# How does pressure change with depth in earth?

I've learned in school that pressure in water changes like

$$p(h) = \rho g h$$

where $h$ is depth in meters, $\rho$ is density (e.g. 1000 $\frac{\text{kg}}{\text{m}^3}$ for water) and $g$ is gravitation acceleration ($\approx 9.81 \frac{\text{m}}{\text{s}^2}$) and $p$ is the pressure in Pascal.

I guess there is no similar law for pressure in earth as it is to different, depending on where you are. But is there a rule of thumb? What do engineers who build tunnels / underground stations do?

• This is a really great question to ask following a lesson on hydrostatic pressure. From a student, it shows they're really thinking about how to apply what they learn; from an instructor, it could be a nice way to stimulate the students to explore more. – Air Sep 28 '15 at 20:17
• This would be an interesting question to pose to a mining engineer, as there are mines in South Africa that are 4 km or so deep, and they are lined in sprayed concrete, which is pumped to location, from memory. Thus hydrostatic pressure would be directly applicable here. – AsymLabs Oct 6 '15 at 13:24

I guess there is no similar law for pressure in earth as it is to different, depending on where you are. But is there a rule of thumb? What do engineers who build tunnels / underground stations do?

I approach this question as an engineer who does a lot of work on buried pipes and occasionally has to qualify buried structures for nuclear power plants. Also, for the sake of brevity, I assume you are talking about only vertical loads on the structure (lateral loads are another complicated topic for foundation engineering).

Soil can act similarly to fluid, depending on the soil type and even the type of structure that is being loaded.

For example, flexible pipes such as PVC, HDPE, and steel can be assumed to be loaded by the soil prism directly above the pipe. Piping is considered flexible if it can sustain a sizable deformation of its cross-section without rupturing. Consider the image below from Moser & Folkman's Buried Pipe Design, 3rd Edition (1):

In this case, since the pipe is considered more flexible than the soil, the pipe deforms under load such that no arching of the soil takes place. As such, the load on the pipe is simply the soil density times the depth of soil, like in your example.

Things get more complicated for so-called rigid pipes, such as concrete pipe or transite (asbestos-cement) pipe. In this case, the rigidity of the pipe is such that the soil on the sides of the pipe settle more than the diameter of the pipe itself and the pipe takes extra loading via soil arching. Below I've pasted another image from Moser & Folkman (1) illustrating this phenomena.

The loading on the pipe depends on how it was buried (positive projection, trench, induced trench, etc.) and is really beyond the scope of this answer. I've included a couple of references at the end of this answer for further reading.

For larger structures such as your tunnels or subway stations, determining the soil load is more complicated. Are there adjacent structures applying load? Has there been something done to stabilize the soil? How are the different soil strata interacting, and how does the relative stiffness of each impact the total load? If tunneling through rock, can the rock support itself without further reinforcement?

All of these considerations and more that I can't think of at the moment come into play when determining the load on a buried structure. There is no true rule of thumb when it comes to designing a buried structure since there are so many considerations when it comes to the actual loading.

1.) Moser, A.P. & Steven Folkman, Buried Pipe Design, 3rd Edition.

3.) Clarke, N.W.B., Buried Pipelines: A Manual of Structural Design and Installation, 1968.

As someone who has been involved with underground infrastructure to depths of at least 1400 metres, there are no rules of thumb. It all comes down to geology and the local conditions.

Soils behave differently to rock and sedimentary rock behave differently to igneous and metamorphised rock. Brittle rock behave differently to ductile rock. Brittle rock in the form of dikes and sills, can fail explosively when over stressed. Some mafic rock can exhibit a creep behaviour over time.

The number, orientation and condition of rock discontinuities is a factor, as is the proximity of faults/shears. The condition of the faults and whether they are active is important as is the width of the fault or fault zone and whether the fault is smooth sided or infilled and if infilled what material fills the fault. Talc on faults only leads to problems.

The juxtaposition of brittle and ductile rock can induce localised stresses due each rock type behaving differently.

Geotechnical holes can give information such as rock quality designation (RQD). Other drill holes in which three dimension stress cells have been placed can be over-cored so that the principle stresses for the rock mass in certain locations can be ascertained.

At depth, lateral stresses can be higher than sub-vertical stresses.

When a tunnel or chamber is excavated underground, the stresses in the rock mass realign. If a system of closely spaced voids is introduced into the rock mass, zones of de-stressed rock can occur, where there rock is no longer under the influence of the virgin rock stress.

In other situations, the lack of confinement introduced when a tunnel or chamber has been excavated can cause the walls of the void to contract; in some cases 50 mm or more.

