Deep Gas Sources in Deformable Porous–Fractured Media: Volcanic and Tectonic Systems

Abstract

Deep gas emissions in volcanic and tectonic environments are commonly interpreted as the surface expression of localized deep emitters. This representation is adequate for first-order description, but it is not physically complete. Deep degassing is more appropriately represented as a coupled source–storage–pathway system in which volatile generation, compressible accumulation, phase change, hydraulic communication, and permeability evolution are dynamically linked. Starting from phase-wise mass conservation in deformable porous–fractured media, reduced equations for gas migration, pore-pressure diffusion, and thermo-poro-mechanical coupling are derived, showing how the distinction between gas-mass transport and pressure propagation provides a unified framework for volcanic and tectonic degassing. Deep pressure gradients are shown to arise from the competition between volatile supply and pathway leakance, while episodic discharge can occur when permeability evolves under effective stress, sealing, and failure. A minimal analytical source–storage–pathway model is further derived, yielding explicit criteria for valve onset, source charging and discharge times, and the distinction between pressure-led and mass-led responses. The framework is then applied to the published Campi Flegrei carbon dioxide (${\mathrm{CO}}_2$) diffuse total output record, providing a real-data illustration of slow storage loading and rapid transient discharge. The analysis considers magmatic exsolution, hydrothermal mediation, metamorphic devolatilization, advective–diffusive near-surface filtering, and the inverse problem through which surface fluxes and gas compositions are used to infer deep source properties. The formulation links magmatic degassing, hydrothermal pressurization, tectonic fluid ascent, and fault-valve behavior within a common continuum-physics perspective and identifies the constitutive assumptions that most strongly control interpretation.

1. Introduction

Deep gas discharge in volcanic and tectonic environments is often described as the surface expression of a localized deep source. Although this representation can be adequate for initial description, it is physically incomplete. In most natural systems, the relevant object is not a point emitter but a coupled source–storage–pathway system in which volatile generation, compressible accumulation, thermal state, and the transmissivity of the surrounding crust evolve together rather than independently [1,2,3,4,5,6]. In volcanic environments, the source term may arise from decompression-driven exsolution, crystallization-driven volatile concentration, and open-system outgassing of shallow reservoirs or crystal-rich mush zones [1,2,3,7]. In tectonic environments, it may arise from metamorphic devolatilization and decarbonation reactions that generate fluids rich in carbon dioxide (${\mathrm{CO}}_2$) at depth, often in coexistence with denser hydrosaline brines of markedly different mobility [5,6]. In both settings, the observable surface flux is therefore not controlled by source strength alone, but by the competition between volatile production, compressible storage, and leakance through a permeability architecture that evolves under the action of stress, heating, alteration, and failure [4,8,9,10]. Recent work on conduit-flow evolution, volcano deformation, and the broader physics of deformable granular and porous ground further emphasizes that volcanic and near-surface geological systems are intrinsically dynamic, multiscale media rather than static containers for fluids [11,12,13].

This broader framing follows from the structure of the governing equations. The equations used to describe deep gas release are not intrinsically volcanic; they belong more generally to the physics of deformable porous–fractured media, for which volcanic and tectonic terrains provide representative geophysical realizations. This perspective clarifies the meaning of a deep source, because what is generated at depth is often not a fully segregated free gas phase already detached from its host, but a volatile inventory that may remain dissolved, partially exsolve, partition between aqueous and gaseous phases, segregate buoyantly, or evolve into superheated or supercritical water–carbon dioxide (${\mathrm{H}}_2$O–CO₂) fluids before any shallow manifestation appears [3,4,7,14]. In volcanic systems, this implies that reservoir overpressure and surface degassing are jointly controlled by volatile loss, host-rock permeability, and thermo-poroelastic response of the surrounding crust [1,4,7]. In tectonic systems, it implies that deep ${\mathrm{CO}}_2$ production is strongly modulated by the structural state of the crust, because faults and fracture networks may either drain the system or compartmentalize it and, thereby, promote overpressure; this role of fault architecture has long been emphasized in studies of fault-zone permeability structure and fault-controlled fluid flow [5,6,9,10,15,16,17]. The documented global association between ${\mathrm{CO}}_2$ discharge and extensional tectonics, the generation of immiscible ${\mathrm{CO}}_2$-rich vapors and brines during hot collisional metamorphism, the structural focusing of hydrothermal upflow at fault damage zones and intersections, the effective-stress dependence of volcanic-rock permeability, and the orders-of-magnitude permeability contrasts measured within single fault-zone architectures all indicate that pathway transmissivity is a state-dependent property rather than a fixed conduit parameter [5,6,8,9,17,18].

A key physical distinction in the analysis developed here is the separation between gas-mass propagation and pressure propagation. A newly generated gas phase migrates according to advection, effective diffusion, buoyancy, capillarity, and phase segregation [14,19]. Pressure, by contrast, propagates as a hydraulic and poro-thermo-mechanical disturbance [10,14,20]. These two processes therefore possess different characteristic timescales, which helps explain why shallow degassing anomalies may record a deep perturbation before the corresponding parcel of newly generated gas reaches the surface [10,19,20,21]. The signal is then further filtered in the shallow domain by advective–diffusive transport in soils, by atmospheric forcing, by boiling and condensation, and by gas–liquid interaction in shallow hydrothermal zones [19,22,23,24]. Several studies have demonstrated this filtering in complementary ways: advective–diffusive soil-gas models show that concentration and isotopic profiles depend strongly on Darcy velocity and on the imposed pressure gradient [19]; thermo-poro-viscoelastic models of mush transport show that pressure and heat may propagate asymmetrically in the presence of thermal gradients [20]; multiphysics reconstructions of fumarolic systems show that two fumaroles supplied by the same reservoir may nonetheless acquire distinct compositions through near-surface mixing with condensed steam [22]; and combined ${\mathrm{CO}}_2$–radon inversions show that even source depth can only be constrained after transport and radioactive decay are modeled explicitly [24].

This study pursues three connected objectives. First, it develops a unified continuum framework for deep gas sources in deformable porous–fractured media by deriving reduced equations for gas transport, pressure diffusion, and thermo-poro-mechanical coupling. Second, it introduces a minimal analytical source–storage–pathway model that makes the onset of valve-like behavior, the associated charging and discharge times, and the separation between pressure transmission and mass transport explicit. Third, it organizes volcanic and tectonic systems into a limited set of source archetypes and transport regimes and examines which combinations of observables can and cannot identify those regimes. The emphasis throughout is on constitutive structure: source strength, storage, transmissivity, and shallow filtering are treated as coupled components of a single inverse problem rather than as independent descriptors. Figure 1 summarizes the physical source–storage–pathway architecture, Figure 2 summarizes the associated regime language, and

Table 1. Principal notation used in this paper.

