Coupled Water and Salt Transport in Porous Materials: Rapid Determination of a Varying Diffusion Coefficient from Experimental Data

A direct method for an accurate and rapid evaluation of a varying salt diffusion coefficient, \(D\) , from experimental data is proposed for a coupled water and salt transport in porous materials. The evaluation uses data on the moisture and salt concentration profiles and is based on a formula obtained from the Boltzmann-Matano method. The coupled transport is described by the diffusion-advection model of Bear and Bachmat. A simple expression for \(D\) in the center of the concentration interval is deduced from the formula to provide a rapid estimate on \(D\) . Possible extensions of this analytical approach are pointed out, suggesting that it can serve as a convenient general tool in engineering calculations. The theoretical results are applied to a laboratory experiment in which a coupled moisture and chloride transport had been investigated in a lime plaster, and the chloride diffusion coefficient had been obtained numerically in dependence on the chloride concentration. The agreement with the numerical results is shown to be rather good, except at low concentrations where our analytical results should be more reliable. It is also shown that the unusually high value of the calculated chloride diffusion coefficient—about three orders of magnitude higher than for free chloride ions in water—cannot be explained by possible inaccuracies in the measurements and/or numerical calculations. The reason is that changes in the measured profiles’ data could cause a change in \(D\) of just the same order of magnitude. This shows that, besides diffusion and advection, additional mechanisms take part in the considered chloride transport.     

Double-Diffusive Natural Convection in Anisotropic Porous Medium Bounded by Finite Thickness Walls: Validity of Local Thermal Equilibrium Assumption

Double-diffusive natural convection in fluid-saturated porous medium inside a vertical enclosure bounded by finite thickness walls with opposing temperature, concentration gradients on vertical walls as well as adiabatic and impermeable horizontal ones has been performed numerically. The Darcy model was used to predict fluid flow inside the porous material, while thermal fields are simulated based on two-energy equations for fluid and solid phases on the basis of a local thermal non-equilibrium model. Computations have been performed for different controlling parameters such as the buoyancy ratio $N$ , the Lewis number Le, the anisotropic permeability ratio $R_\mathrm{p}$ , the fluid-to-solid thermal conductivity ratio $R_\mathrm{c}$ , the interphase heat transfer coefficient $\mathcal{H}$ , the ratio of the wall thickness to its height $D$ , the wall-to-porous medium thermal diffusivity ratio $R_\mathrm{w}$ , and the solid-to-fluid heat capacity ratio $\gamma $ . Thus, the effects of the controlling parameters on heat and mass transfer characteristics are discussed in detail. Moreover, the validity domain of the local thermal equilibrium (LTE) assumption has been delimited for different set of the governing parameters. It has been shown that Le has a noticeable significant effect on fluid temperature profiles and that higher $N$ values lead to a significant enhancement in heat and mass transfer rates. Moreover, for higher $\mathcal{H}, R_\mathrm{c}$ , $R_\mathrm{p}, R_\mathrm{w}$ , or $D$ values and/or lower $\gamma $ values, the solid and fluid phases tend toward LTE.     

Advection–Dispersive Mechanisms of Electrolyte Species in Porous Matrix

Advection–dispersive phenomena inside two geometries of porous media, corresponding to a structured and unstructured network of minichannels, are studied from a combination of analytical, numerical, and electrodiffusion techniques. The instantaneous limiting diffusion current, connected to the concentration of electroactive species flowing in the porous matrix, is recorded and measured during flow experimentation. Transport phenomena investigation consists of the step injection of a tracer of higher concentration than the bulk flow in order to characterize the mixing of electrolyte species inside two arrangements of network, so called \(\times \) _network and T_network. The experimental results are supported by 2D-numerical simulations performed in the \(\times \) _network. A pore model is proposed in order to predict the pore velocity, which is used within the resolution of the diffusion–convection balance. The numerical simulations, based on a second-order finite difference scheme, give rise to a good agreement in terms of mixing index and the methodology employed for the numerical injection concentration appears suitable. The numerical experiments are quite-well representative in laminar regime below the critical Reynolds number ( \(Re_{\mathrm{crit}}\approx \,200\) ). Above \(Re_{\mathrm{crit}}\) , the inertial effects are not negligible and the momentum transfer needs to be taken into account. This latter phenomenon is analyzed at the pore-scale in term of local skin friction measured at the channels crossing, then compared and discussed from analytical solutions.     

