¹ and 0.12 mol L⁻¹, respectively. Aluminium undergoes the most intensive reaction: the introduced Al³⁺ rapidly precipitates as gibbsite (Al(OH)₃), leaving only trace amounts of dissolved aluminium in the aqueous phase. As a result, the aluminium plume appears nearly stationary, giving the impression of limited solute migration despite strong local reactions.

Initially, the injected source solution is mildly alkaline due to the presence of hydroxyl- and carbonate-bearing complexes such as Al(OH)4- and HCO3-. As reactions proceed, the precipitation of gibbsite (Al(OH)₃) and siderite (FeCO₃) releases protons through hydrolysis and carbonate exchange, progressively neutralising the alkalinity and driving the system towards mildly acidic conditions. Calcite (CaCO₃) also precipitates, but plays a dual role by partially buffering the acidity generated through its dissolution equilibrium. Spatially, the right-hand side exhibits a faster pH decline than the left-hand side, primarily because of stronger precipitation reactions and weaker carbonate buffering in that region.

Calcite remains persistently oversaturated throughout the simulation, indicating a continuous but slow precipitation. This sluggish behaviour arises from two limiting factors: low concentration Ca²⁺ and a mildly acidic environment (pH ≈ 6.0). Under such conditions, most dissolved inorganic carbon (DIC) exists as HCO3- rather than CO32-, thus reducing the effective carbonate activity required for calcite growth. Consequently, net calcite precipitation proceeds slowly, achieving only a modest increase of about 15 % of its initial mass by year 50. Although the process consumes small amounts of Ca²⁺ and bicarbonate ions, the resulting buffering capacity is insufficient to neutralise the protons released from metal hydrolysis and gibbsite formation. Hence, calcite acts as a slow but persistent secondary precipitate phase under calcium- and pH-limited conditions.

Siderite forms at a moderate but sustained rate where injected Fe²⁺ overlaps with carbonate species supplied by the carbon source (Y.C, primarily HCO3- / CO32-). Concomitant precipitation of calcite competes for carbonate and thus moderates siderite growth. Compared with gibbsite, the siderite front advances more slowly and extends over a broader region. By year 30, siderite captures roughly 25 %–30 % of the incoming iron along the primary flow paths, increasing to approximately 40 %–45 % by year 50. As a result, dissolved iron concentrations peak near the source, governed by the source strength, and gradually attenuate downstream under reaction control.

Upon arrival of Y.Al, gibbsite precipitates rapidly, causing an abrupt decrease in the dissolved aluminium near the source. Quantitatively, more than 80 % of the incoming Al is removed within the first 10 years in the proximal zone, and the cumulative fraction increases from approximately 5 % initially to nearly 20 % by year 50. Because the precipitation rate greatly exceeds the advective transport rate, most reactions remain confined to the source region, leaving negligible aluminium migration downstream. This behaviour characterises a strongly reaction-controlled regime, where rapid local precipitation suppresses long-range solute propagation.

In general, the solver consistently reproduces these coupled processes: progressive plume expansion over 10–50 years, clear dependence on source strength, moderated pH evolution, and mass-conserving behaviour across all solutes. These results confirm that the model robustly simulates complex, multi-mineral reactive transport involving concurrent precipitation and dissolution fronts.

All simulations were performed on a workstation equipped with an Intel Xeon Silver 4114 CPU (2.20 GHz) and 256 GB RAM. Unless otherwise specified, each case was solved using four parallel MPI processes under the same hardware configuration to ensure consistency and fair comparison of CPU time and memory usage.

To evaluate numerical efficiency and assess the trade-off between computational cost and accuracy, this subsection compares the performance of the full HMC solver with its simplified counterparts, in which either the chemical module is deactivated. The evaluation considers four key indicators: total CPU time, average time per step, maximum RAM usage, and the relative error of solute concentration at the most responsive monitoring point. For each paired comparison, the fully coupled HMC case under identical loading and saturation conditions (Cases A, F, G) serves as the reference, while the simplified (Cases B, K) and multi-mineral (Case L) cases are evaluated against it.

The relative error (RE_Q(%)) quantifies the deviation of a simplified model from its corresponding full HMC reference under the same loading and saturation. For a time-dependent variable Q(t) (e.g., carbon concentration Y.C) measured at a monitoring point, RE_Q(%) is defined using the L2-norm of the temporal series, normalised by the reference: 21REQ(%)=Qmodel(t)-QHMC,ref(t)L2(0,T)QHMC,ref(t)L2(0,T)×100, where ‖X‖L2(0,T) = ∫0TX2(t)dt. In this study, RE_Q(%) is evaluated at the most responsive monitoring point identified from the HMC baseline for each comparison.

Table  summarises the computational cost and relative error evaluated at the top monitoring point (x = 17 m, z = 1 m). All reported performance metrics correspond to the output of the first MPI process (rank 0). Complete HMC simulations (A, F, G) required approximately 25–26 h of CPU time, while deactivation of chemical reactions (B, J, K) reduced total runtime by more than two orders of magnitude, with almost identical memory consumption. The nearly constant use of RAM arises because all cases share the same mesh and field allocations; enabling a module does not significantly alter the solver's data structure or storage requirements. However, this simplification introduces substantial concentration deviations (RE_c ≈ 44 %), demonstrating the importance of retaining chemical coupling to capture realistic solute behaviour. In contrast, the multi-mineral reactive case (L) exhibited the highest computational demand, reflecting the increased cost associated with solving the extended reaction network. Overall, these results confirm that while simplified configurations offer significant computational savings, they do so at the expense of accuracy, underscoring the necessity of HMC modelling for quantitative predictive analyses.

To evaluate the parallel efficiency of the developed solver, the wall time required for a 5-year simulation is analysed as a function of the number of MPI processes. Figure  presents the measured wall time together with the corresponding speedup, which is defined relative to the baseline case with two MPI processes as Speedup = T2/TN.

As the number of MPI processes increases from 2 to 16, the wall time decreases substantially from approximately 3.5 × 10⁴ s to below 5 × 10³ s, demonstrating a marked improvement in computational efficiency. The speedup increases monotonically over the same range and remains close to linear scaling, indicating that the developed solver is well parallelised and can effectively exploit distributed-memory computing for the problem size considered. Overall, these results confirm satisfactory strong-scaling performance up to 16 MPI processes for the 5-year simulation.