Finite element approximation of coupled seismic and by ves88494

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									Finite element approximation of coupled
seismic and electromagnetic waves in gas
       hydrate-bearing sediments
                                   Juan E. Santos†

                †
                                                 ı
                    CONICET, Departamento de Geof´sica Aplicada,
                            ´            ı
 Facultad de Ciencias Astronomicas y Geof´sicas, Universidad Nacional de La Plata
                and Department of Mathematics, Purdue University.


           Joint work with Fabio I. Zyserman and Patricia M. Gauzellino


          2009 SIAM Annual Meeting , Denver, Colorado, July 6-10 2009
                               Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 1/2
Electroseismic Modeling. I


 Within a fluid saturated porous medium there
 exists a nanometer-scale separation of
 electric charge in which a bound charge
 existing on the surface of the solid matrix
 (normally of negative sign) is balanced by
 adsorbed positive ions of the surrounding
 fluid, setting an immobile double (Stern)
 layer, having a thickness of about 10 nm.


                 Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 2/2
Electroseismic Modeling. II
 Further from the surface of the solid matrix
 there exists a distribution of mobile counter
 ions, forming the so called diffuse layer.
 When an electric field is applied to this
 system, the ions in the diffuse layer move,
 dragging the pore fluid along with it because
 of the viscous traction. This is known as
 electro-osmosis and is responsible for the
 electroseismic phenomena.
                   Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 3/2
Electroseismic Modeling. III

                               Electrical double layer
                          Stern Layer                  Diffuse Layer
                  1
                  0
                  0
                  1       −   +   +                          +                         −       +
                  1
                  0
                                                       −
                                                            − +                            −
                  0
                  1
                          −   + +




                                                                      Slipping plane
                                                        +                              +
                  1
                  0       −   +                               +




                                         Shear plane
                                  +
                  1
                  0
          Solid           −   +                          − +                               +

                  0
                  1       −   + +                             +                        − +
                  1
                  0
                                                        −
                                                            −
                  1
                  0
                          −   + +                                                      +
                                                        +                                  −
                  1
                  0       −   +     +                     − +                          −           +

                              11111
                              00000
          Electric                                      Zeta potential

                              11111
                              00000
          potential

                              00000
                              11111
                      0
                              11111
                              00000Debye length
                              11111
                              00000
                              11111
                              00000

                                Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 4/2
2D Model and Sources. I

 Assume that the subsurface is an horizontally
 layered fluid-saturated porus medium.
 Consider two different electromagnetic
 sources, both in the y-direction:
                             ext
 1) an infinite current line Je ,
 2) an infinite magnetic dipole (infinite
            ext
 solenoid) Jm .
                   Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 5/2
2D Model and Sources. II

  ext
 Jm induces electromagnetic fields
 (Ex(x, z), Ez (x, z)), and Hy (x, z), and fluid and
                                  s          s
 solid displacements            (ux (x, z), uz (x, z))                                                  and
 (uf (x, z), uf (x, z)), respectively.
   x          z

 This is known as the PSVTM-mode, in which
 compressional and vertically polarized shear
 seismic waves (PSV-waves) are present.
                      Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 6/2
2D Model and Sources. III

  ext
 Je induces electromagnetic fields
 (Hx (x, z), Hz (x, z)) and Ey (x, z), and fluid and
                               s                                                 f
 solid displacements          uy (x, z)                       and               uy (x, z),
 respectively.

 This is known as the SHTE-mode, where
 only horizontally polarized seismic waves
 (SH-waves) are present.
                    Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 7/2
2D Model and Sources. IV
                 x
             y
                                       ext
                 z                    Je                                           Γt,M
                        ext
                       Jm

                                                          Ω = Ωa ∪ ΩB                       Ωa               Γt,B
                        11111111111
                        00000000000
      11111111111111111111111111111
      00000000000000000000000000000
                        11111111111
                        00000000000
      11111111111111111111111111111
      00000000000000000000000000000
                        00000000000
                        11111111111
      00000000000000000000000000000
      11111111111111111111111111111
                        00000000000
                        11111111111
      00000000000000000000000000000
      11111111111111111111111111111                      1111111111111111111
                                                         0000000000000000000
                        11111111111
                        00000000000
      11111111111111111111111111111
      00000000000000000000000000000

