Unified thermo-compositional-mechanical framework for reservoir simulation

Abstract

We present a reservoir simulation framework for coupled thermal-compositional-mechanics processes. We use finite-volume methods to discretize the mass and energy conservation equations and finite-element methods for the mechanics problem. We use the first-order backward Euler for time. We solve the resulting set of nonlinear algebraic equations using fully implicit (FI) and sequential-implicit (SI) solution schemes. The FI approach is attractive for general-purpose simulation due to its unconditional stability. However, the FI method requires the development of a complex thermo-compositional-mechanics framework for the nonlinear problems of interest, and that includes the construction of the full Jacobian matrix for the coupled multi-physics discrete system of equations. On the other hand, SI-based solution schemes allow for relatively fast development because different simulation modules can be coupled more easily. The challenge with SI schemes is that the nonlinear convergence rate depends strongly on the coupling strength across the physical mechanisms and on the details of the sequential updating strategy across the different physics modules. The flexible automatic differentiation-based framework described here allows for detailed assessment of the robustness and computational efficiency of different coupling schemes for a wide range of multi-physics subsurface problems.

Appendices

Appendix A: Return-mapping algorithm

In this section, we describe the implementation of a return-mapping algorithm for the proposed plasticity model. We assume that the plastic flow potential function has the same form as the plastic yield function (associative plasticity) and consider isotropic hardening/softening behavior in the material of the payzone. For the plasticity, we use the Drucker–Prager model given by Eq. 14 with thermal softening and mechanical hardening. We use the General Closest Point Projection algorithm [56] to determine the plasticity components of the total deformation. The proposed thermo-plasticity model is given in terms of Biot’s effective stresses σ″ = σ + bP. Alternatively, the response can be formulated in terms of Terzaghi’s effective stresses σ′ = σ + P, and this formulation can be performed in the developed framework without significant modifications.

Using the total stress definition given in Eq. 12 and following [73], the plastic strain tensor rate is assumed to be given as a solution of the following system of equations:

$$\begin{array}{@{}rcl@{}} &\mathcal{F}\left( \boldsymbol{\sigma}^{\prime\prime}, c \right) = 0, \end{array} $$

(A.1a)

$$\begin{array}{@{}rcl@{}} &\dot{\boldsymbol{\epsilon}^{p}} = \dot{\lambda} \frac{\partial \mathcal{F}} {\partial \boldsymbol{\sigma}^{\prime\prime}} , \end{array}$$

(A.1b)

$$\begin{array}{@{}rcl@{}} &\dot{\boldsymbol{\sigma}^{\prime\prime}} = \mathbb{C} \cdot \left( \dot{\boldsymbol{\epsilon}} - \dot{\boldsymbol{\epsilon}^{p}} \right) - \mathfrak{a} \dot{T}. \end{array} $$

(A.1c)

where a dot denotes the rate, i.e., the change between loading or time steps. Here, $\mathcal {F}$ is the yield function, λ is the proportionality constant, and $\mathfrak {a} = \mathbf {a} + \frac {\partial \mathbf {a}}{\partial T} \cdot T -\frac {\partial \mathbb {C}}{\partial T} \cdot \left (\boldsymbol {\epsilon } - \boldsymbol {\epsilon }^{p} \right ).$ The system of equations is supplemented by the cohesion increment relationship given as:

$$ \dot c = H_{v} \dot {\epsilon_{v}^{p}} - H_{t} \dot T, $$

(A.2)

where H_v and H_t are two material parameters, ${\epsilon _{v}^{p}}=\text {tr}(\boldsymbol {\epsilon }^{p})$ is the volumetric plastic strain.

As previously, the indices n + 1 and n refer to the current and previous time steps, respectively. A discrete representation of system Eq. A.1a can be written as [25]:

$$\begin{array}{@{}rcl@{}} &&\mathcal{F}\left( \boldsymbol{\sigma}^{\prime\prime}_{n + 1}, c_{n + 1} \right) = 0, \end{array} $$

(A.3a)

$$\begin{array}{@{}rcl@{}} &&-\boldsymbol{\epsilon}^{p}_{n + 1} + \boldsymbol{\epsilon}^{p}_{n} + {\Delta}\lambda \frac{\partial \mathcal{F}} {\partial \boldsymbol{\sigma}^{\prime\prime}_{n + 1}} = 0, \end{array} $$

(A.3b)

$$\begin{array}{@{}rcl@{}} &&\boldsymbol{\sigma}^{\prime\prime}_{n + 1}\,-\,\boldsymbol{\sigma}^{\prime\prime}_{n} \,=\, \mathbb{C} \cdot \left( \boldsymbol{\epsilon}_{n + 1}\,-\,\boldsymbol{\epsilon}_{n} \,-\, \boldsymbol{\epsilon}^{p}_{n + 1}\,+\,\boldsymbol{\epsilon}^{p}_{n} \right) - \mathfrak{a}_{n + 1} \left( T_{n + 1}-T_{n} \right), \end{array} $$

