Univerzita Komenského v Bratislave

Fakulta matematiky, fyziky a informatiky

MSc. Dávid Gregor

Autoreferát dizertačnej práce

The Finite-difference Modelling

of Seismic Wave Propagation

in the Poroelastic Medium – the 2D P-SV Case

na získanie akademického titulu philosophiae doctor

v odbore doktorandského štúdia:

4.1.9 Geofyzika

Bratislava 2020

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Dizertačná práca bola vypracovaná v dennej forme doktorandského štúdia na Katedre

astronómie, fyziky Zeme a meteorológie Fakulty matematiky, fyziky a informatiky, Univerzity

Komenského v Bratislave

Predkladateľ: MSc. Dávid Gregor

Katedra astronómie, fyziky Zeme a meteorológie

Fakulta matematiky, fyziky a informatiky

Univerzita Komenského

Mlynská dolina F1

842 48 Bratislava

Školiteľ: Prof. RNDr. Peter Moczo, DrSc.

Katedra astronómie, fyziky Zeme a meteorológie

Fakulta matematiky, fyziky a informatiky

Univerzita Komenského

Mlynská dolina F1

842 48 Bratislava

Študijný odbor: 4.1.9 Geofyzika

Predseda odborovej komisie: Prof. RNDr. Peter Moczo, DrSc.

Katedra astronómie, fyziky Zeme a meteorológie

Fakulta matematiky, fyziky a informatiky

Univerzita Komenského

Mlynská dolina F1

842 48 Bratislava

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Abstract

The finite-difference (FD) method is the dominant method in numerical modelling of seismic

wave propagation in the poroelastic medium. This is due to the distinct features of the FD

method: the possibility of applying one explicit FD scheme to all interior grid points (points not

lying on a grid border), relative simplicity, and good balance between accuracy and

computational efficiency. At the same time, not all developed and used FD schemes sufficiently

represent the above commonly accepted features. The main missing capability of the developed

schemes is the achievable level of sufficiently accurate discrete representation of material

heterogeneity – both smooth weak and strong including material interface – together with

sufficient computational efficiency.

This Thesis addresses the above crucial aspect of the FD modelling. We present constitutive

relations and equations of motion for the averaged medium reasonably representing smooth

weak and strong material heterogeneity as well as zero, no-zero constant and frequency-

dependent resistive friction. The representation of material interfaces reasonably approximates

boundary conditions at the interface and makes thus a justification for the heterogeneous finite-

difference scheme.

We developed the 2nd-order accurate in time and 4th-order accurate in space, velocity-stress-

pressure staggered-grid finite-difference scheme for numerical modelling of 2D P-SV seismic

wave propagation in heterogeneous poroelastic medium. We demonstrated its accuracy and

efficiency using numerical tests compared with the analytical, semi-analytical and independent

verified numerical methods.

Key words: finite-difference method, seismic waves, poroelastic medium, numerical modelling

Abstrakt

Metóda konečných diferencií (MKD) je dominantnou metódou v numerickom modelovaní

šírenia seizmických vĺn v poroelastickom prostredí. Je tomu tak v dôsledku kľúčových

vlastností MKD: možnosti použiť jednu explicitnú KD schému na všetky vnútorné sieťové

body (body neležiace na hraniciach siete), relatívnej jednoduchosti a dobrej rovnováhy medzi

presnosťou a výpočtovou efektívnosťou. Zároveň však možno konštatovať, že nie všetky

vyvinuté KD schémy dostatočne dobre reprezentujú tieto všeobecne akceptované výhody

MKD. Hlavnými chýbajúcimi vlastnosťami vyvinutých schém sú dosiahnuteľná úroveň

dostatočnej presnosti diskrétnej reprezentácie materiálovej heterogenity – slabej i silnej, ktorá

zahŕňa aj materiálové rozhrania – spolu s dostatočnou výpočtovou efektívnosťou.

Táto dizertačná práca zahŕňa oba tieto kľúčové aspekty KD modelovania. V tejto práci

prezentujeme konštitutívne vzťahy a pohybové rovnice pre spriemerované prostredie, ktoré sú

dostatočne presne reprezentujú slabú alebo silnú materiálovú heterogenitu a tiež nulové,

nenulové konštantné a frekvenčne závislé trenie. Reprezentácia materiálových rozhraní

rozumne aproximuje okrajové podmienky na rozhraní a tvorí tak oprávnený základ pre

heterogénnu KD schému.

Vyvinuli sme 2D P-SV konečno diferenčnú schému na striedavo usporiadanej sieti pre

formuláciu v rýchlosti-napätí a tlaku. Schéma má presnosť 2. rádu v čase a 4. rádu v priestore.

Pomocou numerických testov a porovnania s presnými, semi-analytickými a nezávislými

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numerickými metódami sme preukázali, že schéma umožňuje efektívne a dostatočne presné

modelovanie šírenia seizmických vĺn v heterogénnom poroelastickom prostredí.

Kľúčové slová: metóda konečných diferencií, seizmické vlny, poroelastické prostredie,

numerické modelovanie

Preface The prediction of seismic ground motion during future earthquakes at a site of interest is the

most important task of the earthquake seismology in relation to society. With the exception of

only few populated areas on the Earth, there is a drastic lack of earthquake recordings that could

be used for empirical prediction of the earthquake ground motion. This implies importance and

irreplaceability of theoretical and computational seismology with respect to prediction of the

seismic ground motion during future earthquakes.

The team of computational seismology at Comenius University in Bratislava has long

tradition in developing efficient and accurate numerical methods. Their numerical modelling of

seismic wave propagation and earthquake ground motion based on the finite-difference (FD)

method in realistic complex models of local surface sedimentary structures is internationally

recognized. The team has been for years focused on complex heterogeneous viscoelastic

medium (see, e.g., Moczo et al. 2014 and Kristek et al. 2019). Recently, the team has extended

its modelling to heterogenous poroelastic media

In the first steps towards numerical modelling in the heterogenous poroelastic media we

had to adopt the Biot’s theory of poroelasticity (in the Bachelor Thesis by Gregor 2014),

develop the first computer code and perform numerical tests for models of homogenous

poroelastic medium with zero resistive friction (in the MSc Thesis by Gregor 2016). Then the

next step was to develop method for treating material interfaces inside the smoothly

heterogenous poroelastic medium. Moczo, Gregor, Kristek and De la Puente (2019) developed

the discrete representation of heterogeneous poroelastic medium with constant resistive friction

that makes a part of this Thesis.

