Hydro-Mechanical Evolution of Transport Properties in Porous Media: Constraints for Numerical Simulations

Abstract

During geothermal operations, injection and production of fluid can induce significant pore pressure and temperature changes which impact the stress field thus affecting the reservoir performance. In this context, the transport and mechanical properties of the porous rock can be altered by deformation of the pore and bulk volumes. Different poroelastic formulations for describing the hydro-mechanical behavior of porous rocks are considered. Additionally, a formulation for the drained bulk modulus is introduced which takes into account the poroelastic behavior. This poroelastic formulation has been implemented in the 3D finite element method-based simulator OpenGeoSys to predict porosity changes. Based on the porosity changes, permeability variation under different loading conditions as main property controlling transport of mass and energy in porous media is determined by the Kozeny–Carman relation. Triaxial laboratory experiments conducted on two different kinds of sandstones (Flechtinger and Bentheimer) are used to constrain parameters which control porosity changes and thus permeability changes under drained conditions and to calibrate the numerical simulations. Hydrostatic loading from 1 to 70 MPa in confining stress has been simulated for Flechtinger and Bentheimer sandstones resulting in porosity and permeability decreases. Results are compared to experimental measurements to evaluate the precision of each poroelastic model. Reduction in porosity of 8.1 % with about 1.15 % error for the Flechtinger sandstone has been simulated and of 1.4 % with about 0.54 % error for the Bentheimer sandstone.

Appendices

Appendix 1: Governing Equations for Isotropic Elasticity in the OpenGeoSys Software

This “Appendix” presents the initial governing equations for mechanical processes which have been considered in this work. Deformation of the porous medium is described by the momentum balance equation in terms of effective stress as:

$$\begin{aligned} \nabla \cdot (\varvec{\sigma }_e + \alpha P_{p}I)+\rho _b\varvec{g} = 0 \end{aligned}$$

(24)

where $\varvec{\sigma }_\mathbf{e}$ is the effective stress tensor and $\rho _{s}$ is the solid density. The primary variable to be solved is the displacement vector $\varvec{u}$ by substituting the constitutive law for stress–strain behavior as follow:

$$\begin{aligned} \varvec{\sigma }_\mathbf{e}= & {} \mathbb {C} \varvec{\epsilon } \end{aligned}$$

(25)

$$\begin{aligned} \varvec{\epsilon }= & {} \frac{1}{2}\left( \nabla \varvec{u} + (\nabla \varvec{u})^{T}\right) \end{aligned}$$

(26)

with $\mathbb {C}$ is the forth order material tensor and $\varvec{\epsilon }$ the strain. The subscript T means the transpose of the matrix. In this work, only isotropic elasticity is considered. Before the implementation described in this work, linear elasticity is described by the generalized Hooke’s law:

$$\begin{aligned} \mathbb {C}=\lambda \delta _{ij}\delta _{kl} + 2 \mu \delta _{ik}\delta _{jl} \end{aligned}$$

(27)

where $\delta$ is the Kronecker delta, $\mu$ the shear modulus and $\lambda$ the Lamé constant defined as $\lambda = \frac{2\mu \nu }{(1-2\nu )}$ with $\nu$ the Poisson’s ratio. Equation 27 can also be expressed in terms of elastic strain with the Hookes’s linear elastic laws (Eqs. 28–30):

$$\begin{aligned} \epsilon _{x}= & {} \frac{1}{E}\left( \sigma _{e,x} - \nu (\sigma _{e,y}+\sigma _{e,z})\right) \end{aligned}$$

(28)

$$\begin{aligned} \epsilon _{y}= & {} \frac{1}{E}\left( \sigma _{e,y} - \nu (\sigma _{e,x}+\sigma _{e,z})\right) \end{aligned}$$

(29)

$$\begin{aligned} \epsilon _{z}= & {} \frac{1}{E}\left( \sigma _{e,z} - \nu (\sigma _{e,x}+\sigma _{e,y})\right) \end{aligned}$$

(30)

with

• $\epsilon _{i}$ strains

• $\sigma _{e,i}$ effective stresses

• E Youngs’s modulus

• $\nu$ Poisson’s ratio

OpenGeoSys is based on an object-oriented finite element method concept. In this work, the equations system is solved on quadratic tetrahedral elements. A robust library named LIS (a Library of Iterative Solvers for linear systems, http://ssisc.org/)(Nishida 2010) has been used for the preconditioner and the linear solver.

Appendix 2: Specific Surface Area for a Packed Bed of Spheres

This “Appendix” presents the determination of the specific surface area coefficient used in the permeability determination. These equations are inspired from flow through packed fixed and fluidized beds model for chemical engineering (James 1999). The grains of the sandstone are considered spherical and have all the same radius. Void between these spheres are considered to be all connected (tortuosity $\tau =1$). The specific surface area S is defined as:

$$\begin{aligned} S = \frac{S_{w}}{V_{b}} \end{aligned}$$

(31)

where $S_{w}$ is the wetted surface of the grains and $V_{b}$ the bulk volume. The wetted surface of the grains is equal to the total solid surface area. If there is N grains in a volume V:

$$\begin{aligned} S_{w} = N S_{\mathrm{grain}} = N 4\pi r_{g}^{2} \end{aligned}$$

(32)

The bulk volume $V_{b}$ is linked to the pore volume by: $V_{\phi }=\phi V_{b}=\phi \left( V_{\phi }+V_{s}\right)$ which gives:

$$\begin{aligned} V_{b}=\frac{V_{s}}{1-\phi } \end{aligned}$$

(33)

where $V_{s}$ is the total volume of grains, $V_{s}=N\frac{4}{3}\pi r_{g}^{3}$. Substituting Eqs. 32 and 33 in Eq. 31, gives:

$$\begin{aligned} S = \frac{N 4\pi r_{g}^{2}v}{\frac{1}{1-\phi }N\frac{4}{3}\pi r_{g}^{3}} \end{aligned}$$

(34)

which is simplified to:

$$\begin{aligned} S = \frac{3\left( 1-\phi \right) }{r_{s}} \end{aligned}$$

(35)