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Generating Power Using Geothermal Resources 163

where m is the mass of the fluid, a is the acceleration as the fluid moves up the well, ∑ F is the sum
fl
of all of the forces acting on the fluid, which resolve into the change in pressure over the depth of
the well (dP), the frictional forces acting on the fluid in contact with the pipe over the area of flow
(dF /A), and force due to the overlying column of fluid of density ρ and thickness h (g is the gravi-
b
tational constant). All of these forces are acting in the direction opposite to flow and are therefore
given a negative sign.
For liquid only flow, the density is approximately constant. In addition, since water is virtually
an incompressible fluid under these conditions, there is no acceleration since flow must conserve
both mass and volume. As a result, Equation 9.7 can be rearranged to give the pressure at the top
and bottom of the well as

Δ P = {(2 × f × ρ × v × X )/D} + (g × ρ × X ), (9.8)
2
R
R
where ΔP is the pressure difference between the bottom of the well and the wellhead, f is a friction
factor, ρ is the fluid density, v is the velocity of the fluid, X is the distance from the ground surface to
R
the top of the reservoir, and D is the diameter of the well. The friction factor depends upon the rough-
ness of the steel pipe used in the well (Ω), the well diameter, and the Reynolds number, R , which is a
e
dimensionless number that represents the ratio between inertial forces and viscous forces,
R = (4 × m )/(μ × π × D), (9.9)
v
e
where m is the mass flow rate (in kg/sec) and μ is the absolute viscosity (in kg/(m-sec)). The friction
v
factor is then calculated using the equation

2
f = 0.25 × (1/{log[((Ω/D)/3.7) + (5.74/R e 0.9 )]} ). (9.10)
If we assume that the pressure is 20 MPa (200 bars) at the top of the reservoir, the fluid density
is 837 kg/m at 235°C and 20 MPa, the fluid flow rate is 2.0 m/sec, the pipe diameter is 0.2 m, the
3
distance to the reservoir top is 2000 m, then the pressure difference from the bottom to the top of
the well is

Δ P = 17.5 MPa.

flashinG
Assuming that the fluid will remain liquid during ascent of the well is not necessarily a good
assumption. As the fluid moves up the well, the physical conditions change sufficiently that, under
some circumstances, the fluid may flash while still in the well. The conditions under which that will
happen need to be considered in order to understand the processes that must be accounted for when
designing and managing a generating facility.
To analyze this situation, a number of characteristics of the system must be established, since
they influence how the pressure will change in the well as the fluid moves upward. One of the fac-
tors that must be considered is the extent to which pumping of fluid from the well will affect the
pressure at the bottom of the well. Fluid movement from the reservoir into the well is constrained by
the physical properties of the reservoir, particularly its permeability. As a result, the pressure at the
bottom of the well will be less than that in the geothermal reservoir. The magnitude of this effect is
expressed by the draw-down coefficient (C ), which must be empirically determined for each well.
D
It is computed from the relationship

C = (P – P )/m , (9.11)
R
D
v
B