The water saturated sediments in local surface sedimentary structures can exhibit a high

intrinsic (hydraulic) permeability. The high intrinsic permeability implies the necessity to

incorporate frequency dependence of permeability. Consequently, also the frequency

dependence of the resistive friction. In such poroelastic models, the Biot’s characteristic

frequency can get close to the frequency range of interest in the earthquake seismology and

engineering. The model with constant resistive friction is not applicable. So far, the best

physical model for the frequency-dependent permeability and resistive friction is the JKD

model developed by Johnson, Dashen & Koplik in 1987.

Therefore, we continued our effort and developed discrete representation of the

heterogeneous poroelastic medium and corresponding FD scheme, algorithm and computer

code for simulating wave propagation in models with the JKD model of the frequency-

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dependent permeability and resistive friction. The development of this representation is

presented in this Thesis together with the extensive numerical tests.

Poroelastic Medium For numerical simulation of seismic wave propagation and earthquake motion in the Earth, the

elastic or viscoelastic medium is usually used as an adequate material model of the medium

inside a target domain (volume) of the Earth. In the case of the simplest elastic continuum, the

seismic wave propagation is governed by the equation of motion and linear Hooke’s law.

However, rocks and soils in the Earth are generally composite materials, and hence

inhomogeneous on a microscopic scale. A one of the representatives of such composite

materials is a fluid-saturated porous medium – a poroelastic material formed by two

interpenetrated phases. A solid phase (solid, denoted by the subindex “s” in the following

sections) constitutes a matrix (“m”) (or a frame) of the porous medium, that is a solid with

pores. A fluid phase (“f”) fills in pores of the matrix. The solid matrix of a porous rock can be

constituted of a collection of particles of quartz, feldspar, mica and other minerals of various

sizes, ranging from to 0.1 to1 mm. The fluid phase is typically made of water, air, oil, methane

gas or mixture of these. Porous materials are illustrated in Fig. 1 reproduced from the book

Poroelasticity by Cheng (2016).

Two basic phenomena are typical for physical behaviour of poroelastic material:

• Solid to fluid coupling occurs when a change in applied stress produces a change in

fluid pressure or fluid mass.

• Fluid to solid coupling occurs when a change in fluid pressure or fluid mass produces a

change in the volume of the porous material.

The propagation of waves in a poroelastic medium saturated by a compressible viscous

fluid was first analyzed by Biot in several classical papers (Biot 1956a, Biot 1956b, Biot 1962a,

1962b) and later experimentally confirmed by Plona (1980). Biot's theory has been mainly

utilized for purposes of petroleum industry, in order to perform seismic surveys and determine

the physical properties of rocks inside the reservoirs. Biot assumed that the fluid may flow

relative to the solid matrix causing friction and predicted the existence of two dilatational waves

and one shear wave. The first longitudinal wave, usually denoted as fast P-wave, behaves

similarly to the well-known P-wave in an elastic medium, with relatively large velocity, low

attenuation and very little dispersion. The S-wave also behaves as the well-known S-wave in

an elastic medium and is affected by similar level of attenuation and dispersion as the fast P-

wave. The second longitudinal wave, the slow P-wave, whose existence distinguishes the

poroelastic wavefield from the elastic one, behaves at low frequencies as a diffusion type wave

due to its low phase velocity, very high attenuation and

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Figure 1. Poroelastic materials – reproduced from book Poroelasticity by Cheng (2016).

Rocks: a sand, b sandstone, c volcanic rock, d fractured rock Technical materials: e pervious concrete, f polyurethane foam, g metal foam, j nanoporous

alumina Human body: h bone with osteoporosis, i articular cartilage

dispersion, while at high frequencies becomes a truly propagating wave. At frequencies up to

approximately 410 Hz, in a low-permeability medium, the slow P-wave is significant only very

close to the source or near material heterogeneities. In the same frequency range, the slow P-

wave can propagate to larger distances in a high-permeability medium. The fast P-wave is

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associated with the in-phase fluid and solid compressional motion, while the slow P-wave is

associated with the out-of-phase motions.

Alternative poroelastic theories could have been considered, such as the one developed by

Sahay (2008), who also considers slow S-wave. In his theory, an attenuation mechanism due to

viscous loss within the fluid is taken into account, in addition to Biot's attenuation due to viscous

forces between the solid and fluid phases. The Sahay’s theory seems more advanced than the

but so far it has not received larger attention. In this Thesis we will restrict to Biot’s theory

which is still widely accepted for modelling seismic wave propagation in porous media.

Basically, the wave propagation in poroelastic material may be investigated using two

approaches. The first approach is based on a homogenization procedure (Auriault 1980) and

the second one is based on concepts of continuum mechanics (Biot 1956a). In this work we will

use the second approach called Biot’s theory of poroelasticity. This is because Biot’s theory is

widely used, especially in the field of numerical modelling, and it is more suitable for

investigating wave propagation in media with interface (Sharma 2007). The main assumptions

of the Biot’s theory are

(a) Displacements, strains and particle velocities are small. This allows to neglect the

distinction between the Eulerian and Lagrangian formulations. The constitutive equations,

dissipation forces, and kinetic momenta are linear.

(b) The wavelength is large in comparison with the radius of the pores. This is a requirement

for applying the theory of continuum mechanics and implies that scattering dissipation is

neglected.

(c) The liquid phase is continuous, such that pores are connected and the disconnected pores

are part of the matrix.

(d) Permeability and the material of the matrix are isotropic and the medium is fully saturated.

The theory can be extended to the anisotropic case.

(e) The absence of thermo-mechanical and chemical effects.

Numerical Methods for Simulating Seismic Wave

Propagation in the Poroelastic Medium

Analytical and semi-analytical methods are suitable for spatially homogeneous models or 1D-

heterogenepus models. It is in principle very difficult to address lateral heterogeneity for these

methods. Therefore, grid-based (or element-based) numerical methods are mostly applied. Each

of these methods can discretize 3D complex porous models. Not equally accurately and

computationally efficiently. Each method has its advantages and disadvantages. It is therefore

a matter of a reasonable decision of the numerical-modelling team to choose the best possible

methods for solving specific problems. As in the numerical modelling of seismic wave

propagation in one-component elastic and viscoelastic media, there are four the most important

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methods: pseudospectral method, finite-difference method (FDM), spectral-element method

(SEM) and discontinuous Galerkin method (DGM). The three latter methods dominate.

The articles by Őzdenvar & McMechan (1996) and Sidler et al. (2010) are examples of

application of the pseudospectral method which use global operators.

