Spontaneous Potential - Physics

Physics

Spontaneous potentials (SP) are usually caused by charge separation in clay or other minerals, due to presence of semi-permeable interface impeding the diffusion of ions through the pore space of rocks, or by natural flow of a conducting fluid through the rocks.

The origin of SP across formation can be attributed to two processes involving the movement of ions:

1. Streaming potential (E_k)
2. Electrochemical potential (E_c)

Streaming potential originates from the flow of an electrolyte (water) over naturally charged solids (i.e., surfaces that acquired electrokinetic or zeta potential). The streaming potential appears when mud filtrate is forced into the formation under the differential pressure between mud column and formation. The streaming potential is produced when the flow takes place across mud-cake in front of permeable formations, across permeable formations being invaded, and across shale beds. It is generally accepted that the streaming potential across the mud-cake is compensated by that across the shale. As such, in most cases, the spontaneous potential measured is only related to the electrochemical potential.

Electrochemical potential (E_C) is the sum of liquid junction or diffusion potential (E_J), and membrane potential (E_M)

Liquid junction potential is established at the direct contact of the mud filtrate and formation water at the edge of the invaded formation. Ions Na+ and Cl- diffuse from either solution to the other, but at different rate due to different mobilities. Na+ tends to be less mobile due to its affinity for water molecules.

E_J = K₁ log₁₀(a_w/a_mf)

where:

K₁ = 11.6 mV at 25 °C

a_w = formation water ionic activity

a_mf = mud filtrate ionic activity

Membrane Potential develops when two electrolytes of different ionic concentrations, such as mud and formation water, are separated by shale. The clay minerals in shale are usually made up of atom Al, Si, and O. O2- ions occupy the outer layer and cause a net negative charge. Na+ ions from solution are attracted and allowed to pass through the shale, while Cl- ions are repelled. Na+ ions will migrate between the two solutions, with a net influx from the more saline to the less.

E_M = K₂ log₁₀(a_w/a_mf)

where:

K₂ = 2.3 RT/F, where:

R = ideal gas constant

T = absolute temperature in kelvins

F = Faraday constant

a_w = formation water ionic activity

a_mf = mud filtrate ionic activity

The total electrochemical potential is thus summarized as E_C = E_M + E_J = K log₁₀(a_w/a_mf)

Since spontaneous potential is a measure of electrochemical potential and the ionic activity of a solution is inversely proportional to its resistivity, the above equation can be simplified as SP = E_C = K log₁₀ (R_mfe/R_we), where R_mfe and R_we are equivalent mud filtrate resistivity and equivalent formation water resistivity respectively.

The ideal spontaneous potential across clean bed is known as Static SP (SSP), and defined as follow:

SSP = −K log₁₀ (R_mfe/R_we)