Nonmonotonic plasma potential structures are a common feature of many double layers and sheaths. Steady state plasma potential wells separating regions having different plasma potentials are often found in laboratory experiments. In order to exist, such structures all must find a solution to a common problem. Ions created by charge exchange or ionization in the region of the potential well are electrostatically confined and tend to accumulate and fill up the potential well. The increase in positive charge should eliminate the well. Nevertheless, steady state structures are found in which the wells do not fill up. This means that it is important to take into account processes which ‘pump’ ions from the well. As examples of ion pumping of plasma wells, we consider potential dips in front of a positively biased electron collecting anode in a relatively cold, low density multidipole plasma. Pumping is provided by ion leaks from the edges of the potential dip or by oscillating the applied potential. In the former case the two dimensional character of the problem is shown to be important.