Notation	Definition
$\varphi$	porosity
$S_{\alpha }$	saturation of phase $\alpha$
${\rho }_{\alpha }$	density of phase $\alpha$
${\mathbf{v}}_{\alpha }$	Darcy velocity of phase $\alpha$
k	intrinsic permeability
$k_{r\alpha }$	relative permeability of phase $\alpha$
$P_g$, $P_f$	gas pressure; representative fluid pressure
${\sigma }^{\prime }$	effective stress
$C_P$, $C_T$, $C_{\epsilon }$	pressure storage, thermal, and mechanical coupling coefficients
$D_h$	hydraulic diffusivity
$\mathrm{Pe}$	Péclet number for compositional transport
${\Pi }_{\mathrm{PA}}$	pressure-to-advection timescale ratio
${\Pi }_{\mathrm{SL}}$	source-to-leakance number
${\Pi }_{\mathrm{TP}}$	thermo-pressurization number
${\Pi }_{\mathrm{VF}}$	valve-contrast number
${\Pi }_{\mathrm{SV}}$	source–valve number
$p_c$, $p_r$	opening and reclosing overpressure thresholds
${\tau }_c$, ${\tau }_o$	characteristic charging and open-state relaxation times
$\mathcal{H}$	Heaviside step function
${\beta }_s$	shallow leakage coefficient in the pressure-transfer boundary condition
${\beta }_{\mathrm{reg}}$	regularization weight in the inverse problem
${\zeta }_c$	first-order compositional loss or decay coefficient
${\xi }_P$	complex pressure-transfer wavenumber

summarizes the principal symbols used throughout.

2. Continuum Description of Deep Gas Sources in Deformable Porous–Fractured Media

A continuum description requires a scale choice. Let the representative elementary volume be large enough to average pore-scale and fracture-scale heterogeneity, yet small enough to preserve gradients in pressure, saturation, temperature, and composition. In practice, this immediately produces three distinct modeling languages: an equivalent porous-medium description, a dual-permeability or dual-porosity description, and a discrete-fracture description. The first is appropriate when the dominant transport pathways are sufficiently dense to be homogenized; the second when matrix storage and fracture transmissivity must be retained separately; and the third when a limited number of faults or fractures carry a disproportionate fraction of the flow [14,17,25,26,27,28,29]. In volcanic edifices and hydrothermal terrains, all three regimes may coexist across scales.

In all three descriptions, the basic variables are phase saturation, phase pressure, temperature, porosity, composition, and a mechanical state variable that controls deformation or effective stress. At the phase level, the local balance law reads

$${\displaystyle \frac{\partial }{\partial t}}\left(\varphi S_{\alpha }{\rho }_{\alpha }\right)+\nabla ·\left({\rho }_{\alpha }{\mathbf{v}}_{\alpha }\right)=q_{\alpha }+{\Gamma }_{\alpha },$$

(1)

where $q_{\alpha }$ denotes externally imposed inflow and ${\Gamma }_{\alpha }$ denotes local generation or destruction by exsolution, devolatilization, condensation, boiling, or reaction. At representative-element scale, the phase flux is closed by Darcy’s law,

$${\mathbf{v}}_{\alpha }=-{\displaystyle \frac{k{k}_{r\alpha }}{{\mu }_{\alpha }}}\left(\nabla P_{\alpha }-{\rho }_{\alpha }\mathbf{g}\right),$$

(2)

with k the intrinsic permeability, $k_{r\alpha }$ the relative permeability, ${\mu }_{\alpha }$ the dynamic viscosity, $P_{\alpha }$ the phase pressure, and $\mathbf{g}$ the gravitational acceleration [8,14,19]. For gas–liquid mixtures, $S_{\alpha }$ and $k_{r\alpha }$ depend on capillary pressure and saturation history, and common retention and relative-permeability closures descend from classical unsaturated-flow formulations [30,31]; ${\rho }_{\alpha }$ must generally be closed with a real-fluid equation of state rather than an ideal-gas approximation once near-critical or supercritical conditions are approached [4,14].

The mechanical closure enters through effective stress. In an isotropic Biot-type description, the pore-pressure contribution to total stress can be written

$${\sigma }^{\prime }=\sigma -{\alpha }_B{P}_f\mathbf{I},$$

(3)

where $\sigma$ is total stress, ${\sigma }^{\prime }$ is effective stress, ${\alpha }_B$ is the Biot coefficient, $P_f$ is an appropriate fluid or mixture pressure, and $\mathbf{I}$ is the identity tensor [32,33,34,35]. Throughout the mechanical discussion below, compressive normal stress is taken as positive so that increases in $P_f$ reduce effective compression. Equation (3) expresses the coupling through which pressure modifies permeability and failure thresholds, and therefore, through which source strength, storage, and transmissivity become dynamically coupled.

Three assumptions must be stated explicitly at the outset. First, the treatment below is quasi-static in the hydromechanical sense: inertial terms are neglected in the transport balance, although they may remain relevant for short-timescale wave propagation and for fully dynamic hydromechanical coupling [29,33,34]. Second, the constitutive reductions are local in time and written around a reference state; they are therefore most reliable when state variables evolve continuously over the time window of interest. Third, local thermodynamic equilibrium is assumed at the representative-element scale unless otherwise noted. These assumptions do not trivialize the problem; rather, they define the mesoscale regime in which the dominant geological observables are usually interpreted. The assumptions also delimit the applicability of the reduced model: highly dynamic explosive unrest, fast fragmentation, inertially dominated conduit flow, or strong thermo-chemical disequilibrium would require a different formulation retaining inertia, non-equilibrium kinetics, and explicit fracture propagation.

3. Reduced Equations for Gas Transport, Pressure Diffusion, and Thermo-Poro-Mechanical Coupling

Because the physical distinction between gas-mass transport and pressure propagation is central, it is useful to isolate a reduced pressure equation from the general phase balance. Restricting attention, for clarity, to the gas phase, define the gas mass per unit bulk volume as

$$m_g=\varphi S_g{\rho }_g.$$

(4)

A first-order thermodynamic and mechanical linearization about a reference state gives

$$\mathrm{d}{m}_g={\left({\displaystyle \frac{\partial m_g}{\partial P_g}}\right)}_{T,{\epsilon }_v}\mathrm{d}{P}_g+{\left({\displaystyle \frac{\partial m_g}{\partial T}}\right)}_{P,{\epsilon }_v}\mathrm{d}T+{\left({\displaystyle \frac{\partial m_g}{\partial {\epsilon }_v}}\right)}_{P,T}\mathrm{d}{\epsilon }_v,$$

(5)

where ${\epsilon }_v$ is volumetric strain, positive in dilation. Expanding the total differential yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial m_g}{\partial P_g}}& =\varphi S_g{\rho }_g{c}_g+{\rho }_g{S}_g{\left({\displaystyle \frac{\partial \varphi }{\partial P_g}}\right)}_{T,{\epsilon }_v}+{\rho }_g\varphi {\left({\displaystyle \frac{\partial S_g}{\partial P_g}}\right)}_{T,{\epsilon }_v},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial m_g}{\partial T}}& =-\varphi S_g{\rho }_g{\alpha }_g+{\rho }_g{S}_g{\left({\displaystyle \frac{\partial \varphi }{\partial T}}\right)}_{P,{\epsilon }_v}+{\rho }_g\varphi {\left({\displaystyle \frac{\partial S_g}{\partial T}}\right)}_{P,{\epsilon }_v},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial m_g}{\partial {\epsilon }_v}}& ={\rho }_g{S}_g{\left({\displaystyle \frac{\partial \varphi }{\partial {\epsilon }_v}}\right)}_{P,T}+{\rho }_g\varphi {\left({\displaystyle \frac{\partial S_g}{\partial {\epsilon }_v}}\right)}_{P,T},\hfill \end{array}$$

