Try linear interpolation:
$$\frac{k_E-k_e}{\delta_{e+}} = \frac{k_E-k_P}{\delta_e},$$ $$k_e = fk_P + (1-f)k_E,$$ $$f = \frac{\delta_{e+}}{\delta_e}.$$
Unfortunately, this does not work very well.
If we use a uniform temperature gradient throught the region of interest, then consider the heat fluxes that this graph implies
Hence, using (or implying) the same temperature gradient in both cells (materials) gives different heat fluxes in the two cells when using the respective thermal conductivities.
Find an evaluation of $k_e$, that is physically consistent with heat fluxes through materials of different properties.
That is, find an evaluation of $k_e$ that allows for different temperature gradients to be implied in cells with different conductivities.
Construct $k_e$ by equating fluxes in the two cells (so that energy is conserved):
|---W---|---P---|---E---|
w e
$$\frac{\partial vf}{\partial x}.$$ |---5---|---3---|---5---|---3---|---5---|
|-----|-----|-----|
| 8 | 3 | 8 |
|-----|-----|-----|
| 3 | 5 | 3 |
|-----|-----|-----|
| 8 | 3 | 8 |
|-----|-----|-----|
Or, suppose we take a smooth solution and add a checkerboard pattern (from noise, or errors, etc.), this patter could persist, or even grow.
This happens with pressure P in the fluid equations. The $\nabla P$ term gives $P_e-P_w$ hence $(P_E-P_W)/2$.
Grids are called either staggered or collocated.