Sediments containing gas hydrate dispersed in the pore space are known to show a characteristic seismic anomaly which is a high attenuation along with increasing seismic velocities. Currently, this observation cannot be fully explained albeit squirt-flow type mechanisms on the microscale have been speculated to be the cause. Recent major findings from in situ experiments, using the “gas in excess” and “water in excess” formation method, and coupled with high-resolution synchrotron-based X-ray micro-tomography, have revealed the systematic presence of thin water films between the quartz grains and the encrusting hydrate. The data obtained from these experiments underwent an image processing procedure to quantify the thicknesses and geometries of the aforementioned interfacial water films. Overall, the water films vary from sub-micrometer to a few micrometers in thickness. In addition, some of the water films interconnect through water bridges. This geometrical analysis is used to propose a new conceptual squirt flow model for hydrate bearing sediments. A series of numerical simulations is performed considering variations of the proposed model to study seismic attenuation caused by such thin water films. Our results support previous speculation that squirt flow can explain high attenuation at seismic frequencies in hydrate bearing sediments, but based on a conceptual squirt flow model which is geometrically different than those previously considered.
Important mechanisms of wave attenuation in fluid-saturated porous media from seismic to ultrasonic frequencies, include friction between grain boundaries (Winkler and Nur, 1982), global flow or Biot's mechanism (Biot, 1962), and wave-induced fluid flow at mesoscopic and microscopic scales (e.g., Müller et al., 2010). At the mesoscopic scale, patchy saturation and fractures are the most prominent causes of wave-induced fluid flow (White, 1975; White et al., 1975; Brajanovski et al., 2005; Tisato and Quintal, 2013; Quintal et al., 2014). At the microscopic scale, wave-induced fluid flow is commonly referred to as squirt flow and typically occurs between interconnected microcracks or between grain contacts and stiffer pores (O'Connell and Budiansky, 1977; Murphy et al., 1986; Mavko and Jizba, 1991; Sams et al., 1997; Adelinet et al., 2010; Gurevich et al., 2010). The attenuation caused by global flow as well as that caused by wave-induced fluid flow at microscopic or mesoscopic scales are frequency dependent. While the latter can have a strong effect at seismic frequencies (Pimienta et al., 2015; Subramaniyan et al., 2015; Chapman et al., 2016), global flow will only cause significant attenuation in reservoir rocks at ultrasonic frequencies or higher (e.g., Bourbie et al., 1987). The attenuation caused by friction between grain boundaries is, however, frequency independent and basically depends on the confining pressure and the strain imposed by the propagating wave (Winkler and Nur, 1982). Its effect is expected to be small for the correspondingly small strains caused by seismic waves used in exploration and reservoir geophysics. Furthermore, the attenuation caused by wave-induced fluid flow tends to be linearly superposed to that due to friction between grain boundaries, as shown by Tisato and Quintal (2014).
Gas hydrates (GH) are ice-like structures comprised of gas molecules
entrapped by water molecules (Sloan and Koh, 2008). The widespread global
occurrence of GH and the fact that 1
In the search for GH reservoirs, the attenuation of seismic waves caused by
the pore fluids might be an important survey tool (e.g., Bellefleur
et al., 2007). However, little effort has been directed toward studying its
effects for unconsolidated sediments hosting GH in a rather dispersed manner.
GH forming in the pore space of unconsolidated sediments at given
Review of the established conceptual
models (grains are grey and GH are orange), with
Quantifying GH saturation levels through geophysical exploration techniques is, however, not straightforward as there are still open questions on GH formation, its microstructure, and its distribution in the natural settings. Additionally, the recovery of unaltered natural GH samples is hampered due to their fast decomposition under ambient conditions. Therefore, various researchers have attempted to mimic the natural environment of GH-bearing sedimentary matrices in laboratory experiments (Berge et al., 1999; Ecker et al., 2000; Dvorkin et al., 2003; Yun et al., 2005; Spangenberg and Kulenkampff, 2006; Priest et al., 2006, 2009; Best et al., 2010, 2013; Hu et al., 2010; Li et al., 2011; Zhang et al., 2011; Dai et al., 2012; Schicks et al., 2013). The results of this collective effort established a number of conceptual models for the role of GH embedded in its sedimentary matrix (Fig. 1). Nevertheless, these approximations are currently unsatisfactory. Although it has been suggested that all hydrate habits known from laboratory investigations involving synthetic samples also occur in nature (Spangenberg et al., 2015), none of those simplified models can yield accurate predictions of GH saturations from field electric resistivity or seismic data alone (Waite et al., 2009; Dai et al., 2012).
