Design of convergence criterion for fixed stress split iterative scheme for small strain anisotropic poroelastoplasticity coupled with single phase flow

12/07/2019
by   Saumik Dana, et al.
The University of Texas at Austin
0

We arrive at convergence criterion for the fixed stress split iterative scheme for single phase flow coupled with small strain anisotropic poroelastoplasticity. The analysis is based on studying the equations satisfied by the difference of iterates to show that the iterative scheme is contractive. The contractivity is based on driving a term to as small a value as possible (ideally zero). This condition is rendered as the convergence criterion of the algorithm.

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1 Introduction

Figure 1: Fixed stress split iterative scheme for anisotropic poroelastoplasticity coupled with single phase flow. Our objective is to use the framework of contraction map to design a convergence criterion for the algorithm.

We use the framework of a contraction map to arrive at the convergence criterion for a staggered solution algorithm coupling small strain anisotropic poroelastoplasticity with single phase flow. As shown in Figure 1

, the flow subproblem is solved with stress tensor fixed followed by the poromechanics subproblem in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The analysis is motivated by the results in our previous works as follows

  • In [1], the contraction map for two-grid staggered algorithm lent us closed form expressions for coarse scale moduli in terms of fine scale data. The flow equations were solved on a fine grid and the isotropic poroelasticity equations were solved on a coarse grid.

  • In [2], we used contraction map to demonstrate convergence of staggered solution algorithm for anisotropic poroelasticity coupled with single phase flow. The speciality of this algorithm was that the stress tensor was fixed during the flow solve as an extension to the case with isotropic poroelasticity in which the mean stress was fixed during the flow solve.

2 Model equations

2.1 Flow model

Let the boundary where is the Dirichlet boundary and is the Neumann boundary. The equations are

(2.1)

where is the fluid pressure, is the fluid flux, is the increment in fluid content111[3] defines the increment in fluid content as the measure of the amount of fluid which has flowed in and out of a given element attached to the solid frame, is the source or sink term, is the uniformly symmetric positive definite absolute permeability tensor, is the fluid viscosity, is a reference density, is the porosity, is a measure of the hydraulic conductivity of the pore fluid, is the fluid compressibility and is the time interval.

2.2 Poromechanics model

The important phenomenological aspects of small strain elastoplasticity are

  • The existence of an elastic domain, i.e. a range of stresses within which the behaviour of the material can be considered as purely elastic, without evolution of permanent (plastic) strains. The elastic domain is delimited by the so-called yield stress. A scalar yield function is introduced. The yield locus is the boundary of the elastic domain where and the corresponding yield surface is defined as .

  • If the material is further loaded at the yield stress, then plastic yielding (or plastic flow), i.e. evolution of plastic strains, takes place.

Let the boundary where is the Dirichlet boundary and is the Neumann boundary. The equations are

(2.2)

where is the solid displacement, is the rock density, is the body force per unit volume, is the traction specified on , is the strain tensor, and are the elastic and plastic parts of strain tensor respctively, is the Cauchy stress tensor, is the fourth order symmetric positive definite anisotropic elasticity tensor, is the Biot tensor and is the plastic multiplier satisfying the complementarity condition

(2.3)

The inverse of the constitutive law is

(2.4)

where is a generalized Hooke’s law constant (see [18]) and is the Skempton pore pressure coefficient (see [5]).

2.3 Increment in fluid content

It is given by (see [7])

(2.5)

where is the Biot modulus (see [4], [18]) and is a plastic porosity given by (see [7])

(2.6)

where is a material parameter.

3 Statement of convergence

Figure 2: The contraction map is in terms of quantities and .

