An enhanced sequential fully implicit scheme for reservoir geomechanics

Abstract

In this paper, it is proposed an enhanced sequential fully implicit (ESFI) algorithm with a fixed stress split to approximate robustly poro-elastoplastic solutions related to reservoir geomechanics. The constitutive model considers the total strain effect on porosity/permeability variation and associative plasticity. The sequential fully implicit (SFI) algorithm is a popular solution to approximate solutions of a coupled system. Generally, the SFI consists of an outer loop to solve the coupled system, in which there are two inner iterative loops for each equation to implicitly solve the equations. The SFI algorithm occasionally suffers from slow convergence or even convergence failure. In order to improve the convergence (robustness) associated with SFI, a new nonlinear acceleration technique is proposed employing Shanks transformations in vector-valued variables to enhance the outer loop convergence, with a quasi-Newton method considering the modified Thomas method for the internal loops. In this ESFI algorithm, the fluid flow formulation is defined by Darcy’s law including nonlinear permeability based on Petunin model. The rock deformation includes a linear part being analyzed based on Biot’s theory and a nonlinear part being established using Mohr-Coulomb associative plasticity for geomechanics. Temporal derivatives are approximated by an implicit Euler method, and spatial discretizations are adopted using finite element in two different formulations. For the spatial discretization, two weak statements are obtained: the first one uses a continuous Galerkin for poro-elastoplastic and Darcy’s flow; the second one uses a continuous Galerkin for poro-elastoplastic and a mixed finite element for Darcy’s flow. Several numerical simulations are presented to evaluate the efficiency of ESFI algorithm in reducing the number of iterations. Distinct poromechanical problems in 1D, 2D, and 3D are approximated with linear and nonlinear settings. Where appropriate, the results were verified with analytic solutions and approximated solutions with an explicit Runge-Kutta solver for 2D axisymmetric poro-elastoplastic problems.

Appendix

1.1 A Runge-Kutta solver for poro-elastoplasticity: an axisymmetric approach—the linear poro-elastic case

To construct an Runge-Kutta approximation, it is required to review the poro-elastic equations and find a way to recast the equations as initial value problem, as the Runge-Kutta method structure:

$$ \frac{d\mathbf{y}}{d\mathbf{x}}=\mathbf{f}\left( \mathbf{y}\right) $$

(79)

There are three main considerations for this case:

• 1. The equations are presented in terms of the cylindrical coordinate system.

• 2. The approximation is axisymmetric, leading to a displacement u and pressure p fields that depend only of the radius , i.e., $\mathbf {u}={\Phi }{\left (\mathrm {r}\right )}$ and $p={\Phi }{\left (\mathrm {r}\right )}$.

• 3. Assume steady-state conditions, the initial value problem is described in terms of one independent variable r.

Recalling the linear elastic constitutive law:

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\sigma} &= & 2\mu\left( \boldsymbol{\epsilon}-\boldsymbol{\epsilon}^{\circ}\right)+\lambda \text{tr}\left( \boldsymbol{\epsilon}-\boldsymbol{\epsilon}^{\circ}\right)\mathbf{I}-\sigma^{\circ}\mathbf{I}\\ \boldsymbol{\epsilon} & = &\frac{1}{2}\left( \nabla\mathbf{u}+\nabla^{T}\mathbf{u}\right) \end{array} $$

(80)

Using the considerations above $\mathbf {u}=u_{r}\hat {\textbf {r}}$ and an initial 𝜖° = 0 the effective stress tensor becomes:

$$ \boldsymbol{\sigma}=\left( \mu_{r}+\lambda_{r}\right)\hat{\textbf{r}}\otimes\hat{\textbf{r}}+\left( \mu_{r}+\lambda_{r}\right)\hat{\boldsymbol{\theta}}\otimes\hat{\boldsymbol{\theta}}+\left( \lambda_{r}\right)\hat{\textbf{z}}\otimes\hat{\textbf{z}} $$

(81)

where $\mu _{r}=2\mu \frac {du_{r}}{dr}$ and $\lambda _{r}=\lambda \left (\frac {u_{r}}{r}+\frac {du_{r}}{dr}\right )$. Taking the trace of the expression above can be obtained the following expression for $\frac {du_{r}}{dr}$:

$$ \frac{du_{r}}{dr}=\frac{r\sigma_{rr}-\lambda u_{r}}{r\left( \lambda+2\mu\right)} $$

(82)