Your question is specific to the pressure change with depth in the earth. When that earth consists of soils, the lateral and vertical pressures can be calculated in a number of different ways, depending upon whether your soil is sand, or clay and whether there is ground water present. It can be quite a complex matter, as the following illustrates.

Ratio of Horizontal to Vertical Pressure

Generally speaking, in excavations, under backfilled conditions, and under foundations, horizontal pressure and vertical pressure are not considered to be equivalent, and are dependent upon soil-structure interaction, in terms of active, passive and at rest conditions.

Active conditions are where the structure is moving away from the soil (decreasing pressure on the structure). Passive conditions occur where the structure is moving towards the soil (increasing pressure on the structure) and at rest is where the soil has reached its natural state. You can imagine that all three of these conditions could be observed in a retaining structure, as it may rotate or deform during its lifetime.

Generally, most theories will provide coefficients that can be used to calculate the ratio of horizontal to vertical pressure based upon the state of the soil/structure interaction and the properties of the soils. Some are based upon Poisson's Ratio. I have even used a temperature-based Poisson's ratio to conduct an elastic analysis of horizontal and vertical pressures in bituminous pavement structures using the Boussinesq equations.

Effective Stress

Where groundwater is present the pressure is expressed in terms of effective stress, that is the difference between the total stress and the pore water pressure. This is tricky to understand but has to do with soil buoyancy and other factors.

For example, consider a point of interest 10 m below the ground surface, and uniform sands that have a natural density of 1300 kg/m3, the total stress at the 10 m depth of interest would be 130 kPa. Now consider that the free surface of the groundwater table is at a constant depth of 2 m and assume the density of water to be 1000 kg/m3. The pore pressure at the depth of 10 m would be based upon an 8 m column of water, so that the pore pressure would be 80 kPa at the depth of interest. Thus the effective stress at 10 m becomes 130 kPa - 80 kPa = 50 kPa. This is a very simplified expression as there can be lots of other factors, for example fluctuations in water level, so-called 'quicksand' conditions and for retaining structures such things as drainage, among many other considerations.

Sands (Cohesionless Soils)

For sandy (cohesionless) soils, Rankine Theory (elasticity) is often applied. For this the angle of shear resistance of the soil (friction angle) and the angle of inclination of the excavation/retaining structure becomes critical.

The friction angle of sandy soil is best measured in the laboratory, but it is also considered roughly equivalent to the natural angle of repose of the loose, dry material.

Clays (Frictionless Soils)

For soils with a cohesive element, such as clays and clay silt combinations, Coulombs (Wedge) Theory (plasticity) is commonly applied. Under this analysis, the soil is imagined as a wedge (free-body) behind the structure, and as the solution is non-determinate, the a variety of potential failure surfaces is tried until the solution converges on a maximum soil pressure.

Soils with Friction and Cohesion

Coluomb's Theory can be used on soils that exhibit both friction and cohesion. Rankine's method is not suitable for cohesive soils. However, determining the ratio of horizontal to vertical stress may require further analysis.

Often the ratio can be established by determining the states of stress as represented by a Mohr's Circle. These properties are often measured by Triaxal Shear Tests where a column of soil is tested in the laboratory under a range of confining pressures. This can establish the cohesive strength and friction angle of the material and the ratio of horizontal to vertical stress according to depth.

General Elastic Theory

There are other theoretical methods that are often used to compute the horizontal and vertical pressures beneath a point of a foundation. Commonly two methods are applied: 1) Westergaard Theory and 2) Boussinesq Theory. The ratio of horizontal to vertical pressure at some point beneath the surface is largely a function of the estimated value of Poisson's Ratio.

Westergaard Theory is elastic theory applied to layered media. This is the case in most conditions typically found in practice.

Boussinesq Theory is elastic theory applied to a homogeneous elastic half space. Whereas this may not be so applicable to all soils, it does find frequent application under simplifying assumptions.

Closure

This is just a taste of the more common analysis techniques that are used to evaluate earth pressures in excavations, under foundations and behind retaining structures. There are others, for example Log Spiral Analysis for braced excavations, which is frequently used. While the theories can be complex, when one considers the great difficulty in establishing the true composition of subsurface soil conditions (ie the existence of layers, layer thicknesses and the variability of the properties of the soils), it becomes clear that pressure/stress analysis requires a great deal of experience and skill.

In simple terms earth pressure is both very similar and very different.

The vertical earth pressure is given by: Density x height x gravity. Here the density depends on the material, which varies with type of soil.

The horizontal earth pressure is where it diverges from the simple water model. The percentage of the vertical force applied horizontally depends on the ability of the soil to support and transfer load. Usually this is a simple coefficient for granular material (around 0.5) and for cohesive takes account of the shear strength.

There are theories, such as silo theory, that reduce the volume of soil acting on an point base on failure planes.