(8)

with

$$c_g={\displaystyle \frac{1}{{\rho }_g}}{\left({\displaystyle \frac{\partial {\rho }_g}{\partial P_g}}\right)}_T,{\alpha }_g=-{\displaystyle \frac{1}{{\rho }_g}}{\left({\displaystyle \frac{\partial {\rho }_g}{\partial T}}\right)}_{P_g}.$$

(9)

Introducing the reduced coefficients

$$\begin{array}{cc}\hfill C_P& =\varphi S_g{\rho }_g{c}_g+{\rho }_g{S}_g{\left({\displaystyle \frac{\partial \varphi }{\partial P_g}}\right)}_{T,{\epsilon }_v}+{\rho }_g\varphi {\left({\displaystyle \frac{\partial S_g}{\partial P_g}}\right)}_{T,{\epsilon }_v},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill C_T& =\varphi S_g{\rho }_g{\alpha }_g-{\rho }_g{S}_g{\left({\displaystyle \frac{\partial \varphi }{\partial T}}\right)}_{P,{\epsilon }_v}-{\rho }_g\varphi {\left({\displaystyle \frac{\partial S_g}{\partial T}}\right)}_{P,{\epsilon }_v},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill C_{\epsilon }& ={\rho }_g{S}_g{\left({\displaystyle \frac{\partial \varphi }{\partial {\epsilon }_v}}\right)}_{P,T}+{\rho }_g\varphi {\left({\displaystyle \frac{\partial S_g}{\partial {\epsilon }_v}}\right)}_{P,T},\hfill \end{array}$$

(12)

and defining the gas transport conductivity coefficient as

$$K_g={\rho }_g{\displaystyle \frac{k{k}_{rg}}{{\mu }_g}},$$

(13)

substitution of Darcy’s law into the gas-phase mass balance leads to

$$C_P{\displaystyle \frac{\partial P_g}{\partial t}}=\nabla ·\left[K_g\left(\nabla P_g-{\rho }_g\mathbf{g}\right)\right]+q_g+{\Gamma }_g+C_T{\displaystyle \frac{\partial T}{\partial t}}-C_{\epsilon }{\displaystyle \frac{\partial {\epsilon }_v}{\partial t}}.$$

(14)

Equation (14) is the core reduced equation of the study. Gas pressure rises when mass is added locally, when heating expands the fluid more efficiently than the pore volume, or when compaction reduces storage; it relaxes when transmissivity is sufficiently large for hydraulic gradients to dissipate [4,7,14,20]. Equations (1)–(14) are derived under local thermodynamic equilibrium at the representative-element scale, small perturbations about a reference state, negligible inertial terms in the transport balance, and a quasi-static hydromechanical limit. The reduction to the diffusion form below further assumes weak density gradients, modest buoyancy variations over the representative scale, and transport coefficients that vary slowly over the characteristic scale of interest. The algebraic derivation of Equation (14) is given in Appendix A.

In the isothermal, weakly deforming limit, one obtains

$${\displaystyle \frac{\partial P_g}{\partial t}}=D_h{\nabla }^2{P}_g+{\displaystyle \frac{q_g+{\Gamma }_g}{C_P}},D_h={\displaystyle \frac{K_g}{C_P}},$$

(15)

which implies the characteristic pressure-propagation time

$$t_P\sim {\displaystyle \frac{L^2}{D_h}}$$

(16)

over a length scale L. Two further reductions follow immediately. Under undrained conditions at fixed bulk strain, $\mathrm{d}{m}_g=0$ gives

$${\left.{\displaystyle \frac{\partial P_g}{\partial T}}\right|}_{m_g,{\epsilon }_v}={\displaystyle \frac{C_T}{C_P}},$$

(17)

whereas at fixed temperature and gas mass, one finds

$${\left.{\displaystyle \frac{\partial P_g}{\partial {\epsilon }_v}}\right|}_{m_g,T}=-{\displaystyle \frac{C_{\epsilon }}{C_P}},$$

(18)

so that dilation lowers pore pressure and compaction raises it [4,14,20].

Pressure propagation, however, is not equivalent to gas transport. To describe the migration of a compositional anomaly or tracer species i, a separate advection–diffusion balance is required,

$${\displaystyle \frac{\partial \left(\varphi S_g{C}_i\right)}{\partial t}}+\nabla ·\left(C_i{\mathbf{v}}_g-\varphi S_g{D}_i\nabla C_i\right)=R_i,$$

(19)

where $C_i$ is the gas-phase concentration of species i, $D_i$ an effective diffusivity, and $R_i$ a source, sink, or phase-partition term [19,22,24]. Thermal evolution may be represented, at the same level of reduction, by

$$C_{\mathrm{th}}{\displaystyle \frac{\partial T}{\partial t}}+{\rho }_f{c}_{pf}{\mathbf{v}}_f·\nabla T=\nabla ·(\kappa \nabla T)+Q_T,$$

(20)

where $C_{\mathrm{th}}$ is an effective volumetric heat capacity, $\kappa$ is thermal conductivity, and $Q_T$ is a volumetric heat source. The combination of Equations (14), (19) and (20) defines the minimal dynamic core of the problem. It also shows why the same deep perturbation may generate multiple observational time signatures: pressure may respond first, gas composition later, and temperature later still.

4. Dimensionless Groups and Transport Regimes

The framework becomes predictive when a small number of dimensionless groups is used to separate the principal dynamical regimes. Let $u_g=\left|{\mathbf{v}}_g\right|/\left(\varphi S_g\right)$ be the interstitial gas speed, L a characteristic transport length, D an effective species diffusivity, and $\Delta P_{\mathrm{ref}}$ a representative pressure scale. The characteristic advection and diffusion times are

$$t_{\mathrm{adv}}\sim {\displaystyle \frac{L}{u_g}},t_{\mathrm{diff}}\sim {\displaystyle \frac{L^2}{D}},$$

(21)

from which the Péclet number follows as

$$\mathrm{Pe}={\displaystyle \frac{u_{g}L}{D}}.$$

(22)

The ratio between the pressure-propagation time and gas-advection time is

$${\Pi }_{\mathrm{PA}}={\displaystyle \frac{t_P}{t_{\mathrm{adv}}}}={\displaystyle \frac{u_{g}L}{D_h}}.$$

(23)

A source-to-leakance number can be defined as

$${\Pi }_{\mathrm{SL}}={\displaystyle \frac{{\dot{M}}_s}{{\lambda }_s\Delta P_{\mathrm{ref}}}},$$

(24)

where ${\dot{M}}_s$ is a characteristic source mass flux (mass per unit time), and ${\lambda }_s$ an effective leakance. Thermal pressurization may be summarized by

$${\Pi }_{\mathrm{TP}}={\displaystyle \frac{C_T\Delta T}{C_P\Delta P_{\mathrm{ref}}}},$$

(25)

and the permeability contrast associated with valve-like behavior by

$${\Pi }_{\mathrm{VF}}={\displaystyle \frac{k_o}{k_c}},$$

(26)

where $k_c$ and $k_o$ are the closed and open permeabilities, respectively.

The Péclet number distinguishes advective from diffusive species transport. The ratio ${\Pi }_{\mathrm{PA}}$ compares pressure propagation with gas advection. When ${\Pi }_{\mathrm{PA}}\ll 1$, the system is pressure led: hydraulic reorganization, shallow flux transients, and even permeability changes may occur before a newly generated gas parcel has traversed the transport domain [10,20,21]. When ${\Pi }_{\mathrm{PA}}\gtrsim 1$, mass transfer and pressure transfer occur on comparable timescales. The quantity ${\Pi }_{\mathrm{SL}}$ compares source loading with drainage capacity, while ${\Pi }_{\mathrm{TP}}$ measures the importance of thermal expansion relative to pressure storage. Finally, ${\Pi }_{\mathrm{VF}}$ quantifies how dramatically the transport operator changes when a sealing threshold is crossed.

