Review of Preferential Flow: Stokes Approach to Infiltration and Drainage by Peter Germann

The book is written by one of the leading authorities on macropore flow and kinematic wave treatment of unsaturated flow. This 199 page treatise is a masterpiece on preferential flow, with particular reference to macropore flow. Some of the attractive features of the book are (1) this is perhaps the only book devoted exclusively to the treatment of preferential flow using the kinematic wave theory; (2) the book reflects the author’s expertise and vast experience; and (3) the book is a rigorous treatment of preferential flow.

The subject matter of the book is divided into two parts spanning 13 chapters. The first part, comprising seven chapters, deals with Stokes flow applied to preferential infiltration, whereas the second part deals with viscous flow applied to infiltration and drainage.

In the preface, the author writes: “The book offers a unified treatment of concepts based on first principles of hydrodynamics, leading to viscous flow in permeable media.” This book is more than true to this statement.

Chapter 1 introduces the concept of preferential flow. Considering the simultaneous flow of water and air in permeable media, which soils are, the foundations of capillary flow and viscous are revisited. Dealing with major flow-controlling forces, including gravity, pressure, capillary, and viscous, during infiltration, the chapter postulates a dual process approach to preferential flow characterized by macropore flow and finger flow, which may occur simultaneously in the soil. Macropore flow occurs in fissures, cracks, and voids that are so wide that capillary forces are not exerted on flow, whereas finger flow may occur in homogeneous matrices in zones of higher water contents that are separated from their drier surroundings. Macropore flow follows in discrete structural voids, whereas hydrodynamic conditions induce finger flow in diffuse pore systems. Gravity adds momentum to and viscosity abstracts momentum from the moving water, leading to the kinematic view of infiltration. The chapter concludes with a statement of assumptions and limitations.

Chapter 2 discusses fundamentals. Starting with the Navier-Stokes equation, it presents, from the first principles, the derivations of inertia; gravity; and the potential due to gravity, viscosity, and shear force. Then, the chapter presents Stokes flow, including parabolic velocity profile, volume flux of the flow, mobile water volume, the velocity of shock front, and the limit to the applicability of Stokes flow. It then goes on to discussing surface tension, capillarity, capillary rise, maximum pore radius, relation between aperture and capillary head, the widest aperture just able to exert the capillarity, capillary head, the water retention curve, and specific water capacity. The chapter is an excellent introduction to the physics of unsaturated water flow and is highly illustrative and rich in information.

The water-content wave (WCW) constitutes the subject matter of Chapter 3 that entails a discussion of one rectangular pulse across a porous medium considering Stokes flow. The Stokes flow representing infiltration into porous media is referred to as viscous permeable media flow or viscous flow. Enumerating four conditions under which the WCW is treated, the chapter begins with the adaptation of Stokes flow to infiltration and derives the depth and time it takes the wetting front to intercept the draining front, velocity of the rear end, space-time distribution of the mobile water content, temporal position of the wetting front, mobile water content, and volume flux density at the wetting front. It then presents profiles of mobile water content, position of wetting front, mobile water content, and volume flux density within three time intervals. It then considers the time series of the wetting and draining fronts during the three time intervals. The chapter is concluded with a discussion of one initial and four boundary conditions. This is an excellent chapter providing a lot of geometric insights.

Chapter 4 extends the discussion of the preceding chapter to the routing of input pulses in porous media as kinematic waves. It discusses the application of kinematic wave theory to viscous flow based on three considerations: (1) macropore flow restriction, (2) equivalence with viscous flow, and (3) the velocity of shock. First, the propagation of a single pulse as a special case is dealt with, including the depth and time at which the shock front intercepts the draining front, mobile water content at the wetting front, and interception function. The discussion is extended to a sequence of two input pulses, one running behind the other, referred to as a decreasing jump. The mobile water content wetting front position and depth and the time and depth at which the wetting front intercepts the draining front are derived. The propagation of an increasing jump is discussed next. The chapter is concluded with a treatment of superposition of kinematic waves and kinematic weaves with a sink term. This is an outstanding chapter.