Chaouachi et al. (2015) performed in situ experiments based on different
formation mechanisms, including the “water in excess” and the “gas in
excess” methods, to form gas hydrates in various sedimentary matrices. The
in situ experiments coupled with high-resolution synchrotron-based X-ray
micro-tomography (SRXCT) yielded 3-D images of sub-micrometer spatial
resolution. Using the “gas in excess” method, the water present in the
samples weds the grain surfaces and transforms into GH at the required
For this study, the SRXCT data presented by Chaouachi et al. (2015) underwent an image processing workflow in order to quantify the thicknesses of the thin interfacial water films. Based on the obtained results, we introduce a conceptual model for GH-bearing sediments to numerically study squirt flow. Our numerical simulations allow for the dispersion of the P wave modulus and the frequency-dependent P wave attenuation. The results demonstrate the high levels of seismic attenuation/dispersion that a range of variations of our conceptual model can cause. Additionally, our results support the suggestions that the estimation of GH saturation for GH occurring in a rather dispersed manner could be accomplished by using seismic wave attenuation as a tool for indirect geophysical quantification (Guerin and Goldberg, 2002; Priest et al., 2006; Best et al., 2013; Marin-Moreno et al., 2017).
Chaouachi et al. (2015) conducted various in situ experiments coupled with synchrotron-based tomography at the TOMCAT beamline of the Paul Scherrer Institute in Villigen, Switzerland. The aim was to study the formation process and distribution of gas hydrates in various matrices, such as pure quartz sand and glass beads, as well as mixtures of quartz sand with clay minerals. These in situ experiments were conducted using an experimental setup that allowed for high pressures and low temperatures. Further details are given by Chaouachi et al. (2015), Falenty et al. (2015), and Sell et al. (2016).
Raw (unfiltered) 2-D image in
For this study, the SRXCT data obtained from the abovementioned in situ
experiments focused on samples containing pure natural quartz sand sieved at
a 200–300
Volume-rendered phases in
a representative image sample. For a better visualization the phases are
introduced step-by-step, with
The broad range of grey scale values of the filtered images were classified using watershed segmentation combined with region growing tools from the Avizo Fire 7 (FEI, France) and Fiji software packages. In the present study, we determined the thickness variation and geometry of the water film (Fig. 3). Following the image enhancement and segmentation process described by Sell et al. (2016), the segmented data illustrate the characteristics and appearance of the phases distributed in the samples (Fig. 4). Moreover, the high resolution of the data enables us to obtain 3-D images in which particular details, such as water bridges connecting two interfacial water films, are detectable (Fig. 5). With information collected from the 3-D data, our proposed conceptual model involves spherical grains covered by a homogenous water film which is, in turn, embedded in non-porous hydrate. The conceptual model can be adjusted to include water bridges connecting the water films (Fig. 6) and/or isolated water pockets within the hydrate and separated from the water films.
Volume-rendered image of a representative
region of interest (ROI) of
Schemes of
To estimate frequency-dependent attenuation in the GH systems described above
we employ a hydromechanical approach (Quintal et al., 2016) based on the
conservation of momentum.
Using this general mathematical formulation (Eqs. 1 and 2), a heterogeneous
medium can be described as having an isotropic, linear elastic solid frame
and fluid-filled cavities or pores, to which a specific choice of material
parameters can be assigned. Equation (2) reduces to Hooke's law by setting
the shear viscosity
When the aforementioned heterogeneous medium is deformed, fluid pressure differences between neighbor regions induce fluid flow or, more accurately, fluid pressure diffusion, which in turn results in energy loss caused by viscous dissipation (Quintal et al., 2016). At the microscopic scale, this attenuation mechanism is commonly referred to as squirt flow (e.g., O'Connell and Budiansky, 1977; Murphy et al., 1986) and is the sole cause of attenuation in our simulations, as we neglect the inertial terms in Eqs. (1) and (2).