After the pioneering work by De la Puente et al. (2008) and De la Puente (2008), Ward et

al. (2017, 2020), Shukla et al. (2019, 2020), Zhan et al. (2018, 2020) applied the discontinuous

Galerkin method. In the very recent article, Ward et al. (2020) developed a DG methodology

for a coupled 3D poroelastic/elastic isotropic model incorporating Biot's LF and HF regimes.

They noted that the exact Riemann-problem-based numerical flux implementation naturally

resolves all material discontinuities. DGM is a very powerful and universal numerical-

modelling method applicable to a large variety of problems in seismology. The crucial critical

aspects are mesh generation and computational efficiency. The latter may pose a problem for a

class of wavefield-medium configurations. Based on the published articles the method seems

suitable for modelling seismic wave propagation in poroelastic media.

The finite-element method was used by, e.g., Bermudez et al. (2006) and Santos & Sheen

(2007). The classical finite-element method is 2nd-order accurate in both time and space being

equivalent in this aspect to the classical displacement-formulation FDM on the conventional

grid. Much more powerful and accurate is SEM that was successfully applied to poroelastic

media by Morency & Tromp (2008) who also elaborated excellent fundamental formulation of

the theory of seismic wave propagation in the poroelastic media. As in the case of DGM, the

mesh generation and computational efficiency can be a problem especially in case of

complicated geometry of material interfaces.

For further reading on computational poroelasticity and numerical modelling we refer to

review by Carcione et al. (2010).

The most used method so far is FDM. Given this and the goal of this Thesis, we will focus

on the FDM modelling in this overview. We will include the so-called FDTD – the Finite-

difference Time-domain approach. This is to be distinguished from FDFD – the Finite-

difference Frequency-domain approach. We will use the introduction in the article by Moczo,

Gregor, Kristek and De la Puente (2019) and also include more recent articles.

The FD method has been used for modelling wave propagation in the elastic medium since

late 60s – probably the best-known example being the article by Alterman & Karal (1968). The

method was applied for simulating wave propagation in the porous medium later. One of the

first studies was the article by Garg et al. (1974) who investigated propagation of compressional

waves in fluid‐saturated elastic porous media using the computer code POROUS (Riney et al.

1972, 1973).

More studies applying the FD method appeared in the 90s. Hassanzadeh (1991) used it to

solve the Biot’s poroacoustic equations in a homogeneous fluid-saturated porous medium. Zhu

& McMechan (1991) developed a 2nd-order explicit conventional FD scheme, following Kelly

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et al. (1976), for solving the Biot’s poroelastic equations in a homogeneous porous medium.

Dai et al. (1995) extended the FD method to Biot’s poroelastic wave equations in 2D

heterogeneous porous media. Considering a 1st-order hyperbolic system equivalent to Biot's

equations, they developed a MacCormack predictor-corrector (4, 2) scheme (4th-order accurate

in space and 2nd-order accurate in time), based on a spatial splitting technique. The scheme on

the collocated grid updates the vector consisting of the solid and fluid particle-velocity

components, the solid stress-tensor components, and the fluid pressure.

Schemes used by Hassanzadeh (1991) and Zhu & McMechan (1991) are based on the

homogeneous approach - wave equations for each homogeneous layer are solved separately and

the boundary conditions must be satisfied explicitly on the interfaces between different layers.

Dai et al. (1995) applied the heterogeneous approach – one scheme is used for all interior grid

points and the heterogeneity of medium is accounted for by values of material parameters

assigned to the grid positions. Dai et al. explicitly said about the heterogeneous approach what

was then considered correct: the boundary conditions are satisfied implicitly and more

complicated geometries can be accommodated with no extra effort. It has been later recognized

that this is not so (Moczo et al. 2002, 2014; Kristek et al. 2017).

Zhang (1999) developed a quadrangle-grid velocity-stress FD scheme for simulating wave

propagation in 2D heterogeneous porous media. In his non-orthogonal spatial grid, two solid

particle-velocity components and two components of velocity of the pore fluid relative to that

of the solid share one grid position whereas the three solid-matrix stress components and the

fluid pressure share another grid position. Apart from the non-orthogonal geometry and fluid-

related field quantities, the grid is the partly-staggered grid (first used by Andrews in 1973 and

later called a rotated staggered grid by Saenger and his collaborators).

Zeng et al. (2001) applied the perfectly matched layer in their FD numerical modelling of

wave propagation in poroelastic media. Wang et al. (2003) improved efficiency of the FD

modelling of wave propagation in poroelastic media by developing an (8, 2) explicit velocity-

stress staggered-grid (SG) scheme. Wang et al. (2003) pointed out, referring to Levander

(1988), that with the SG schemes strong velocity-contrast interfaces can be handled with high

accuracy because derivatives of material parameters are not required. As already mentioned,

the representation of interfaces seems easy and implicit with the heterogeneous schemes solving

the first-order equations of motion but, in fact, it is not the case – again we refer to Moczo et

al. (2002, 2014) and Kristek et al. (2017).

Saenger et al. (2004) applied the rotated staggered FD grid (the basic variant of the partly-

staggered grid) technique to calculate elastic wave propagation in 3D porous media. Saenger et

al. (2005) extended the approach to porous media saturated with a Newtonian (viscous) fluid

using a generalized Maxwell body.

Sheen et al. (2006) presented a parallel implementation of the 2D P-SV velocity-stress SG

FD scheme, 4th-order in space and 2nd-order in time, for fluid-saturated poroelastic media. They

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also addressed the aspect of the effective media parameters. They averaged shear modulus

according to Graves (1996). They applied harmonic averaging to density, effective fluid density,

fluid density and mobility of the fluid. Harmonic averaging of density, however, violates the

boundary conditions at an interface (Moczo et al. 2002). The authors correctly suggest the need

to properly address the problem of the effective grid material parameters in the subsequent

research.

Masson et al. (2006) applied a 2D (4, 2) velocity-stress SG FD scheme. They analysed

stability conditions and demonstrated that over a wide range of porous material properties

typical of sedimentary rock and despite the presence of fluid pressure diffusion (Biot slow

waves), the usual Courant condition governs the stability as if the problem involved purely

elastic waves. In relation to our study we may note that they averaged shear modulus according

to Graves (1996). Masson & Pride (2007) applied FD modelling to investigate seismic

attenuation and dispersion due to mesoscopic-scale heterogeneity. Specifically, they simulated

one loss mechanism that can be important across the seismic band of frequencies when

heterogeneity is present within fluid-saturated porous samples.

Krzikalla & Müller (2007a, b) applied the rotated spatial FD operators (Saenger et al. 2000)

to improve stability in modelling high-contrast materials compared to standard SG scheme.