                  11111111
                  00000000
                                                         1111111111111111111
                                                         0000000000000000000
                        11111111111
                        00000000000
      11111111111111111111111111111
      00000000000000000000000000000
                        11111111111
                        00000000000
      00000000000000000000000000000
      11111111111111111111111111111                      1111111111111111111
                                                         0000000000000000000
                                                         1111111111111111111
                                                         0000000000000000000                                            Γr,M
                                                         0000000000000
                                                         1111111111111
                        11111111111
                        00000000000
      00000000000000000000000000000
      11111111111111111111111111111

                  00000000
                  11111111
                        00000000000
                        11111111111
      00000000000000000000000000000
      11111111111111111111111111111                      0000000000000000000
                                                         1111111111111111111
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                        00000000000
      11111111111111111111111111111
      00000000000000000000000000000                      0000000000000000000
                                                         1111111111111111111

                                                         0000000000000
                                                         1111111111111
                        00000000000
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      00000000000000000000000000000
      11111111111111111111111111111

                  11111111
                  00000000
                        11111111111
                        00000000000
      00000000000000000000000000000
      11111111111111111111111111111
                        00000000000
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      11111111111111111111111111111
      00000000000000000000000000000

                                                         1111111111111
                                                         0000000000000
                        00000000000
                        11111111111
      11111111111111111111111111111
      00000000000000000000000000000

                  11111111
                  00000000
                        11111111111
                        00000000000
      11111111111111111111111111111
      00000000000000000000000000000
      11111111111
      00000000000       00000000000
                        11111111111
      11111111111111111111111111111
      00000000000000000000000000000
      0000000000000000000
      1111111111111111111                                                                                             Γr,B
                                                         1111111111111
                                                         0000000000000
                        00000000000
                        11111111111
      1111111111111111111
      0000000000000000000

                  00000000
                  11111111
                        00000000000
                        11111111111
      1111111111111111111
      0000000000000000000
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      00000000000       00000000000
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      0000000000000000000                                1111111111111111111
                                                         0000000000000000000
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                        00000000000
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      0000000000000000000

                  00000000
                  11111111
      11111111111
      00000000000       00000000000
                        11111111111
      0000000000000
      1111111111111
      1111111111111111111
      0000000000000000000                                0000000000000000000
                                                         1111111111111111111
                        00000000000
                        11111111111
      0000000000000000000
      1111111111111111111
      00000000000
      11111111111       00000000000
                        11111111111                              ΩB
                  11111111
                  00000000
                        11111111111
                        00000000000                      1111111111111111111
                                                         0000000000000000000
      0000000000000
      1111111111111
      11111111111
      00000000000       00000000000
                        11111111111
                        00000000000
                        11111111111                      1111111111111111111
                                                         0000000000000000000

                  00000000
                  11111111
      11111111111
      00000000000       00000000000
                        11111111111                      1111111111111111111
                                                         0000000000000000000

      0000000000000
      1111111111111
      00000000000
      11111111111       00000000000
                        11111111111                      1111111111111111111
                                                         0000000000000000000
                        00000000000
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                                                         1111111111111111111

                  00000000
                  11111111
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                        00000000000                      1111111111111111111
                                                         0000000000000000000

      0000000000000
      1111111111111
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      00000000000       00000000000
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                        00000000000
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                                                         1111111111111111111
                                                         0000000000000000000
                                                         1111111111111111111
                                                         0000000000000000000

                  00000000
                  11111111
      00000000000000000000000000000
                0
                1
      11111111111111111111111111111
                        00000000000
                        11111111111

      0000000000000
      1111111111111
                1
                0       00000000000
                        11111111111
      00000000000000000000000000000
      11111111111111111111111111111
                        00000000000
                        11111111111l,M
                        00000000000
                        11111111111
                        Γ
      00000000000000000000000000000
      11111111111111111111111111111
      11111111111
      00000000000
                0
                1       11111111111
                        00000000000
                        11111111111
                        00000000000
                        00000000000
                        11111111111
                        11111111111
                        00000000000
                                                                       Γb,M , Γb,B
                        00000000000
                        11111111111
      1111111111111111111
      0000000000000000000
                        00000000000
                        11111111111
      0000000000000000000
      1111111111111111111
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                        00000000000
      0000000000000000000
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                        00000000000
                        11111111111
      1111111111111111111
      0000000000000000000      l,B
                        00000000000
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      0000000000000000000
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                        00000000000
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      0000000000000000000
      1111111111111111111
                               Γ

 3D layered subsurface, 2D model Ω = Ωa ∪ ΩB and
 electromagnetic sources.

                               Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 8/2
Formulation in Terms of Scattered Fields E s , H s . PSVTM-mode.

