2. Rock physics

Gassmann [4] first addressed rock physics by defining theoretical rocks based upon fractions of a sample of solid, liquid and gas. Later, more sophisticated approaches (effective medium theories: EMTs) represented rocks by additionally including pore shape (including fractures) and more quantitative descriptions (summarized by Berryman [5, 6]). These theories assume that response of a fractured or porous rock are heterogeneous on a macro-scale, but can be reproduced in a homogeneous rock that is equivalent to the former in the regime of static deformation. These additions better define the relationship among elastic parameters and material properties and therefore how they influence the propagation of seismic waves from microearthquakes. Of course, here and in subsequent discussions, we refer to general results, which sometimes vary.

Rock physics has been used to interpret recordings of active seismic sources primarily for oil and gas studies [6, 7]. Additionally, several authors have previously interpreted microearthquake studies with rock physics [8–14]. The effectiveness of seismic measurements to obtain reservoir properties has been successfully demonstrated for pore fluid pressures in country rock from injected CO2 [15, 16] and fracture densities in underground repositories [17]. Authors have also used effective medium theory to further interpret tomography [1, 18–20]. Here we attempt to utilize a more comprehensive relationship between microearthquake recordings and material and fluid properties in a geothermal environment. The basic theory of rock physics, laboratory studies, and field observations that can be applied to interpreting tomographic solutions from microearthquakes for geothermal reservoir properties is outlined.

#### 2.1 Seismic velocities

Hooke's Law shows that the strain field resulting from a generalized stress can be separated into a volume change with amplitude λ and a shape change with amplitude μ:

$$
\kappa\_{\rm ij} = \lambda \varepsilon\_{\rm ii} \partial\_{\rm ij} + 2\mu \varepsilon\_{\rm ij} \tag{1}
$$

where sij is stress and ϵii is strain.

The solution to the wave equation in terms of λ and μ specifies field observables Vp and Vs:

$$
\omega\_p = \sqrt{\frac{\lambda + 2\mu}{\rho}} = \sqrt{\frac{\mathbf{K} + \frac{4}{3}\mu}{\rho}}\tag{2}
$$

$$
\omega\_s = \sqrt{\frac{\mu}{\rho}} \tag{3}
$$

where ρ is bulk density of the material and K is bulk modulus (below).

Rock Physics Interpretation of Tomographic Solutions for Geothermal Reservoir Properties DOI: http://dx.doi.org/10.5772/intechopen.81226

#### 2.2 Elastic parameters

The elastic parameters used in this discussion can be described in terms of seismic velocity and density [7]. These solutions include density. We provide a density value of the aggregate state of the country rock with depth. Density can be removed by division from the solution for shear velocity (Eq. (3)). The parameters are as follows:

Poisson's ratio σ; the ratio of compressional or tensional strain to strain in orthogonal directions.

$$\sigma = \frac{V\_p^2 - 2V\_s^2}{2\left(V\_p^2 - V\_s^2\right)}.\tag{4}$$

' s ratio is a valuable tool because it can be calculated from velocities alone and is insensitive to density variations caused by lithology. Seismologists have s ratio interchangeably. However, it is evident s ratio, but over the ' Poisson' traditionally used Vp/Vs and Poisson that there is not a linear relation between Vp/Vs and Poisson range of values generally observed in seismic data (0.2 < σ < 0.3) the relationship is essentially linear. Poisson's ratio can range as low as 0.1 for foam, concrete, and dry, gas-saturated sands and as high as 0.5 for a perfectly elastic material, such as rubber at low strain. Cork has a value of 0.0. Effective medium theories (EMT) suggest that Poisson's ratio tends to vary smoothly with rock microstructure and elasticities of porous materials show that different pore shapes produce characteristic values for Poisson's ratio [6].

Shear modulus (μ): the relation of a shear stress to a shear strain in the same direction.

$$
\mu = \nu\_s^2 \rho. \tag{5}
$$

Lambda (λ): the ratio of compressional or tensional stress to strains in orthogonal directions:

$$
\lambda = \rho \left(\mathbf{V}\_p^2 - 2\mathbf{V}\_s^2\right). \tag{6}
$$

Lambda is the off-diagonal component of the isotropic stiffness tensor in the absence of shearing effects and is referred to as incompressibility [5]. It is independent of μ. In the Gassmann model, λ is elastically dependent on fluid properties, while μ is not [5].

Young's modulus (E): the relation between the stress applied and the resulting strain in the same direction:

$$\mathbf{E} = \rho V\_s^2 \left\{ \frac{\mathbf{3R}^2 - 4}{R^2 - 1} \right\} \tag{7}$$

with

$$\mathbf{R} = \frac{V\_p}{V\_s} \tag{8}$$

Little research has been done on the relation of Young's modulus to reservoir properties, but we execute an analysis in this study to see if any effects can be identified.

