Equivalent viscoelastic solids for heterogeneous fluid-saturated porous rocks

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. N1-N13 ◽  
Author(s):  
J. Germán Rubino ◽  
Claudia L. Ravazzoli ◽  
Juan E. Santos
Keyword(s):  
Monte Carlo ◽  
Seismic Waves ◽  
Fluid Pressure ◽  
Porous Rocks ◽  
Quality Factors ◽  
Shear Tests ◽  
Viscoelastic Solids ◽  
Shear Moduli ◽  
Phase Velocities ◽  
Biot’S Equations

Different theoretical and laboratory studies on the propagation of elastic waves in real rocks have shown that the presence of heterogeneities larger than the pore size but smaller than the predominant wavelengths (mesoscopic-scale heterogeneities) may produce significant attenuation and velocity dispersion effects on seismic waves. Such phenomena are known as “mesoscopic effects” and are caused by equilibration of wave-induced fluid pressure gradients. We propose a numerical upscaling procedure to obtain equivalent viscoelastic solids for heterogeneous fluid-saturated rocks. It consists in simulating oscillatory compressibility and shear tests in the space-frequency domain, which enable us to obtain the equivalent complex undrained plane wave and shear moduli of the rock sample. We assume that the behavior of the porous media obeys Biot’s equations and use a finite-element procedure to approximate the solutions of the associated boundary value problems. Also, because at mesoscopic scales rock parameter distributions are generally uncertain and of stochastic nature, we propose applying the compressibility and shear tests in a Monte Carlo fashion. This facilitates the definition of average equivalent viscoelastic media by computing the moments of the equivalent phase velocities and inverse quality factors over a set of realizations of stochastic rock parameters described by a given spectral density distribution. We analyzed the sensitivity of the mesoscopic effects to different kinds of heterogeneities in the rock and fluid properties using numerical examples. Also, the application of the Monte Carlo procedure allowed us to determine the statistical properties of phase velocities and inverse quality factors for the particular case of quasi-fractal heterogeneities.

Attenuation and dispersion of compressional waves in fluid‐filled porous rocks with partial gas saturation (White model)—Part II: Results

Geophysics ◽  
1979 ◽  
Vol 44 (11) ◽  
pp. 1789-1805 ◽  
Author(s):  
N. C. Dutta ◽  
H. Odé
Keyword(s):  
Attenuation Coefficient ◽  
Seismic Waves ◽  
Fluid Pressure ◽  
Porous Solids ◽  
Type I ◽  
Approximate Theory ◽  
Type Ii ◽  
Porous Rocks ◽  
Gas Saturation ◽  
Quantitative Results

In this investigation, Biot’s (1962) theory for wave propagation in porous solids is applied to study the velocity and attenuation of compressional seismic waves in partially gas‐saturated porous rocks. The Physical model, proposed by White (1975), is solved rigorously by using Biot’s equations which describe the coupled solid‐fluid motion of a porous medium in a systematic way. The quantitative results presented here are based on the theory described in Dutta and Odé (1979, this issue). We removed several of White’s questioned approximations and examined their effects on the quantitative results. We studied the variation of the attenuation coefficient with frequency, gas saturation, and size of gas inclusions in an otherwise brine‐filled rock. Anomalously large absorption (as large as 8 dB/cycle) at the exploration seismic frequency band is predicted by this model for young, unconsolidated sandstones. For a given size of the gas pockets and their spacing, the attenuation coefficient (in dB/cycle) increases almost linearly with frequency f to a maximum value and then decreases approximately as 1/√f. A sizable velocity dispersion (of the order of 30 percent) is also predicted by this model. A low gas saturation (4–6 percent) is found to yield high absorption and dispersion. An analysis of all of the field variables (stresses and displacements) is presented in terms of Biot’s type I (the classical compressional) wave and type II (the diffusion) wave. It is pointed out that the dissipation of energy in this model is mainly due to the relative fluid flow from the type II wave. From our formulation, many of White’s equations can be derived as suitable approximations, and it is shown that the discontinuity in fluid pressure assumed by White at the gas‐water interface is the discontinuity in the fluid pressure contribution by the type II wave. Our quantitative results are in reasonably good agreement with White’s (1975) approximate theory. However, the phase velocities computed by White’s approximate treatment do not approach the correct zerofrequency limit (Gassmann‐Wood) when compared to the present theory. Most of these disagreements disappear if the corrections to White’s theory as suggested by Dutta and Seriff (1979, this issue) are incorporated.

