Water that stays on the fields

If the source of the water is rain on its way to the river on flat saturated ground and not an overflow from the river it is called ponding or ponded field, it is one of several types of flooding

Ponding is a type of flooding that can happen in relatively flat areas. Rain water falling in an area is normally stored in the ground, in canals or lakes, or is drained away, or pumped out. When more rainwater enters a water system than can be stored, or can leave the system, flooding occurs. In this case, rain is the source of the flood: not water coming from a river, but water on its way to the river. That's why it is also called "pluvial flood".
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It is the term used in scientific literature

An analytical solution is provided for predicting time dependent seepage into an array of equally spaced parallel ditch drains in a homogeneous and anisotropic soil medium underlain by an impervious layer and receiving water from a ponded horizontal field of infinite extent. The solution can account for both unequal levels of water in the adjacent drains and variable depths of ponding at the soil surface.
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'Marsh' is my first instinct, but that would be like a rice paddy. What you are looking for is a temporary type of a state for a normally dry terrain.

standing water

comes to mind as the actual term.

It's called inundation

flooding

[ODO]

Or you can simply refer to it as stagnant water/ water stagnation.

The state of not flowing or moving

[ODO]

Usage:

The field is inundated with flood water

I would suggest the adjectival form flooded.

Stagnated land suggests terrain which is soaked with water, but stagnated waters means water that is still. The term inundation is more about volume. Google images seems to agree that flooded fields is in common use.

• Several farmers had to rescue animals from flooded fields (BBC)

Cambridge Dictionaries use the same example in their definition

covered with ​water: flooded ​fields

Still water formed e.g. by the pooling of fallen rain in an area with poor drainage can be called standing water:

3. (Of water) stagnant or still. OED

It's used both as a description of stillness as well as a description of the stagnancy typically associated with still, small bodies of water like puddles.