Hydrodynamic Stability of a Water Vessel

The flow of the geological fluid at the point of formation of a water vessel is steady state or strong near to this mode (it is all the more steady state as the water vessel is located at great depth). So that the characteristics structural (morphology 3D) of a water vessel can be maintained beyond its point of formation, it is necessary that a stable hydrodynamic balance is established in each point of its way. This stability is determined by analyzing the reaction of this hydrodynamic system to any fluctuation of a variable of hydraulic state (the stability of the thermal variables of state concerns the steady state character of the flow).

Stability in space (or structural stability)

The diagram below illustrates some of the possible fluctuations around the point of balance:

Effect of a fluctuation on hydrodynamic balance in a point of a water vessel

The hydrodynamic confinement ceiling presented on this diagram corresponds to the maximum gradient of pressure beyond whose a ramification is formed. This ramification gives rise to a water capillary. The flow within the water capillary is always very weak (<1 %) compared to the flow within the water vessel from which it is resulting. The hydrodynamic confinement ceiling is function of the compact character of the crossed geological layer. This concept covers at the same time the hardness of material and the low porosity of this one. The more compact the geological layer crossed by the water vessel is, the more one needs a high gradient of pressure before a ramification can be formed. When a water vessel passes from a compact geological layer towards a less compact layer, a junction in two or several branches of comparable importance is formed. It is not then any more the formation of a water capillary, but of a branch presenting a flow comparable with that of the principal flow.

One distinguishes two mechanisms from stabilization of the water flow:

• discrete stabilization (within the meaning of discontinuous) and
• stabilization continues.

These two mechanisms are described hereafter:

Discrete Stabilization (or discontinuous)

The hydrodynamics of water vessel is governed, on the one hand, by the conservation equation of the momentum and, on the other hand, by the equation of Darcy. The solution of these equations is such as, in the absence of loss of flow,

• the gradient of pressure,
• rate of flow,
• the slope…

evolve all in an increasing and similar monotonous way. This progressive increase in the values taken by these variables of state leads the gradient of pressure to reach the hydrodynamic confinement ceiling (to which it was already close at the point to formation of the water vessel). The crossing of this ceiling of confinement results in the formation of a ramification producing a water capillary. The small reduction of the flow of flow associated with this phenomenon slightly makes decrease the slope of the way followed by the water vessel. This reduction in the flow results, on the diagram represented above, in a lowering of the curve representing the gradient of hydraulics of impulse (which passes by the origin) according to the rate of flow. This displacement thus brings back the point of balance under the hydrodynamic confinement ceiling.

This mechanism of stabilization occurs by jolts (with each formation of a water capillary) and maintains the variables of flow in the zone close to the point of balance. This zone can move appréciablement at the time of the transition from a geological layer to another. The flow then undergoes an important fluctuation by formation of a junction whose induced water vessels are of more comparable size (than between a water capillary and the water vessel from which it is resulting).

Continuous Stabilization

Stabilization continues flow within a water vessel results from the combined effect of the two laws fixing hydrodynamic balance:

• the rise in the hydraulic gradient of impulse along the way induces a joint increase rate of flow,
• the increase this rate of flow increases the pressure losses (the hydraulic gradient of resistance) which slow down the flow.

Thus, for example, if:

∇p_i ↑ ⇒ v ↑ ⇒ ∇p_r ↑ ⇒ v ↓

where ∇p_i and ∇p_r correspond respectively to the hydraulic gradient of impulse and the hydraulic gradient of resistance.

These antagonistic effects on the rate of flow maintain the hydrodynamics of the water vessel in steady balance. If the rate of flow and the gradient of pressure are stable in a given point of the way, all the other variables of state are it too. With constant flow, i.e. on the sections between ramifications:

• the cross-section of stream discharge evolves in an opposite way at the speed: an increase speed along the way of the water vessel produces a constrictive effect on its section,
• the slope of the way evolves in a way similar to the hydraulic gradient and the rate of flow.

The hydrodynamics of the water vessels is such as they form stable structures in space. These structures can be exploited from a hydrogeological point of view.

The angiogeoscopy detects in a precise, reliable and fast way these stable hydrodynamic structures, therefore also the optimal collecting locations of the water bearing sites.

The water bearing sites are associated with a all the more steady state flow as they are located at an important depth. Thus, the efferent flow (outgoing) of all the geothermal reservoirs and all the hydrothermal reservoirs is steady state. Since the flow of a water vessel is steady state, all its other variables of state are it too.

It is instructive to consider the change of the temperatures and on the evolution of the pressures along ascending underground water flow.