06.05.2021

Matthew Profit, 6^{th} May 2021

Many geotechnical problems require solving a coupled set of partial differential equations to determine both stress and pore fluid pressure evolution. Typical applications include land settling, wellbore stability, and the forward modelling of geological structures.

These are typically very large problems, especially when we’re dealing with the depletion of an oil reservoir or the deformation of geological structures by plate tectonics.

The differential equations, which are often referred to in literature as “porous media” governing equations, can be solved in a variety of ways.

One common approach is to use an explicit-implicit solver, also known as a staggered coupled solver. This determines the material stresses using an explicit solver and pore fluid pressures with an implicit solver. Primary variables are passed back and forth between the two solvers and updated using predictor-corrector methods.

Although this approach uses explicit material stress algorithms that are traditionally more stable, it still has its drawbacks. For example:

- Difficulty determining the degree of solution error, particularly for highly nonlinear systems
- Confusion caused by pore fluid pressures being solved on both the explicit and implicit solver, yielding two solutions for the same variable
- Difficulty determining an appropriate mass scaling factor, which is key for industrial scale problems where scaled models are used to achieve acceptable run times
- Excessive run time, since even when using mass scaling methods, time steps can be very small, particularly for stiff, competent rocks
- Difficulty selecting suitable damping coefficients, even for a quasi-static solution

This leaves the analyst with numerous tricky modelling decisions to make around domain size, material characterization, loading conditions, boundary conditions, and in-situ stresses.

To avoid this situation, we are working to develop a fully implicit solver – i.e., one that solves both governing equations implicitly.

Figures 1-3, below, show

- an exact solution (Fig.1, Terzaghi),
- Terzaghi solution using staggered explicit-implicit coupling scheme (Fig.2, Elfen),
- Terzaghi solution using monolithic fully implicit coupling scheme (Fig.3, Elfen)

Figure 1 – Exact Solution (Terzaghi)

Figure 2 – Terzaghi solution using staggered explicit-implicit coupling scheme (Elfen)

Figure 3 – Terzaghi solution using monolithic fully implicit coupling scheme (Elfen)

The first step is to develop implicit material models that accurately and stably integrate the stress response in an environment where stress-softening deformation is ubiquitous.

The soft rock material model in Elfen, known as SR3, has a strong track record simulating the mechanical response of rocks such as sandstones and shales across a large range of stress states but it only operates within the explicit solver. Our latest development work brings the SR3 material model into the ambit of the implicit solver.

Both mechanical stresses and pore fluid pressures will now be solved using a fully implicit algorithm with a unified time-stepping scheme.

Many geotechnical problems traditionally solved using the explicit-implicit approach can now be solved using the fully implicit solver, significantly reducing the number of input parameters required and solving the governing equations exactly.

Classes of problem we plan to simulate with the new software include wellbore instability, variable sedimentation rates (and subsequent consolidation of geomaterials), and the formation of complex geological structures such as faults and folds due to plate tectonics.

We hope this new technology will be of great interest and value to your business. If you are simulating problems within one of the application areas we’ve mentioned or you’re facing a different challenge that the fully implicit simulator can help, we’d love to hear about it!

Share your thoughts by dropping us a line at john.cain@rockfieldglobal.com

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