Evoke the total stress σ_t equilibrium and using the Biot decomposition of the total stress:

$$ \text{div}\left( \boldsymbol{\sigma}-\sigma^{\circ}\mathbf{I}-\alpha(p-p^{\circ})\mathbf{I}\right)=0 $$

(83)

The expression for $\frac {d\sigma _{rr}}{dr}$ is obtained from the momentum conservation directly:

$$ \frac{d\sigma_{rr}}{dr}=\frac{-\sigma_{rr}+\left( \frac{2\mu u_{r}}{r}+\lambda\left( \frac{u_{r}}{r}+\frac{r\sigma_{rr}-\lambda u_{r}}{r\left( \lambda+2\mu\right)}\right)\right)}{r}-\alpha\frac{\eta}{k}q_{r} $$

(84)

Also, the quantities σ_θθ and σ_zz are:

$$ \sigma_{\theta\theta}=\frac{2\mu u_{r}}{r}+\lambda\left( \frac{u_{r}}{r}+\frac{r\sigma_{rr}-\lambda u_{r}}{r\left( \lambda+2\mu\right)}\right) $$

(85)

$$ \sigma_{zz}=\frac{\lambda\left( \lambda\left( \sigma_{rr}+\sigma_{\theta\theta}\right)+2\mu\left( \sigma_{rr}+\sigma_{\theta\theta}\right)\right)}{2\left( \lambda+\mu\right)\left( \lambda+2\mu\right)} $$

(86)

Reinstate that Darcy constitutive expression provides the expression for $\frac {dp}{dr}$:

$$ \frac{dp}{dr}=-\frac{\eta}{\kappa}q_{r} $$

(87)

Finally, the mass conservation equation affords the expression for $\frac {dq_{r}}{dr}$:

$$ \frac{dq_{r}}{dr}=-\frac{q_{r}}{r} $$

(88)

Regarding the initial value problem, the spatial derivative for variable y is clearly:

$$ \frac{d\mathbf{y}}{d\mathbf{x}}=\left\{ \begin{array}{c} \frac{du_{r}}{dr}\\ \frac{d\sigma_{rr}}{dr}\\ \frac{dp}{dr}\\ \frac{dq_{r}}{dr} \end{array}\right\} \text{and} \mathbf{f}\left( \mathbf{y}\right)=\left\{ \begin{array}{c} \frac{r\sigma_{rr}-\lambda u_{r}}{r\left( \lambda+2\mu\right)}\\ \frac{-\sigma_{rr}+\left( \frac{2\mu u_{r}}{r}+\lambda\left( \frac{u_{r}}{r}+\frac{r\sigma_{rr}-\lambda u_{r}}{r\left( \lambda+2\mu\right)}\right)\right)}{r}-\alpha\frac{\eta}{k}q_{r}\\ -\frac{\eta}{\kappa}q_{r}\\ -\frac{q_{r}}{r} \end{array}\right\} $$

(89)

For the completeness of the initial value problem, the data y° is evaluated at the permeability reservoir radius r_e. It is important to point that the permeability can be one function of the state variable y in a nonlinear sense.

1.2 The poro-elastoplastic case

The expression (89) can be rewritten in terms of strain and stress data:

$$ \frac{d\mathbf{y}}{d\mathbf{x}}=\left\{ \begin{array}{c} \frac{du_{r}}{dr}\\ \frac{d\sigma_{rr}}{dr}\\ \frac{dp}{dr}\\ \frac{dq_{r}}{dr} \end{array}\right\} \text{and} \mathbf{f}\left( \mathbf{y}\right)=\left\{ \begin{array}{c} \epsilon_{rr}\\ \frac{-\sigma_{rr}+\sigma_{\theta\theta}}{r}-\alpha\frac{\eta}{k}q_{r}\\ -\frac{\eta}{\kappa}q_{r}\\ -\frac{q_{r}}{r} \end{array}\right\} $$

(90)

Thus, the approximation above can be recast as elastoplastic problem by delaying α, K_dr, and the elastoplastic strain between two consecutive points in order to consider the nonlinear effects of plasticity during the RK process. The payoff is a very similar implementation for the RK solver with the need of additional discrete points to reach a reasonable approximation.

For the completeness of the initial value problem, the data y° is evaluated at the permeability reservoir radius r_e where it is assumed to be linear poro-elastic data. For that reason, it is expected that as the number of discrete points is incremented the poro-elastoplastic approximation became more precise. It makes the RK solver a suitable approximation for comparison and/or verification purposes.