6. Theoretical formulation of hydromechanical coupling#

The laws introduced can be based on coupled hydromechanical modeling, by elements XXX_JOINT_HYME. In this part we will talk about the hydraulic part of the law, as well as about the coupling itself; all the details on the mechanical part of the law have been described previously.

6.1. Hydraulic modeling#

Fluid flows from areas of high pressure to areas of low pressure. A theoretical way to take into account steady flow is to associate an energy with the given hydraulic state [13] _ \(H(p(x))\) dependent on pressure distribution. The first hypothesis is to assume that energy depends explicitly on the change in pressure and not on the pressure itself \(H=H(\nabla p(x))\). By taking the simplest possible convex form of this gradient dependence, we thus obtain energy \(H=C{(\overrightarrow{\nabla }p)}^{2}/2\) where \(C\) is a parameter of the law, which does not depend on pressure.

By calculating the generalized forces corresponding to the pressure gradient field we obtain Fick’s first law. Hydraulic flow is proportional to the pressure gradient:

\[\]

: label: eq-62

overrightarrow {w} =frac {partial H} {partialoverrightarrow {nabla} p} =Coverrightarrow {nabla} p

In this energy formalism we seek the pressure field at equilibrium by minimizing hydraulic energy \(\underset{p(\overrightarrow{x})}{\mathrm{min}}\underset{\Omega }{\int }H(\overrightarrow{\nabla }p(\overrightarrow{x}))d\Omega\). This gives an equilibrium equation similar to that of mechanics \(\mathrm{div}\overrightarrow{w}=0\). In the framework of this model, solving the hydraulic equilibrium equation is equivalent to solving a mechanical problem in a quasistatic manner, where the hydraulic flow is equivalent to stresses \(\overrightarrow{w}\iff \sigma\), the pressure field corresponds to the displacement field \(p(\vec{x})\iff u(\vec{x})\) and finally the pressure gradient is similar to the deformation field \(\overrightarrow{\mathrm{\nabla }}p\mathrm{\iff }\varepsilon\).

6.2. Influence of hydraulics on mechanics#

The presence of fluid in the joint adds hydrostatic stress and this fact changes the normal mechanical stress \({\sigma }_{n}\to {\sigma }_{n}-p\). By applying significant pressure, it is possible to cause the joint to break by a simple hydraulic effect. We can shift the mechanical law downwards according to the pressure value \(p\) at each point to take into account the effects of pressure, see.

6.3. Influence of mechanics on hydraulics#

In the case of fluid flow through a crack the hydraulic flow must increase with the opening (\({\delta }_{n}\)) of this last one (\(\overrightarrow{w}~O({\delta }_{n})\overrightarrow{\nabla }p\)). In Poiseuille’s law, which was found empirically for the laminar flow of a viscous and incompressible fluid, the flow dependence on opening is cubic (the law is often called the cubic law). The hydraulic part of the law uses this type of coupling. The equations to be solved are written as follows:

(6.1)#\[ \ text {div}\ overrightarrow {w} =0;\ overrightarrow {w} =\ frac {\ rho} {12\ stackrel {} {\ mu}} {\ mu}} {\ delta}}} {\ delta}} _ {n} _ {n} _ {n}} ^ {3}\ overrightarrow {\nabla} p\]

In the case of a flow of fluid through the junctions of a dam, there are significant flows even for closed joints. It is then necessary to define a minimum thickness \({\epsilon }_{\mathrm{min}}\), keyword OUV_MIN, below which the flow reaches its minimum value. We regularize the flow equations in the following way:

(6.2)#\[ \ vec {w} =\ frac {\ rho} {12\ stackrel {\ rho} {\ rho} {12\ stackrel {oung}} {\ mathit {rho}} {\ mathit {rho}} {12\ stackrel {\ rho}} {12\ stackrel {oung}} {12\ stackrel {ounty}} {\ rho} {12} {\ stackrel {\ rho}} {12\ stackrel {ounty}} {\ rho} {12} {\ stackrel {\ rho}} {12\ stackrel {ounty}} {\ rho} {12} {\ stackrel {\]

For a non-zero pressure gradient the flow never reaches the zero value, \(\text{min}\overrightarrow{w}~{\epsilon }_{\mathrm{min}}^{3}\overrightarrow{\nabla }p\), which corresponds to the flow through the permeable walls of the closed joint.