Attenuation and dispersion of compressional waves in fluid‐filled porous rocks with partial gas saturation (White model)—Part I: Biot theory

Geophysics ◽  
1979 ◽  
Vol 44 (11) ◽  
pp. 1777-1788 ◽  
Author(s):  
N. C. Dutta ◽  
H. Odé
Keyword(s):  
Seismic Waves ◽  
Equations Of Motion ◽  
Fluid Pressure ◽  
Present Theory ◽  
Biot Theory ◽  
Type I ◽  
Porous Rocks ◽  
Water Contact ◽  
Coupled Equations ◽  
Attenuation And Dispersion

An exact theory of attenuation and dispersion of seismic waves in porous rocks containing spherical gas pockets (White model) is presented using the coupled equations of motion given by Biot. Assumptions made are (1) the acoustic wavelength is long with respect to the distance between gas pockets and their size, and (2) the gas pockets do not interact. Thus, the present theory essentially is quite similar to that proposed by White (1975), but the problem of the radially oscillating gas pocket is solved in a more rigorous manner by means of Biot’s theory (1962). The solid‐fluid coupling is automatically included, and the model is solved as a boundary value problem requiring all radial stresses and displacements to be continuous at the gas‐brine interface. Thus, we do not require any assumed fluid‐pressure discontinuity at the gas‐water contact, such as the one employed by White (1975). We have also presented an analysis of all of the field variables in terms of Biot’s type I (the classical compressional) wave and, type II (the diffusion) wave. Our quantitative results are presented in Dutta and Odé (1979, this issue).

scholarly journals Seismically induced changes in groundwater levels and temperatures following the ML5.8 (ML5.1) Gyeongju earthquake in South Korea

2021 ◽  
Author(s):  
Soo-Hyoung Lee ◽  
Jae Min Lee ◽  
Sang-Ho Moon ◽  
Kyoochul Ha ◽  
Yongcheol Kim ◽  
...  
Keyword(s):  
South Korea ◽  
Groundwater Level ◽  
Seismic Waves ◽  
Groundwater Resources ◽  
Fluid Pressure ◽  
Water Levels ◽  
Aquifer System ◽  
Groundwater Levels ◽  
Monitoring Wells ◽  
Gyeongju Earthquake

AbstractHydrogeological responses to earthquakes such as changes in groundwater level, temperature, and chemistry, have been observed for several decades. This study examines behavior associated with ML 5.8 and ML 5.1 earthquakes that occurred on 12 September 2016 near Gyeongju, a city located on the southeast coast of the Korean peninsula. The ML 5.8 event stands as the largest recorded earthquake in South Korea since the advent of modern recording systems. There was considerable damage associated with the earthquakes and many aftershocks. Records from monitoring wells located about 135 km west of the epicenter displayed various patterns of change in both water level and temperature. There were transient-type, step-like-type (up and down), and persistent-type (rise and fall) changes in water levels. The water temperature changes were of transient, shift-change, and tendency-change types. Transient changes in the groundwater level and temperature were particularly well developed in monitoring wells installed along a major boundary fault that bisected the study area. These changes were interpreted as representing an aquifer system deformed by seismic waves. The various patterns in groundwater level and temperature, therefore, suggested that seismic waves impacted the fractured units through the reactivation of fractures, joints, and microcracks, which resulted from a pulse in fluid pressure. This study points to the value of long-term monitoring efforts, which in this case were able to provide detailed information needed to manage the groundwater resources in areas potentially affected by further earthquakes.

Seismic wave attenuation due to fluid pressure diffusion at the mesoscopic scale: an experimental and numerical study

2021 ◽  
Author(s):  
Samuel Chapman ◽  
Jan V. M. Borgomano ◽  
Beatriz Quintal ◽  
Sally M. Benson ◽  
Jerome Fortin
Keyword(s):  
Seismic Wave ◽  
Seismic Waves ◽  
Wave Attenuation ◽  
Fluid Pressure ◽  
Fluid Distribution ◽  
Mesoscopic Scale ◽  
Seismic Wave Attenuation ◽  
X Ray ◽  
Pressure Diffusion ◽  
Attenuation And Dispersion