Wenzlau & Müller (2009) implemented a velocity-stress FD scheme for modelling wave

propagation and diffusion in porous media. The scheme is 2nd-order accurate in time and

includes high-order spatial differentiation operators and is parallelized using the domain-

decomposition technique. The spatial discretization combines the standard and rotated SG

operators. Wenzlau & Müller presented several tests to estimate the accuracy of poroelastic

wave-propagation schemes for the high-frequency case where the Biot slow wave is a

propagating wave mode and for the low-frequency case when this wave mode becomes

diffusive. They introduced a diffusion wavelength in relation to the numerical dispersion. They

also addressed the stability for strongly heterogeneous medium and compared the standard and

rotated staggered-grid schemes. Masson & Pride (2010) expanded their scheme to Biot’s

equations across all frequencies. O’Brien (2010) presented 3D (4, 2) standard and rotated SG

schemes. They analysed stability and numerical dispersion.

Chiavassa & Lombard (2011, 2013) made an important contribution to numerical

modelling of wave propagation in heterogeneous 2D fluid/poroelastic media mainly by

applying an immersed interface method to discretize the interface conditions and to introduce

a subcell resolution. They also implemented a 4th-order ADER scheme with Strang splitting for

time-marching, and a space-time mesh-refinement to capture the slow compressional wave.

With respect to the discontinuous material heterogeneity the immersed-interface method is the

most sophisticated approach applied so far. Arbitrary-shaped interfaces can be handled, and

accuracy is ensured by a subcell resolution on a Cartesian grid. Its only disadvantage is the

relatively high computational cost.

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Chiavassa & Lombard (2013) made this explicit statement: “… arbitrary-shaped

geometries with various interface conditions … are badly discretized by FD methods on

Cartesian grids”. Well, the “FD methods” is in fact a large family of FD schemes differing

strongly in their accuracy and efficiency. The statement by Chiavasa & Lombard is relevant

mainly for those FD modellers who were misled by the absence of derivatives of material

parameters in the velocity-stress or displacement-stress formulation of the equation of motion

and constitutive relation. Those modellers assumed that boundary conditions are implicitly

satisfied at boundaries. Consequently, majority of the FD schemes for seismic wave propagation

had been using for years grid material parameters inconsistent with the boundary conditions at

the interface; see Moczo et al. (2014).

Blanc (2013) presented a numerical approach to the complete Biot-DA (diffusive

approximation) system in which the equations of evolution are split into two parts: a

propagative part is discretized using a 4th-order FD scheme, and a diffusive part is solved

exactly. She also implemented the immersed-interface method to account for the jump

conditions and for the geometry of the interfaces on a Cartesian grid.

Itzá et al. (2016) applied the global optimal implicit staggered-grid FD scheme based on

least-squares, proposed originally by Liu (2014) for the elastic medium, to 2-D porous media.

Balam et al. (2018) incorporated the above FD scheme into the self-organizing maps of the

neural network algorithm to obtain synthetic waveforms in poroelastic media for reservoir

modelling. Zhang et al. (2017) developed explicit flat-free-surface boundary condition for

modelling seismic wave propagation in poroelastic media using the staggered-grid FD scheme.

Yang & Mao (2017) presented simulation of seismic wave propagation in 2D poroelastic media

using weighted-averaging FD stencils in the frequency–space domain. Their scheme is an

extension of the 25-point weighted-averaging FD scheme proposed by Min et al. (2000). They

used the optimal weighting coefficients in order to account for medium heterogeneity without

rotating the coordinate system and the grid points.

A recent and very valuable contribution to the FD modelling of seismic wave propagation

in porous media is the article by Sun et al. (2019). Sun et al. developed a modification of the

traction-imaging scheme presented originally by Zhang & Chen (2006) and Zhang et al. (2012)

for modelling seismic wave propagation in models with nonplanar free surface. Sun et al.

(2019) extended their approach to poroelastic media. They expressed equations of Biot’s theory

in curvilinear coordinates and solved on a collocated grid by utilizing the 4th-order accurate

Runge–Kutta scheme with alternating applications of forward and backward MacCormack FD

approximations. They derived formulas of the free-surface boundary conditions for poroelastic

media in the curvilinear coordinates and subsequently implemented with the traction-imaging

method.

The FD method is the dominant method in numerical modelling of seismic wave

propagation in the poroelastic medium. This is due to the distinct features of the FD method:

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the possibility of applying one explicit FD scheme to all interior grid points (points not lying

on a grid border), relative simplicity, and good balance between accuracy and computational

efficiency. At the same time, it is true that not all developed and used FD schemes sufficiently

represent the above commonly accepted aspects. Moczo, Gregor, Kristek and de la Puente

(2019) explained what is missing in the published work on FD schemes – sufficiently accurate

discrete representation of material heterogeneity and sub-cell resolution. Moczo, Gregor,

Kristek and de la Puente (2019) presented such FD scheme for the poroelastic medium with

zero or non-zero constant resistive friction. The latter article makes a part of this Thesis.

Therefore, next we formulate goals of the Thesis including the contribution presented in the

article.

Goals

The main goal of the Thesis is to develop an FD tool for modelling 2D P-SV seismic wave

propagation in the poroelastic medium. This includes the following major partial goals:

• development of the constitutive law for a) smoothly and weakly heterogeneous medium

and b) strongly heterogeneous medium with material interfaces

• development of the equations of motion for smoothly-and-weakly and strongly

heterogeneous medium a) with zero resistive friction a) with non-zero constant resistive

friction and c) frequency-dependent permeability and resistive friction,

• adaptation of the partition method to equations of motion for the averaged medium with

the non-zero resistive friction,

• development of the nd2 -order accurate in time and th4 -order accurate in space, velocity-

stress-pressure FD scheme,

• development of computational algorithm and computer code based on the FD scheme,

• numerical tests of the computational algorithm and computer code.

Conclusions

In the dissertation we presented

• derivation of the constitutive relations for the poroelastic medium following mainly

Carcione (2015),

• equations of motion in the time domain for the poroelastic medium with the constant

resistive friction according to Biot’s theory (Biot 1956a),

• equations of motion in the frequency domain for the poroelastic medium with the

frequency-dependent permeability and resistive friction based on the model by Johnson

et al. (1987).