6.4. Hydromechanical coupling#

Hydromechanical coupling involves the two mechanisms described above: on the one hand, the fluid acts by pressure on the joint lips; on the other hand, the more open the crack is, the easier the fluid flow is. In the absence of external forces, the hydromechanical calculation is presented schematically in this form:

(6.3)#\[\begin{split} \ {\ begin {array} {cc}\ vec {w} =\ vec {w} =\ vec {w} (\ vec {\ delta} (u),\ vec {\nabla} p); &\ text {div}\ vec {w} =0\\ vec {w} =0\\ vec {w} =\ vec {\ sigma} (u), p); &\ text {div}\ vec {w} =0\\ vec {\ sigma} =\ vec {\ sigma} =\ vec {\ sigma} (u), p); &\ text {div}\ vec {w} =0\\ vec {w} =0\\ vec {\ sigma} =\ vec {\ sigma} (u), p); &\ text {div}\ vec {w} =0\\ vec {w}} =0\ end {array}\ text {}\ Rightarrow\ text {}\ vec {Y} =\ vec {Y} (\ vec {X});\ text {div}\ div}\ vec {Y} =0\end{split}\]

Solving hydromechanical equilibrium equations is equivalent to solving the mechanical problem in a quasistatic manner, where we introduce generalized constraints \(\overrightarrow{Y}=(\overrightarrow{w},\overrightarrow{\sigma })\), and the vector field of unknowns \(\overrightarrow{X}=(p,u)\).

6.5. Tangent matrix#

Since the generalized efforts depend on \(u\) only through \(\overrightarrow{\delta }(u)\), to calculate the tangent hydromechanical coupling matrix, it is necessary to know only the following four terms:

(6.4)#\[ \begin{align}\begin{aligned} \ frac {\ partial\ overrightarrow {\ sigma}} {\ partial\ overrightarrow {\ delta}}\\ \ textrm {,}\\ \ frac {\ partial\ overrightarrow {\ sigma}} {\ partial p}}} {\ partial p} `, :math:`\frac{\partial \overrightarrow{w}}{\partial \overrightarrow{\nabla }p}` and:math:`\ frac {\ mathrm {\ partial}\ overrightarrow {w}}} {\ mathrm {\ partial}}\ overrightarrow {\ delta}}\end{aligned}\end{align} \]

The first term is the same as in pure mechanics, it is given in equation (). The second term is trivial, as the only non-zero component is equal to \(\partial {\sigma }_{n}/\partial p=-1\). The term diagonal hydraulics takes a simple form because hydraulic flow only depends on the pressure gradient:

(6.5)#\[ \ frac {\ partial\ vec {w}}} {\ partial\ vec {\nabla} p} =\ frac {\ rho} {12\ stackrel {] {\ mu}}\ text {max} {({{}} {({}} _ {} _ {\ mathit {min}}} + {\ delta} {\ mu}}}\ text {max} {{}} {{n}} {({{n}}) {({}} _ {n}) {({} _ {n}) _ {n} _ {n}) _ {n} _ {n}) _ {3} _ {n})} ^ {3}\]

In the last term only the derivative with respect to the normal opening is not zero:

(6.6)#\[ \ frac {\ partial\ overrightarrow {w}}} {\ partial {\ delta}} _ {n}} =\ frac {\ rho} {4\ stackrel {} {\ mu}}} {({\ epsilon}}} {({\ epsilon}} _ {\ epsilon}} _ {n}}}} {\ stackrel {} {\ mu}}}} {({\ epsilon}} _ {\ epsilon} _ {\ epsilon} _ {min}} _ {n}} =\ frac {\ rho} {4\ stackrel {} {\ mu}}}} {({\ epsilon}}}} {({\ epsilon}} _ {\ epsilon} _ {\]

This quantity is equal to zero for a closed crack \({\delta }_{n}<0\). The tangent matrix thus formulated is not symmetric.