2. Benchmark solution#

2.1. Calculation method#

It is an analytical solution. The fluid pressure \({p}_{f}\) imposed in the cohesive interface is transmitted to the pore pressure at the level of the lips of the cohesive interface (cf Documentation [R7.02.18]). Taking into account the boundary conditions, everything happens as if a pore pressure step were imposed at the end of the two sub-columns formed by the cohesive interface. So the problem is unidirectional. These two sub-columns are frozen because the movements are blocked. By neglecting gravity, the equation for the conservation of mass in the porous matrix is written as follows:

\(\frac{\partial {m}_{w}}{\partial t}+\nabla \cdot M=0\)

with \({m}_{w}=\mathrm{\rho }\mathrm{\varphi }(1+{\mathrm{ϵ}}_{v})\) the mass inputs into the porous matrix and \(M=\frac{-\mathrm{\rho }{K}^{\text{int}}}{{\mathrm{\mu }}_{w}}\nabla p\) the flow of Darceen fluid into the porous matrix. Since the movements are blocked, the volume deformation \({\mathrm{ϵ}}_{v}\) is zero. Moreover, \(d\mathrm{\varphi }=(b-\mathrm{\varphi })(d{\mathrm{ϵ}}_{v}+\frac{\mathit{dp}}{{K}_{s}})\) with \({K}_{s}\) the compressibility modulus of the solid matrix. However, the solid matrix is supposed to be incompressible because we have taken \(b=1\) (and \(b=1-\frac{{K}_{m}}{{K}_{s}}\) with \({K}_{S}\) the compressibility module of the porous medium). So finally \(d\mathrm{\varphi }=0\). Finally \(\frac{d\mathrm{\rho }}{\mathrm{\rho }}=\frac{\mathit{dp}}{{K}_{w}}\). The mass conservation equation can therefore be rewritten:

\(\frac{\mathrm{\rho }\mathrm{\varphi }}{{K}_{w}}\frac{\partial p}{\partial t}-\frac{{K}^{\text{int}}}{{\mathrm{\mu }}_{w}}[\nabla p\cdot \nabla \mathrm{\rho }+\mathrm{\rho }\mathrm{\Delta }p]=0\)

Overlooking the second-order term \(\nabla p\cdot \nabla \mathrm{\rho }\), it comes from:

\(\frac{\mathrm{\varphi }{\mathrm{\mu }}_{w}}{{K}_{w}{K}^{\text{int}}}\frac{\partial p}{\partial t}=\mathrm{\Delta }p\)

It is a diffusion equation. The coefficient \(\frac{\mathrm{\varphi }{\mathrm{\mu }}_{w}}{{K}_{w}{K}^{\text{int}}}\) has the dimension of time over a length squared. In our problem, the characteristic dimension for broadcasting is \(\frac{\mathit{LZ}}{2}\). A characteristic diffusion time \(\mathrm{\tau }=\frac{\mathrm{\varphi }{\mathrm{\mu }}_{w}}{{K}_{w}{K}^{\text{int}}}\ast {(\frac{\mathit{LZ}}{2})}^{2}\) is deduced from this. This characteristic time is worth \(\mathrm{\tau }=\mathrm{0,459842}s\). At the end of \(t=10s\), the diffusion is therefore complete, the pore pressure in the porous matrix then verifies \(\mathrm{\Delta }p=0\).

According to the boundary conditions, below the interface:

  • the pore pressure is \(p(y)=\frac{2y}{\mathit{LZ}}{p}_{f}\) and the fluid flow in the matrix is \(M=\frac{-2\mathrm{\rho }{K}^{\text{int}}{p}_{f}}{\mathit{LZ}{\mathrm{\mu }}_{w}}\).

And at the top of the interface:

  • the pore pressure is \(p(y)=(1-\frac{y}{\mathit{LZ}})2{p}_{f}\) and the fluid flow in the matrix is \(M=\frac{2\mathrm{\rho }{K}^{\text{int}}{p}_{f}}{\mathit{LZ}{\mathrm{\mu }}_{w}}\).

2.2. Reference quantities and results#

The value of the outflows from the cohesive interface to the lower and upper parts of the porous column as well as the value of the pore pressure in \(z=4m\) and \(z=6m\) are tested.

Tested sizes

Value

LAG_FLI (Flow leaving the interface to the bottom of the column)

4.07748 kg·m²/s

LAG_FLS (Flow leaving the interface to the top of the column)

4.07748 kg·m²/s

PRE1 (at Z=4m)

8Mpa

PRE1 (at Z=6m)

8Mpa

2.3. Uncertainties about the solution#

The reference solution is analytical.

2.4. Bibliographical references#

  1. Reference documentation R7.02.18 (Hydromechanical elements coupled with the Extended Finite Element Method).