Dynamic Linear Thermo-Poroelasticity Model

• Dynamic linear thermo-poroelasticity is a model that integrates mechanical, hydraulic, and thermal processes in porous media for simulating wave propagation and coupled multiphysics phenomena.
• The approach employs a frequency-domain variational formulation with T-coercivity analysis to ensure well-posedness and robust finite element discretization.
• Numerical benchmarks confirm first-order convergence and locking-free performance even under nearly incompressible and degenerate parameter regimes.

A dynamic linear thermo-poroelasticity model describes the interaction of mechanical, hydraulic, and thermal processes in a deformable porous medium, integrating inertia, thermal relaxation, and coupling effects with second-order time derivatives. This model is fundamentally relevant for simulating wave propagation in saturated media, analyzing heat and mass transfer in geomechanical structures, and designing robust numerical discretizations for computational mechanics. The following exposition details the governing equations, functional framework, well-posedness theory, mixed finite element discretization, error analysis, and numerical validation of this class of models, focusing principally on the frequency-domain analysis and locking-free discretization strategy as established in recent literature (Li et al., 22 Nov 2025).

1. Governing Equations and Constitutive Laws

The dynamic linear thermo-poroelasticity model comprises three tightly coupled partial differential equations for the displacement of the solid skeleton $(u)$, pore fluid pressure $(p)$, and temperature $(T)$. Constitutive relations are based on Biot’s poroelasticity and classical linear thermoelasticity. Explicitly,

• Unknowns: $u(x,t) \in \mathbb{R}^d$, $p(x,t) \in \mathbb{R}$, $T(x,t) \in \mathbb{R}$.
• Physical coefficients:
  • $\rho$ : mixture density
  • $\mu, \lambda$ : Lamé constants
  • $\alpha$ : Biot–Willis constant
  • $b$ : thermal stress constant
  • $s_0$ : specific storage
  • $\kappa$ : permeability/viscosity tensor
  • $\kappa_T$ : thermal conductivity
  • $\tau_p, \tau_T$ : relaxation times (Darcy-inertia, Lord–Shulman heat law)
  • $c_0$ : heat capacity

Total Cauchy stress: $$\sigma(u,p,T) = 2\mu \epsilon(u) + \lambda\,(\nabla \cdot u)\,I - \alpha\,p\,I - b\,T\,I, \qquad \epsilon(u) = \frac12(\nabla u + \nabla u^T)$$

Strong-form PDE system:

• Solid momentum:

$$\rho\,\partial_{tt}u - \nabla\cdot\sigma(u,p,T) + \alpha\,\nabla p + b\,\nabla T = f$$

• Pore-fluid conservation (with inertial/relaxational correction):

$$s_0\,\partial_t p + \alpha\,\nabla\cdot(\partial_t u)-\nabla\cdot(\kappa\,\nabla p) + \tau_p\,\partial_{tt} p = g_p$$

• Energy balance (Lord–Shulman thermal relaxation):

$$c_0\,\partial_t T + b\,\nabla\cdot(\partial_t u) - \nabla\cdot(\kappa_T\,\nabla T) + \tau_T\,\partial_{tt} T = g_T$$

Homogeneous boundary conditions are imposed on distinct boundary segments for displacement, pressure, and temperature.

2. Frequency-Domain Variational Formulation

The model is reformulated in the frequency domain by assuming time-harmonic fields, $e^{i\omega t}$, as

$$u(x,t) = \operatorname{Re}[\,\hat{u}(x)\,e^{i\omega t}\,], \quad p(x,t) = \operatorname{Re}[\,\hat{p}(x)\,e^{i\omega t}\,], \quad T(x,t) = \operatorname{Re}[\,\hat{T}(x)\,e^{i\omega t}\,].$$

The transformed equations yield a coupled system for $(u, w, p, T)$ with auxiliary variable $w$ (Darcy velocity in mixed formulation), yielding the abstract operator equation: $$\mathcal{A}(u, w, p, T; v, z, q, s) = L(v, z, q, s)$$ on complex-valued spaces $V=H^1_0(\Omega)^d$, $Z=H(\text{div}; \Omega)$, $Q=L^2(\Omega)$, $S=H^1_0(\Omega)$, with essential boundary conditions per field.