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The results of the dissertation include

• derivation of the equations of motion in the time domain for the JKD model of the

poroelastic medium based on the Laplace-Carson transform,

• development of the constitutive law for the averaged medium, that is, the discrete

representation of the strongly heterogeneous (including material interfaces) poroelastic

medium (presented in the Geophys. J. Intl. article by Moczo, Gregor, Kristek and de la

Puente 2019),

• development of the equations of motion for the averaged medium with constant resistive

friction (presented in the article by Moczo, Gregor, Kristek and de la Puente 2019) and

equations of motion for the averaged medium with the JKD model of the frequency-

dependent permeability and resistive friction,

• adaptation of the diffusive approximation developed by Blanc et al. (2012) and Blanc

(2013) to the convolutional term representing resistive friction in the JKD model of the

poroelastic medium including a novel optimization procedure for determination of the

whole-model abscissae and spatially varying weights of the discrete diffusive

approximation,

• adaptation of the partition method developed by Carcione & Quiroga-Goode (1995) and

applied to the diffusive representation by Blanc et al. (2012) and Blanc (2013),

• development of the n d2 -order accurate in time and th4 -order accurate in space,

velocity-stress-pressure finite-difference (FD) scheme for the averaged poroelastic

medium with a) zero resistive friction, b) non-zero constant resistive friction and c) JKD

model of the frequency-dependent permeability and resistive friction,

• generalization of the stress-imaging approach and AFDA approach (based on Kristek et

al. 2002 and Moczo et al. 2004), PML (based on Kristek et al. 2009) and source (based

on Carcione 1998 and Carcione & Quiroga-Goode 1996) to the poroelastic medium,

• development of the FORTRAN-95 computer code based on the discrete representation

of the heterogeneous medium and corresponding FD scheme with the option for

simulating wave propagation in models a) with zero resistive friction, b) non-zero

constant resistive friction and c) JKD model of the frequency-dependent permeability

and resistive friction,

• development of the FORTRAN-95 computer codes based on the analytical or semi-

analytical solutions by Dai et al. (1995), Karpfinger et al. (2005) and Carcione &

Quiroga-Goode (1996); numerical tests using the codes for models of the unbounded

homogeneous poroelastic medium with zero resistive friction and frequency-dependent

permeability and resistive friction served the basis for the ultimate comparisons with

the semi-analytical method by Mesgouez & Lefeuve-Mesgouez (2009),

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• extensive numerical tests of the FD algorithm and computer code for heterogeneous

models with zero resistive friction against the exact method by Diaz & Ezziani (2008);

(presented in the article by Moczo, Gregor, Kristek and de la Puente 2019),

• numerical tests of the FD algorithm and computer code for the model of a lens with zero

resistive friction against the discontinuous Galerkin method by de la Puente et al.

(2008); (presented in the article by Moczo, Gregor, Kristek and de la Puente 2019),

• extensive numerical tests of the FD algorithm and computer code for the JKD models

of the homogeneous halfspace and layer over halfspace against the semi-analytical

solution by Mesgouez & Lefeuve-Mesgouez (2009).

The main conclusion:

We have developed the FD scheme capable of sub-cell resolution in the strongly heterogeneous

poroelastic medium with a) zero resistive friction, b) non-zero constant resistive friction and c)

JKD model of the frequency-dependent permeability and resistive friction.

Numerical example documenting the sub-cell resolution

of the developed FD scheme

The example shows very good level of agreement between the FD seismograms and semi-

analytical (SA) seismograms for both the Biot’s and JKD models, and all interface positions in

the FD grid. The latter fact is due to the sub-cell resolution of the developed FD scheme.

Figure 31. Source-receiver configuration in the model of a layer over halfspace.

15

Figure 36. 5 positions of the horizontal layer-halfspace interface in the grid labelled A, B, C, Dand E. Labels A and E are framed in order to indicate that the distance between the two positionsis just one grid spacing h. The source and receivers are at the same positions in the grid in all 5cases.

Figure 48. The same as in Fig. 47 but for the interface positions D and E.

16

Figure 47. The zv seismograms at receiver R2 in the models of the layer over halfspace for theBiot’s and JKD models calculated by our FD scheme and semi-analytical (SA) method byMesgouez and Lefeuve-Mesgouez. Labels A, B and C indicate both the layer thicknesses andpositions of the layer-halfspace interface in the FD grid (the grid being one and the same for allthicknesses/interface positions, see Fig. 36).

17

List of publications ADC Vedecké práce v zahraničných karentovaných časopisoch ADC01 Moczo, Peter [UKOMFKAFZM] (37%) - Gregor, Dávid [UKOMFKAFZM] (33%) - Kristek, Jozef

[UKOMFKAFZM] (25%) - de la Puente, Josep (5%): A discrete representation of material heterogeneity for the finite-difference modelling of seismic wave propagationin a poroelastic medium Lit.: 50 zázn. In: Geophysical Journal International. - Roč. 216, č. 2 (2019), s. 1072-1099. - ISSN (print) 0956-540X Registrované v: wos Indikátor časopisu: IF (JCR) 2017=2.528 Ohlasy (2): [o1] 2019 Ou, W. - Wang, Z.: Simulation of Stoneley wave reflection from porous formation in borehole using FDTD method. In: Geophysical Journal International, Vol. 217, No. 3, 2019, s. 2081-2096 - SCI ; SCOPUS [o1] 2019 Sun, Y. C. - Ren, H. - Zheng, X. Z. - Li, N. - Zhang, W. - Huang, Q. - Chen, X.: 2-D poroelastic wave modelling with a topographic free surface by the curvilinear grid finite-difference method. In: Geophysical Journal International,Vol. 218, No. 3, 2019, s. 1961-1982 - SCI ; SCOPUS

AFG Abstrakty príspevkov zo zahraničných vedeckých konferencií AFG01 Moczo, Peter [UKOMFKAFZM] (34%) - Gregor, Dávid [UKOMFKAFZM] (33%) - Kristek, Jozef

[UKOMFKAFZM] (33%): The finite-difference modelling of seismic waves in poroelastic medium: A new efficient and accurate discrete representation of a strongdiscontinuous and smooth material heterogeneity [elektronický dokument] In: Geophysical Research Abstracts [elektronický dokument] : 20. - Göttingen : Copernicus, 2018. - S. 1-1, Art. No. EGU2018-7488 [online]. - ISSN 1607-7962 [EGU General Assembly 2018. 15, Vienna, 09.04.2018 - 13.04.2018] URL: http://meetingorganizer.copernicus.org/EGU2018/EGU2018-7488.pdf

AFG02 Moczo, Peter [UKOMFKAFZM] (34%) - Gregor, Dávid [UKOMFKAFZM] (33%) - Kristek, Jozef [UKOMFKAFZM] (33%): A unified discrete representation of the elastic, viscoelastic, and poroelastic interface and strong material heterogeneity in thefinite-difference modeling of seismic wave propagation [elektronický dokument] In: Seismological Research Letters : 2018 Seismology of the Americas Meeting [elektronický dokument]. - Roč. 89, č. 2B (2018), s. 804-804 [print]. - ISSN (print) 0895-0695 [Seismology of the Americas 2018. Miami, 14.05.2018 - 17.05.2018] URL: https://seismology2018.org/wp-content/uploads/2018/03/SSALACSC-Program-2018.pdf