<p>Monitoring of the subsurface with seismic methods can be improved by better understanding the attenuation of seismic waves due to fluid pressure diffusion (FPD). In porous rocks saturated with multiple fluid phases the attenuation of seismic waves by FPD is sensitive to the mesoscopic scale distribution of the respective fluids. The relationship between fluid distribution and seismic wave attenuation could be used, for example, to assess the effectiveness of residual trapping of carbon dioxide (CO2) in the subsurface. Determining such relationships requires validating models of FPD with accurate laboratory measurements of seismic wave attenuation and modulus dispersion over a broad frequency range, and, in addition, characterising the fluid distribution during experiments. To address this challenge, experiments were performed on a Berea sandstone sample in which the exsolution of CO2 from water in the pore space of the sample was induced by a reduction in pore pressure. The fluid distribution was determined with X-ray computed tomography (CT) in a first set of experiments. The CO2 exosolved predominantly near the outlet, resulting in a heterogeneous fluid distribution along the sample length. In a second set of experiments, at similar pressure and temperature conditions, the forced oscillation method was used to measure the attenuation and modulus dispersion in the partially saturated sample over a broad frequency range (0.1 - 1000 Hz). Significant P-wave attenuation and dispersion was observed, while S-wave attenuation and dispersion were negligible. These observations suggest that the dominant mechanism of attenuation and dispersion was FPD. The attenuation and dispersion by FPD was subsequently modelled by solving Biot’s quasi-static equations of poroelasticity with the finite element method. The fluid saturation distribution determined from the X-ray CT was used in combination with a Reuss average to define a single phase effective fluid bulk modulus. The numerical solutions agree well with the attenuation and modulus dispersion measured in the laboratory, supporting the interpretation that attenuation and dispersion was due to FPD occurring in the heterogenous distribution of the coexisting fluids. The numerical simulations have the advantage that the models can easily be improved by including sub-core scale porosity and permeability distributions, which can also be determined using X-ray CT. In the future this could allow for conducting experiments on heterogenous samples.</p>

Toward phenomenological universality of the mechanical behavior of arbitrarily anisotropic porous rocks

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA167-WA183 ◽  
Author(s):  
Patrick N. J. Rasolofosaon
Keyword(s):  
Mechanical Behavior ◽  
Empirical Relation ◽  
Fluid Pressure ◽  
Porous Rocks ◽  
Porous Network ◽  
Hooke’S Law ◽  
Fluid Content ◽  
Elastic Hysteresis ◽  
Hydraulic Hysteresis ◽  
Hooke's Law

The great diversity of the microstructures of rocks impedes the use of a universal rock physics model with idealized geometry to correctly describe the mechanical behavior of rocks. In this quest for universality, by ignoring the detailed description of the causes of the observed phenomenon and only focusing on the empirical relation between the cause (applied stress) and the effect (resulting strain), phenomenological models such as the linear elastic Hooke’s law roughly describe the mechanical behavior of rocks of contrasted microstructures. However, in detail, numerous laboratory experiments covering broad frequency and strain ranges (both typically more than eight orders of magnitude) on various types of rocks have also shown deviations from Hooke’s law due to anisotropy, frequency dependence, nonlinearity, possibly with the presence of hysteresis, and poroelasticity. A phenomenological model has been recently proposed that synthesizes all these behaviors in a single model, but unfortunately does not integrate the porous nature of rocks. The new model is based on a reformulation in modified spectral decomposition of the previous work using the 7D poroelastic tensor linking the dynamic parameters (i.e., the six stress components and fluid pressure) and the kinematic parameters (i.e., the six strain components and the local increase of fluid content ζ). In addition to the elastic hysteresis of the stress-strain curves, the model also predicts the existence of a second hysteresis, or hydraulic hysteresis, of the curve fluid pressure p versus fluid content ζ, qualitatively similar to the first one. Indeed, the elastic hysteresis is due to the opening and the closure of some compliant pores at different stress levels. These pores represent possible access radii for the saturating fluid; the hysteresis in the geometry of the porous network also induces the hydraulic hysteresis in the p-ζ curves.

scholarly journals Fluid pressure diffusion effects on the excess compliance matrix of porous rocks containing aligned fractures

2020 ◽  
Vol 222 (1) ◽  
pp. 715-733
Author(s):  
Gabriel A Castromán ◽  
Nicolás D Barbosa ◽  
J Germán Rubino ◽  
Fabio I Zyserman ◽  
Klaus Holliger
Keyword(s):  
Seismic Velocity ◽  
Fractured Rock ◽  
Finite Thickness ◽  
Practical Importance ◽  
Fluid Pressure ◽  
Porous Rocks ◽  
Compliance Matrix ◽  
Pressure Diffusion ◽  
Reservoir Permeability ◽  
Wide Range

