Splashing twigs away while in a body of water:
The rule: It is forbidden for a person to splash an item away from oneself or towards oneself.
The reason for this is because: doing so is also included in the decree against swimming on Shabbos.
Twigs: Therefore twigs which are on the surface of water are forbidden to be splashed away to the sides in order to clean up the water that they be clear.
To which bodies of water does the prohibition apply? [However] all the above applies to a river, however in a vessel or a pool with an [external] rim this decree was not made at all, as was explained above.
Whenever it remains permitted to enter into a body of water on Shabbos [such as for Mikveh or on hot day], it remains forbidden to splash away from oneself, or towards oneself, dirt or other objects, unless it is a pool which contains an external rim, as explained above by swimming, in which case doing so is permitted.
May one splash away the dirt that floats on the surface of a Mikvah? 
One may splash the dirt to the sides even with his hands, even if the Mikveh is filled to the top. Doing so does not involve a Borer prohibition [even if the dirt covers over the entire Mikveh and one thus would not desire to immerse without splashing the dirt away] being that nothing is being removed from the Mikveh but is rather being moved to the sides. [If however the Mikveh is filled to the top one is not allowed to splash the dirt outside of the Mikveh].
Some however write that it is nevertheless proper to take care to splash the dirt away to the sides together with a lot of water.
May one use a net to remove dirt from a Mikveh on Shabbos?
No. Doing is Biblically forbidden due to the Borer prohibition.
 Ketzos Hashulchan 146 footnote 34/17 and 35; Shabbos Kehalacha Vol. 2 p. 242.
Splashing away dirt may involve the “Swimming” prohibition as well as the Borer prohibition .
 And certainly with ones legs and elbow in which one is doing so with an irregularity. [Ketzos Hashulchan ibid 35].
 In which case there is no room to say that it is considered to have a rim. This does not involve the swimming prohibition mentioned above as a) Perhaps the decree only applied to pools with an earth floor and not to a tiled floor, which is not exchangeable with a river, as well b)The Nodah Beyehuda rules that the decree against swimming in pools does not apply at all today being that we no longer make the swimming rafts, as well as c)The prohibition against splashing twigs is less severe than the swimming prohibition. Thus with a joint of all the above one may be lenient to splash dirt away in a Mikveh, however not to swim or tread in. [Ketzos Hashulchan ibid 34/17].
 So is implied from Ketzos Hashulchan and so rules Shabbos Kehalacha ibid.
 Ketzos Hashulchan ibid 35
 An opinion brought down in the Ketzos Hashulchan ibid, and so rules Shabbos Kehalacha ibid that it is proper to do so as this strengthens the allowance even more, although it is not required from the letter of the law. However the Ketzos Hashulchan himself sides that there is no need at all to do so.