' Bulk modulus (K): a measure of how compressible a material is. It relates the volume s change in shape resulting from triaxial or hydrostatic stress:

Applied Geophysics with Case Studies on Environmental, Exploration and Engineering Geophysics

$$\mathbf{K} = \mathbf{V}\frac{\partial P}{\partial V} = \rho \left\{ V\_p^2 - \frac{4\mathbf{V}\_s^2}{3} \right\}. \tag{9}$$

where V is volume and P is pressure. So, K approaches 0.0 for a fully rigid body. Hereafter lambda, bulk modulus, Young's modulus, and Poisson's ratio are referred to as λ, K, E, and σ, respectively.

#### 2.3 Attenuation

Attenuation is the loss of energy with wave propagation, and Q is the quality quotient that describes the amount of attenuation (Aki and Richards, p. 220). Q has the reciprocal effect of attenuation, i.e., high Q is low attenuation and low Q is high attenuation. Attenuation is generally assumed to be due to inter-crack motion or fluid flow between pores, and is generally called intrinsic attenuation. Extrinsic attenuation is apparent attenuation when seismic energy is scattered due to small fractures. Energy is not actually lost, but a wavefront pulse of a propagating arrival broadens with distance, as would be observed with actual attenuation. Menke [21] fit observations for dry competent rock, and explains how attenuation decreases with depth due to crack closure and stiffening. Tokzoz and Johnson [22] also say this explains why many laboratory studies show attenuation to be frequency independent. In our analysis we do not distinguish pulse broadening due to intrinsic or extrinsic Q. This is determined by comparisons to other attributes. We also assume frequency independent Q.

### 2.4 Cracks, fractures and faults

Several mechanisms exist for the creation and destruction of permeability at depth; all of these involve cracks, fractures or faults. Cracks are presumed to be associated with weak grain boundaries, fractures are at the scale of multiple cracks, and faults are dislocations from earthquakes. In the most general case, the nucleation and propagation of cracks may increase the connectivity between cracks and fractures, and thus permeability [23]. Fractures alone can be conduits of permeability, as observed in the Salton Sea [19]. Faults are often conduits for fluid flow as well and thus also affect permeability. Faults are considered tectonic in nature and are often aligned with regional stress patterns.

Accurate locations of microearthquakes can often denote permeable zones while their moment tensors can identify the orientation and type of fractures being formed. Guilham et al. [24], Johnson [25] and Julian [26] interpreted moment tensors to obtain focal mechanism solutions that indicate the existence of isotropic and deviatoric dislocation events at The Geysers.

The rupture process and resulting permeability from microearthquakes may not be identical at all scales. Microearthquakes that rupture an entire crack or fracture may have an end effect, i.e., deformation at the end of a fracture to accommodate slip. Johnson [27] theorizes that this gives rise to orthogonal tensile crack opening at the end of cracks or fractures. Microearthquakes associated with faults may rupture only part of a larger feature created by previous earthquakes, thus not create tensile cracks.

Rocks with parallel crack or fracture systems will cause anisotropic wave propagation and shear-wave splitting will occur. In geothermal environments, shearwave velocity anisotropy has been observed to as high as 10% [28, 29]. Several authors attempt to address anisotropy as a tomography problem in order to identify where it occurs [30, 31]. Rose diagrams [28, 32] only map observed shear-wave splitting and do not identify locations of its occurrence.

#### Rock Physics Interpretation of Tomographic Solutions for Geothermal Reservoir Properties DOI: http://dx.doi.org/10.5772/intechopen.81226

Cracks and fractures (C&F) decrease μ without significantly reducing ρ and thus decrease both Vp and Vs. In the presence of C&F a geologic material is expected to have low Vp and Vs near the surface due to low μ and generally increase monotonically with depth due to the closure of cracks and fractures as pressure increases from the lithostatic load [33, 34]. Basement rocks at lithostatic pressures consistent with depths of greater than 3 km can be expected to have very low permeability (< 10–3 μ Darcy) due to closure of fractures [35]. At these depths permeability can be due to micro-fractures. Once C&F close, velocity will no longer increase with depth. O'Connell and Budiansky [36] relate a C&F density parameter to the effective Poisson's ratio, which is a direct reflection of their effect on Vp and Vs. λ, B, and E will also increase with depth due to compliance from open C&F.

In hydrothermal environments, grain-scale geochemical reactions can cement micro-fractures and stiffen the rock matrix. As a result, their relative effect on velocity will be reduced. This is observed in laboratory measurements of velocities in recovered core from The Geysers [33]. This suggests that much of the observed depth dependence of Vp and Vs in The Geysers reservoir rocks is due to closure of larger-scale C&F. Large permeable fractures might then be identified by regions of high velocity gradients in field data with a high density of compliant fractures. This may be particularly true in geothermal areas where healed micro-fractures will contribute less to observed gradients [33, 37].

Attenuation is also significantly affected by C&F. C&F increase extrinsic attenuation and reduce Q p and Qs. The lithostatic load tends to decrease extrinsic attenuation with depth due to closure of C&F. Extrinsic attenuation is potentially different for P-wave propagation and thus Q p than for S-wave propagation and Qs if C&F are aligned, since particle motion is transverse.

