ABSTRACT

In classical finite difference simulators a well is only accounted for by a constant withdrawal rate in the well-cell material balance equations.

This representation is over-simplified because:

1- It assumes the well is always flowing at the specified rate, whatever the pressure

2- It is unable to properly split the production between the different aquifer horizons

3- It is unable to account for crossflow between aquifers, through the well-bore.

In order to overcome these oversimplifications a model was developed which includes a Well Inflow Performance Routine that allows the simulator to calculate the contribution of each aquifer taking into consideration the productivity index of each layer. For this purpose the model uses the Peaceman steady-state formula that relates the well rate to the difference between the block model pressure and the bottom hole flowing pressure.

The model investigates the ability of each aquifer to maintain target production rates for a specified field area, allows the optimization by Dynamic Programming techniques of a aquifer performance and provides the automatic well equipment selection based on specific performance curves of the type of pumps installed.

1 INTRODUCTION

In order to rationalise the full utilisation of aquifer potentialities in waterflood operations aiming at a better control of the fluids movement and displacement, there is a growing need to search for new procedures and methodologies to overcome the limitations of the classical techniques. The mathematical model proposed here was designed as a very flexible tool able to be applied to complex and heterogeneous hydrologic units overcoming the limitations of the most common classical groundwater simulators. The tool was tested in a Middle East aquifer.

2 OVERSIMPLIFICATION OF THE CLASSICAL GROUNDWATER MODELS

In the modelling of an aquifer by numerical methods it is necessary to use grid blocks whose horizontal dimensions are much larger than the diameter of a well. As a result, the pressure calculated for a block containing a well is greatly different from the flowing bottom-hole pressure of the well. Usually the groundwater simulators apply the equations of Darcy and continuity to handle the flow of water in a porous medium. Some limitations arise in these models namely:

a) Classical simulators are originally rate-specified. More precisely the production injection data are put in as volumes of water withdrawn from (or added to) the well block during the simulated time step.

This method can be used for the history match period, in which the actual water production per well is known. However, for the prediction period this method is not suitable: the simulator originally calculates blockpressures, but wells are not represented. Furthermore it does not take into account variations of production caused by pressure variations.

b) Classical simulators are unable to properly split the production between the different layers aquifer harizons.

The assumption of a split in the ratio of transmissivities is only valid if the aquifer horizons have the same potential (pressure at datum). This is not always the case: even if it is true the beginning of the simulation, it will case to be true after the aquifers have produced for some time. The reason is that the amount of water produced by each aquifer depends not only at the respective hydraulic properties but the existing pressure as well. Consequently it can be stated also that classical simulators are unable to account for cross-flow from one aquifer eo another through the well bore.

3 THE PROPOSED MODEL

To overcome these drawbacks we proposed to build a simulator that shonld also be able to account for crossflow and inter-unit communication and for an automatic well equipment selection from a appropriate data base configuration. We have based on the architecture of the 3D groundwater flow model by Trescott', modifying it afterwards incorporating the adequate modules in orderto achieve theproposed objectives. Discretization process used is a finite difference technique with a particular robust implicit procedure as iterative numerical algorithm (SIP - Strongly Implicit Procedure). Simulation options includ heterogeneous and anisotropic mediums, boundaries of different kinds, watertable calculations' and discharge/recharge source terms.

3.1 Well inflow performance modole

A Well Inflow Performance (WIP) routine was developed in order to be incorporated into the simulator. The basis and the methodology of the WIP routine are the following:

I) A relationship between rate (Q) and the bottom hole flowing pressure (BHFP) is defined for a set adifferent wells with different equipment. It is based on: - the type of pump and power availability for pumped wells; - the standard completion for each well type at an average depth for an average well head elevation; - the standard well head pressure.

