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JWST will be orbiting L2 which is unstable equilibrium thus will require propellant for station keeping. JWST will carry propellant for 5-10 years.

What will happen once it runs out of propellant? Will it drift in a random direction? Is some direction more probable than others?

PS: why is JWST not parked at L4/L5 instead? They are stable equilibriums and JWST would not require much propellant to stay there indefinitely. Yes, it would require extra shielding from Earth but to a layperson that seems negligible compared to the sun shield it already carries.

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    $\begingroup$ For your PS: It will be at the Sun-Earth L2, about 1.5 million km from Earth. Sun-Earth L4/L5 are 1AU from Earth, or about 100 times farther away. It would not require "extra shielding from Earth" at SEL4/L5, but it would be a lot more trouble trying to maintain high-speed comms for all of the data it will generate. $\endgroup$
    – uhoh
    Oct 3, 2017 at 13:48
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    $\begingroup$ Pure speculation; if it does run out while it is still producing useful science, it will get refueled. See Is Landsat-7's propellant resupply port “robot-ready”? (Restore-L mission). Also see Besides HST, JWST and stations, are there any examples of satellites designed for service in space?. ...its launch interface ring will carry optical targets that could be sought out by a servicing robot’s machine vision as it approaches to attach itself in order to make ... $\endgroup$
    – uhoh
    Oct 3, 2017 at 17:15
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    $\begingroup$ Another comment for your PS: As uhoh already mentioned L4/5 is much farther away. So yould need many orbital manouvers to reach this point and the JWST has a.) not enough fuel for this and I don't know if there is a rocket thats large enough to bring an even bigger JWST into orbit. b.) It would take years to bring it in position c.) It can be refueled if we develop the technology within the next years. But docking in space is already hard and docking at L4/5 would mean that we need fully automated approaching and connection due to signal delays $\endgroup$
    – CKA
    Jun 21, 2018 at 11:06

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This Northrop Grumman video (starting at 09:31) illustrates JWST's orbit in a non-rotating (normal) frame. It's really in an orbit around the Sun about 1% farther than Earth's, but the weak tug of Earth pulls it along a bit faster so that it remains in 1:1 resonance with the Earth. The orbit is called a "Halo orbit" because in a rotating frame it looks like it's a ring around the L2 point.

Regular, but very small station keeping propulsive maneuvers keep it in this otherwise unstable configurations. The more frequent the adjustments, the lower overall fuel consumption per year. According to this and James Webb Space Telescope Initial Mid-Course Correction Monte Carlo Implementation using Task Parallelism and Station Keeping Monte Carlo Simulation for the James Webb Space Telescope there will be a small propulsive station-keeping event every 21 days. Wikipedia says this will consume 2-4 m/s of delta-v per year from a budget of 150 m/s, so the lifetime could conceivably be much longer than 5-10 years, although I believe about half of that (~67 m/s) will be used in mid-course corrections on it's way out to the Halo orbit.

If/when station-keeping maneuvers were to stop, JWST will wander away from this resonance and go into a more (or less) chaotic orbit around the sun. It will not be unpredictable though. The understanding of the orbital mechanics is well developed, and if there is no refueling, the final orbit will be chosen and the final propulsive maneuvers will be carefully executed to put the JWST in it's chosen final path into Heliocentricity.

However, there is talk about making the JWST approachable for service missions, and this might make refueling possible. See for example Besides HST, JWST and stations, are there any examples of satellites designed for service in space?, refueling technology for spacecraft is also discussed in Is Landsat-7's propellant resupply port “robot-ready”? (Restore-L mission).

NASA's Propellant Transfer Technologies, Robotic Refueling Mission, and RRM Task : Refueling webpages have more photos and information.

Considering that the Hubble Space Telescope is still highly productive after 27 years, it's hard to imagine that there is no contingency for extending the JWST's lifetime beyond ten years, even though it many not be a formal part of it's current mission. Throwing away a billions-of-dollar science instrument simply for want of some propellant (or other consumables) is hard to imagine.

In JPL Horizons there is a sample calculation of a Halo orbit for JWST that was done in 2014. It orbits for about eleven years, and presumably includes small station-keeping maneuvers to do so. This real-world orbit is more complex than a pure CR3BP (Circular Restricted Three-body problem) halo orbit because the Earth is in an elliptical orbit around the Sun, and the Moon adds further perturbations.

The first figure below shows the complete, eleven-year orbit with it's final wander away from L2 when station keeping ends. This is a "top-down" view of the Ecliptic plane, with the Sun to the left.

The next three images are just one year's worth, showing the two loops of the six-month period halo orbit. The Blue dot is the Earth, the green torus is the Moon's orbit, and the red line is the orbit of JWST. The coordinates are in the rotating frame of the Earth-Moon barycenter around the Sun.

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above left: Top-down view, eleven years. right: Top-down view, one year. (Sun to the left)

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above left: View from the Sun, one year. right: View from the side, one year. (Sun to the left)

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    $\begingroup$ @HephaestusAetnaean "Throwing away a billions-of-dollar science instrument simply for want of some propellant (or other consumables) is hard to imagine." In ten years time after the future launch date for JWST, it may not cost "billions of dollars" to bring some additional propellant for station keeping. Let's keep the focus on the topic of the question. Refueling is not the same thing as replacing subsystems, and in general cost for space operations is going down. $\endgroup$
    – uhoh
    Oct 7, 2017 at 1:07
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    $\begingroup$ After a second inspection, you're right, my mistake. I read too much into the question. Your first comment makes a lot more sense to me now. Cheers. $\endgroup$
    – Hephaestus Aetnaean
    Oct 7, 2017 at 3:06
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    $\begingroup$ Personally, IMHO instead of refueling, which is a very complex operation in space, JWST should have a grapple fixture, and within 10 years a very rudimentary spacecraft - something like common satellites except without the whole "mission-specific payload" should be sent, equipped with a grappling device, dock to the port and serve as an external propulsion module. $\endgroup$
    – SF.
    Jun 20, 2018 at 21:36
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    $\begingroup$ @uhoh - A photo of the JWST's LVIR. PS: Did the edit, minimal changes. $\endgroup$
    – Rob
    Jun 21, 2018 at 19:38
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    $\begingroup$ @uhoh - What was Restore-L is now called OSAM-1. Planning started in 2016. It was projected to launch in 2020. It is now projected to launch in late 2023, maybe, and is projected to cost over \$1.3 billion (US). To make matters worse, OSAM-1 is now a mission to a vehicle in LEO, not necessarily Landsat-7. At some point, (and \$1.3 billion is well past that point), the sunk cost fallacy kicks in. $\endgroup$
    – David Hammen
    Dec 11, 2021 at 22:47

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