Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media

Round 1

Reviewer 1 Report

The authors should compare results with standard mixed finite elements in order to show the advantage of IGA analysis. Moreover they shoud cite  recent progress in IGA analysis based in Mixed integration Points (MIP) and reduced integration presented for K-L and Solid Shells models

Author Response

We have added a discussion focusing on the comparison of our IGA analysis results with results from standard mixed finite elements. We refer to the extensive finite element study on pressure oscillations presented by Haga et el. 2011 (Ref. 28) where the low permeability layer problem is one of the numerical examples investigated. Haga et al. presented a systematic case study where they considered three different formulations for the mathematical model of poroelasticity and several combinations of finite elements. The mathematical formulations considered are 1) two-field formulation (displacement and fluid pressure), similar to our paper, 2) three-field formulation (displacement, velocity and fluid pressure) and 3) four-field formulation (displacement, soild pressure, velocity and fluid pressure). These different formulations were numerically investigated using several combinations of standard triangular and quadrilateral finite elements with different combinations of polynomial orders or continuities for the field variables. Lagrangian, Raviart-Thomas, Crouzeix-Raviart (triangular) and Rannacher-Turek (quadrilateral) elements were used for the numerical studies. For the low permeability problem, equal order triangular and quadrilateral Lagrange finite elements (both first- and second-order) resulted in a solution with pressure oscillations, as expected. Mixed triangular and quadrilateral Lagrange elements (second-order for displacement and first-order for fluid pressure) were found to improve the pressure solution but local pressure spikes were still exhibited. Satisfactory solutions were obtained only when using three- and four-field formulations with different element types for the field variables, for example when representing the velocity field with linear Raviart-Thomas elements where the displacement and fluid pressure use second- and lowest-order triangular or quadrilateral Lagrangian elements, respectively. Our IGA analysis results produced satisfactory results just based on the two-field formulation, without introducing additional field variables and combinations of element types for the field variables. This comparison and detailed discussion is added in Section 4.3 (line 350-373 in revised manuscript). Regarding recent progress in IGA analysis based on mixed integration points (MIP) and reduced integration, we have updated the introduction section by citing three additional relevant publications (line 130-133 in revised manuscript).

Reviewer 2 Report

1. Please add the labels of coordinate system along with Fig.3 and Fig. 7. It could help readers to correlate the boundary conditions with the results showed in Fig.4, Fig.5, Fig.8, and Fig.9. Also, it would great if the difference in coordinate systems among this research and former works could be mentioned (for example, y axis in this research is defined as z axis in Ref. 28).
2. The results showed that the pressure oscillation is greatly reduced by adopting the proposed model. However, more explanations could be made to clarify the differences between the original model and new model. In Fig.5, the curve calculated from mix order formulation had only a small pressure oscillation started at the location when y/h~1.0, and that from the equal order formulation started earlier. Since there is no interlayers in the model of Terzaghi’s Problem, It is easy to understand the contribution of the mix order formulation contributes to the reduction of pressure oscillation in this case.
   However, when a low permeability layer was added in between two normal layers, the difference between the mix model and equal order model is not that intuitive. In equal order model, the pressure oscillation initiated as soon as the point reached 0.25 (y/h), and ended at the boundary position (0.75). As for the mix model, pressure oscillation was suppressed and initiated around 0.6 (y/h) visually. If this phenomenon is due to the contribution of some terms in the formula, it could be great to add some discussions in related paragraphs.
3. Discussions on the major outcomes of this work and differences from former works could be added in the end of introduction (around line 130 to 134).
4. Obeying poroelesticity behavior is the basic assumption for all the models in this research. It would be great if the authors could address the applicable range of the proposed model. In the selected verification cases, the boundary of material model is significantly different (for example, external load and young's modulus). Would it be possible to define the applicable range of the proposed model? In other words, is the model applicable if a certain material proves to be poroelastic under a given external loading? Slip line field method could also be adopted to calculated for the force and deformation conditions of material, but is mostly suitable for the cases when material underwent plastic deformation.

Author Response

1. The labels of the coordinate systems are added in Fig. 3 and Fig. 7. As remarked, we use 'y' for the vertical direction in this manuscript. We have added a sentence (in the paragraph before Fig. 3, line 263-266 in revised manuscript) clarifying the difference between the coordinate systems used in our manuscript and similar references like Ref. 28.
2. Where the pressure oscillations start is affected by a combination of factors including the mathematical nature of the equations, the boundary conditions of the problem and the order and continuity of elements used for the numerical solution. For both the Terzaghi and low permeability layer problems the pressure boundary condition is such that pressure dissipates only at the top boundary i.e., at y/h=1. To explain where the pressure oscillations occur, we refer to the mass balance equation i.e., Eq. 14. For short and early time steps coupled with a very low permeability, the second and third terms in Eq. 14 become negligible and the solution to Eq. 14 requires divergence-free displacements close to the dissipation boundary and/or in the layer with the low permeability. Requiring to satisfy this with equal order elements for displacement and pressure locks out most of the displacement degrees of freedom leading to oscillations in the pressure solution. Thus, equal order elements result in significant pressure oscillations in the low permeability layer. Note also that the pressure in the high permeability layer dissipates very quickly making the interface between the two layers the main dissipation boundary. We have included additional explanation on this in Section 4.3, line 324-330 in revised manuscript.
3. We have added the main contribution of our paper in the last paragraph of the introduction section (line 135-138 in revised manuscript) in comparison with existing works. The following sentences are added: "We demonstrate through numerical examples that mixed isogeometric analysis can overcome pressure oscillations with the right choice of continuity and meshing, without introducing additional field variables in the mathematical formulation. The results are discussed in comparison with related numerical studies."
4. Poroelastic simulations are often used in application areas where plastic deformations are negligible or do not occur. Examples of such application areas include geomechanics and reservoir engineering where the deformations are small compared to the scale of the model or where the loading level is low to cause plastic deformations. This is discussed in the introduction section with additional clarification. (line 17-18 in revised manuscript)

Reviewer 3 Report

All authors made great efforts to clarify the mixed formulation for isogeometric analysis of poroelasticity on Terzaghi’s classical one-dimensional consolidation problem and consolidation of a layered soil with a middle low permeability layer.

Some parts are clear and the present form could be accepted. For the readers, all authors had better understand this paper regarding the following question.

Why not compare the result between numerical and physical test results?

Do you have a plan to compare with physical test results? If yes, please add it to the text.

Author Response

Physical testing is not considered within the scope of our manuscript. The primary reason for this is that our aim was to investigate the pressure oscillations that arise during the numerical solution of the problems of poroelasticity. These pressure oscillations arise purely due to numerical reasons and are non-physical in nature. Hence, we haven't considered comparing the numerical results here with physical tests.

Round 2

Reviewer 1 Report

The paper, in this reviewer opinion, is ready for publication.