The connection with natural geological scales is made through ratios rather than universal parameter values because permeability and storage vary over many orders of magnitude in crustal porous–fractured media [9,26,29]. For shallow hydrothermal and fault-controlled pathways, characteristic lengths of hundreds of meters to kilometers, hydraulic diffusivities controlled by fracture transmissivity, and interstitial gas speeds inferred from site-specific fluxes may place the same system in different regions of the regime map during different phases of unrest. Thus ${\Pi }_{\mathrm{PA}}$ is to be interpreted as a diagnostic measure of whether the observed signal is expected to be pressure led or mass led, whereas ${\Pi }_{\mathrm{SL}}$ and ${\Pi }_{\mathrm{SV}}$ measure whether source loading is likely to remain below drainage capacity or cross a threshold for rapid discharge. This scaling interpretation is particularly important when comparing field sites, because the same ${\mathrm{CO}}_2$ output can correspond to a high-source/low-leakance system or to a moderate-source/high-transmissivity pathway.

A further useful quantity is a reaction-to-transport ratio. If ${\tau }_{\mathrm{rxn}}$ denotes a characteristic timescale for reaction, including exsolution or devolatilization, one may define a Damkohler-like number,

$$\mathrm{Da}={\displaystyle \frac{t_{\mathrm{adv}}}{{\tau }_{\mathrm{rxn}}}}.$$

(27)

Large $\mathrm{Da}$ indicates that volatile generation is locally fast compared with transport and that source zones are reaction controlled; small $\mathrm{Da}$ indicates that volatile generation is slow and that migration dominates over in situ production. This dimensionless perspective, with additional scaling details given in Appendix B, clarifies what is, and is not, diagnostic in surface observations. A sudden increase in ${\mathrm{CO}}_2$ flux can reflect an increase in source strength, a decrease in leakance followed by delayed breakthrough, or a transient jump in permeability; without an independent constraint on at least one of the governing dimensionless groups, these alternatives remain non-unique.

The regime classification is summarized by the two-parameter sketch in Figure 2, in which ${\Pi }_{\mathrm{PA}}$ and the source–valve number ${\Pi }_{\mathrm{SV}}$ introduced below define the primary separation between pressure-led, leakage-dominated, mass-led, and valve-prone behavior, with ${\Pi }_{\mathrm{VF}}$ entering as a secondary condition for strong threshold contrast. Low ${\Pi }_{\mathrm{PA}}$ and high ${\Pi }_{\mathrm{SV}}$ identify systems in which rapid pressure communication coexists with threshold-controlled source loading, a combination characteristic of hydrothermal systems above volatile-rich reservoirs. High ${\Pi }_{\mathrm{VF}}$ indicates strong valve contrast and, therefore, susceptibility to pulsed discharge, whereas low ${\Pi }_{\mathrm{VF}}$ is characteristic of systems whose permeability evolves more continuously. This representation summarizes the asymptotic structure of the governing equations and illustrates why similar surface signals may arise from physically distinct deep states. The reduced analytical model developed next makes the onset condition for threshold-controlled discharge and the separation between pressure transmission and mass transport explicit. The dimensionless reduction and limiting regimes are summarized in Appendix B.

5. A Minimal Analytical Source–Storage–Pathway Model

The regime classification introduced above can be made explicit by deriving a minimal analytical model in which source accumulation, threshold opening, and pathway transmission can all be written in closed form. Let $p_s=P_s-P_u$ denote the source overpressure relative to the shallow hydraulic reference pressure $P_u$, and let $q\left(t\right)$ denote the net integrated mass supply to the source region, including deep inflow and internally generated volatiles. The reduced source-storage balance can then be written as

$${\mathcal{S}}_s{\displaystyle \frac{\mathrm{d}{p}_s}{\mathrm{d}t}}=q\left(t\right)-\lambda \left(p_s\right)p_s,$$

(28)

with a threshold-controlled leakance

$$\lambda \left(p_s\right)={\lambda }_c+\left({\lambda }_o-{\lambda }_c\right)\mathcal{H}(p_s-p_c),$$

(29)

where ${\mathcal{S}}_s$ is the effective source storage, ${\lambda }_c$ and ${\lambda }_o$ are the closed- and open-state leakances, $p_c$ is the opening threshold, and $\mathcal{H}$ denotes the Heaviside step function. This minimal model is not intended to replace full multiphase formulations; its purpose is to isolate the smallest analytical structure capable of representing long-timescale charging, short-timescale discharge, and threshold-controlled transmissivity in the sense long associated with fault-valve behavior [10,36,37,38].

For constant forcing $q\left(t\right)=q_0$ and subcritical storage $p_s<p_c$, Equations (28) and (29) reduce to a linear charging problem with solution

$$p_s\left(t\right)={\displaystyle \frac{q_0}{{\lambda }_c}}+\left(p_{s0}-{\displaystyle \frac{q_0}{{\lambda }_c}}\right)e^{-t/{\tau }_c},{\tau }_c={\displaystyle \frac{{\mathcal{S}}_s}{{\lambda }_c}},$$

(30)

where $p_{s0}$ is the initial overpressure. The system reaches the opening threshold only if the closed-state equilibrium pressure exceeds $p_c$; that is,

$${\displaystyle \frac{q_0}{{\lambda }_c}}>p_c.$$

(31)

This suggests a source–valve number,

$${\Pi }_{\mathrm{SV}}={\displaystyle \frac{q_0}{{\lambda }_c{p}_c}},$$

(32)

for which ${\Pi }_{\mathrm{SV}}>1$ is the necessary condition for threshold opening. If $p_{s0}<p_c$ and ${\Pi }_{\mathrm{SV}}>1$, the charging time required to reach the threshold is

$$t_c={\tau }_{c}ln\left({\displaystyle \frac{q_0/{\lambda }_c-p_{s0}}{q_0/{\lambda }_c-p_c}}\right).$$

(33)

Once the pathway opens, the same balance evolves with the larger leakance ${\lambda }_o$. If reclosing occurs at a lower threshold $p_r<p_c$, the open-state relaxation is

$$p_s\left(t\right)={\displaystyle \frac{q_0}{{\lambda }_o}}+\left(p_c-{\displaystyle \frac{q_0}{{\lambda }_o}}\right)e^{-t/{\tau }_o},{\tau }_o={\displaystyle \frac{{\mathcal{S}}_s}{{\lambda }_o}},$$

(34)

with discharge duration

$$t_o={\tau }_{o}ln\left({\displaystyle \frac{p_c-q_0/{\lambda }_o}{p_r-q_0/{\lambda }_o}}\right),$$

(35)

provided $q_0/{\lambda }_o<p_r<p_c$. The inequality ${\lambda }_o\gg {\lambda }_c$, therefore, implies ${\tau }_o\ll {\tau }_c$: the same system charges slowly and discharges over a short timescale without any change in the deep forcing itself. This provides a compact reduced expression of valve-like behavior and shows explicitly that episodic degassing need not require episodic volatile production. Appendix C gives the corresponding analytical charging and discharge solutions.