Our 2-D problem is equivalent to a 3-D case under plain strain conditions,
which means no strain outside the modeling plane is allowed to develop. For
the corresponding simulations, we consider the directions
Fundamental block of an idealized periodic medium representing sediment grains which are separated from the embedding GH background by a thin interfacial water film.
The triangular mesh used for the numerical model shown in Fig. 7. To distinguish between the phases: quartz is denoted with #1, GH is denoted with #2, and the interfacial water film is depicted in light blue.
The numerical solution is based on a finite-element approach in the frequency domain. We employ an unstructured triangular mesh which allows for an efficient discretization of slender heterogeneities having large aspect ratios, such as the thin interfacial water films, by strongly varying the sizes of the triangular elements (e.g., Quintal et al., 2014). A few elements across the thin interfacial water film are necessary to accurately capture the viscous dissipation in this region, while much larger elements are sufficient in the solid elastic domains. The sizes of smallest and largest elements in our meshes differ by three orders of magnitude.
To assess the P wave attenuation and modulus dispersion caused by
squirt flow, we subject a rectangular numerical model to an oscillatory test.
A sinusoidal downward displacement is applied homogeneously at the top
boundary of the numerical model. At the bottom, the displacement in the (
Similar to the 2-D problem, the solution to our 3-D problem is based on the application of an unstructured mesh with tetrahedral elements. The element sizes in our 3-D meshes also vary by about three orders of magnitude.
Many sources of squirt flow might coexist in unconsolidated sediments hosting GH, such as those resembling the conventional squirt flow models introduced by O'Connell and Budiansky (1977) for interconnected microcracks and by Murphy et al. (1986) for microcracks or grain contacts connected to spherical pores. Marin-Moreno et al. (2017) describes an integrated approach that combines the effects of some squirt flow models and other attenuation mechanisms. Here our objective diverges from that. We instead aim at studying the squirt flow phenomenon and the resulting frequency-dependent attenuation associated with a specific model, which is geometrically different from the previously mentioned conventional squirt flow models and is based on the thin interfacial water films. We thus neglect all other potential sources of attenuation.
Material properties used in the numerical simulations. The properties of quartz are based on the work of Bass (1995) and those of hydrate on Helgerud (2003).
Our 2-D numerical model domain corresponds to a fundamental block of
a periodic distribution of unconsolidated circular quartz grains dispersed in
a continuous GH background and separated from the latter by a thin
interfacial water film (Fig. 7). The subdomain representing the thin
interfacial water film is described by the corresponding properties of this
viscous fluid, while the other subdomains are described by properties of two
different elastic solids (quartz and GH). These properties are given in
Table 1 and the numerical mesh is shown in Fig. 8. We consider thicknesses of
the interfacial water film ranging from 0.1 to 1
Real part of P wave modulus,
Fluid pressure
Zoom-in of the top right quadrant of the
model shown in Fig. 9 displaying the fluid velocity components (
Zoom-in of the top right quadrant of the
model shown in Fig. 7 showing the local attenuation (
Fundamental blocks of two periodic media
representing loose sandstone grains which are separated from the embedding GH
background by a thin interfacial water film. In
Real part of P wave modulus,
The 3-D counterpart of the model shown in Fig. 7: a fundamental block of a periodic medium representing unconsolidated quartz grains which are separated from the embedding GH background by a thin interfacial water film.
Real part of P wave modulus (
The numerical results are expressed as the real part of the P wave modulus
and the P wave attenuation (
The geometry of the introduced model (Fig. 7) is different to the classical
squirt-flow geometries involving interconnected plane cracks or a plane crack
connected to a low aspect ratio pore.
To better understand how dissipation
occurs for this type of geometry, we initially focus on the fluid pressure
field
In Fig. 11 we observe the text book (e.g., Jaeger et al., 2007) parabolic profile of the fluid velocity across the interfacial water film, with larger fluid velocity in the center of the film, governed by Navier–Stokes equations. This fluid velocity is associated with an energy dissipation caused by viscous friction, shown in Fig. 12. At the boundaries of the interfacial water film, larger viscous friction explains the lower fluid velocity and larger energy dissipation, in comparison to the center of the film. The attenuation is strongly reduced towards the center of the film by a few orders of magnitude. Looking at how these fields change along the interfacial water film, we observe that the maximal velocity and attenuation (compare Figs. 11 and 12) coincide with the maximal pressure gradient (Fig. 10). In contrast, in the middle of the higher pressure and lower pressure regions, the pressure gradient is minimal causing the fluid velocity and attenuation to drop drastically.
