Balance hydrodynamic flow within a water vessel

The formation of water vessels - or angiogeogenesis - rests on a balance between:

• on the one hand, a hydraulic gradient of impulse to which the flow is subjected which escapes from the water reservoir. This gradient of pressure starts the flow within the water vessel starting from the leakage point - or not of exurgence - water reservoir from which the water vessel is resulting. It results from the overpressure which reigns within this water reservoir and which cannot be contained by the lithospheric context of this one;
• in addition, a hydraulic gradient of resistance which results from the pressure loss undergone by the flow within the water vessel.

The presentation of hydrodynamic balance below wants to be primarily descriptive and heuristic. It does not present the equations of the hydrodynamics associated, but draft their interpretation.

Hydraulic Gradient of impulse

The hydraulic gradient of impulse is produced starting from a leakage within a water reservoir, itself started by the water and thermal feeding with this reservoir. With the right of this leakage a flow presenting is formed a gradient of pressure called hydraulic gradient of impulse. This One causes of:

• to put the fluid moving, i.e. to increase its kinetic energy,
• to raise the fluid, i.e. to increase its potential energy,
• to fight frictions of the fluid against its walls of contact,

The first two terms are conservative, while the third term is dissipative. The first term is weak compared to both others which are of the same order of magnitude.

The hydraulic gradient of impulse influences the evolution of the other variables of state through the conservation equation of the momentum.

As indicated on the graph below, the rate of flow is proportional to this hydraulic gradient of impulse within a water vessel.

Hydraulic Gradient of resistance

The hydraulic gradient of resistance is produced by the frictions undergone by the flow of the geological fluid against its walls within the porous environment which it crosses.

The pressure loss is all the more important as the rate of flow is high.

There exists a value of resistance to the flow for which the speed of this one is worthless. The More this value of resistance to the flow decreases, the more the rate of flow increases. The slope of the right-hand side corresponds, at first approximation, with the proportionality factor of the equation of Darcy.

Balance

The flow of flow in a point of the way of a water vessel is established with a value corresponding to balance between the hydraulic gradient of impulse and the hydraulic gradient of resistance. With this particular value of the gradient of pressure associated with a point with the way with a water vessel, the values taken by the other variables of state characterizing correspond the flow, in particular:

• rate of flow,
• the cross-section of stream discharge,
• the slope…

For the facility of the representation, only the two more important variables of state are represented: the gradient of pressure: ∇p (S) and rate of flow v (S).

It is key to carry out that all variables of state characterizing the flow within a water vessel:

• are defined in a particular point of its way and vary along this way (i.e. according to the curvilinear X-coordinate, S, which follows this way)
• are, on the other hand, constants in time, because a steady state flow was postulated (what is all the more true as one is located at more a great depth).

The diagram below presents the general aspect of this point of balance which is at the crossing between the curve defining the hydraulic gradient of impulse and that defining the pressure loss in function the rate of flow:

Balance between hydraulic gradient and pressure loss for a given depth

This balance is not enough to explain the angiogeogenesis. So that water vessels can be formed and be especially maintained in time and space, it is necessary still that a state of steady balance can be reached on all the way of the water vessel.