Sesquilinear forms (see detailed forms in (Li et al., 22 Nov 2025)):

• $a_1, a_2, a_3$ comprise the principal elasticity, Darcy, and heat equations with inertial, viscous, and relaxation terms.
• Coupling bilinears $b_1$, $b_2$, $b_3$ (poroelastic and Darcy), $c_1$, $c_2$ (thermo-mechanical), and $d$ (storage).
• The system is parameterized by $\omega > 0$; analysis is restricted to $\omega < \omega_\text{crit}$ for well-posedness.

3. Functional Analysis and Well-Posedness

The operator $\mathcal{A}$ is block-structured on $X=V \times Z \times Q \times S$, decomposed into principal (Fredholm) and compact coupling terms. The analysis proceeds as follows:

• T-coercivity is established for the elasticity block $a_1$ via a spectral operator involution $T$, enabling a Gårding-type inequality that yields Fredholm index zero.
• Fredholm theory and Lax–Milgram: The main block (mechanics plus Darcy) is Fredholm and injective, hence bijective. Compactness of cross-couplings and coercivity of the thermal block (via $a_3$) ensure the full operator $\mathcal{A}$ is invertible.
• Uniform inf-sup stability and injectivity hold independently of mesh parameter $h$ and for feasible frequency ranges, guaranteeing uniqueness and continuous dependence of the solution.

4. Locking-Free Mixed Finite Element Discretization

Discretization is performed with specialized FE spaces designed to prevent volumetric locking and to control spurious oscillations:

• $V_h$ : Bernardi–Raugel $P_1$ enriched by facet bubbles ($H^1$-conforming for elasticity)
• $Z_h$ : RT$_0$ ($H(\text{div})$ for Darcy velocity)
• $Q_h$ : $P_0$ (cellwise constant pressure)
• $S_h$ : $P_1$ conforming temperature

Discrete sesquilinear forms utilize $L^2$-projections $P_h$ on divergence terms (reduced integration), which is crucial for locking avoidance as $\lambda \to \infty$. Stabilization for temperature is available: $$S_h((u_h, w_h, p_h, T_h), (v_h, z_h, q_h, s_h)) := \delta \sum_K h_K^2 (\nabla T_h, \nabla s_h)_K.$$ The stabilized discrete operator $\mathcal{A}_{h,\text{stab}}$ is shown to replicate the continuous T-coercivity and Fredholm structures, again yielding uniform inf-sup bounds $\beta_1 > 0$ and unique well-posed FE solutions.

5. Error Estimates and Robustness

Under regularity assumptions $u \in H^2$, $w \in H^1(\text{div})$, $p \in H^1$, $T \in H^2$, optimal a priori error estimates are established: $$\|u-u_h\|_{H^1} + \|w-w_h\|_{H(\text{div})} + \|p-p_h\|_{L^2} + \|T-T_h\|_{H^1} \leq C h ( |u|_2 + \lambda|\nabla\cdot u|_1 + |w|_1 + |\nabla\cdot w|_1 + |p|_1 + |T|_2 )$$ This convergence is robust to high incompressibility ($\lambda \to \infty$), nearly vanishing thermal/poroelastic couplings ($a_0, b_0, c_0 \to 0$), and degenerate physical parameters. No volumetric locking, pressure, or temperature oscillations arise, and the stability is maintained up to high frequencies subject to $\omega<\omega_\text{crit}$.

6. Numerical Benchmarks and Validation

Numerical studies on the unit square and engineering benchmarks (cantilever bracket, layered permeability) confirm:

• First-order convergence in all norms and unknowns, including for nearly incompressible domains ($\lambda=10^6$) and degenerate couplings.
• Stability and accuracy for wave numbers $\omega$ up to at least 50 (subcritical regime).
• Temperature stabilization ($\delta>0$) corrects oscillatory artifacts for highly vanishing $\Theta$ (thermal conductivity).
• Absence of spurious wiggles for pressure and temperature is demonstrated in the presence of challenging parameter regimes.

7. Context and Relevance

The presented model and discretization approach are pivotal in modern computational poromechanics, enabling the simulation of wave propagation, multiphysics coupling, and robust solution of practical engineering and geoscience problems. The analytical framework, characterized by T-coercivity and variational Fredholm theory, together with the locking-free mixed FE implementation, sets the standard for high-fidelity models of dynamic linear thermo-poroelastic phenomena. Applications encompass seismic wave analysis, subsurface energy extraction, and coupled thermal-hydraulic-structural design. Extensions to nonlinear or inelastic regimes, higher-order discretizations, and reduced-order modeling remain active research directions. The separation and stabilization of coupled fields are critical for computational reliability in real-world operations (Li et al., 22 Nov 2025).