The pathway can be treated analytically as well. Let $z\in [0,L]$ denote distance from the deep source to the shallow observation level. A minimal pressure-transmission problem is

$${\displaystyle \frac{\partial p}{\partial t}}=D_h{\displaystyle \frac{{\partial }^{2}p}{\partial z^2}},0<z<L,$$

(36)

with boundary conditions

$$p(0,t)=p_s\left(t\right),-D_h{\left.{\displaystyle \frac{\partial p}{\partial z}}\right|}_{z=L}={\beta }_{s}p(L,t),$$

(37)

where ${\beta }_s$ is a shallow leakage coefficient representing the hydraulic conductance of the upper boundary, including soil–air interface resistance, shallow hydrothermal drainage, and atmospheric coupling. The limit ${\beta }_s\to 0$ corresponds to an effectively sealed or no-flux boundary, whereas ${\beta }_s\to \infty$ approaches a drained or fixed-pressure boundary. For harmonic forcing $p_s\left(t\right)=\Re \left\{{\widehat{p}}_s{e}^{i\omega t}\right\}$, the pressure amplitude at the observation level is

$$\begin{array}{cc}\hfill \widehat{p}(L,\omega )& =H_P\left(\omega \right){\widehat{p}}_s\left(\omega \right),\hfill \\ \hfill H_P\left(\omega \right)& ={\left[cosh\left({\xi }_{P}L\right)+{\displaystyle \frac{{\beta }_s}{D_h{\xi }_P}}sinh\left({\xi }_{P}L\right)\right]}^{-1},\hfill \\ \hfill {\xi }_P& ={\left({\displaystyle \frac{i\omega }{D_h}}\right)}^{1/2},\hfill \end{array}$$

(38)

with $\omega$ denoting the frequency. Equation (38) is a hydraulic transfer function: pressure variations are transmitted quasi-statically for $\omega \ll D_h/L^2$ and strongly attenuated for $\omega \gg D_h/L^2$.

A compositional anomaly or conservative tracer obeys a different transfer law. If the concentration $c(z,t)$ satisfies

$${\displaystyle \frac{\partial c}{\partial t}}+u{\displaystyle \frac{\partial c}{\partial z}}=D_c{\displaystyle \frac{{\partial }^{2}c}{\partial z^2}}-{\zeta }_{c}c,$$

(39)

with $D_c$ the effective compositional diffusivity and ${\zeta }_c$ a first-order compositional loss or decay coefficient, then, retaining the decaying branch of the harmonic solution for forcing $c(0,t)=\Re \left\{{\widehat{c}}_s{e}^{i\omega t}\right\}$, one obtains

$$\widehat{c}(L,\omega )=H_C\left(\omega \right){\widehat{c}}_s\left(\omega \right),H_C\left(\omega \right)=exp\left[{\displaystyle \frac{u-\sqrt{u^2+4D_c({\zeta }_c+i\omega )}}{2D_c}}L\right],$$

(40)

where a hat denotes a complex amplitude, $c_s$ is the source concentration amplitude at $z=0$, and u denotes the interstitial transport speed. In the advective limit $\mathrm{Pe}\gg 1$, Equation (40) reduces to

$$\widehat{c}(L,\omega )\approx {\widehat{c}}_s\left(\omega \right)exp\left[-{\displaystyle \frac{({\zeta }_c+i\omega )L}{u}}\right],$$

(41)

which identifies a mass-transport lag ${\tau }_M=L/u$ and an attenuation factor $e^{-{\zeta }_{c}L/u}$. The corresponding pressure timescale is ${\tau }_P\sim L^2/D_h$, so the pressure-led criterion becomes

$${\Pi }_{\mathrm{PA}}={\displaystyle \frac{{\tau }_P}{{\tau }_M}}={\displaystyle \frac{uL}{D_h}}\ll 1.$$

(42)

The analytical model, therefore, supplies two explicit thresholds that are only implicit in the general discussion above: ${\Pi }_{\mathrm{SV}}>1$ for threshold opening and ${\Pi }_{\mathrm{PA}}\ll 1$ for pressure-led surface response. Together, they form a compact bridge between source mechanics, pathway transmission, and observable timing; the corresponding analytical steps are summarized in Appendix C.

6. Deep Source Archetypes

The dimensionless framework motivates a classification of deep gas sources into a small set of archetypes, summarized in

Table 2. Deep source archetypes and their principal observables.

Archetype	Reduced Physical Description	Most Sensitive Observables	Primary Interpretive Ambiguity
Distributed reactive source	Volatile generation distributed within a finite volume	Broad pressure anomalies, structurally focused surface discharge, geochemical gradients tied to reaction or exsolution	Source depth versus permeability architecture; volumetric source versus focusing
Leaky pressurized reservoir	Compressible compartment feeding one or more pathways	Storage-controlled delay, transient pressure communication, pressure–flux hysteresis	Source strength versus storage versus leakance
Threshold-valve source	Pressure-dependent transmissivity with abrupt opening after failure threshold	Pulsed discharge, swarm-like transients, episodic degassing	True source increase versus transient permeability increase
Multi-reservoir/multi-pathway source	Deep generation, intermediate storage, shallow phase partitioning and mixing	Composite time series, compositional decoupling between outlets, strong non-uniqueness	Deep source geometry versus shallow filtering and partitioning

. The first is the distributed reactive source. In this case gas is generated throughout a finite volume $V_s$ and the local generation term can be represented as

$${\Gamma }_g(\mathbf{x},t)=Q_{\Gamma }\left(t\right){\displaystyle \frac{{\chi }_{V_s}\left(\mathbf{x}\right)}{V_s}},$$

(43)

where ${\chi }_{V_s}$ is the indicator function of the source volume and $Q_{\Gamma }\left(t\right)$ is the total generation rate integrated over $V_s$. This archetype is physically appropriate for crystal-rich mush zones, devolatilizing metamorphic aureoles, or laterally extensive crustal domains in which volatile generation is volumetrically distributed rather than localized [2,6].

The second archetype is the leaky pressurized reservoir. Here a compressible source region accumulates fluid mass and releases it through one or more transmissive pathways. If $P_s$ is source pressure and $P_u$ the pressure of the overlying domain, the governing balance is

$${\mathcal{S}}_s{\displaystyle \frac{\mathrm{d}{P}_s}{\mathrm{d}t}}={\dot{M}}_{\mathrm{in}}+{\int }_{V_s}\Gamma \mathrm{d}V+{\Theta }_s{\displaystyle \frac{\mathrm{d}{T}_s}{\mathrm{d}t}}-{\lambda }_s(P_s-P_u),$$

(44)

where ${\mathcal{S}}_s$ is the effective storage, ${\dot{M}}_{\mathrm{in}}$ denotes imposed mass inflow and $T_s$ is the source temperature, ${\Theta }_s$ is the thermal storage coupling, and ${\lambda }_s$ is an effective leakance. This archetype is the natural representation of a magmatic–hydrothermal system in which volatile supply from depth feeds a shallower, storage-dominated reservoir [4,7].

The third archetype is the threshold-valve source. In this case the storage equation alone is insufficient because the transport operator changes when pore pressure reaches a critical value. The minimal constitutive expression is

$$k_{\mathrm{eff}}\left(p_f\right)=k_c+(k_o-k_c)\mathcal{H}(p_f-p_c),$$

(45)

where $p_f=P_f-P_u$ is the relevant fluid overpressure, $p_c$ is the critical opening threshold. The threshold may represent shear reactivation, tensile opening, or both. Physically, this archetype describes systems in which transmissivity is not merely pressure sensitive, but discontinuously reorganized by failure [10,36,37,38].