#### 2.5 Effects of fluids and steam

Following Eqs. (2) and (3), the inclusion of fluids into either pores or fractured material increases density and decreases Vp and Vs. Fluids also increase the bulk modulus, which increases Vp. In low porosity rocks, increased bulk modulus generally dominates the increase in density, whereas the reverse is true for high porosity rocks. The shear modulus (μ) is independent of fluids in the absence of geochemical effects and is determined by the porous rock matrix [6]. Therefore, density changes are the only effect that fluids have on Vs.

Injection of fluids can cause micro-fractures and/or microearthquakes due to thermal contraction or hydro-fracturing. Fluids also significantly affect attenuation. Partial saturation increases intrinsic attenuation. However, full saturation should lower intrinsic attenuation by inhibiting diffusion. Diffusion is also different for P-wave propagation and thus Q p than for S-wave propagation and Qs because particle motion is transverse. We note that Qs decreases and Q p increases with saturation (Figures 5 and 6).

Berryman et al. [8] emphasized that lambda and density contain information about saturation, while both combined with shear modulus contain information about porosity. Berryman et al. recast λ (Eq. (6)) as λ/μ, removing density ˜ <sup>ρ</sup> <sup>¼</sup> <sup>μ</sup>=ν<sup>2</sup> ° <sup>s</sup> and showing lambda's application in identifying the degree of saturation and type of saturation, i.e., the arrangement of fluids in a rock (inhomogeneous or homogeneous). They showed that for homogeneous saturation, λ/μ remains low for partial saturation and high for full saturation, and for inhomogeneous saturation there is a monotonic increase in λ/μ as saturation increases (equal at full saturation).

Typically, seismologists use high values of Poisson's ratio (or Vp/Vs) to indicate fluid saturation. There is a dramatic increase in Poisson's ratio with saturation by

the replacement of vapor by water, which increases the effective bulk modulus, while Vs is unaffected by fluid content.

Laboratory work suggests that using a combination of attenuation and velocity can improve discrimination of pore fluid content. Winkler and Nur [38] measured moduli and attenuation for longitudinal and torsional modes in porous and cracked rock near 1 kHz, where intrinsic attenuation dominates extrinsic attenuation. Plotting the data in Q p/Qs and Vp/Vs coordinates separates dry, partially saturated, and fully saturated conditions. These data cannot be used quantitatively until more measurements are done with low frequency data. The strong trends in Winkler and Nur's data suggest, however, that field data should reflect similar effects. Thus, values above a slope of 1.0 for Q p/Qs versus Vp/Vs plot indicate saturation and lower values indicate drier conditions. Hutchings et al. [3] found that The Geysers' data supported Winkler and Nur's theory.

Gritto [39] theorized that when injected water contacts reservoir rock, heat is drawn from the reservoir rock until the water vaporizes. The resulting cooling and contraction of the rock generates tensile (mode I) cracks and subsequent microseismicity. Once all water is converted to steam, the rock remains at a stable temperature with no further seismicity and the reservoir has reached maximum steam concentration. Permeability can be measured by monitoring the spatial and temporal migration of the micro-seismic cloud associated with fluid injection [40]. The resulting cooling may also result in increased shear moduli.

#### 2.6 Temperature and pressure effects

Rocks subjected to high temperatures and pressures undergo a transition from brittle to crystalline plastic behavior. The temperature of this transition ranges from around 300°C (quartz) to 400–450°C (feldspar) and also depends on pressure and strain rate [41, 42]. When stressed, some rocks may undergo cataclastic flow, which is characterized by ductile stress-strain behavior as well as cracking and frictional sliding.

The effect of temperature can oppose the effect of pressure [34]. In a dynamic situation, heating or cooling of fluids within pores can cause fractures and increase permeability while pressure can close fractures. Experiments where granite was cooled from high temperatures up to 646°C showed permeability increased up to a 1000 times over original values [43]. Darot et al. also found permeability decreased rapidly with confining pressure, being effectively zero for confining pressures over 30 MPa.

#### 2.7 Summary: porosity, permeability, and saturation

There are seven primary interpretations of porosity, permeability and saturation from observable microearthquake data. Comparisons are made relative to normal geology at similar depths and temperatures, meaning geology that has a monotonic increase in velocity and Q as a function of depth, and saturation, porosity and temperature that is considered average for the geologic condition of the study area. The interpretations are as follows:

1. Dry competent geology with low porosity might be identified by increased Vp and Vs due to high shear modulus due to few C&F; high Q p and Qs due to lack of diffusion and little extrinsic attenuation due to few C&F; Poisson's ratio near 0.25; lambda, bulk and Young's modulus are relatively high due to incompressibility of stiff material. No fracturing or permeability is assumed.

Rock Physics Interpretation of Tomographic Solutions for Geothermal Reservoir Properties DOI: http://dx.doi.org/10.5772/intechopen.81226

