If the problem of fluid flow in a reservoir toward a well cannot beapproximated as a two-dimensional one, the analytical treatment becomes verydifficult. This is especially true when horizontal wells are involved and theend-effect cannot be neglected.
There are, in literature mathematical expressions to calculate productivity ofhorizontal wells for various geometric configurations and many of these includethe effect of flow convergence. Some of them appear to be significantly inerror.
The paper describes experiments with an electrolytic model. This approach usesthe analogy between the flow of fluid particles, driven by fluid flowpotential, in a porous medium and movement of ions, driven by electricalpotential, in an electrolyte. A simple and low-cost experimental setup permitsthe testing of various theoretical equations and the results arepresented.
In many cases the flow of fluids in a homogeneous reservoir toward a well, vertical or horizontal, can be analyzed simply whenthe problem is two-dimensional. In some situations this approach may besufficient; but when, because of more complicated geometry, the effect ofconvergence toward the tip of a well (as, for example, in the case of avertical well partially penetrating a liquid-bearing porous matrix layer) hasto be taken into account, the difficulties of establishing the flow patternand, consequently, calculations of the well productivity, may besignificant.
A common simplification is the assumption of a uniform flux along the activewell length. But this approach, as pointed out by Muskat 1, does notgive the correct results, even for the relatively simple case of a partiallypenetrating vertical well and steady-state flow.
The situation becomes more difficult for horizontal wells of finite lengthdraining bounded volumes. There is usually a problem of convergent flow towarda well in a plane perpendicular to the well's axis, but on top of this there isa three-dimensional convergence toward the tips of the well. Probably in somesituations when, for example, the well is "Very long and draining a thin layerand narrow pattern, this additional convergence can be neglected. But in anycase, an engineer should be at least aware of the order of error thisapproximation can create.
There are several known formulae that are used for horizontal wellsproductivity predictions and which take into account various aspects of thegeometry of the problem, including well-end convergence. In some cases theformulae published by different authors differ significantly.
Unfortunately, there is not much information available about the experimentalor field data regarding these problems. There is no= doubt that obtaining suchdata, especially in the field situation, may be difficult. It would requireflux measurements at various sections of a well, treated as isolated segments, but at the same time the flow inside the wellbore would have to be maintained.However, there is a possibility to model these conditions, at least forsteady-state flow in some situations, and to obtain experimental data. This canbe achieved by using the analogy between fluid flow in porous medium andelectric current flow in electrolyte.