Geometry of flow is described in Chapter 5. In addition to free-surface flow, the three flow geometries of (1) plane Poiseuille flow between two parallel plates, (2) Hagen-Poiseuille flow along a vertical cylinder, and (3) corner flow along a vertical corner are dealt with. For each flow geometry, the velocity profile across the water film, volume flux, volume density, mobile water content, wetting front velocity, and hydraulic radius are derived. The chapter discusses the impact of presumed flow geometry on viscous flow and is concluded with a discussion on the dominance of viscosity.

Chapter 6 deals with Darcy’s law and the Richards equation in view of viscous flow. It first discusses Darcy’s law for saturated porous media and then compares Darcy’s law with viscous flow for two common conditions: (1) complete water saturation that envelops the maximum mobile water content and (2) gravity as the only flow-driving force. Comparing with plane Poiseuille flow, it derives the hydraulic conductivity at saturation, flow velocity as a function of pressure head, and discharge. It extends the saturated hydraulic conductivity to three cases: (1) viscous flow, (2) macropore flow redistribution, and (3) contradiction.

The next chapter discusses the Richards equation for partially saturated porous media. It goes on to discuss dimensionless capillary number and Bond number that help define the thresholds of domineering forces acting in a particular flow regime. The capillary number relates to the capillary force with the viscous force, whereas the Bond number is the ratio of gravity force to the capillary force. The chapter is concluded with a discussion of domains of flow in porous media. Three domains are identified: (1) the domain of exclusive sequential capillary in the sense of Richards equation, where tensile strength of water is strong and capillarity dominates flow, (2) the domain of laminar viscous flow that follows from Stokes irrotational flow; the tensile strength of water is weak allowing for the coexistence of capillary and viscous flow, and (3) the tensile strength is feeble and flow becomes unstable. Chapter 7 concludes Part 1, summarizing achievements and previewing Part II on viscous flow applied to infiltration and drainage.

Experimental protocol, data acquisition, and interpretation are dealt with in Chapter 8. For classification of water content waves, four levels of complexity are introduced: (1) procedures for flow path level, (2) procedures at the local level, (3) procedures at the profile level, and (4) procedures at the system level. For experimental determination of parameters film thickness ($F$) and contact length of WCW ($L$), pairs of data sets are specified. Then, estimation of these parameters from drainage flow is discussed. The chapter is concluded with a summary.

Chapter 9 deals with procedures for estimating parameters $F$ and $L$ from data measured at the local level. It deals with wetting front arrival, rivulet approach for conceptualizing the water content as a superposition of a number of rivulets; water abstraction from a low, restricted drainage flow; minimum interval length for analyzing volume of moisture content and soil moisture; relationship of wetting front velocity and volume flux density; and viscous flow in fissured rocks. The chapter is concluded with a summary.

Chapter 10 discusses procedures for estimating parameters $F$ and $L$ at the profile level. It deals with two features of viscous flow: (1) rapid flow increase and (2) concave decrease of water content due to an infiltrating pulse. It first discusses viscous flow initiation in dry soils, viscous flow restriction at the interface, viscous flow and finger flow, and viscous flow and dye staining. The chapter is concluded with a summary of viscous flow at the profile level.

Procedures at the system level are presented in Chapter 11. The discussion deals with the Kiel sand tank viscous flow matching to data, spatial structures of viscous flow, and capillary head. The chapter is concluded with a summary.

Chapter 12 deals with procedures at the flow path level. It discusses superposition of kinematic waves-Lorin’s candle, plausibility of wetting front velocity and parameter $L$ in unconsolidated sands, viscous flow and capillary heads, and viscous flow and structural voids.

The concluding Chapter 13 dwells upon scales and extents of viscous flow. Viscous flow separates signal velocity from signal magnitude. The specific contact length $L$ of the input initiated water content wave is the parameter linking flow with the structure of the porous medium when film thickness $F$ remains thin in the widest continuous flow carrying voids. The chapter presents temporal scaling of front velocities and temporal scaling of volume flux densities. It then discusses spatial scaling of viscous flow, macropore flow; wetting front velocities; and temporal and spatial extents of viscous flow. The chapter is concluded with viscous flow scaling in hillslope hydrology.

The book is well-written and is highly useful. It is a must read for those who are interested in preferential flow. The author deserves a lot of applause for writing this treatise and sharing his rich wisdom.