Elucidating the Role of Interfacial Tension for Hydrological Properties of Two-Phase Flow in Natural Sandstone by an Improved Lattice Boltzmann Method

We investigated the interfacial tension (IFT) effect on fluid flow characteristics inside micro-scale, porous media by a highly efficient multi-phase lattice Boltzmann method using a graphics processing unit. IFT is one of the most important parameters for carbon capture and storage and enhanced oil recovery. Rock pores of Berea sandstone were reconstructed from micro-CT scanned images, and multi-phase flows were simulated for the digital rock model at extremely high resolution (3.2  \(\upmu \) m). Under different IFT conditions, numerical analyses were carried out first to investigate the variation in relative permeability, and then to clarify evolution of the saturation distribution of injected fluid. We confirmed that the relative permeability decreases with increasing IFT due to growing capillary trapping intensity. It was also observed that with certain pressure gradient \(\Delta P\) two crucial IFT values, \(\sigma _{1}\) and \(\sigma _{2}\) , exist, creating three zones in which the displacement process has totally different characteristics. When \(\sigma _{1}< \sigma < \sigma _{2}\) , the capillary fingering patterns are observed, while for \(\sigma < \sigma _{1}\) viscous fingering is dominant and most of the passable pore spaces were invaded. When \(\sigma > \sigma _{2}\) the invading fluid failed to break through. The pore-throat-size distribution estimated from these crucial IFT values ( \(\sigma _{1 }\) and \(\sigma _{2})\) agrees with that derived from mercury porosimetry measurements of Berea sandstone. This study demonstrates that the proposed numerical method is an efficient tool for investigating hydrological properties from pore structures.     

Unsteady MHD double-diffusive convection boundary-layer flow past a radiate hot vertical surface in porous media in the presence of chemical reaction and heat sink

This paper presents an analytical study of the unsteady MHD free convective heat and mass transfer flow of a viscous, incompressible, gray, absorbing-emitting but non-scattering, optically-thick and electrically conducting fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot vertical plate with constant velocity. We employ a Darcian viscous flow model for the porous medium the Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The homogeneous chemical reaction of first order is accounted in mass diffusion equation. The governing equations are solved in closed form by Laplace-transform technique. A parametric study of all involved parameters is conducted and representative set of numerical results for the velocity, temperature, concentration, shear stress function $\frac{\partial u}{\partial y} \vert_{y=0}$ , temperature gradient $\frac{\partial \theta }{ \partial y}\vert_{y=0}$ , and concentration gradient $\frac{ \partial \phi }{\partial y}\vert_{y=0}$ is illustrated graphically and physical aspects of the problem are discussed.     

Micro-structural Impact of Different Strut Shapes and Porosity on Hydraulic Properties of Kelvin-Like Metal Foams

Various ideal periodic isotropic structures of foams (tetrakaidecahedron) with constant ligament cross section are studied. Different strut shapes namely circular, square, diamond, hexagon, star, and their various orientations are modeled using CAD. We performed direct numerical simulations at pore scale, solving Navier–Stokes equation in the fluid space to obtain various flow properties namely permeability and inertia coefficient for all shapes in the porosity range, \(0.60<\varepsilon <0.95\) for wide range of Reynolds numbers, \(10^{-6} . We proposed an analytical model to obtain pressure drop in metallic foams in order to correlate the resulting macroscopic pressure and velocity gradients with the Ergun-like approach. The analytical results are fully compared with the available numerical data, and an excellent agreement is observed.     

Miscible and Immiscible Foam Injection for Mobility Control and EOR in Fractured Oil-Wet Carbonate Rocks