The fourth archetype is the multi-reservoir or multi-pathway system. Here, neither source nor pathway can be represented by a single compartment. Instead, deep generation, intermediate storage, shallow phase partitioning, and multiple outlets interact to produce composite signals. This class is common in active volcanic systems, where deep magmatic volatiles, hydrothermal reservoirs, condensed steam, and structurally distinct fumarolic outlets all contribute to the observed signal [4,22]. It is also common in tectonic settings where compartmentalized faults, relay zones, and connectivity changes generate multiple effective storage levels [9,17,18].

7. Pressure Buildup, Leakance, and Evolving Permeability

The origin of overpressure can now be written explicitly in terms of source generation and leakance. In volcanic systems, the most natural source term is thermodynamic. Let $M_m$ denote magma mass, $X_t$ the total volatile mass fraction, and $X_{\mathrm{sat}}(P,T,\chi )$ the volatile concentration at saturation, where $\chi$ denotes composition, crystallinity, and other state variables. The exsolved volatile inventory is then

$$M_{\mathrm{ex}}=M_m{[X_t-X_{\mathrm{sat}}(P,T,\chi )]}_{+},$$

(46)

and differentiation in the oversaturated regime gives

$${\Gamma }_{\mathrm{ex}}={\displaystyle \frac{\mathrm{d}{M}_{\mathrm{ex}}}{\mathrm{d}t}}=-M_m\left[\left({\displaystyle \frac{\partial X_{\mathrm{sat}}}{\partial P}}\right)\dot{P}+\left({\displaystyle \frac{\partial X_{\mathrm{sat}}}{\partial T}}\right)\dot{T}+\sum _j\left({\displaystyle \frac{\partial X_{\mathrm{sat}}}{\partial {\chi }_j}}\right){\dot{\chi }}_j\right].$$

(47)

Equation (47) makes the principal volcanic source mechanisms transparent. Because volatile solubility generally increases with pressure, decompression implies $\dot{P}<0$ and therefore ${\Gamma }_{\mathrm{ex}}>0$ [1,2,3,7]. Crystallization modifies $\chi$ and concentrates volatiles in the residual melt, thereby driving second-boiling behavior [2,3]. Recharge, heating, and mixing perturb both temperature and composition and may shift the system abruptly toward vigorous gas release [1,3,7]. In physical terms, critical degassing, mush outgassing, and volatile-overpressure coupling are all special realizations of Equation (47).

In tectonic settings, the deep source term is more naturally reaction driven. If $r_j(P,T,\mathbf{a})$ denotes the rate of metamorphic devolatilization or decarbonation reaction j, with stoichiometric volatile yield ${\nu }_j$, the bulk generation term can be written as

$${\Gamma }_{\mathrm{rxn}}=\sum _j{\nu }_j{r}_j(P,T,\mathbf{a}).$$

(48)

The source may therefore occupy an extended crustal volume rather than a geometrically simple cavity [5,6,38]. Once magmatic exsolution or metamorphic devolatilization is embedded in a source region $V_s$, the appropriate reduced description is again the source-storage balance of Equation (44). Here, ${\dot{M}}_{\mathrm{in}}$, $\Gamma$, and ${\lambda }_s(P_s-P_u)$ are all expressed consistently in mass units, $\Gamma >0$ denotes net volatile generation within $V_s$, and ${\lambda }_s$ has dimension of mass · ${\mathrm{time}}^{-1}$ · ${\mathrm{pressure}}^{-1}$.

For a pathway of length L, cross-sectional area A, permeability $k_{\mathrm{eff}}$, density ${\rho }_f$, and viscosity $\mu$, the effective leakance scales as

$${\lambda }_s\sim {\rho }_f{\displaystyle \frac{k_{\mathrm{eff}}A}{\mu L}},$$

(49)

so that the source overpressure obeys the leading-order estimate

$$\Delta P_s\sim {\displaystyle \frac{\mu L}{{\rho }_f{k}_{\mathrm{eff}}A}}\left[{\dot{M}}_{\mathrm{in}}+{\int }_{V_s}({\Gamma }_{\mathrm{ex}}+{\Gamma }_{\mathrm{rxn}})\mathrm{d}V+{\Theta }_s{\dot{T}}_s\right].$$

(50)

Deep pressure gradients therefore arise whenever volatile supply and thermal pressurization exceed pathway leakance [4,5,6,26,27]. Although Equation (50) is deliberately reduced, it shows that any decrease in $k_{\mathrm{eff}}$ amplifies overpressure, even if the deep source term itself remains unchanged.

Permeability evolution is therefore a primary constitutive component. Laboratory studies on volcanic rocks show that transmissivity depends strongly on porosity, fracture state, confining pressure, and damage history; field observations and geophysical inversions likewise indicate that permeability can vary by orders of magnitude during transient stress perturbations [8,28,39,40]. This history dependence shows that pressure-threshold models are limiting approximations to valve dynamics. A real system may display hysteresis between opening and closing, delayed sealing by mineral precipitation, and connectivity changes not captured by a single scalar threshold [9,10]. A minimal continuous representation can be written in terms of a damage or connectivity variable $\mathcal{D}\in [0,1]$,

$$k_{\mathrm{eff}}(p_f,\mathcal{D})=k_c+(k_o-k_c)\mathcal{D},$$

(51)

with loading–unloading hysteresis represented by

$${\displaystyle \frac{\mathrm{d}\mathcal{D}}{\mathrm{d}t}}={\displaystyle \frac{1-\mathcal{D}}{{\tau }_{\mathrm{open}}}}\mathcal{H}(p_f-p_c)-{\displaystyle \frac{\mathcal{D}}{{\tau }_{\mathrm{seal}}}}\mathcal{H}(p_r-p_f),p_r<p_c.$$

(52)

Here, ${\tau }_{\mathrm{open}}$ and ${\tau }_{\mathrm{seal}}$ are opening and resealing timescales. The condition $p_r<p_c$ makes the unloading path different from the loading path, so that permeability remains elevated until pressure has fallen below a lower reclosing threshold. Equations (51) and (52) reduce to the Heaviside valve model in the limit of instantaneous opening and sealing, but they also allow gradual damage accumulation, delayed healing, and partial connectivity.

Capillarity and relative permeability further complicate the picture. In two-phase ${\mathrm{H}}_2$O–CO₂ systems, permeability evolution is not only a matter of fracture aperture or connected porosity, but also of saturation state and the mobility contrast between wetting and non-wetting phases. In such cases, storage is shared between elastic pore-volume change and phase redistribution, and the pathway cannot be separated cleanly from the fluid’s thermodynamic state. This contributes to the pronounced nonlinearity observed in volcanic and hydrothermal systems, even where the geometry is geometrically simple.

