Composite Effective Porosity for Multi-Unit Wells

Background

The Problem With Real Wells

The standard Wilkinson equation calculates a single effective porosity (N_e) value from a single specific capacity (Q/s) measurement:

Eqn. 1 always returns a result, regardless of how the well is completed — but what that result represents depends on the well. When a well is screened across a single hydrogeologic unit, one specific capacity measurement corresponds to one aquifer, and Eqn. 1 returns the effective porosity of that aquifer directly. Most real production wells, however, are not built this way. A typical municipal or agricultural well is often completed across, and draws water from, two or more distinct hydrogeologic units — for example, an upper sand layer and a deeper gravel layer, separated by a thin clay lens, all open to the same well bore. In that case, Eqn. 1 still returns a valid result, but that result is inherently a composite of every contributing layer, not the effective porosity of any single one.

When a pumping test is run on a well like this, the specific capacity that comes out of the test is a single, composite number — a blend of the hydraulic response of every unit the well is screened across. This page explains the reasoning behind that composite value and provides a calculator for the case where a user separately knows, or can estimate, the specific capacity and thickness of each individual layer.

Theory

A Flow-Weighted Composite

The reasoning starts from a simple physical idea: if each hydrogeologic unit within a well's open interval has its own true effective porosity, and each unit contributes some share of the total flow that the well produces, then the composite effective porosity measured at the well bore should be a weighted average of the individual units' porosities — weighted by how much of the total flow each unit contributes, not by how much of the open interval it occupies. A thick but poorly transmissive layer and a thin but highly transmissive layer do not contribute equally just because they are both screened; the layer that yields more water under the same drawdown has more influence on the composite result.

Formally, for a well screened across n layers, the composite effective porosity is defined as:

In practice, this means calculating N_e(i) independently for each layer — by applying Eqn. 1 to that layer's own specific capacity, not a single blended value for the whole well — and then combining those individual results into a single composite number, weighted by each layer's share of the well's total specific capacity. Specific capacities add directly across layers screened in the same well bore under a shared drawdown, so a layer's share of the total specific capacity is a direct measure of its share of the well's total yield. A layer that produces 70% of the well's flow contributes 70% of the weight to the final composite value, regardless of how much or how little of the open interval it occupies.

Applicability

What Data Does This Require?

This calculator is different from, and more demanding than, the standard single-well calculator. It requires a separate specific capacity (Q/s) value and a separate saturated thickness value for each individual layer the well is screened across — not a single blended measurement for the whole well. Most production wells only ever produce one composite pumping test result, so this per-layer data is only available in specific circumstances, such as:

Packer testing that isolates individual intervals during drilling; flowmeter logging that profiles inflow contribution by depth; test wells completed and tested one interval at a time before being converted to a multi-unit production well; or exploratory or research boreholes with detailed zonal testing.

If you only have a single composite Q/s value for a multi-unit well, use the standard calculator instead — the result you get there is already the composite N_e for the well as a whole, calculated correctly under the assumption described above. This page's calculator is for the less common case where you want to see how each individual layer contributes to that composite figure.

Calculator

Calculate Your Composite N_e

Enter the saturated thickness and specific capacity for each layer the well is screened across. Add rows for additional layers as needed.

Example

A Worked Example

Consider a well screened across two layers: an upper sand unit with 12 m of saturated thickness and a specific capacity of 25 m²/day, and a lower gravel unit with 8 m of saturated thickness and a specific capacity of 180 m²/day. Applying Eqn. 1 to each layer independently gives N_e(1) ≈ 0.197 for the sand and N_e(2) ≈ 0.232 for the gravel. The well's total specific capacity is 25 + 180 = 205 m²/day, so the sand layer carries a weight of 25/205 ≈ 0.122 and the gravel layer carries a weight of 180/205 ≈ 0.878. The composite effective porosity is therefore (0.122 × 0.197) + (0.878 × 0.232) ≈ 0.228 — much closer to the gravel layer's value, because the gravel transmits far more water than the sand even though it occupies less of the total screened thickness (8 m versus 12 m).

Note that this composite value is not simply the average of the two individual N_e values (which would be 0.215), nor is it weighted by thickness (which would give ≈ 0.211) — it is weighted toward whichever layer contributes more of the well's total flow. A thin, highly transmissive layer can dominate the composite result even when a thicker, less transmissive layer occupies most of the open interval.