Foam injection is a proven enhanced oil recovery (EOR) technique for heterogeneous reservoirs, but is less studied for EOR in fractured systems. We experimentally investigated tertiary \(\text {CO}_{2}\) injections, and \(\text {N}_{2}\) - and \(\text {CO}_{2}\) -foam injections for enhanced oil recovery in fractured, oil-wet limestone core plugs. Miscible \(\text {CO}_{2}\) and \(\text {CO}_{2}\) -foam was compared with immiscible \(\text {CO}_{2}\) - and \(\text {N}_{2}\) -foam as tertiary recovery techniques, subsequent to waterfloods, in fractured rocks with different wettability preferences. At water-wet conditions waterfloods produced approximately 40 % OOIP, by spontaneous imbibition. Waterflood oil recovery at oil-wet conditions was below 20 % OOIP, due to suppressed imbibition where water predominantly flowed through the fractures, unable to mobilize the oil trapped in the matrix. Tertiary, supercritical \(\text {CO}_{2}\) -mobilized oil trapped in the matrix, particularly at weakly oil-wet conditions, by diffusion. Recovery by diffusion was high due to small core samples, high initial oil saturation and a continuous oil phase at oil-wet conditions. Both immiscible \(\text {CO}_{2}\) - and \(\text {N}_{2}\) -foams and miscible, supercritical \(\text {CO}_{2}\) -foam demonstrated high ultimate oil recoveries, but immiscible foam was less efficient (30 pore volumes injected) compared to miscible foam (2 pore volumes injected) to reach ultimate recovery. This is explained by the capillary threshold pressure preventing the injected \(\text {N}_{2}\) gas from entering the matrix, verified by computed X-ray tomography, and the mobilized oil was displaced by the aqueous surfactant in the foam. At miscible conditions, there exists no capillary entry pressure between the oil-saturated matrix and the injected \(\text {CO}_{2}\) , allowing foam to invade the matrix for efficient oil recovery.     

Modeling the Chlorides Transport in Cementitious Materials By Periodic Homogenization

In this work, we develop a macroscopic model for diffusion–migration of ionic species in saturated porous media, based on periodic homogenization. The prior application is chloride transport in cementitious materials. The dimensional analysis of Nernst–Planck equation lets appear dimensionless numbers characterizing the ionic transfer in porous media. Using experimental data, these dimensionless numbers are linked to the perturbation parameter ${\varepsilon}$ . For a weak-imposed electrical field, or in natural diffusion, the asymptotic expansion of Nernst–Planck equation leads to a macroscopic model coupling diffusion and migration at the same order. The expression of the homogenized diffusion coefficient only involves the geometrical properties of the material microstructure. Then, parametric simulations are performed to compute the chloride diffusion coefficient through different complexity of the elementary cell to go on as close as possible to experimental diffusion coefficient of the two cement pastes tested.     

A Fast Algorithm for Invasion Percolation

We present a computationally fast Invasion Percolation (IP) algorithm. IP is a numerical approach for generating realistic fluid distributions for quasi-static (i.e., slow) immiscible fluid invasion in porous media. The algorithm proposed here uses a binary-tree data structure to identify the site (pore) connected to the invasion cluster that is the next to be invaded. Gravity is included. Trapping is not explicitly treated in the numerical examples but can be added, for example, using a Hoshen–Kopelman algorithm. Computation time to percolation for a 3D system having $N$ total sites and $M$ invaded sites at percolation goes as $O(M \log M)$ for the proposed binary-tree algorithm and as $O(M N)$ for a standard implementation of IP that searches through all of the uninvaded sites at each step. The relation between $M$ and $N$ is $M = N^{D/E}$ , where $D$ is the fractal dimension of an infinite cluster and $E$ is Euclidean space dimension. In numerical practice, on finite-sized cubic lattices with invasion structures influenced by the injection boundary and boundary conditions lateral to the flow direction, we observe the scaling $M = N^{0.852}$ in 3D (valid through the second decimal place) instead of $M= N^{0.843}$ based on the infinite cluster fractal dimension $D=2.53$ .     

Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion

We prove various decay bounds on solutions (f _n : n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ^? f _n in terms of a suitable average of the moments of the initial data for every positive ?. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of ${L^p(\mathbb{R}^d \times [0, T])}$ norms of the moments ${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$ , ( ${\int_0^{\infty} m^a f_m(x, t)dm}$ in the case of continuous Smoluchowski’s equation) for every ${p \in [1, \infty]}$ . In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function ${\phi(n)}$ that is closely related to the total increase of the diffusion coefficient in the interval (0, n].     

Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

Drainage of Capillary-Trapped Oil by an Immiscible Gas: Impact of Transient and Steady-State Water Displacement on Three-Phase Oil Permeability

In a previous paper (Dehghanpour et al., Phys Rev E 83:065302, 2011a), we showed that relative permeability of mobilized oil, $k_\mathrm{ro}$ , measured during tertiary gravity drainage, is significantly higher than that of the same oil saturation in other tests where oil is initially a continuous phase. We also showed that tertiary $k_\mathrm{ro}$ strongly correlates to both water saturation, $S_\mathrm{w}$ , water flux (water relative permeability), $k_\mathrm{rw}$ , and the change in water saturation with time, $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ . To develop a model and understanding of the enhanced oil transport, identifying which of these parameters ( $S_\mathrm{w},\,k_{\mathrm{rw}}$ , or $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ ) plays the controlling role is necessary, but in the previous experiments these could not be deconvolved. To answer the remaining question, we conduct specific three-phase displacement experiments in which $k_{\mathrm{rw}}$ is controlled by applying a fixed water influx, and $S_\mathrm{w}$ develops naturally. We obtain $k_{\mathrm{ro}}$ by using the saturation data measured in time and space. The results suggest that steady-state water influx, in contrast to transient water displacement, does not enhance $k_{\mathrm{ro}}$ . Instead, reducing water influx rate results in excess oil flow. Furthermore, according to our pore scale hydraulic conductivity calculations, viscous coupling and fluid positioning do not sufficiently explain the observed correlation between $k_{\mathrm{ro}}$ and $S_{\mathrm{w}}$ . We conclude that tertiary $k_{\mathrm{ro}}$ is controlled by the oil mobilization rate, which in turn is linked to the rate of water saturation decrease with time, $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ . Finally, we develop a simple model which relates tertiary $k_{\mathrm{ro}}$ to transient two-phase gas/water relative permeability.     

Concentration fluctuations in a stirred reactor

Turbulent mixing of tap-water with salt-water in a 6.3 dm³ Continuous Stirred Tank Reactor is analyzed from concentration measurements obtained by a conductivity micro-probe. The frequency response of the instrument has been tested previously by specific dynamic trials. As well as the field of mean concentration C and the degree of segregation \(\sqrt {c\prime ^2 } /\bar C\) the Eulerian length macroscales of concentration in the tank and characteristics of the turbulent diffusion in the discharge flow are deduced from spectral analysis of the fluctuations and from the turbulent velocity field measured by laser Doppler velocimetry.     

Experimental Characterization of Porosity Structure and Transport Property Changes in Limestone Undergoing Different Dissolution Regimes

Limestone dissolution by $\hbox {CO}_2$ -rich brine induces critical changes of the pore network geometrical parameters such as the pore size distribution, the connectivity, and the tortuosity which govern the macroscopic transport properties (permeability and dispersivity) that are required to parameterize the models, simulating the injection and the fate of $\hbox {CO}_2$ . A set of four reactive core-flood experiments reproducing underground conditions ( $T = 100\,^{\circ }\hbox {C}$ and $P = 12$ MPa) has been conducted for different $\hbox {CO}_2$ partial pressures $(0.034 < P_{\mathrm{CO}_2}< 3.4\; \hbox {MPa})$ in order to study the different dissolution regimes. X-ray microtomographic images have been used to characterize the changes in the structural properties from pore scale to Darcy scale, while time-resolved pressure loss and chemical fluxes enabled the determination of the sample-scale change in porosity and permeability. The results show the growth of localized dissolution features associated with high permeability increase for the highest $P_{\mathrm{CO}_2}$ , whereas dissolution tends to be more homogeneously distributed for lower values of $P_{\mathrm{CO}_2}$ . For the latter, the higher the $P_{\mathrm{CO}_2}$ , the more the dissolution patterns display ramified structures and permeability increase. For the lowest value of $P_{\mathrm{CO}_2}$ , the preferential dissolution of the calcite cement associated with the low dissolution kinetics triggers the transport that may locally accumulate and form a microporous material that alters permeability and produces an anti-correlated porosity–permeability relationship. The combined analysis of the pore network geometry and the macroscopic measurements shows that $P_{\mathrm{CO}_2}$ regulates the tortuosity change during dissolution. Conversely, the increase of the exponent value of the observed power law permeability–porosity trend while $P_{\mathrm{CO}_2}$ increases, which appears to be strongly linked to the increase of the effective hydraulic diameter, depends on the initial rock structure.     

Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number $10^{-5}\le \hbox {Da}\le 10^{-3}$ , porous layer height ratio $0\le d/L\le 1$ , thermal conductivity ratio $1\le k_{1,3}\le 20$ , and dimensionless time $0\le \tau \le 1000$ on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.     