 Let

                    σ(x, z) = σ p (z) + σ s (x, z),

 σ p (z): background conductivity, σ s (x, z): conductivity anomaly.
 Let E s = E − E p , H s = H − H p : scattered fields.
 For the PSVTM-mode, solve for the primary fields E p , H p
 solution of:
                                        s
                   ∇ × E p = −iωµH p + Jm ,                                        in         Ω,
                   ∇ × H p = σp E p,                      in Ω.

 Analytical expressions for E p , H p are known for the case that Ω is
 the whole space R3 and σp is constant.

                            Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 9/2
Differential Model for the PSVTM-mode. I
 In terms of the scattered electromagnetic fields the equations for
 PSVTM electroseismic modeling are (Pride, 1994):
             σE s − curlHy = −σ s E p ,
                         s
                                                               Ω,
                           s
             curlE s + iωµHy = 0,                     Ω,
             −ω 2 ρb us − ω 2 ρf uf − ∇ · τ (u) = 0,                                      ΩB ,
             −ω 2 ρf us + η(κ(ω))−1 iωuf + ∇pf
                  = η(κ(ω))−1 L(ω) (E p + E s ) , ΩB ,


             τlm (u) = 2N εlm (us ) + δlm (λc ∇ · us − αKav ξ) ,
             pf (u) = −αKav ∇ · us + Kav ξ.

 τlm (u): stress tensor,   pf (u): fluid pressure.
                            Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 10/2
Differential Model for the PSVTM-mode. II
 L(ω): complex coupling coefficient, depending on the effective
 permeability, porosity, fluid permittivity and salinity among other
 parameters.

 κ(ω) : Dynamic permeability:
                                          1                       −1
                        ω 4               2      ω
     κ(ω) = κ0      1+i                       +i                         = κr (ω) − iκi (ω),
                        ωc m                     ωc

 κ0 : effective permeability
 m: dimensionless parameter in the range 4 ≤ m ≤ 8.
 The electroseismic equations were solved employing absorbing
 boundary conditions at the artificial boundaries.
                               Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 11/2
Weak Formulation for the PSVTM-mode. I


              s
 Find (E s , Hy , us , uf ) ∈ H(curl, Ω) × L2 (Ω) × [H 1 (ΩB )]2 × H(div, ΩB ) such that

        (σE s , ψ) − (Hy , curlψ) + a(1 − i)E s · χ, ψ · χ + (curlE s , ϕ) + (iωµHy , ϕ)
                       s                                                          s


                = − (σ s E p , ψ) ,    (ψ, ϕ) ∈ H(curl, Ω) × L2 (Ω)

                                 “        ”           “     ”      “                 ”
            2         s   s          f 2s           2   s f                 −1   f f
        −ω (ρb u , v )ΩB − ω ρf u , v          − ω ρf u , v      + η(κ(ω)) iωu , v
                                            ΩB                ΩB                      ΩB
          X                          “                ”
            (τlm (u), εlm (v s ))ΩB − pf (u), ∇ · v f
                                                            ˙                 ¸
        +                                               + iω DSΓp (u), SΓP (v) Γ
                                                                              ΩB                                                      p
          l,m
                  “                                              ”
                                           p        s        f
            = L(ω)η(κ(ω)) (E + E ) , v−1
                                                                          ,     (v s , v f ) ∈ [H 1 (ΩB )]2 × H(div, ΩB ).
                                                                   ΩB
                 “                    ”
                   s      s      f
        SΓp (u) = u · ν, u · χ, u · ν

 ν: unit outer normal on ΓB = ∂ΩB , χ: a unit tangent on ΓB oriented counterclockwise,
                                  “    ”
                                    σ
 D: a positive definite matrix, a = 2ωµ



                                               Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 12/2
A Finite Element Procedure for the PSVTM-mode. I

 T h (Ω): nonoverlapping partition of Ω = Ωa ∪ ΩB into rectangles
 Ωj of diameter bounded by h

                                                      s
 For the electric vector E s and the magnetic scalar Hy we use
                                       h
       V h = {ψ ∈ H(curl, Ω) : ψ|Ωj ∈ Vj ≡ P0,1 (Ωj ) × P1,0 (Ωj )},

                                   h
       W h = {ϕ ∈ L2 (Ω) : ϕ|Ωj ∈ Wj ≡ P0 (Ωj )}.



 DOF: tangential componets of E s at the mid points of the sides of
                           s
 each Ωj and the value of Hy at the center of Ωj .

