Challenges of Coupled Geomechanical Modeling: II. Connecting Different Worlds

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A scene of the movie, Interstellar by Christopher Nolan shown the spaceship, Endurance.
Figure 1. A scene of the 2014 sci-fi movie, Interstellar directed by Christopher Nolan showing the spaceship, Endurance trying to travel through a wormhole.

We may prefer to think different modules in coupled models are like matching pieces of a jigsaw puzzle and, by coupling, we are just fitting them to figure out the big picture. In reality, this is not probably the most accurate image of the problem as, usually, different modules in these models are for different physics with different field equations, different discretization methods, and even sometimes different time steps, a fact that modelers ought to remember to acquire precise results from their models. In practice, these unfitting pieces need to be stitched to each other by interfaces, i.e., the mediums with the role of transferring and translating data between these modules. Like any language translation, if translated loosely or if some data get lost in translation, misinterpretations and errors will dominate the solution. In this post, I will discuss some of the common communication issues that can threaten the accuracy of coupled modeling. Some others will be discussed in another post of this series to be published soon on geometry and gridding.

Cell-To-Node Projection, A Big Deal!
A general schematic showing the cells of a fluid flow/heat transfer model versus the elements and nodes of a geomechanical problem.
Figure 2. A general schematic showing the cells of a fluid flow/heat transfer model versus the elements and nodes of a geomechanical problem (source: Soltanzadeh, 2015).

This is a less noticed (or better to say ‘less considered’) issue that has the potential of leading to significant errors especially in locations close to the boundaries of the permeable and low permeability rocks (for instance, a reservoir and its caprock). Most likely, for many justifiable reasons, your fluid flow/heat transfer module uses a version of finite volume method (FVM) for solving its field equations while your geomechanics model uses finite element method (FEM). These two methods are significantly different in the ways they discretize geometry and in how they treat the field equations, set aside the different physics involved. While in FVM the equilibrium equations are written for a volume (cell), in FEM, they are written for nodes of an element. As a result, in FVM, properties (e.g., pore pressure, temperature, or stresses) are assigned to the volume of discretized cells but, in FEM, they are assigned to the nodes of elements. As its name makes it evident, the cell-to-node projection becomes a concern when the common properties between these modules are transferred or projected between the cells of one module to the nodes of another. To get an idea of this process, see the simple example below.


Example

Assume a basic fluid flow – geomechanics coupled model with very simple geometry as shown in Figure 3a for a reservoir and its caprock. For simplicity, both modules have the same mesh.

A simple example showing how cell-to-node projection problem appears in coupled modeling.
Figure 3. A simple example showing how cell-to-node projection problem appears in coupled modeling (source: Soltanzadeh, 2015).

1. Imagine, as a result of production, pore pressure change of dp is given by the fluid flow module within the reservoir and, as expected, no pressure change is predicted by this module in the caprock (see Figure 3b).

2. The given cellular pressure needs to be translated into nodal properties of the geomechanics module. Using  a simple technique, pore pressure at each node is calculated by averaging pore pressure in the cells around it (Figure 3c).

3. Now, let’s see how these nodal pressure changes are understood by the geomechanics module. This module simply calculates pore pressure change in each of its element by averaging the nodal properties and gives what we see in Figure 3d.

You see how a pore pressure change of dp/4 has been artificially introduced into the bottom layer of the caprock by just data transfer from one module to another. Also. pore pressure change within the top layer of the reservoir is 3dp/4 in the FEM model lower than the fluid flow module prediction .  It works similarly for heat transfer. 


To get a sense of the importance of this issue, imagine the model given in the example above has been developed for a caprock integrity assessment of an in-situ thermal project (e.g., SAGD). Having artificially created pore pressure and temperature in the riskiest location of the caprock easily leads to significant increase in the predicted chance of caprock failure and predicts lower maximum allowable pressure (MOP), something a producer does not like to hear. Goumiri and Prevost (2010) tried to quantify the error rising from cell-to-node projection.

Remedies

Dealing with the cell-to-node projection is not probably the easiest task in modeling. One way to solve the problem is using specific numerical techniques such as using piecewise shape functions and low order integration in the FEM module. This is not the modeler’s job and must had been formulated in the software by the developer. This, of course, will take the load off your shoulders, so, if you are using a full thermo-poro-mechanical software suit, always check your software’s technical manuals to see how its interface treats the issue of data transfer between different modules. However, several modeling softwares have not been designed with this consideration, or in some cases, the modelers have to write their own interfaces to facilitate data transfer between different modules. In these cases, a solution is using very fine gridding at the vulnerable boundaries so you will be able to ignore the wrongly behaving elements with more confidence and, instead, track your model behaviour at the adjacent observation cells that are not significantly affected by the issue (see Figure 4, for an example). Splitting your thick caprock in 4 equal layers and tracking the response in the bottom cells, for instance, very likely will give you inaccurate results.

Figure 5. Gridding for a Low-temperature CO2 injection study. Notice the gridding of the caprock close to its boundary with the reservoir and the location of the observation cell in the caprock (soltanzadeh and Jafari, 2013).
Figure 4. (a) pore pressure change and (b) temperature change profiles for a low-temperature CO2 injection case study. Notice the fine gridding of the caprock close to its boundary with the reservoir and the location of the observation cell in the caprock (source: soltanzadeh and Jafari, 2013).

Strength of Coupling

Different coupling approaches may be used in modeling such as partly coupled, one-way, two-way, and iterative two-way. Serious problems might occur when coupling degree between different modules is not strong enough and it neglects or underestimates the effect of different physics on each other. For instance, for a mechanically-sensitive reservoir, implementing a one-way or even a simple two-way coupling approach might lead to significant errors in the results. Examples are unconsolidated rocks such as shallower reservoirs or oil sands or shale gas reserves and fractured reservoirs. I fully acknowledge this fact that we still have a long way to understand the detailed physics of fluid flow/heat transfer and geomechanics interaction for different reasons such as the science being young and research being costly and expensive but there is no excuse for ignoring what we already know.

Expertise Coupling

Figure 6
Figure 5

Naturally, for a modeler coming from one discipline, it is a common tendency to underestimate the influences or complexities of other disciplines and ‘take it easy’ when it comes to ‘them’. Flipping through the literature, many examples may be found, e.g., reservoir engineers who see geomechanics as one of the ‘add ons’ of their reservoir modeling softwares and also geomechanics specialist with similar mentality about fluid flow modeling. We may even encounter experts from other disciplines (like mathematicians) who, without a solid understanding of any of these physics, still insist on doing all different parts of the job themselves assuming the story is just about solving some equations or using a software package. Occasionally, a similar attitude may even be observed in some software developing companies. Of course, in this diverse world of science, nobody can or expected to master every branch of science and coupled geomechanical modeling, as an inter-disciplinary practice, is not an exception and it, definitely,  needs the collective knowledge of a diverse team of experts or let’s call it ‘expertise coupling’ in addition to coupling of different physics for its success.

To be continued.

Read Part I of This Series

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