8. Surface Observables, Filtering, and the Inverse Problem

Once deep gas sources are formulated in this way, the interpretation of surface measurements becomes a forward problem rather than a direct reading of depth. Let $\mathbf{m}$ denote the model vector containing source pressure $P_s$, integrated generation rate $\Gamma$, effective source depth H, permeability field $k(\mathbf{x},t)$, thermal field $T(\mathbf{x},t)$, pathway geometry, and the parameters governing shallow filtering. Let $\mathbf{d}$ denote the vector of observables, including diffuse ${\mathrm{CO}}_2$ flux, fumarolic composition, ${\delta }^{13}$C, radon flux, temperature, and structural or geophysical constraints. The observation equation may be written abstractly as

$$\mathbf{d}=\mathcal{F}\left[\mathbf{m}\right]+\mathit{\epsilon },$$

(53)

where $\mathcal{F}$ is the forward transport operator and $\mathit{\epsilon }$ collects measurement and model error [22,23,24]. Equation (53) formalizes a fundamental filtering property: the surface signal is filtered by transport, phase partitioning, and mixing. Two fumaroles may therefore be supplied by the same shallow reservoir and yet acquire different compositions because they interact differently with condensed steam or liquid-dominated zones near the surface [22]. Diffuse soil emissions may contain both deep and shallow contributions, and the shallow component may vary seasonally enough to obscure the deep one unless additional tracers are used [23].

The shallow domain acts as a physical filter because it modifies different species in different ways. Consider one-dimensional steady upward transport in the vadose zone with constant gas-filled porosity ${\theta }_g=\varphi S_g$, constant upward interstitial speed u, effective diffusivity $D_i$, and vertical coordinate z positive upward. After division of the steady bulk balance by ${\theta }_g$, and with reaction terms expressed per unit gas-filled volume, the transport equation for species i becomes

$$D_i{\displaystyle \frac{{\mathrm{d}}^2{C}_i}{\mathrm{d}{z}^2}}-u{\displaystyle \frac{\mathrm{d}{C}_i}{\mathrm{d}z}}+R_i\left(C_i\right)=0.$$

(54)

For nearly conservative ${\mathrm{CO}}_2$ in a strongly degassing domain, one may approximate $R_i\simeq 0$, whereas for radon-222 (²²²Rn) radioactive decay must be retained, so that

$$D_{\mathrm{Rn}}{\displaystyle \frac{{\mathrm{d}}^2{C}_{\mathrm{Rn}}}{\mathrm{d}{z}^2}}-u{\displaystyle \frac{\mathrm{d}{C}_{\mathrm{Rn}}}{\mathrm{d}z}}-{\lambda }_{\mathrm{Rn}}{C}_{\mathrm{Rn}}=0.$$

(55)

For definiteness, Equations (54) and (55) are interpreted under prescribed concentration or flux at the deep entry level $z=-H$ and mixed or prescribed boundary conditions at the ground surface $z=0$. The corresponding characteristic roots are

$$r_{\pm }={\displaystyle \frac{u\pm \sqrt{u^2+4D_{\mathrm{Rn}}{\lambda }_{\mathrm{Rn}}}}{2D_{\mathrm{Rn}}}},$$

(56)

and the flux per unit bulk surface area is therefore

$$J_i^{\mathrm{bulk}}\left(0\right)={\theta }_g\left[u{C}_i\left(0\right)-D_i{\left.{\displaystyle \frac{\mathrm{d}{C}_i}{\mathrm{d}z}}\right|}_{z=0}\right].$$

(57)

These relations explain the diagnostic value of ${\mathrm{CO}}_2$–radon pairing [19,24]. ${\mathrm{CO}}_2$ may be transported advectively over considerable depth ranges with limited attenuation, whereas radon experiences a strong depth penalty governed by advection, diffusion, and radioactive decay. Surface radon can therefore constrain effective source depth only when the transport model is explicitly included.

The inverse problem follows by linearization about a reference model ${\mathbf{m}}_0$,

$$\delta \mathbf{d}=\mathbf{G}\delta \mathbf{m},\mathbf{G}={\left.{\displaystyle \frac{\partial \mathcal{F}}{\partial \mathbf{m}}}\right|}_{{\mathbf{m}}_0},$$

(58)

and a regularized objective function can be written as

$$\Phi \left(\mathbf{m}\right)={∥{\mathbf{W}}_d\left({\mathbf{d}}_{\mathrm{obs}}-\mathcal{F}\left[\mathbf{m}\right]\right)∥}^2+{\beta }_{\mathrm{reg}}{∥{\mathbf{W}}_m\left(\mathbf{m}-{\mathbf{m}}_{\mathrm{prior}}\right)∥}^2.$$

(59)

Here, $\mathbf{G}$ is the sensitivity matrix, ${\mathbf{W}}_d$ and ${\mathbf{W}}_m$ weight the data and prior information, and ${\beta }_{\mathrm{reg}}$ controls regularization strength [22,23,24]. Non-uniqueness arises because source depth, pathway permeability, upward velocity, source strength, and shallow mixing parameters can all trade off against one another in the production of similar surface fluxes or concentrations. Flux alone does not determine source depth, composition alone does not determine pathway geometry, and even isotopes alone may not uniquely discriminate deep and shallow components if transport fractionation and mixing are left unconstrained. Recent reduced-inference work on radon–thoron observations has formalized a similar source-to-observable cascade, distinguishing anomaly reporting, mechanism discrimination, and local inversion according to the information content of the observation package [41].

The practical treatment of this non-uniqueness requires probabilistic or sequential data-assimilation approaches rather than deterministic fitting alone. Bayesian Markov Chain Monte Carlo methods are well suited to sampling the posterior distribution of source, storage, and transmissivity parameters when the forward model is moderately expensive but explicit [42]. Ensemble Kalman filtering and related ensemble smoothers are natural alternatives when the objective is to assimilate transient observations, such as repeated flux surveys, deformation, pressure proxies, or seismicity, into an evolving state vector [43]. In either case, the role of the reduced framework is to constrain the parameterization: it identifies which quantities should be treated as state variables, which quantities should be treated as transfer variables, and which trade-offs are structurally unavoidable.

The inverse problem is therefore a central test of the physical formulation. A deep-source theory that does not predict which parameters are identifiable from which observables remains incomplete. From an observational perspective, it is practical to distinguish between state variables and transfer variables. Pressure, saturation, and permeability are internal state variables that are rarely observed directly. Flux, composition, temperature, radon, and transient response are transfer variables that carry indirect information about those internal states. A model must not only reproduce these observables, but also identify which combinations of them constrain source strength, storage, and transmissivity independently.

9. Illustrative Application to the Campi Flegrei ${\mathbf{CO}}_{\mathbf{2}}$ Degassing Record

The preceding sections develop a reduced analytical framework rather than a site-specific numerical simulator. To demonstrate its use with observations, the framework is applied to a published degassing record from Campi Flegrei (Campania, Italy). The aim is not to perform a complete inversion of the caldera, nor to claim uniqueness of the inferred parameters. The objective is limited and testable: to show how a filtered surface observable can be decomposed into slow source–storage loading and faster transient discharge within the same analytical language used above.

The application uses the carbon dioxide (${\mathrm{CO}}_2$) diffuse total output (DTO) estimates reported by Carlo Cardellini and colleagues [44] for the Solfatara–Pisciarelli degassing structure. That study provides thirty diffuse ${\mathrm{CO}}_2$ surveys between 1998 and 2016, based on 13,158 accumulation-chamber measurements and geostatistical estimation of the total diffuse output. The published DTO spans approximately 745–2815 t ${\mathrm{d}}^{-1}$, with survey-dependent uncertainty. The DTO is not a direct measurement of source pressure; in the terminology of Section 8, it is a filtered observable that depends on source strength, storage, leakance, shallow mixing, and the active area of degassing. It is nevertheless an appropriate quantity for illustrating the forward operator because it integrates the surface expression of the Solfatara–Pisciarelli pathway.