AFG03 Moczo, Peter [UKOMFKAFZM] (30%) - Kristek, Jozef [UKOMFKAFZM] (25%) - Stripajová, Svetlana [UKOMFKAFZM] (15%) - Gregor, Dávid [UKOMFKAFZM] (13%) - Bard, Pierre-Yves (9%) - Kristeková, Miriam (8%): Effects of structural parameters on earthquakeground motion characteristics in sedimentary basins [elektronický dokument] In: Geophysical Research Abstracts [elektronický dokument] : 21. - Göttingen : Copernicus, 2019. - S. 1-1, Art. No. 7521 [online]. - ISSN 1607-7962 [EGU General Assembly 2019. Viedeň, 07.04.2019 - 12.04.2019] URL: https://meetingorganizer.copernicus.org/EGU2019/EGU2019-7521.pdf

AFG04 Moczo, Peter [UKOMFKAFZM] (30%) - Kristek, Jozef [UKOMFKAFZM] (25%) - Stripajová, Svetlana [UKOMFKAFZM] (15%) - Gregor, Dávid [UKOMFKAFZM] (13%) - Bard, Pierre-Yves (9%) - Kristeková, Miriam (8%): Effects of structural parameters on characteristicsof earthquake ground motion in water-saturated sedimentary basins In: Seismological Research Letters. - Roč. 90, č. 2 (2019), s. 943-945. - ISSN (print) 0895-0695 [SSA Annual Meeting 2019. Seatle, 23.04.2019 - 26.04.2019] URL: https://www.seismosoc.org/wp-content/uploads/2019/03/SSA-Program-2019.pdf

AFH Abstrakty príspevkov z domácich vedeckých konferencií AFH01 Gregor, Dávid [UKOMFKAFZM] (40%) - Moczo, Peter [UKOMFKAFZM] (30%) - Kristek, Jozef

[UKOMFKAFZM] (25%) - de la Puente, Josep (5%): FDTD Modelling of seismic wave propagation in porous media In: NMEM 2019 : Workshop on Numerical Modeling of Earthquake Motions: Waves and Ruptures. - Bratislava : Knižničné a edičné centrum, 2019. - S. 33-33. - ISBN 978-80-8147-090-5 [NMEM 2019 : Numerical Modeling of Earthquake Motions: Waves and Ruptures : Workshop. Smolenice, 30.06.2019 - 04.07.2019] URL: http://www.nuquake.eu/NMEM2019/

BFA Abstrakty odborných prác zo zahraničných podujatí (konferencie, ...) BFA01 Gregor, Dávid [UKOMFKAFZM] (40%) - Moczo, Peter [UKOMFKAFZM] (30%) - Kristek, Jozef

[UKOMFKAFZM] (25%) - de la Puente, Josep (5%): Accurate and efficient finite-difference modelling of seismic waves in a strongly heterogeneous poroelastic medium[elektronický dokument]

18

Lit.: 4 zázn. In: YIC 2019 : 5th ECCOMAS Young Investigators Conference [elektronický dokument]. - [Krakow] : University of Science and Technology, 2019. - S. 262-262 [YIC 2019 : Eccomas Young Investigators Conference. Krakow, 01.09.2019 - 06.09.2019] URL: https://yic2019.agh.edu.pl/wp-content/uploads/2019/08/YIC2019_book_of_abstracts.pdf

BFB Abstrakty odborných prác z domácich podujatí (konferencie, ...) BFB01 Gregor, Dávid [UKOMFKAFZM] (40%) - Moczo, Peter [UKOMFKAFZM] (40%) - Kristek, Jozef

[UKOMFKAFZM] (20%): Numerical testing of a new discrete representation of a poroelastic interface the first results Popis urobený 17.1.2018 Lit. 3 zázn. In: 12. Slovak Geophysical Conference : Abstracts [elektronický zdroj]. - [S.l.] : [s.n.], 2017. - S. 53 [online] [Slovak Geophysical Conference 2017. 12th, Bratislava, 28.-29.9.2017] URL: http://www.kaeg.sk/wp-content/uploads/2017/04/SGC2017_Abstracts.pdf

Štatistika kategórií (Záznamov spolu: 8): ADC Vedecké práce v zahraničných karentovaných časopisoch (1) AFG Abstrakty príspevkov zo zahraničných vedeckých konferencií (4) AFH Abstrakty príspevkov z domácich vedeckých konferencií (1) BFA Abstrakty odborných prác zo zahraničných podujatí (konferencie, ...) (1) BFB Abstrakty odborných prác z domácich podujatí (konferencie, ...) (1) Štatistika ohlasov (2): [o1] Citácie v zahraničných publikáciách registrované v citačných indexoch (2) 12.5.2020

Bibliography

Alterman, Z.S., F.C. Karal 1968. Propagation of elastic waves in layered media by finite difference methods. Bull.

Seism. Soc. Am. 58, 367-398. Andrews, D.J., 1973. A numerical study of tectonic stress release by underground explosions. Bull. Seism. Soc.

Am. 63, 1375-1391. Auriault, J. L. 1980. Dynamic behavior of a porous medium saturated by a Newtonian fluid. Int. J. Eng. Sci. 18,

775-785. Balam, R.I., U. Iturrarán-Viveros, J.O. Parra 2018. Modeling poroelastic wave propagation in a real 2-D complex

geological structure obtained via self-organizing maps. Pure Appl. Geophys. 175(8), 2975–2986. Bermudez, A., J.L. Ferrín, A. Prieto 2006. Finite element solution of new displacement/pressure poroelastic

models in acoustics. Comput. Methods Appl. Mech. Eng. 195(17–18), 1914–1932. Biot, M. A 1956a. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range.

J. acoust. Soc. Am. 28, 168-178. Biot, M. A 1956b. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency

range. J. acoust. Soc. Am. 28, 179-191. Biot, M. A 1962a. Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33, 1482-

1498. Biot, M. A 1962b. Generalized theory of acoustic propagation in porous dissipative media. J. acoust. Soc. Am. 34,

1254-1264. Blanc, E. 2013. Theory of propagation of elastic waves in a fluid-saturated porous solid. PhD Thesis, Aix-Marseille

Université. Blanc, E., G. Chiavassa, B. Lombard 2012. Biot-JKD model: simulation of 1D transient poroelastic waves with

fractional derivatives. J. Comput. Phys. 237, 1-20. Brown, R. J. S. 1980. Connection between formation factor for electrical resistivity and fluid solid coupling factor

in Biot’s equations for acoustic waves in fluid‐filled porous media. Geophysics 45, 1239-1319. Carcione, J.M. 1998. Viscoelastic effective rheologies for modelling wave propagation in porous media. Geophys.

