Earth mat analysis in two-layer soil

The IEEE 80-2013 equations of the earth mat take one resistivity. In a two-layer soil the design analyses the mat numerically: it divides the mat into segments, finds the current each one leaks into the layered soil, and calculates the grid resistance and the voltages on the surface from them.

Read first: Two-layer soil · Mesh voltage and touch voltage

1Why the equations are not enoughCopy link to Why the equations are not enough

The equations of Mesh voltage and touch voltage and Step voltage were derived for a uniform soil. In a two-layer soil the current divides between the layers, and the surface potential depends on both resistivities and the depth of the interface (Two-layer soil). The analysis takes the layers as fitted, so the design takes its result for the verdict in a two-layer soil and reports the equations, with the Wenner average, beside it.

2The mat as segmentsCopy link to The mat as segments

The design divides the mat into straight segments: one per side of each mesh, and one per electrode, split where an electrode crosses the interface. Each segment leaks a uniform current into the soil, and the mat is bonded, so each segment is at the rise of earth potential. The potential a segment of length $L$ carrying a current $I$ sets up at a point is the line-source integral, $\rho I/(4\pi L)\ln \left((r_1+r_2+L)/(r_1+r_2-L)\right)$, where $r_1$ and $r_2$ are the distances to its ends. The interface adds images of each segment, weighted by powers of the reflection factor $K$, as in the apparent resistivity of Two-layer soil, Eq. (1).

Setting the potential at the middle of each segment equal to the rise of earth potential (GPR) gives one equation per segment. Their solution gives the segment currents, whose sum is the grid current. The grid resistance is the GPR per ampere of that current.

3The voltages on the surfaceCopy link to The voltages on the surface

With the currents known, the design calculates the potential at points of the surface over the mat and 10 m beyond it. The mesh voltage is the GPR less the lowest surface potential over the mat. The step voltage is the largest difference between surface points 1 m apart (IEEE Std 80-2013 §8.3). Both are checked against the tolerable voltages, with $C_s$ taken on the upper layer (Surface layer and Cs). Step voltage, Figure 2 shows the surface potential along a line across the edge of the default mat in the example soil.

The analysis runs for a mat of up to 11 000 segments. Above that, or when the soil needs more calculation than the analysis takes, the equations give the verdict and the Notes tab says so.