                           Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 13/2
A Finite Element Procedure for the PSVTM-mode. II


 For each component of the solid displacement us we use the
 space N C h :

       R = [−1, 1]2 ,      N C(R) = Span{1, x, z , α(ˆ) − α(ˆ)}
                                            ˆ ˆ x           z

                                      5 4          2
                            α(ˆ) = x − x .
                              x    ˆ    ˆ
                                      3
  DOF: values of us at the midpoint of each edge of R.


   N C h = {v : vj = v|Ωj ∈ N C h (Ωj ), vj (ξjk ) = vk (ξjk ) ∀(j, k)}.

 The space N C h yields about half the numerical dispersion than
 that of standard bilinear elements.
                             Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 14/2
A Finite Element Procedure for the PSVTM-mode. III




 For the fluid displacement uf vector we use:

          Mh = {v ∈ H(div, ΩB ) : vj = v|Ωj ∈ P1,0 × P0,1 (Ωj )}

 DOF: value of uf · ν at the mid points of each side of Ωj .




                           Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 15/2
A Finite Element Procedure for the PSVTM-mode. IV

 Find (E s,h , Hy , us,h , uf,h ) ∈ V h × W h × [N C h ]2 × Mh such that
                s,h



     (σE s,h , ψ) − (Hy , curlψ) + a(1 − i)E s,h · χ, ψ · χ
                      s,h

                             s,h
     +(curlE s,h , ϕ) + (iωµHy , ϕ)
          = − (σ s E p , ψ) ,       (ψ, ϕ) ∈ V h × W h


     −ω 2 ρb us,h , v s   ΩB
                               − ω 2 ρf uf,h , v s                    ΩB
                                                                              − ω 2 ρf us,h , v f                          ΩB

     + η(κ(ω))−1 iωuf,h , v f         ΩB
                                               +                  τlm (uh ), εlm (v s )                        ΩB
                                                      l,m

     − pf (uh ), ∇ · v f   ΩB
                                 + iω DSΓp (uh ), SΓP (v)                                         Γp

     = L(ω)η(κ(ω))−1 E p + E s,h , v f                                ΩB
                                                                              , (v s , v f ) ∈ [N C h ]2 × Mh .

                                   Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 16/2
A Priori Error Estimates for the PSVTM-mode.

 A priori   error estimate:

 Theorem:
 for ω > 0 and sufficiently small h > 0,

             E s − E s,h   0   + curl(E s − E s,h )                         0
                                                                                   s    s,h
                                                                                + Hy − Hy                                0

                 + us − us,h       1,h,ΩB       + uf − uf,h                           0,ΩB

                 + (E s − E s,h ) · χ            0,Γ     + us − us,h                         0,ΓB

                      ≤ C(ω) h                Es        1 + curl E s                         1
                                                                                                  s
                                                                                               + Hy                  1

                           + h1/2        us       2,ΩB        + uf                1,ΩB        + ∇ · uf                     1,Ωb            .




                                    Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 17/2
Numerical Electroseismic Modeling for the PSVTM-mode. I



 Gas hydrates (GH) are crystalline molecular complexes composed of
 water and natural gas, mainly methane, which form under certain
 conditions of low temperature, high pressure and gas concentration. They
 are found in permafrost regions and seafloor sediments along the
 continental margins.



  GH are ice-like structures within the pore space that cause strong
 changes in the conductivity and phase velocities of the seismic waves,
 making possible to detect its presence using electroseismic data.




                             Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 18/2
Numerical Electroseismic Modeling for the PSVTM-mode. II
                                       x

                                 z
                                                                               Vp=2200 m/s
  800                                                                                      Sigma=1 S/m
                                                                               Vs=1250 m/s
               Sandstone − Brine

                                                                               Vp=4100 m/s
  200          Sandstone − Brine + GH                                          Vs=2300 m/s Sigma=0.008 S/m

  200         Sandstone − Brine


 Brine-saturated sandstone subsurface model including a layer containing BRINE + GH in the pore space. The
 black dot at the top indicates the infinite solenoid source in the y− direction located at (x                               = 0, z = 0). The
 geophones are located near the earth surface, indicated with inverted triangles. Distances are given in meters.
 The model was discretized with a 2241 × 1121 mesh and solved for 100 frequencies with a main source
 frequency of 20 Hz. The reference mesh size was h∗          = 1.75 m and the diffusion length at 20 Hz was 10 m.
 The CPU time running with 20 processors in the steele parallel computer at Purdue University was 7 hours.
                                            Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 19/2
z-component of solid acceleration (left) and fluid pressure (right) at t=0.2 s. GH-saturation Sgh = 0.8


