Question

Calculate pressure as a function of depth in a vapor-dominated geothermal system consisting of a near-surface liquid layer \(400 \mathrm{~m}\) thick overlying a wet steam reservoir in which the pressurecontrolling phase is vapor. Assume that the hydrostatic law is applicable and that the liquid layer is at the boiling temperature throughout. Assume also that the steam reservoir is isothermal.

Short answer

Pressure increases with depth in the liquid layer but stays constant in the isothermal steam reservoir.

Step-by-step solution

Understand the Hydrostatic Law

The hydrostatic law states that the pressure at each depth in a fluid at rest is given by the equation \( P = P_0 + \rho gh \), where \( P \) is the pressure at depth, \( P_0 \) is the atmospheric pressure, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity, and \( h \) is the depth below the surface.

Pressure in the Liquid Layer

In the liquid layer, the pressure increases linearly with depth due to the weight of the overlying liquid. Assuming the density \( \rho_l \) for the liquid, the pressure at the bottom of this layer (just above the steam reservoir), \( P_{liquid} \), is given by the hydrostatic equation: \[ P_{liquid} = P_0 + \rho_l g h_l \] where \( h_l = 400 \ \text{meters} \).

Consider the Steaming Reservoir

For the steam reservoir, since it is isothermal and vapor-dominated, the pressure remains constant with depth. This is due to the fact that vapor equality implies the pressure is primarily dependent on temperature and not depth. Thus, \( P_{steam} = P_{liquid} \).

Combine to Express Overall Pressure Profile

For a point below the liquid layer, the pressure remains as determined at the boundary since it's an isothermal vapor system. Therefore, for a depth \( z \) within the liquid layer (up to 400 m), \( P(z) = P_0 + \rho_l g z \), and for a depth beyond 400 m, \( P(z) = P_{liquid} \), where \( P_{liquid} \) is the constant pressure in the steam reservoir.

Key concepts

Hydrostatic Law

The hydrostatic law is a fundamental principle that describes how pressure increases within a fluid at rest due to the weight of the fluid above. This law can be expressed using the formula: \[ P = P_0 + \rho gh \]where:

• \(P\) is the pressure at a given depth.
• \(P_0\) is the atmospheric pressure at the surface.
• \(\rho\) is the density of the fluid.
• \(g\) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\) on Earth).
• \(h\) is the depth below the fluid's surface.

This equation indicates that pressure within the fluid increases linearly with depth. The deeper you go, the greater the pressure due to the weight of the fluid column above. This principle is vital for calculating how pressure changes within the geothermal systems.

Vapor-Dominated Geothermal System

A vapor-dominated geothermal system is a type of geothermal reservoir where steam is the primary phase that influences the pressure. In these systems, the steam's presence is so dominant that it dictates the pressure behavior, unlike in liquid-dominated systems where both liquid and vapor coexist. These systems are important because:

• Steam's ability to rise allows for efficient energy extraction.
• Temperature and pressure conditions within the vapor region are uniform, simplifying calculations.
• They typically exhibit superheated conditions, enhancing energy output.

In our exercise, the steam reservoir is described as being vapor-dominated, meaning the pressure is assumed constant with depth, primarily controlled by the steam at the boiling point.

Isothermal Steam Reservoir

An isothermal steam reservoir maintains a constant temperature throughout the system. In geothermal contexts, this means that:

• Temperature does not vary with depth, unlike in liquid-dominated reservoirs.
• The pressure in the steam remains constant due to its direct dependence on temperature rather than depth.
• Energy extraction can be more predictable due to stable temperature conditions.

In the given problem, the isothermal condition implies that below the liquid layer, the temperature remains consistent, thus keeping the pressure stable based on the vapor's saturated conditions.

Pressure Function of Depth

Understanding pressure as a function of depth is crucial in hydrostatic and geothermal calculations. In this scenario:

• In the upper liquid layer (400 m thick), pressure increases steadily with depth according to the hydrostatic law: \[P(z) = P_0 + \rho_l g z\]
• Beneath this layer, within the steam reservoir, pressure remains constant beyond the depth of 400 m as \[P(z) = P_{liquid}\].
• This constancy reflects the equalizing effect of steam in an isothermal environment.

This dual behavior—a linear increase in pressure in the liquid followed by a constant level in the vapor—allows for a clear depiction of pressure throughout this geothermal system. Students should understand this profile's implications for geothermal energy extraction.