Spatiotemporal complexity of a three-species ratio-dependent food chain model

In this paper, we investigate the complex dynamics of a ratio-dependent spatially extended food chain model. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for global stability. On the basis of bifurcation analysis, we present the evolutionary process of pattern formation near the coexistence equilibrium point $(N^*,P^*,Z^*)$ via numerical simulation. And the sequence cold spots $\rightarrow $ stripe–spots mixtures $\rightarrow $ stripes $\rightarrow $ hot stripe–spots mixtures $\rightarrow $ hot spots $\rightarrow $ chaotic wave patterns controlled by parameters $a_1$ or $c_1$ in the model are presented. These results indicate that the reaction–diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics.     

Correlation of Water Vapor Permeability with Microstructure Characteristics of Building Materials Using Robust Chemometrics

This paper studies various microstructure parameters of natural and artificial building materials and aims to their correlation to the water vapor permeability. Three categories of building materials were investigated: stones, bricks, and plasters. Mercury intrusion porosimetry was applied in order to obtain the materials microstructure characteristics, a variety of pore size distributions and pore structure measurements, such as total porosity. The water vapor permeability of materials was determined experimentally according to ASTM standard E96-00. A robust principal component regression approach, coupled with multiple outlier detection, was applied in order to correlate water vapor permeability values to pore size distributions. A good quality correlation model was found by utilizing relative specific pore volume and relative specific pore surface distributions, whereas using pore structure measurements, such as total porosity, the correlation results were very poor. From the results, specific ranges of pore size distribution, corresponding to pores radius sizes greater than $10\,\upmu \text{ m }$ 10 μ m and between 1.778 and $0.421\,\upmu \text{ m }$ 0.421 μ m , contribute to the water vapor permeability of the materials.     

Predicting Vertical Flow Barriers Using Tracer Diffusion in Partially Saturated,Layered Porous Media

Sudden changes in isotopic tracer concentration in pore waters have been interpreted as indicating barriers to vertical advective flow through porous rocks in the subsurface, e.g. step changes in \(^{87}\hbox {Sr}/^{86}\) Sr ratio are often used in the oil and gas industry as a signature of reservoir compartmentalisation. This study shows that this is not necessarily the case. It can take millions of years for such step changes to equilibrate by diffusion if there is no flow resulting from pressure or density gradients even in high permeability, high porosity rocks, particularly if the water saturation is low. Changes in tracer concentration gradients can be good indicators of changes in porosity (or water saturation) between layers. In contrast changes in sorption without a change in porosity are almost impossible to identify. The time taken for concentration gradients to equilibrate is affected by the layer properties but can be quickly estimated from the harmonic average of the effective diffusion coefficient for each layer and a simple analytical expression for a homogeneous system. This was achieved by performing a sensitivity analysis on different layer properties (porosity contrast, saturation contrast, sorption contrast, thickness ratio) using existing analytical solutions for diffusion in layered systems.     

Characteristics of Wave Propagation in The Saturated Thermoelastic Porous Medium

Took into consideration the coupling effect of thermo, hydraulics and mechanics, a set of thermo–hydro-mechanical coupled wave equations for fluid–saturated soil are developed. In these wave equations, the $P_{3}$ -wave in solid phase and $P_{4}$ -wave in fluid phase are coupled into $T$ -wave in fluid–saturated soil by the assumption that the temperature of the solid phase is equal to the temperature of liquid phase at the same position. The dispersion equations for the thermo-elastic wave, which can be degraded to the equations for elastic wave in fluid–saturated soil, are derived from the above equations by introducing four potential functions. Then, these equations are solved numerically. The characteristics of wave phase velocity, attenuation and the effect of thermal expansion, initial temperature and porosity, etc., on phase velocities of $P_{1}$ -, $P_{2}$ -, and $T$ -wave are discussed. As a reference, the characteristics of the propagation of elastic waves in fluid–saturated soil are also studied. The computation results show that (1) the phase velocity of $P_{1}$ -wave obtained by the theory of thermoporoelascity (THM) is faster than that by the theory of poroelasticity (HM); (2) the attenuation of $P_{1}$ -wave obtained by either the theory of THM or HM are consistent; (3) the dissemination characteristics of $P_{2}$ -wave are almost consistent; (4) the phase velocity of $T$ -wave is the slowest among the three compressional waves; and (5) The attenuation versus frequency characteristic of $T$ -wave is similar to that of $P_{2}$ -wave.