Water Rate Constraint Crossflow Calculations

Let us define the water rate constraint as q_c. When we hit this rate constraint, the sum of the water rate for each layer; q_w from all the layers (i to n) must be equal to the rate constraint.

Now, for each layer, we have a reservoir pressure (P_r) and well flowing pressure (P_wf). The relationship for water rate is defined by the following equation:

We call this the inflow performance relationship (IPR), where J stands for the productivity index. KOLDUN: CSG Monte-Carlo makes the following simplification for the wellbore; the well flowing pressure for each layer is constant. This well flowing pressure is calculated assuming some static column of water and gas inside the wellbore. Therefore, we can add some constant on to the tubing head pressure of the well.

This constant we can call the delta pressure from the wellbore gradient (P_wg).

The unknowns in our set of equations are; the water rates for each layer, and the tubing head pressure. The number of unknowns is equal to the number of equations so we are able to solve this set of linear equations.

equation 1
equation 2 to n

We will end up with the following matrix equation:

If the matrix is not singular, the solution is:

If it is singular, we remove the non-producing layers.