For the application interval 2010–2016, when the record is sufficiently dense for a time-domain illustration, the observed DTO is represented as

$$F_{\mathrm{mod}}\left(t\right)=F_0+a(t-t_0)+A_{v}exp\left[-{\displaystyle \frac{t-t_v}{{\tau }_v}}\right]\mathcal{H}(t-t_v),$$

(60)

where $F_0+a(t-t_0)$ is the leading-order response of a leaky storage system to slowly increasing effective source input or leakance-weighted supply, and the exponential term is the open-state relaxation associated with a short transient discharge episode. Equation (60) is the observable-space counterpart of the source balance in Equation (28): the first term captures multi-year loading, while $A_v$ and ${\tau }_v$ (where the subscript v denotes the transient valve-like discharge component) represent the amplitude and relaxation time of a rapid transmissivity or discharge perturbation. The parameter $t_0$ is fixed at 2010 and $t_v$ is set to the date of the maximum observed DTO in the application interval. The parameters $F_0$, a, $A_v$, and ${\tau }_v$ are then estimated by weighted least squares using the published DTO uncertainties.

The fit gives $F_0=936\pm 74$ t ${\mathrm{d}}^{-1}$, a slow-loading coefficient $a=97\pm 21$ t ${\mathrm{d}}^{-1}{\mathrm{yr}}^{-1}$, a transient amplitude $A_v=1393\pm 322$ t ${\mathrm{d}}^{-1}$, and a relaxation time ${\tau }_v=0.13\pm 0.06$ yr, with the goodness-of-fit $R^2=0.86$ and reduced statistic ${\chi }_{\nu }^2=1.66$ (with $\nu$ degrees of freedom) for the 2010–2016 subset (Figure 3). The physical significance of the illustrative calculation lies in the short value of ${\tau }_v$ relative to the multi-year growth of the background DTO. It indicates that the 2015 high-output episode can be represented as a rapid discharge or transmissivity perturbation superposed on slower storage loading, rather than as a uniformly increasing trend in deep volatile production. In the language of the reduced model, this is a pressure- and leakance-sensitive response, not a unique measurement of new gas mass arriving from depth.

This interpretation is consistent with independent Campi Flegrei studies showing that hydrothermal pressure, ${\mathrm{CO}}_2$ emission, and seismicity share closely related temporal evolutions during the 2010–2020 unrest [45]. It is also compatible with recent geochemical reconstruction of the Solfatara magmatic–hydrothermal system, in which shallow, intermediate, and deep reservoirs are connected through a deep fault–fracture zone and the intermediate to deep reservoirs undergo marked pressure–temperature evolution through time [46]. The purpose of this calculation is not to present Equation (60) as a complete Campi Flegrei model. Its diagnostic role is to show that the same data can be described in terms of a slow loading branch and a short transient discharge branch, making explicit which part of the signal is attributed to storage and which part to pathway transmissivity.

10. Discussion

Several implications follow from the framework assembled above. First, deep degassing is more consistently described through the triplet source–storage–transmissivity than through the notion of a deep emitter alone. The source term by itself does not determine the surface signal. Storage delays and smooths transmission; permeability focuses, redistributes, or intermittently suppresses it; and the shallow domain modifies the signal once more before it is observed. This layered structure explains why surface gas anomalies may precede, lag, or even decouple from geodetic or seismic indicators, depending on which part of the coupled system is perturbed first. The reduced analytical model makes this statement quantitative: ${\Pi }_{\mathrm{SV}}>1$ is the threshold for valve-like opening under constant forcing, whereas ${\Pi }_{\mathrm{PA}}\ll 1$ is the criterion for pressure transmission to outrun mass transport. The Campi Flegrei application illustrates the same point with a real degassing record: the DTO time series can be represented as slow storage loading plus a short transient discharge episode, rather than as a single undifferentiated increase in source flux.

Second, volcanic and tectonic systems must not be treated as physically disjoint categories. Volcanic terrains emphasize exsolution, multiphase thermodynamics, and hydrothermal mediation, whereas tectonic terrains emphasize structural focusing, compartmentalization, and reaction-driven fluid production. Yet the reduced equations, constitutive couplings, and dimensionless groups required to describe the two are fundamentally the same. This commonality supports the porous–fractured-media framing: it does not erase geological differences, but expresses them through source terms, storage laws, and transmissivity evolution rather than through unrelated vocabularies.

Third, permeability is the least constrained constitutive quantity in the problem. Source terms can often be parameterized to first order from thermodynamics or reaction stoichiometry, whereas permeability depends on stress path, fracture geometry, sealing, capillarity, hysteresis, and connectivity in ways that remain only partially constrained [26,27,28,39,40]. This explains why different models with similar source terms may still predict substantially different shallow responses.

A final implication concerns model design. Excessively simplified models collapse source, storage, and permeability into a single effective flux and therefore risk interpretive overreach; excessively complex models become weakly identifiable and difficult to falsify. Suitable reduced models should preserve the source–storage–pathway distinction while retaining diagnostic links to observable quantities.

11. Conclusions and Future Directions

Deep gas sources in volcanic and tectonic systems can be represented as coupled source–storage–pathway systems in deformable porous–fractured media. Gas mass propagates by advection, diffusion, buoyancy, and phase segregation; pressure propagates by hydraulic diffusion and poro-thermo-mechanical coupling; and deep pressure gradients arise whenever volatile supply exceeds pathway leakance. Once this distinction is recognized, critical magmatic degassing, hydrothermal pressurization, metamorphic ${\mathrm{CO}}_2$ release, fault-valve behavior, and transient surface emissions above reinjection or tectonic perturbations can all be treated within a common theoretical class.

The framework developed here contributes five elements relevant to future work. First, it provides a regime classification based on dimensionless groups that separate pressure-led, mass-led, valve-dominated, and shallow-filter-dominated behavior. Second, it provides a minimal analytical source–storage–pathway model yielding explicit criteria for valve onset (${\Pi }_{\mathrm{SV}}>1$), charging and discharge times, and pressure-led response (${\Pi }_{\mathrm{PA}}\ll 1$). Third, this framework provides a compact classification of source archetypes into distributed, leaky, threshold, and multi-reservoir systems. Fourth, it provides an inversion framework that clarifies why source depth, permeability architecture, and shallow mixing remain non-unique unless multiple observables are combined. Fifth, it provides a real-data illustration at Campi Flegrei, showing how the reduced model decomposes a published ${\mathrm{CO}}_2$ output record into slow loading and rapid transient discharge components.

Future progress will require both additional site characterization and refinement of the constitutive laws. In particular, future work should incorporate better real-fluid closure for ${\mathrm{H}}_2$O–CO₂ systems, more realistic history-dependent permeability laws, and joint assimilation strategies that combine gas geochemistry, thermal observations, structural data, transient response, and uncertainty-aware inversion tools such as Bayesian sampling and ensemble filtering. A primary objective is to derive parsimonious, testable laws through which deep volatile production, storage, and transmissivity can be inferred from filtered surface observables.

The detailed algebraic derivation of the reduced pressure equation, the nondimensional scaling, and the analytical solutions of the minimal source–storage–pathway model are provided in Appendix A, Appendix B and Appendix C.