SUMMARY The presence of sets of open fractures is common in most reservoirs, and they exert important controls on the reservoir permeability as fractures act as preferential pathways for fluid flow. Therefore, the correct characterization of fracture sets in fluid-saturated rocks is of great practical importance. In this context, the inversion of fracture characteristics from seismic data is promising since their signatures are sensitive to a wide range of pertinent fracture parameters, such as density, orientation and fluid infill. The most commonly used inversion schemes are based on the classical linear slip theory (LST), in which the effects of the fractures are represented by a real-valued diagonal excess compliance matrix. To account for the effects of wave-induced fluid pressure diffusion (FPD) between fractures and their embedding background, several authors have shown that this matrix should be complex-valued and frequency-dependent. However, these approaches neglect the effects of FPD on the coupling between orthogonal deformations of the rock. With this motivation, we considered a fracture model based on a sequence of alternating poroelastic layers of finite thickness representing the background and the fractures, and derived analytical expressions for the corresponding excess compliance matrix. We evaluated this matrix for a wide range of background parameters to quantify the magnitude of its coefficients not accounted for by the classical LST and to determine how they are affected by FPD. We estimated the relative errors in the computation of anisotropic seismic velocity and attenuation associated with the LST approach. Our analysis showed that, in some cases, considering the simplified excess compliance matrix may lead to an incorrect representation of the anisotropic response of the probed fractured rock.

Inversion for Biot-consistent material properties in subresonant oscillation experiments with fluid-saturated porous rock

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. MR67-MR79 ◽  
Author(s):  
Igor B. Morozov ◽  
Wubing Deng
Keyword(s):  
Traveling Wave ◽  
Seismic Waves ◽  
Rock Specimen ◽  
Low Frequency ◽  
P Wave ◽  
Axial Deformation ◽  
Porous Rock ◽  
Frequency Dependent ◽  
Shear Moduli ◽  
Young’S Moduli

To quantitatively interpret the results of a subresonant laboratory or numerical experiment with wet porous rock, it is insufficient to merely state the measured frequency-dependent viscoelastic moduli and [Formula: see text]-factors. The measured properties are apparent, i.e., dependent on the experimental setup such as the length of the sample and boundary conditions for pore flows. To reveal the true properties of the material, all experimental factors need to be accurately modeled and corrected for. Here, such correction is performed by developing an effective Biot’s model for the material and using it to predict driven oscillations of a cylindrical rock specimen. The model explicitly describes elastic and inertial effects, Biot’s flows, and viscous internal friction within the solid frame and pore fluid, and it approximates squirt and other wave-induced flow effects. The model predicts the dynamic permeability of the specimen, fast (traveling) and slow (diffusive) P- and axial-deformation waves, and it allows accurate modeling of any other ultrasonic or seismic-frequency experiments with the same rock. To illustrate the approach, attenuation and dispersion data from two laboratory and numerical experiments with sandstones are inverted for effective, frequency-dependent moduli of drained sandstone. Several observations from this inversion may be useful for interpreting experiments with porous rock. First, Young’s moduli measured in a short rock cylinder differ from those in a traveling wave within an infinite rod. In particular, for the modeled 8 cm long rock specimen, modulus dispersion and attenuation ([Formula: see text]) are approximately 10 times greater than for a traveling wave. Second, P-wave moduli cannot be derived from the measured Young’s and shear moduli by using conventional (visco)elastic relations. Third, because of wavelengths comparable with the size of the specimen, slow waves contribute to its quasistatic and low-frequency behaviors. Similar observations should also apply to seismic waves traveling through approximately 10 cm layering in the field.

Incorporating mechanisms of fluid pressure relaxation into inclusion‐based models of elastic wave velocities

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1173-1181 ◽  
Author(s):  
S. Richard Taylor ◽  
Rosemary J. Knight
Keyword(s):  
Elastic Wave ◽  
Water Saturation ◽  
Laboratory Data ◽  
Fluid Pressure ◽  
P Wave ◽  
Porous Rocks ◽  
Low Frequencies ◽  
Wave Velocities ◽  
Exact Agreement ◽  
Flow Through

Our new method incorporates fluid pressure communication into inclusion‐based models of elastic wave velocities in porous rocks by defining effective elastic moduli for fluid‐filled inclusions. We illustrate this approach with two models: (1) flow between nearest‐neighbor pairs of inclusions and (2) flow through a network of inclusions that communicates fluid pressure throughout a rock sample. In both models, we assume that pore pressure gradients induce laminar flow through narrow ducts, and we give expressions for the effective bulk moduli of inclusions. We compute P‐wave velocities and attenuation in a model sandstone and illustrate that the dependence on frequency and water‐saturation agrees qualitatively with laboratory data. We consider levels of water saturation from 0 to 100% and all wavelengths much larger than the scale of material heterogeneity, obtaining near‐exact agreement with Gassmann theory at low frequencies and exact agreement with inclusion‐based models at high frequencies.