As elucidated in Figure 2, we use the notations and for the change in the quantity during the flow and poromechanics solves respectively over the coupling iteration and for the change in the quantity over the coupling iteration at any time level such that

The problem statement is: find , and such that

(3.7)
(3.8)
(3.9)

where the finite dimensional spaces , and are

where represents the space of constants, represents the space of trilinears and the details of are given in [2]. The details of (3.7), (3.8) and (3.9) are given in Appendices A, B and C respectively. The fixed stress split iterative scheme is a contraction map given by

where the term is driven to a small value by the convergence criterion. Step 1: Flow equations
Testing (3.7) with and (3.8) with , we get

(3.10)
(3.11)

From (3.10) and (3.11), we get

(3.12)

Step 2: Poromechanics equations
Testing (3.9) with , we get

(3.13)

We now invoke (2.4) to arrive at . Substituting in (3.13), we get

(3.14)

Step 3: Combining flow and poromechanics equations
Adding (3.12) and (3.14), we get

(3.15)

Step 4: Variation in fluid content
In lieu of (2.5), we write

(3.16)

which can be written as

Dividing throughout by , we get

(3.17)

From (3.15) and (3.17), we get

(3.18)

Adding and subtracting to the LHS of (3.18) results in

Multiplying throughout by results in

(3.19)

Step 5: Invoking the fixed stress constraint
In lieu of (2.5) and the fixed stress constraint during the flow solve, we get

Further, since the pore pressure is frozen during the poromechanical solve, we have . As a result, we can write

(3.20)

Subtracting (3.20) from (3.16), we can write

(3.21)

which implies that

(3.22)

In lieu of (3.22), we can write (3.19) as

(3.23)

Step 6: Invoking the Young’s inequality
We invoke the Young’s inequality (see [8]) for the RHS of (3.23) as follows

(3.24)

In lieu of (3.24), we write (3.23) as

4 Convergence criterion

We desire to drive the following quantity to zero

(4.25)

In lieu of (3.21), we can write

(4.26)

In lieu of (4.26), we can write (4.25) as

which can also be written as

As a result, we pose the convergence criterion as

(4.27)

where is a pre-specified tolerance and represents a small value.

4.1 Computation of quantities of interest

Figure 3: Algorithm with convergence criterion.
Figure 4: Return mapping.

In lieu of (2.5) and (2.6), we can write

(4.28)
(4.29)
(4.30)

where we keep in mind that the pore pressure is frozen during the poromechanics solve and stress tensor is fixed during the flow solve. The quantity is obtained after the flow solve while the quantities and are obtained after the poromechanical solve.

To understand why , we present the basic algorithmic framework for the solution of elastoplastic equations: The system of equations (2.2) is first solved with for a trial stress state .

  • If , then we proceed with .

  • If , then the system (2.2) is solved with thereby lending us plastic strain. The procedure to solve (2.2) with is refered to as a return mapping algorithm (see [10], [11], [19], [13], [14], [15], [12], [16], [17], [9]). The solution of the return mapping algorithm is such that and we proceed with .

In summary, the solution is such that . During the subsequent flow solve, since the stress tensor is fixed, the value of the yield function does not change i.e. still. This implies that during the flow solve and the porous solid does not accumulate any plastic strain during the flow solve.

Substituting (4.28)-(4.30) in (4.27), we get

(4.31)

The algorithm with the convergence criterion is given in Figure 3 and the return mapping is elucidated in Figure 4.

Appendix A Discrete variational statement of mass conservation

In lieu of (2.5), we write mass conservation equation as

(1.32)

The discrete in time form of (1.32) in the time step is written as

where is the time step. The fixed stress split constraint implies that gets replaced by as is fixed during the flow solve. The modified equation is written as

As a result, the discrete weak form of mass conservation is given by

Replacing by and subtracting the two equations, we get

Appendix B Discrete variational statement of Darcy’s law

The weak form of Darcy’s law is given by

(2.33)

where is given by

and is given by

We use the divergence theorem to evaluate the first term on RHS of (2.33) as follows

(2.34)

where we invoke on . In lieu of (2.33) and (2.34), we get

Replacing by and subtracting the two equations, we get