        -1000    -500       0       500       1000                      -1000           -500               0              500            1000
        0                                               4e-17           0                                                                          8e-09

                                                        3e-17                                                                                      6e-09
      200                                                           200
                                                        2e-17                                                                                      4e-09
      400                                                           400
                                                        1e-17                                                                                      2e-09

      600                                               0           600                                                                            0

                                                      -1e-17                                                                                      -2e-09
      800                                                           800
                                                      -2e-17                                                                                      -4e-09
     1000                                                          1000
                                                      -3e-17                                                                                      -6e-09

     1200                                             -4e-17 1200                                                                                 -8e-09

  The snapshots show upward and downward travelling wavefronts generated by the presence of the the
  GH-bearing layer.




                                          Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 20/2
z-component of solid acceleration(left) and fluid pressure (right) at t=0.3 s. GH-saturation Sgh = 0.8




        -1000    -500       0       500        1000                      -1000           -500               0              500            1000
        0                                                3e-17           0                                                                          6e-09


      200                                                2e-17       200                                                                            4e-09


      400                                                1e-17       400                                                                            2e-09


      600                                                0           600                                                                            0


      800                                              -1e-17        800                                                                           -2e-09


     1000                                              -2e-17 1000                                                                                 -4e-09


     1200                                              -3e-17 1200                                                                                 -6e-09

  The snapshots show upward travelling wavefronts generated by the presence of the the GH-bearing layer.



                                           Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 21/2
Traces of z-component of solid acceleration for GH-saturation Sgh =0.8

                                                          Arrival from the
                                                          bottom of the
                                                          GH−bearing layer



                                                                                         R15

                                                                                         R13

                                                                                         R11

                                                                                         R9
                             [m/s2]




                                                                                        R7

                                                                                        R5

                                                                                        R3
                                                                                        R1



                                      0   0.2              0.4                0.6                0.8
                                                                           time [s]

                                        Arrival from the
                                        top of the
 Notice that the amplitude of the arrivals increase (smaller receiver numbers) as we move away from the location
                                        GH−bearing layer
 of the infinite solenoide source located at x=0.




                                            Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 22/2
Traces of z-component of solid acceleration for different GH saturation


         The electroseismic experiment is able to discriminate between different GH saturations




          8e-13                                                                        8e-13
                                  GH Sat = 0.8                                                                               GH Sat = 0.8
                                  GH Sat = 0.4                                                                               GH Sat = 0.4
          6e-13                                                                        6e-13

          4e-13                                                                        4e-13

          2e-13                                                                        2e-13
[m/s2]




                                                                             [m/s2]
               0                                                                            0

         -2e-13                                                                       -2e-13

         -4e-13                                                                       -4e-13

         -6e-13                                                                       -6e-13

         -8e-13                                                                       -8e-13
                   0     0.2     0.4      0.6     0.8                                           0.2     0.25        0.3       0.35        0.4 0.45              0.5
                                           time [s]                                                                                        time [s]




                                                   Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 23/2
Traces of z-component of solid acceleration for different layer widths.


                5e-13                                                                                6e-13
                                             Layer 200 m                                                                             Layer 200 m
                4e-13                         Layer 50 m                                                                              Layer 50 m
                                              Layer 10 m                                                                              Layer 10 m
                3e-13                                                                                4e-13

                2e-13
                                                                                                     2e-13
                1e-13
       [m/s2]




                                                                                            [m/s2]
                    0
                                                                                                         0
                -1e-13
                -2e-13
                                                                                                     -2e-13
                -3e-13
                -4e-13                                                                               -4e-13
                -5e-13
                -6e-13                                                                               -6e-13
                         0       0.2       0.4    0.6     0.8                                                  0.25       0.3         0.4
                                                                                                                                   0.35     0.45                 0.5
 The arrival at t ≈          .3 sec. is   associated with wavefronts
                                                   time [s]                     generated at the top of the                  GH-saturated layer.
                                                                                                                                        time [s]              The 10
 and 50 m width layers are seen as single reflectors. The second arrival at about .38 for the 200 m layer width
 case corresponds to the wavefront generated at the bottom of the GH bearing layer. Notice that a seismic
 source of the same frequency can not see the 10 m width layer.




                                                           Finite element approximation of coupled seismic and electromagnetic waves in gas hydrate-bearing sediments – p. 24/2

								
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