(A.3c)

$$\begin{array}{@{}rcl@{}} &&c_{n + 1} = c_{n} + H_{v}\left( \boldsymbol{\epsilon}^{p}_{v,n + 1} -\boldsymbol{\epsilon}^{p}_{v,n} \right) - H_{t}\left( T_{n + 1}- T_{n} \right), \end{array} $$

(A.3d)

where Δλ = λ_n+ 1 − λ_n. The values T_n+ 1, 𝜖_n+ 1 are fixed during the return mapping step. System Eq. A.3d is nonlinear and can be solved [24] using Newton’s method with respect to the primary unknowns (σ″_n+ 1, Δλ). However, this requires the calculation of the Hessian of the yield function, $\mathcal {F}$, which is a challenging procedure because of the need to compute the second derivatives of a complex function.

The solution (σ″_n+ 1, Δλ) of system (A.3d) implicitly depends on 𝜖 and T. Consequently, the plastic strain 𝜖^p also depends on 𝜖 and T. Thus, the incremental form of Eq. A.1c may be written as [46]:

$$\dot{\boldsymbol{\sigma}^{\prime\prime}} = \mathbb{C}^{ep} \cdot \dot{\boldsymbol{\epsilon}} - \mathbf{a}^{ep} \dot{T}. $$

(A.4)

To ensure a complete linearization of the global equations, the derivatives $\left (\frac {\partial \boldsymbol {\sigma }^{\prime \prime }}{\partial \boldsymbol {\epsilon }}, \frac {\partial \boldsymbol {\sigma }^{\prime \prime }}{\partial T} \right )$ must be derived, which is equivalent to the calculation of consistent elasto-plastic tensors $\left (\mathbb {C}^{\text {ep}}, \mathbf{a} ^{\text {ep}} \right )$. This is a critical aspect of the plasticity return-mapping algorithms [6, 13]. For this purpose, we employ an inverse-theorem approach implemented in AD-GPRS [61]. The system Eq. A.3d can be written in matrix notation as:

$$ \mathbb{R}_{7} = \mathbb{M}_{7 \times 7} \cdot \mathbb{X}_{7} = 0, $$

(A.5)

where $\mathbb {X}_{7}$ is the vector of unknowns (six components of σ″_n+ 1 plus Δλ). After we obtain the solution $\mathbb {X}_{7}$, we compute the matrix of derivatives $\mathbb {J}_{7 \times 7}$ as follows:

$$ \mathbb{J}_{7 \times 7} = \frac{\partial \mathbb{X}_{7}}{\partial \mathbb{Y}_{7}} = \left( \frac{\partial \mathbb{R}_{7}}{\partial \mathbb{X}_{7}} \right)^{-1} \cdot \frac{\partial \mathbb{R}_{7}}{\partial \mathbb{Y}_{7}}, $$

(A.6)

where $\mathbb {Y}_{7} = \left ({\boldsymbol{\epsilon}}_{n + 1}, T_{n + 1} \right )$. Then, the elasto-plastic tensors are simply the components of the matrix $\mathbb {J}_{7 ~\times ~7}$:

$$ \left[\mathbb{C}^{\text{ep}} \quad \mathbf{a}^{\text{ep}} \right] = \mathbb{J}_{7 \times 6} $$

(A.7)

Next, we consider the solution procedure for a given time interval (t_n, t_n+ 1). The state of the material is known at time step t_n, and we need to compute the stress σ_n+ 1 and the tensors a_n+ 1 and $\mathbb {C}_{n + 1}$ at time step t_n+ 1. For this purpose, we use an implicit integration algorithm. The fundamental idea relies on performing a predictor step in which the loading increment is elastic (plastic increment is zero). This allows us to estimate the current stress (trial solution) and evaluate the yield function. If the resulting yield function is less than zero, the trial solution is accepted and all internal state variables are updated assuming elastic deformation; otherwise, the trial state is incorrect and the material state should be plastic. In this case, we iteratively project stresses to the yield surface by solving Eq. A.3d. A high-level view of the basic steps is outlined in Algorithm 1.

Appendix B: Solution strategies

In this section, we provide details of the implementation of the FI and SI schemes. For FI, the mass, energy, and momentum conservation equations are solved simultaneously, and the corresponding system of nonlinear equations is solved using Newton’s method with a damped update. For the SI method, we partition the problem and solve each subproblem sequentially, iterating between the solutions. As previously, we define the fluid variables R = (p, T, x) (pressure p, temperature T, and phase compositions x) and the mechanics variables U = (u, 𝜖^p) (displacement u and plastic strain 𝜖^p). We use the backward Euler time-integration scheme, where the simulation time step Δt is defined by Δt = t_n+ 1 − t_n and the indices n + 1, n refer to the current and previous time steps, respectively. We always use the solution from the previous converged time step as the initial guess for the current time step (Algorithm 2¹).