Drained pore modulus and Biot coefficient from pore-scale digital rock simulations

https://doi.org/10.1016/j.ijrmms.2018.12.019Get rights and content

Abstract

We present a digital rock workflow to determine poroelastic parameters which are difficult to extract from well-log or laboratory measurements. The drained pore modulus is determinant in the compaction problem. This modulus represents the ratio of pore volume change to confining pressure when the fluid pressure is constant. In laboratory experiments, bulk volume changes are accurately measured by sensors attached to the outer surface of the rock sample. In contrast, pore volume changes are notoriously difficult to measure because these changes need to quantify the pore boundary deformation. Hence, accurate measures of the drained pore modulus are challenging. We simulate static deformation experiments at the pore-scale utilizing a digital rock image. We model an Ottawa F-42 sand pack obtained from an X-ray micro-tomographic image. This image is segmented into a network of grains and pore space. The network of grains is taken as an elastic, isotropic and homogeneous continuum. We then compute the linear momentum balance for the network of grains. We calculate the change in pore volume using a post-processing algorithm, which allows us to compute the local changes in pore volume due to the applied load. This process yields an accurate drained pore modulus. We then use an alternative estimate of the drained pore modulus. We exploit its relation to the drained bulk modulus and the solid phase bulk modulus (i.e., Biot coefficient) using the digital rock work- flow. Finally, we compare the drained pore modulus values obtained from these two independent analyses and find reasonable agreement.

Introduction

A standard model for the effective elastic behaviour of porous, fluid-saturated rocks is the theory of linear poroelasticity. As it is a macroscopic theory, only an average of the textural and compositional complexities at pore-scale enter the poroelastic description. Since the introduction of the poroelasticity theory,1 the definition of the deformation parameters for the constitutive relations has been sought.2, 3, 4 We use volume averaging of the pore-scale equations to quantify the parameters appearing in the constitutive equations of linear poroelasticity that include the effects of pore-scale heterogeneities.5, 6, 7, 8

A poroelastic parameter particularly important for geomechanical reservoir analysis is the Biot coefficient. It is used to estimate the reservoir stress path.9 The Biot coefficient represents the fluid volume change induced by the bulk volume change in drained conditions, which is difficult to measure in the laboratory. An alternative definition of the coefficient is as the effective pressure coefficient for the bulk that is measured in constant bulk volume tests.10 However, this interpretation is not correct in micro-inhomogeneous rocks.11 Another relevant parameter is the drained pore modulus used in geotechnical applications such as reserve estimation.12, 13 The drained modulus measures the change of pore volume as the confining pressure increases while the fluid pressure is constant. This drained coefficient is as difficult to measure in the laboratory as the Biot coefficient.14

Given these problems, we explore new avenues to estimate the Biot coefficient and the drained pore modulus. For idealized, two-dimensional porous structures the Biot coefficient has been computed numerically15 and it has been found that microstructural properties such as the arrangement of elliptical cracks are important. Therefore, we aim at an approach that handles the complexities of a three-dimensional network of grains. In this paper we numerically determine the drained poroelastic parameters using digital rock technology (DRT). DRT technology uses 3D pore-scale images of rocks (i.e., the geometry of the pore-scale features is known and only limited by the resolution of the imaging process) in conjunction with numerical solvers for the pore-scale governing equations. However, thus far, attempts to quantify poromechanical parameters using DRT have not been convincing and remain a major challenge.16

The outline is as follows. First, we present the poroelastic framework with particular focus on the interrelations between parameters characterizing the drained rock behaviour. Then, we numerically simulate static deformation experiments at the pore-scale utilizing digitized rock images of sandstone. From the pore-scale simulations we infer the macroscopic deformation moduli by volume-averaging. We interpret these moduli in the poroelastic framework which accounts for micro-inhomogeneities in a generic way and which explicitly links the change of porosity to the constitutive equations of poroelasticity.8 Finally, we compare the numerical estimates with lab- oratory measurements reported in the literature and obtain a reasonable agreement with these measurements.

Section snippets

Constitutive equations of linear poroelasticity and the porosity perturbation equation

We consider an isotropic poroelastic solid and restrict the analysis to volumetric changes only. Then, there are the following kinematic variables of interest: The bulk volumetric strain (ε) and the increment of fluid content (ζ) according to the consolidation theory of Biot,1 where we adopt the notation suggested in.17 In addition, since pore boundaries move during deformation (Fig. 1) the proportion of solid to fluid volume changes and amounts to a change of porosity η−η0.5 Here η is the

General workflow description and properties of the network of grains

Typically, there are two types of workflows in DRT. In the first one, micro-structural information is obtained by analysing micro CT scan images with the help of image processing methods. The second workflow consists of multi-physics modelling where 2D micro CT scan images are stacked to create a 3D digital sample replicating the real micro-structure. We implement the second work flow. This includes segmentation of the images into separable pore and grain spaces. Pore spaces are used to

Comparison between the direct and indirect determination of Kp and α

The direct and indirect estimates of drained pore modulus and Biot coefficient differ by 0.75 GPa and 0.05, respectively. These differences are considered to be within the range of numerical errors. There are different sources of error and limitations in our modelling setup. For the determination of K0 the bulk volume changes are computed from the displacements of the outer surfaces of the sample. Given that one of this outer surface is fixed in space (to make the ABAQUS simulations feasible)

Conclusion

There have been always queries about the meaning of the poroelastic constants that appear in the macroscopic constitutive equations of a porocontinuum which, in the present case, is meant to represent a real rock sample. The volume averaging framework of poroelasticity establishes a clear link between microscopic fields and their macroscopic counterparts. This provides an opportunity to estimate the macroscopic poroelastic constants if the pore-scale geometry is exactly known and representative

Acknowledgments

We appreciate the constructive feedback and encouragement of Serge Shapiro and Gry Couples during the 6th Biot conference on poromechanics on an earlier version of this work.

References (29)

  • J. Altmann et al.

    Poroelastic contribution to the reservoir stress path

    Int J Rock Mech Min Sci

    (2010)
  • X. Tan et al.

    Numerical study of variation in Biot's coefficient with respect to microstructure of rocks

    Tectonophysics

    (2014)
  • M.A. Biot

    General theory for three-dimensional consolidation

    J Appl Phys

    (1941)
  • M.A. Biot et al.

    The elastic coefficients of the theory of consolidation

    J Appl Mech

    (1957)
  • R.Y. Makhnenko et al.

    Elastic and inelastic deformation of fluid-saturated rock

    Philos Trans R Soc A

    (2016)
  • A.H.D. Cheng

    Poroelasticity

    (2016)
  • P.N. Sahay et al.

    Seismic wave propagation in inhomogeneous and anisotropic porous media

    Geophys J Int

    (2001)
  • P.N. Sahay

    Biot constitutive relation and porosity perturbation equation

    Geophysics

    (2013)
  • T.M. Müller et al.

    Porosity perturbations and poroelastic compressibilities

    Geophysics

    (2013)
  • T.M. Müller et al.

    Generalized poroelasticity framework for microinhomogeneous rocks

    Geophys Prospect

    (2016)
  • Omdal E, Madland M, Breivik H, Naess K, Korsnes R, Hiorth A, Kristiansen T. Experimental investigation of the effective...
  • T.M. Müller et al.

    Biot coefficient is distinct from effective pressure coefficient

    Geophysics

    (2016)
  • Schutjens P, Heidug W. On the pore volume compressibility and its application as a petrophysical parameter. In:...
  • G.P. De Oliviera et al.

    Pore volume compressibilities of sandstones and carbonates from helium porosimetry measurements

    J Petrol Sci Eng

    (2016)
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