II) A productivity index (PI) is calculated for the well, based on local transmissivity. The calculation uses a steady-state type formula due to Peaceman2, modified for numerical purposes:

where:

Note that Peaceman defined in eqn (1) the concept of equivalent radius of well block as the radius at which the steady-state flowing pres sure for the actual well is equal to the numerically calculated pressure for the well block. In particular for the case of an orthogonal grid the equivalent radius is given by the following expression:

where:

The modification allows use of the PI to relate the rate to the differencebetween the well blockmodel pressure andthe BHFP according to the expression:

III) Formultinode wells the individual node rates (contribution of each well) are calculated taking into account: - the PI of each node; - the block pressure of each node; - the constraint of a well-bore flowing potential common to all nodes.

IV) The common flowing pressure calculated above is compared to the minimum BHFP define in (I). If the actual BHFP is larger than the minimum, the simulation process is carried to the next time step, otherwise the specified rate is reduced by a proper amount in order to obtain an actual BHFP just above the minimum, and the current time-step calculation is re-iterated.

V) This process of reducing the flow rate to keep BHFP above the minimam is carried out until a further reduction will bring the rate out of range (below the minimum rate). At this point the well is shut in (rate = 0).

3.2 Crossflow and inter-unit communication

Other feature of the software modules developed in this model is associated with crossflow and inter-unit communication. An algorithm was included to enable the simulator to calculate the contribution of each layer and properly accounts for crossflow when present. This is done using the eqn (3) that splits the total specified rate between layers in a proper manner.

3.3 Automatic well equipment selection

Concerning the automatic well equipment selection it is worth to note that the WIP routine evaluates the possibility of each well to maintain the required rate during the simulation time-step given a constraint of minimum BHFP. This constraint depends on the type of submersible pump and on other technical characteristics.

3.4 Optimization of the aquifer performance

Aquifers can play many roles in the overall development of the water resources of an area. The most obvious use of an aquifer is to supply water to wells. The amount of ground-water that can be produced will vary under varied patterus of pumping and development. In order to give an efficient support to the optimization of an aquifer performance, Dynamic Programming techniques were used to defined the best number and location of water supply wells.

3.5 The algorithm

In short the algorithm for each individual time step
encompasses the following steps:
a) The simulator calculates block pressures.
b) Flowing pressure is computed with the Peaceman formula (eqn 1) in a multinode well, using the rate specified, the absolute height difference and the average gradient between nodes.
c) Verify if the flowing pressure calculated is feasible for the specific performance curve of the equipment installed, (see figure 1) which means checking if it is greater than the
minimum admissible.
d) If it is not the case, it is performed a rate reduction following the specific performance curve of the equipment.
e) After reducing the rate, the time step should be re-iterated and a new process starts from item a leading to a satisfactory BEIFP at the end of the time step.
f) If the new rate falls outside the range (below the minimam rate) the equipment could be shut in,regarding the operational guidelines.

Figure 2 shows, for a specific well, the BHFP and water production vs time plot where it is visible how the operations rules (change of equipment) affect the predictions.

Finally the figure 3 shows an example of output of WIP routine: It gives two messages -before and after using performance curves-, checking the calculated BHFP by means of indications "OK" or " below minimum of . . . ", and the amount of well rate reduced for a specified type of pump (WT).

4 CONCLUSIONS

The obvious limits of the classical models can be overcome by the develcpment of a mathematical model which includes a Well Inflow Performance routine that allows the simulator to calculate the contribution of each layer, and properly accounts for crossflow (when present), taking into consideration the productivity index of each layer and investigating the ability of each aquifer to maintain target production rates for each field area by making an automatic selection of the optimal well equipment. Besides, the model improves the accuracy of the history match and refines the performance prediction, because it introduces a modified Peaceman correction formula that allows an accurate comparison between pressures calculated by the model and pressures measured in the wells. The implementation of Dynamic Programming Techniques brings in the mathematical model a flexible tocl able to be applied to different purpose in the aquifer management.

5 REFERENCES

1. Trescott P. - Docwnentation of finite-difference model for simulation of three-dimensional groundwaterflow, U.S.G.S. Open Eile Report 75-438, 1975. 2. Peaceman D. W. - Interpretation of well-block pressures in numerical reservoir simulation with non square grid blocks anisotropic permeobility, SPE 10528,1982,553-569.