Effective Pressure

Part 8, Part 8: Pressure, Stress, and Time-Lapse

Learning objectives

State Terzaghi's law: the frame feels the confining pressure minus the pore pressure
Build the depth ladder at 3 km: lithostatic about 67.7 MPa, hydrostatic about 30.3 MPa, effective about 37.4 MPa
Show that 10 MPa of overpressure lowers the effective pressure to 27.4 MPa without touching the depth
Read the effective-stress coefficient and see why the frame, not the depth, is what the rock responds to

What the Grains Actually Feel

A rock at depth is squeezed two ways at once. The whole column of overburden presses down, the confining pressure $P_c$ c, and the fluid filling the pores presses back out, the pore pressure $P_p$ p. The mineral frame does not feel either one alone; it feels the difference. Terzaghi wrote it in one line, the effective pressure $P_{eff} = P_c - n\,P_p$ eff=Pc−n,Pp, with $n$ an effective-stress coefficient that is 1 unless the rock says otherwise. The physical picture is simple: the fluid holds the grains apart, carrying part of the load itself, so the grain contacts bear only what is left over. Everything in this part, every velocity that moves with production, moves because $P_{eff}$ eff moves, and $P_{eff}$ eff is the one number the frame responds to.

A Ladder at Three Kilometers

Put numbers on it at a depth of 3 km. Take a mean overburden density of 2.3 g/cc and integrate it down: with the gravity factor of 0.00981 MPa per meter per unit density, the lithostatic (confining) pressure is $P_c = 67.7$ c=67.7 MPa. Now the pore fluid. If the pore system is connected to the surface through the rock above, it sits at the hydrostatic pressure of a brine column of density 1.03 g/cc, which at the same depth is $P_p = 30.3$ p=30.3 MPa. Subtract, and the frame feels $P_{eff} = 37.4$ eff=37.4 MPa. That is the normal, or hydrostatic, case, and it is the anchor every other case is measured against. The confining pressure is fixed by how much rock lies above; the effective pressure is what is left after the pore fluid takes its share.

Overpressure Unloads the Frame

Now break the connection. Seal the pore fluid off, load it faster than it can drain, or generate fluid in place, and the pore pressure climbs above hydrostatic. Add just 10 MPa of overpressure, so $P_p$ p rises to 40.3 MPa while the overburden is unchanged at 67.7, and the frame now feels only $P_{eff} = 27.4$ eff=27.4 MPa. The rock is at the same 3 km, under the same weight of overburden, yet its grain contacts are as loaded as a normally pressured rock two-thirds as deep. Nothing about the depth changed; the fluid simply took over more of the load. This is the whole reason pressure matters to seismic: the next section shows that velocity tracks $P_{eff}$ eff, not depth, so an overpressured rock is a slow rock, and Part 8.3 turns that slowness into a way to see the overpressure before the drill bit reaches it.

References

Terzaghi, K. (1943). Theoretical Soil Mechanics. John Wiley & Sons.
Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.