In this subsection, a few alterations are added to the basic model illustrated in Fig. 7. These alterations are based on more detailed observations obtained from SRXCT, such as water pockets in non-porous GH or a water bridge which might occur and connect two neighboring interfacial water films (Fig. 13). The effect of these abovementioned features on the P wave modulus dispersion and attenuation (Fig. 14) is studied and compared to results obtained from corresponding models where these features have not been considered.
The inclusion of water pockets has a modest effect on the attenuation and dispersion, while it reduces the overall value of the P wave modulus, as a certain volume of GH is replaced by a much less stiff material (water). The modest increase in attenuation is associated with a more compressible effective background; no attenuation occurs within the water pockets.
The connecting water bridge introduces an additional length scale for the
dissipation process, as fluid flow and dissipation will also occur through
this relatively short and wide path. This explains the additional attenuation
peak observed at higher frequencies, while the previous peak at
This subsection considers a comparison between the results of the simulation
illustrated in Figs. 10–12, for the 2-D model shown in Fig. 7, and those of
a simulation performed on its 3-D counterpart. Our 3-D model consists of
a sphere in the middle of a cube (Fig. 15), for which a centered cross
section matches the 2-D model shown in Fig. 7. The thickness of the water
film is 1
Interfacial water films between sediment grains and the embedding GH matrix
were recently observed in GH-bearing sediments through synchrotron-based
micro-tomography at a spatial resolution down to 0.38
The numerical scheme is based on a set of coupled equations that reduce to Hooke's law in the subdomains of the model corresponding to the elastic solid materials (grains and GH) and to the quasi-static, linearized Navier–Stokes equations in the subdomains corresponding to the fluid (water). The results for our conceptual model show that the P wave attenuation peak is shifted to lower frequencies with decreasing thickness of the interfacial water film and with increasing grain size (or the length of the film), as analogously known for the microcrack aperture and length in classical squirt flow models. Furthermore, we tested the effect of inserting water pockets in an embedding GH matrix and the effect of connecting two neighboring interfacial water films through a water bridge. In general, the water bridges have a stronger effect on energy dissipation than the water pockets. Introducing such connections between neighboring interfacial water films causes a broadening of the P wave attenuation spectrum towards higher frequencies. Conversely, the presence of water pockets in the GH background only causes a slight overall increase in P wave attenuation. Although the majority of our simulations were performed for 2-D models, results of a 3-D simulation showed that 3-D effects are small for the basic 2-D models that we have considered.
Our results represent a strong base to explain fundamental processes in GH-bearing sediments and support previous speculations (Guerin and Goldberg, 2002; Dvorkin and Uden, 2004; Priest et al., 2006) that squirt flow is an important attenuation mechanism in such media, even at frequencies as low as those in the seismic range. This strengthens the perception that P wave attenuation may be used as an indirect geophysical attribute to estimate GH saturation. Nevertheless, further studies considering more realistic geometries for the microstructure of GH bearing sediments are necessary for the development of a successful strategy to estimate GH saturations, where hydrate is distributed in a dispersed manner instead of massive layers. This study represents the first attempt at understanding P wave attenuation in unconsolidated sediments which have large GH saturations. Following work will be aimed at implementing the segmented 3-D images obtained from synchrotron-based micro-tomography as a direct model input for numerical investigations, whereby realistic grain-to-grain contacts will be taken into account. The step towards more realistic structures as a model input is challenging due to the corresponding large computational demand. Furthermore, such model input requires additional segmentation steps for the 3-D images that allow for a smoothing of the stair-like resolution artifacts at the boundaries of the interfacial water films.
The data can be accessed upon request from any of the authors.
The authors declare that they have no conflict of interest.
The authors thank the staff of the GZG crystallography group headed by Werner F. Kuhs of the Georg August University, Göttingen, for their collaboration during the in situ experiments at the TOMCAT beamline (Paul Scherrer Institute in Villigen, Switzerland) in 2012 and 2013. The presented work was co-funded by the German Science Foundation (DFG grant Ke 508/20 and Ku 920/18). Edited by: Charlotte Krawczyk Reviewed by: two anonymous referees