Geostatistical Rock Physics Inversion for Predicting the Spatial Distribution of Porosity and Saturation in the Critical Zone

Abstract

Understanding the subsurface structure and function in the near-surface groundwater system, including fluid flow, geomechanical, and weathering processes, requires accurate predictions of the spatial distribution of petrophysical properties, such as rock and fluid (air and water) volumetric fractions. These properties can be predicted from geophysical measurements, such as electrical resistivity tomography and refraction seismic data, by solving a rock physics inverse problem. A Bayesian inversion approach based on a Monte Carlo implementation of the Bayesian update problem is developed to generate multiple realizations of porosity and water saturation conditioned on geophysical data. The model realizations are generated using a geostatistical algorithm and updated according to the ensemble smoother approach, an efficient Bayesian data assimilation technique. The prior distribution includes a spatial correlation function such that the model realizations mimic the geological spatial continuity. The result of the inversion includes a set of realizations of porosity and water saturation, as well as the most likely model and its uncertainty, that are crucial to understand fluid flow, geomechanical, and weathering processes in the critical zone. The proposed approach is validated on two synthetic datasets motivated by the Southern Sierra Critical Zone Observatory and is then applied to data collected on a mountain hillslope near Laramie, Wyoming. The inverted results match the measurements, honor the spatial correlation prior model, and provide geologically realistic petrophysical models of weathered rock at Earth’s surface.

Appendix: Rock Physics Model

The elastic component of the rock physics model computes P-wave velocity properties from petrophysical properties of rocks and fluids. The mineral phase is assumed homogeneous and constant. The model variables are porosity and water saturation, and the model prediction is P-wave velocity. For a porous rock saturated with a mixture of two fluid components (water and air), the density $\rho$ is computed according to Eq. (7) and the P-wave velocity according to Eq. (6).

For the saprolite, Dvorkin’s model (Dvorkin and Nur 1996) is adopted to compute P-wave velocity $V_{{\text{P}}}$ as a function of the saturated rock elastic moduli, $K_{{{\text{sat}}}}$ and $G_{{{\text{sat}}}}$, by combining Hertz–Mindlin equations, modified Hashin–Shtrikman lower bounds, and Gassmann’s equations as

$$ K_{{{\text{sat}}}} (\phi ,S_{{\text{w}}} ) = \frac{{K_{{{\text{sol}}}}\left( {\alpha - \frac{4}{3}G_{{{\text{HM}}}} \beta \phi + \gamma } \right)}}{{\alpha + K_{{{\text{sol}}}} \beta \phi + \gamma }}, $$

(A1)

where

$$ \begin{aligned} \alpha & = K_{{{\text{fl}}}} \left( {S_{{\text{w}}} } \right)\left( {K_{{{\text{sol}}}} + \frac{4}{3}G_{{{\text{HM}}}} } \right)\left( {K_{{{\text{sol}}}} - K_{{{\text{HM}}}} } \right), \\ \beta & = \left( {K_{{{\text{sol}}}} - K_{{{\text{HM}}}} } \right)\left( {K_{{{\text{sol}}}} - K_{{{\text{fl}}}} \left( {S_{{\text{w}}} } \right)} \right), \\ \gamma & = K_{{{\text{sol}}}} \left( {K_{{{\text{HM}}}} + \frac{4}{3}G_{{{\text{HM}}}} } \right)\left( {K_{{{\text{sol}}}} - K_{{{\text{fl}}}} \left( {S_{{\text{w}}} } \right)} \right)\phi_{{\text{c}}} , \\ \end{aligned} $$

(A2)

and

$$ G_{{{\text{sat}}}} \left( {\phi } \right) = \frac{{\phi \left( {G_{{{\text{HM}}}} - G_{{{\text{sol}}}}} \right)\xi + \delta G_{{{\text{sol}}}} }}{{\phi \left( {G_{{{\text{sol}}}} - G_{{{\text{HM}}}} } \right) + \delta }}, $$

(A3)

where

$$ \begin{aligned} \xi & = \frac{1}{6}G_{{{\text{HM}}}} \frac{{9K_{{{\text{HM}}}} + 8G_{{{\text{HM}}}} }}{{K_{{{\text{HM}}}} + 2G_{{{\text{HM}}}} }}, \\ \delta & = \left( {\xi + G_{{{\text{HM}}}} } \right)\phi_{{\text{c}}} , \\ \end{aligned} $$

(A4)

with $K_{{{\text{sol}}}}$ being the solid-phase bulk modulus, $G_{{{\text{sol}}}}$ the solid-phase shear modulus, $K_{{{\text{fl}}}}$ the fluid-phase bulk modulus, $\phi_{{\text{c}}}$ the critical porosity, $K_{{{\text{HM}}}}$ the Hertz–Mindlin bulk modulus, and $G_{{{\text{HM}}}}$ the Hertz–Mindlin shear modulus (Dvorkin and Nur 1996). The Hertz–Mindlin elastic moduli depend on the solid-phase elastic moduli, the critical porosity, the coordination number, and the effective pressure. In the proposed approach, the solid-phase elastic moduli $K_{{{\text{sol}}}}$ and $G_{{{\text{sol}}}}$ are assumed to be constant and known, whereas in the general case they are computed using Voigt–Reuss–Hill averages. The fluid-phase bulk modulus $K_{{{\text{fl}}}}$ is computed using the Reuss average for homogeneous mixtures and Voigt average for patchy mixtures.

For the bedrock, an inclusion model is adopted based on the self-consistent approximation model proposed in Berryman (1995) and Te Wu (1966), which provides the elastic moduli for a porous rock with a single inclusion type equal to the pore volume

$$ K_{{{\text{sat}}}} (\phi ,S_{{\text{w}}} )= K_{{{\text{sol}}}} + \phi \left( {K_{{{\text{fl}}}}(S_{{\text{w}}})- K_{{{\text{sol}}}} } \right)P, $$

(A5)

$$ G_{{{\text{sat}}}}(\phi) = G_{{{\text{sol}}}} - \phi G_{{{\text{sol}}}} Q, $$

(A6)

where $P$ and $Q$ are geometrical factors. The fluid-phase bulk modulus $K_{{{\text{fl}}}}$ is computed using Reuss or Voigt mixing laws. The geometrical factors $P$ and $Q$ depend on the aspect ratio of the pores (Berryman 1995; Mavko et al. 2020; Grana et al. 2021).