Second‐order theory of flow in three‐dimensional deforming media

Abstract

A basic assumption in groundwater flow theory is that consolidation occurs in the vertical direction only. In real three‐dimensional media this condition is not satisfied a priori. In the present paper a second‐order theory of flow is developed including the consideration of horizontal soil displacements. The ‘strain nucleus’ or ‘tension center’ approach, well‐known in thermoelasticity, is used. The formulation of the theory shows that volume strain in a point P of the system may be viewed as being distinctly contributed by pore pressure variations in and outside P. The latter contribution, which vanishes identically in one‐dimensional media, is called the ‘three‐dimensional effect’. Solutions applied to a semi‐infinite mechanically homogeneous medium show that the three‐dimensional effect is negligible and the diffusion equation is good as long as the ratio W between the average depth and the thickness of aquifer is ≥2, whatever its areal exent. If W < 2 (shallow and thick aquifers), the above effect produces an additional compression of soil during an intermediate stage after pumping has begun and hence a slowing down of the rate of pressure head decline, as has been predicted by standard solutions. The most critical conditions occur for W=1/2, i.e., in unconfined aquifers. The present theory also allows for a short qualitative analysis of the ‘Noordbergum effect’ or reverse water level fluctuation.