Prosp. 46, 249-270. Carcione, J.M 2015. Wave Fields in Real Media: Wave propagation in Anisotropic, Anelastic, Porous media and

Electromagnetic Media. Amsterdam: Elsevier. Carcione, J.M., C. Morency, J.E. Santos 2010. Computational poroelasticity – A review. Geophysics 75, A229-

A243. Carcione, J.M., G. Quiroga-Goode 1995. Some aspects of the physics and numerical modelling of Biot

compressional waves. J. Comput. Acoust. 3(4), 261-280. Carcione, J.M., G. Quiroga-Goode 1996. Full frequency-range transient solution for compressional waves in a

fluid-saturated viscocoustic porous medium. Geophys. Prospec. 44, 99-129. Cheng, A. H.-D. 2016. Poroelasticity. Springer. Chiavassa, G., B. Lombard 2011. Time domain numerical modeling of wave propagation in 2D heterogeneous

porous media. J. Comput. Phys. 230, 5288–5309.

19

Chiavassa, G., B. Lombard 2013. Wave Propagation Across Acoustic/Biot’s Media: A Finite-Difference Method. Commun. Comput. Phys. 14(4), 985-1012.

Dai, N., A. Vafidis, E. Kanasewich 1995. Wave propagation in heterogeneous porous media a velocity-stress, finite-difference method. Geophysics 60(2), 327–340.

Diaz, J., A. Ezziani 2008. Analytical Solution for Wave Propagation in Stratified Poroelastic Medium. Part I: the 2D Case. Technical Report 6591, INRIA.

De la Puente, J. 2008. Seismic Wave Propagation for Complex Rheologies: Discontinuous Galerkin Methods for

the Simulation of Earthquakes. Saarbrücken: VDM Verlag Dr. Müller. De la Puente, J., M. Dumbser, M., Käser, H. Igel 2008. Discontinuous Galerkin methods for wave propagation in

poroelastic media. Geophysics 73, T77-T97. Garg, S. K., A. H. Nayfeh, A. J. Good 1974. Compressional waves in fluid-saturated elastic porous media. J. Appl.

Phys 45, 1968–1974. Graves, R. W., 1996. Simulating seismic wave propagation in 3D elastic media using staggered-grid finite

differences. Bull. Seism. Soc. Am. 86, 1091-1106. Gregor, D. 2014. Physics of seismic wave propagation in poroelastic media. Bachelor Thesis. Comenius

University, Bratislava. Gregor, D. 2016. Finite-Difference modelling of seismic waves in poroelastic medium. MSc Thesis. Comenius

University, Bratislava. Hassanzadeh, S. 1991. Acoustic modeling in fluid saturated porous media. Geophysics 56(4), 424–435. Itzá, R., U. Iturrarán-Viveros, J.O. Parra 2016. Optimal implicit 2-D finite differences to model wave propagation

in poroelastic media. Geophys. J. Int. 206(2), 1111–1125. Johnson, D. L., J. Koplik, R. Dashen. 1987. Theory of dynamic permeability and tortuosity in fluid-saturated

porous media. J. Fluid Mech. 176, 379-402. Karpfinger, F., T.M Müller, B. Gurevich 2005. Radiation patterns of seismic waves in poroelastic media. Annual

International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, 1791–1795. Kelly, K.R., R.W. Ward, S.R. Treitel, M. Alford 1976. Synthetic seismograms: A finite‐ difference approach.

Geophysics 41, 2-27. Kristek, J., P. Moczo, R. J. Archuleta 2002. Efficient methods to simulate planar free surface in the 3D 4th-order

staggered-grid finite-difference schemes. Studia Geoph. Geod. 46, 355–381. Kristek, J., P. Moczo, E. Chaljub, M. Kristekova 2017. An orthorhombic representation of a heterogeneous

medium for the finite-difference modelling of seismic wave propagation. Geophys. J. Int. 208, 1250-1264. Kristek, J., P. Moczo, E. Chaljub, M. Kristekova 2019. A discrete representation of a heterogeneous viscoelastic

medium for the finite-difference modelling of seismic wave propagation. Geophys. J. Int. 217, 2021–2034. Kristek, J., P. Moczo, M. Galis 2009. A brief summary of some PML formulations and discretizations for the

velocity–stress equation of seismic motion. Studia Geoph. Geod. 53, 459–474. Lefeuve-Mesgouez, G., A. Mesgouez, G. Chiavassa, B. Lombard 2012. Semi-analytical and numerical methods

for computing transient waves in 2D acoustic/poroelastic stratified media. Wave Motion 49(7), 667-680. Levander, A. R. 1988. Fourth-order finite-difference P-SV seismograms. Geophysics 53, 1425-1436. Liu, Y., 2014. Optimal staggered-grid finite-difference schemes based on least-squares for wave equation

modelling. Geophys. J. Int. 197, 1033–1047. Masson, Y.J., S.R. Pride 2007. Poroelastic finite difference modeling of seismic attenuation and dispersion due to

mesoscopic-scale heterogeneity. J. geophys. Res. 112, B03204. Masson, Y.J., S.R. Pride. 2010. Finite-difference modeling of Biot’s poroelastic equations across all frequencies.

Geophysics 75(2), N33–N41. Masson, Y.J., S.R. Pride, K.T. Nihei 2006. Finite-difference modeling of Biot’s poroelastic equations at seismic

frequencies. J. Geophys. Res. 111, B10305. Mesgouez, A., G. Lefeuve-Mesgouez 2009. Transient solution for multilayered poroviscoelastic media obtained

by an exact stiffness matrix formulation. Int. J. Numer. Anal. Meth. Geomech. 33, 1911-1931. Min, D.J., C. Shin, B.D. Kwon, S. Chung 2000. Improved frequency-domain elastic wave modeling using

weighted-averaging difference operators. Geophysics 65(3), 884–895. Moczo, P., D. Gregor, J. Kristek, J. de la Puente 2019. A discrete representation of material heterogeneity for the

finite-difference modelling of seismic wave propagation in a poroelastic medium. Geophys. J. Int. 216(2), 1072-1099.