The effect of pore pressure depletion and injection cycles on ultrasonic velocity and quality factor in a quartz sandstone

Geophysics ◽  
2007 ◽  
Vol 72 (2) ◽  
pp. E43-E51 ◽  
Author(s):  
P. Frempong ◽  
A. Donald ◽  
S. D. Butt
Keyword(s):  
Pore Pressure ◽  
Effective Stress ◽  
Axial Strain ◽  
Fluid Pressure ◽  
Porous Rocks ◽  
Confining Stress ◽  
Viscoelastic Deformation ◽  
Experimental Conditions ◽  
Pore Pressures ◽  
Pressure Depletion

Passing seismic waves generate transient pore-pressure changes that influence the velocity and attenuation characteristics of porous rocks. Compressional ultrasonic wave velocities [Formula: see text] and quality factors [Formula: see text] in a quartz sandstone were measured under cycled pore pressure and uniaxial strain conditions during a laboratory simulated injection and depletion process. The objectives were to study the influence of cyclical loading on the acoustic characteristics of a reservoir sandstone and to evaluate the potential to estimate pore-fluid pressure from acoustic measurements. The values of [Formula: see text] and [Formula: see text] were confirmed to increase with effective stress increase, but it was also observed that [Formula: see text] and [Formula: see text] increased with increasing pore pressure at constant effective stress. The effective stress coefficient [Formula: see text] was found to be less thanone and dependent on the pore pressure, confining stress, and load. At low pore pressures, [Formula: see text] approached one and reduced nonlinearly at high pore pressures. The change in [Formula: see text] and [Formula: see text] with respect to pore pressure was more pronounced at low versus high pore pressures. However, the [Formula: see text] variation with pore pressure followed a three-parameter exponential rise to a maximum limit whereas [Formula: see text] had no clear limit and followed a two-parameter exponential growth. Axial strain measurements during the pore-pressure depletion and injection cycles indicated progressive viscoelastic deformation in the rock. This resulted in an increased influence on [Formula: see text] and [Formula: see text] with increasing pore-pressure cycling. The value [Formula: see text] was more sensitive in responding to the loading cycle and changes in pore pressures than [Formula: see text]; thus, [Formula: see text] may be a better indicator for time-lapse reservoir monitoring than [Formula: see text]. However, under the experimental conditions, [Formula: see text] was unstable and difficult to measure at low effective stress.

scholarly journals Biot's equations-based reservoir parameter inversion using deep neural networks

2021 ◽  
Vol 18 (6) ◽  
pp. 862-874
Author(s):  
Fansheng Xiong ◽  
Heng Yong ◽  
Hua Chen ◽  
Han Wang ◽  
Weidong Shen
Keyword(s):  
High Efficiency ◽  
Synthetic Data ◽  
Rock Physics ◽  
Seismic Attributes ◽  
Inversion Method ◽  
Reservoir Properties ◽  
Shear Moduli ◽  
Parameter Inversion ◽  
Reservoir Parameter ◽  
Biot’S Equations

Abstract Reservoir parameter inversion from seismic data is an important issue in rock physics. The traditional optimisation-based inversion method requires high computational expense, and the process exhibits subjectivity due to the nonuniqueness of generated solutions. This study proposes a deep neural network (DNN)-based approach as a new means to analyse the sensitivity of seismic attributes to basic rock-physics parameters and then realise fast parameter inversion. First, synthetic data of inputs (reservoir properties) and outputs (seismic attributes) are generated using Biot's equations. Then, a forward DNN model is trained to carry out a sensitivity analysis. One can in turn investigate the influence of each rock-physics parameter on the seismic attributes calculated by Biot's equations, and the method can also be used to estimate and evaluate the accuracy of parameter inversion. Finally, DNNs are applied to parameter inversion. Different scenarios are designed to study the inversion accuracy of porosity, bulk and shear moduli of a rock matrix considering that the input quantities are different. It is found that the inversion of porosity is relatively easy and accurate, while more information is needed to make the inversion more accurate for bulk and shear moduli. From the presented results, the new approach makes it possible to realise accurate and pointwise inverse modelling with high efficiency for actual data interpretation and analysis.

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