Case 1b Poro-elasto-plastic simulation (SR4)

The result files for the project are in directory: Mat_003\Case1\Results  

This example is identical to Case1a except that SR4 plasticity is included in the constitutive response of the matrix. Most of the figures will highlight the differences between the two cases.

In the figures below the evolution of stresses and pore pressure is shown. It can be noted that:

1.As in the poroelastic case the stresses show a negative correlation with pore pressure

2.As in the poroelastic case effective stress in Y direction is lower than effective stress in X direction due to the contribution of the fractures (which are perpendicular to Y axis and therefore accommodate some of the stress in the form of fracture displacement)

3.The stress path in the p'-q space (effective mean stress - deviatoric stress space) shows that:

a.The initial stress state is on the initial yield surface (black thin line) near the peak strength (point a).

b.From t=1.0 to t= 2.0 days pore pressure decreases. The stress path moves initially from point "a" to "b" towards the k0 path (stress path under burial conditions in which generally σ'z > σ'x = σ'y for intact samples). This is expected as the decrease in pore pressure causes plasticity (compaction) driven by the overburden stress. Note that the plastic path is not exactly on the k0 path due to the influence of the fractures that result in σ'x > σ'y. Then the path moves from point "b" to "c" following the burial path. During that period the strength of the material has increased due to plastic hardening as shown by the yield surface size increase (from initial yield surface which is the thin black line to the blue yield surface).

c.From time t=2.0 to t=2.5 days pore pressure increases leading to stress decrease due to unloading. The stress path moves following an elastic path with decreasing p' and q from "c" to "d".

d.From time t=2.5 to t=3.0 days pore pressure decreases and hence stresses increase. Initially the stress path moves elastically from "d" to "c" following the same unloading path until pore pressure reaches the previous minimum value at time c.a. t=2.8. Then the stress path moves from "c" to "e" and there is more plastic compaction with hardening with a yield surface size increase (from the blue surface to the thick black surface).

Evolution of stresses and pore pressure

Comparison of the evolution of vertical displacement, porosity and permeability for poroelastic (Case1a) and poro-elasto-plastic cases (present case) is shown in the next figure. As expected plasticity leads to larger displacements and larger porosity and permeability reduction. It can be noted that during the elastic unloading/reloading stage changes in displacement, porosity and permeability are smaller than in plastic phases as only the elastic strains are recovered (see for example the change in slope in porosity curve at time c.a. t=2.8 days).

Comparison of displacement, porosity and permeability for Case1a (poroelasticity) and Case1b (SR4 plasticity)

The evolution of the different compressibilities is shown on the following figures (see Case1a results section for an explanation on the different compressibilities). It can be observed that:

1.In the first figure on the left it can be seen that the pore volume compressibility (D(phi)/D(pf)) and the pore volume compressibility as a function of p' (D(phi)/D(peff)) are much larger than the elastic compressibility (Elastic phi(k) )during the plastic phases (time t=1.0 to t=2.0 days and t=2.8 to t=3.0 days). This is consistent with the larger porosity loss observed for the present case considering plasticity compared to the case considering poroelasticity only. During the elastic phase (t=2.0 to t=2.8 days) it can be seen that the pore volume compressibility coincides with the elastic compressibility because all the porosity change is elastic. The pore volume compressibility relative to the effective mean stress does not coincide because as explained in the results section of Case1a the change in effective mean stress is different than the change in pore pressure.

2.In the center figure it can be seen that the elastic compressibility in ParaGeo (which is defined as a function of the current porosity) differs substantially to the elastic compressibility as defined in the literature (which is defined as a function of the initial porosity, e.g. Dean et al. 2006) due to the large plastic porosity loss.  

3.For the poroelastic and plastic cases there is a difference in the elastic compressibility as defined in the literature even though the initial porosity is the same for both cases (0.15). This difference is attributed therefore to the difference in Young's modulus that will be discussed below in the next figures below.

Evolution of the different compressibilities (left and center). Comparison of elastic compressibilities (based on initial porosity) for Case1a (poroelasticity) and Case1b (SR4 plasticity)

As seen in the following figures Young's modulus is different for poroelastic and plastic cases. In both cases Bulk modulus (and hence Young's modulus) is a function of the effective mean stress so that the larger the p' the larger the Young's modulus. Thus the larger values of the Young's modulus for the case with SR4 plasticity must imply larger values of the effective mean stress as shown in the central figure. Also it is noteworthy to note that the difference in p' results from the larger horizontal stresses in the plastic case. This is expected as in the poroelastic case horizontal stresses can only develop due to the Poisson's ratio effect on strain upon elastic vertical strain whereas in the plastic case larger horizontal stresses can develop due to the plastic consolidation. It should be noted that the slight difference in the initial Young's modulus and effective mean stress value between the poroelastic and plastic cases (at time t=1.0 days) is attributed to little plasticity which occurred during the initialisation stage.

Comparison of the evolution in Young modulus and effective stresses

Comparison of the evolution of fracture properties for poroelastic and plastic cases is shown in the figures below. As mentioned above, in the poroelastic case horizontal strains (and hence stresses) develop due to Poisson's ratio effect. On the other hand in the case considering plasticity horizontal strains resulting from vertical plastic consolidation are much larger. This has also an impact in fracture properties as in the plastic case there is more fracture displacement which decreases fracture aperture. Consequently fracture permeability (which depends on fracture porosity) in reduced more drastically. For the same reason fracture normal stiffness (which increases as fracture aperture decreases) and fracture normal stress are larger in the case with plasticity than in the case considering only poroelasticity.

Comparison of the evolution of fracture properties for cases 1a (poroelasticity) and 1b (SR4 plasticity)

In the figures below comparison of matrix porosity and permeability and fracture permeability as a function of pore pressure for the two cases is provided. The differences between the two cases have already been discussed above.

Comparison of porosity and matrix and fracture permeabilities for Case1a (poroelasticity) and Case1b (SR4 plasticity)

Dean, R.H., Gai,X., Stone, C.M. and Minkoff, S.E. (2006). A comparison of techniques for coupling porous flow and geomechanics. SPE Journal, 11(1), 132–140.