Moczo, P., J. Kristek, M. Gális 2004. Simulation of planar free surface with near-surface lateral discontinuities in the finite-difference modeling of seismic motion. Bull. Seism. Soc. Am. 94(2), 760-768.

Moczo, P., J. Kristek, M. Gális 2014. The Finite-Difference Modelling of Earthquake Motions: Waves and

Ruptures. Cambridge: Cambridge University Press.

Moczo, P., J. Kristek, V. Vavryčuk, R. J. Archuleta, L. Halada 2002. 3D heterogeneous staggered-grid finite-difference modeling of seismic motion with volume harmonic and arithmetic averaging of elastic moduli and densities. Bull. Seism. Soc. Am. 92, 3042-3066.

Morency, Ch., J. Tromp 2008. Spectral-element simulations of wave propagation in porous media. Geophys. J.

Int. 175, 301-345. O’Brien, G., 2010. 3D rotated and standard staggered finite-difference solutions to Biot’s poroelastic wave

equations: Stability condition and dispersion analysis. Geophysics 75, T111-T119.

20

Plona, T. J., 1980. Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies.

Appl. Phys. Lett. 36, 259. Riney, T.D, J.K Dienes, C.A Plazier, S.K. Garg, J.W. Kirsch, D.H. Brownell, A.J. Good 1972. Ground Motion

Models and Computer Techniques, Formal Report under Contract No. DASA 01-69-C-0159(P0Ü003), S Report 3SR-1071 (unpublished).

Riney, T.D., G.A. Frazier, S.K Garg, A.J Good, R.G. Herrmann, L.W. Morland, J.W. Pritchett, M.H. Rice, J. Sweet 1973. Constitutive models and computer techniques for gorund motion predictions. Systems, Science and Software Report No. SSS-R-73-1490, 357 pp. (unpublished).

Saenger, E. H., N. Gold, S. A. Shapiro 2000. Modeling the propagation of elastic waves using a modified finite-difference grid. Wave Motion 31, 77-92.

Saenger, E.H., O.S. Krüger, S.A. Shapiro 2004. Numerical considerations of fluid effects on wave propagation: Influence of the tortuosity. Geophys. Res. Lett. 31, L21613.

Saenger, E.H., S.A. Shapiro, Y. Keehm 2005. Seismic effects of viscous Biot-coupling: Finite difference simulations on micro-scale. Geophys. Res. Lett. 32, L14310.

Sahay, P.N. 2008. On the Biot slow S-wave. Geophysics 73(4), N19-N33. Santos, J.E., D. Sheen 2007. Finite element methods for the simulation of waves in composite saturated

poroviscoelastic materials. SIAM J. Numer. Anal., 45(1), 389–420. Sharma, M. D. 2007. Wave propagation in a general anisotropic poroelastic medium: Biot’s theories and

homogenization theory. J. Earth Syst. Sci. 116, 357–367. Sheen, D.-H., K. Tuncay, C.-E. Baag, P.J. Ortoleva 2006. Parallel implementation of a velocity-stress staggered-

grid finite-differences method for 2D poroelastic wave propagation. Computers & Geosciences 32, 1182–1191. Shukla K., J. Chan, M. V. de Hoop, P. Jaiswal 2020. A weight-adjusted discontinuous Galerkin method for the

poroelastic wave equation: Penalty fluxes and microheterogeneities. J. Comput. Phys. 403, 109063. Shukla K., J. S. Hesthaven, J. M. Carcione, R. Ye, J. de la Puente, P. Jaiswal 2019. A nodal discontinuous Galerkin

finite element method for the poroelastic wave equation. Computat. Geosci. 23, 595-615. Sidler, R., J.M. Carcione, K. Hollinger 2010. Simulation of surface waves in porous media. . Geophys. J. Int.

183(2), 820–832. Sun, Y.-C., R. Hengxin, X.-Z. Zheng, N. Li, W. Zhang, Q. Huang, X. Chen 2019. 2-D poroelastic wave modelling

with a topographic free surface by the curvilinear grid finite-difference method. Geophys. J. Int. 218(3), 1961–1982.

Wang, X., H. Zhang, D. Wang 2003. Modelling seismic wave propagation in heterogeneous poroelastic media using a high-order staggered finite-difference method. Chinese J. Geophysics 46, 1206-1217.

Ward N.D., S. Eveson, T. Lähivaara 2020. A Discontinuous Galerkin method for three-dimensional poroelastic wave propagation: forward and adjoint problems. arXiv:2001.09478v1 [physics.comp-ph] 26 Jan 2020.

Ward N.F.Dudley, T. Lähivaara, S. Eveson 2017. A discontinuous Galerkin method for poroelastic wave propagation: The two-dimensional case. J. Comput. Phys. 350, 690-727.

Wenzlau, F. T. M Müller, 2009. Finite-difference modeling of wave propagation and diffusion in poroelastic media. Geophysics 74(4), T55-T66.

Yang, Q., W. Mao 2017. Simulation of seismic wave propagation in 2-D poroelastic media using weighted-averaging finite difference stencils in the frequency–space domain, Geophys. J. Int. 208, 148-161.

Zeng, Q.Z., Q.H. Liu 2001. A staggered-grid finite-difference method with perfectly matched layers for poroelastic wave equations. J. Acoust. Soc. Am. 109, 2571-2580.

Zhan Q., M. Zhuang, Q. H. Liu. 2018. A compact upwind ux with more physical insight for wave propagation in 3-d poroelastic media. IEEE Trans. on Geosci and Remote Sensing 56(10), 5794-5801.

Zhan Q., M. Zhuang, Y. Mao, Q. H. Liu 2020. Unified Riemann solution for multi-physics coupling: Anisotropic poroelastic/elastic/fluid interfaces. J. Comput. Phys. 402, 108961.

Zhang, J., 1999. Quadrangle-grid velocity-stress finite difference method for poroelastic wave equations. Geophys.

J. Int. 139, 171–182. Zhang, W., X. Chen 2006. Traction image method for irregular free surface boundaries in finite difference seismic

wave simulation. Geophys. J. Int. 167(1), 337–353. Zhang, Y., P. Ping, S.-X. Zhang 2017. Finite-difference modeling of surface waves in poroelastic media and stress

mirror conditions, Appl. Geophys. 14(1), 105–114. Zhang, W., Z. Zhang, X. Chen 2012. Three-dimensional elastic wave numerical modelling in the presence of

surface topography by a collocated-grid finite-difference method on curvilinear grids. Geophys. J. Int. 190(1), 358–378.

Zhu, X., G. A. McMechan 1991. Numerical simulation of seismic responses of poroelastic reservoirs using Biot theory. Geophysics 56, 328-339.