Abstract
Internal erosion is a major cause of embankment dam failure. Unravelling and instability of the downstream slope, initiated by internal erosion and leakage through the dam core, is one of the most likely breach mechanisms for large, zoned embankment dams. To be able to model this mechanism, the relationship between the hydraulic gradient and the flow velocity for the coarse rockfill material must be understood. Because most studies of this topic have focused on the flow parameters in gravelsize materials with Reynolds (Re) numbers lower than 25,000, permeability measurements are needed coarser rockfill material under heavily turbulent flow regimes prevailing in rockfill material under certain design flow scenarios. This paper presents the setup and results of a series of field and laboratory experimental studies and the subsequent data interpretation, from which relevant hydraulic conductivity parameters, defined in applicable flow laws, were extracted. This study demonstrates that the exponent of a power flow law relating the hydraulic gradient and the flow velocity is Re number dependent for pore Re numbers \(<\)60,000. The power remains constant (Re number independent) above this Re number threshold for the fully developed turbulent regime. This validity threshold as well as the constant behaviour also applies if the flow law is written in a quadratic form. The aforementioned threshold lies beyond the ranges investigated experimentally by previous researchers. The experiments in this study examined Re numbers as large as 220,000 for graindiameter distributions in the range 100–160 mm and as large as 320,000 in the range 160–240 mm.
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Abbreviations
 \(a\) :

Energy dissipation index (–)
 \(A_{\mathrm{vs}}\) :

Volumespecific surface area (\(\hbox {m}^{2}\))
 \(b\) :

Coefficient depending on the fluid and porous medium properties (\(\hbox {m}^{1\mathrm{a}}\,\hbox { s}^{\mathrm{a}1}\))
 \(\acute{b}=b/n^{a}\) :

Permeability parameter used for field test ODE (\(\hbox {m}^{1\mathrm{a}}\,\hbox { s}^{\mathrm{a}1}\))
 \(c_{\mathrm{q}}\) :

Medium’s constants for the quadratic flow law (–)
 \(c_{\mathrm{p}}\) :

Medium’s constants for the power flow law (–)
 \(C_{\mathrm{E}}\) :

Ergun’s constant (\(\hbox {m}^{1}\,\hbox {s}^{2}\))
 \(d\) :

Characteristic length parameter of the medium (m)
 \(d_{i}\) :

Grain diameter at i % passing (m)
 \(d_{\mathrm{H}}\) :

Harmonic mean particle diameter (m)
 \(Fr\) :

Froude number (–)
 \(\mathop {F}\limits ^{\rightharpoonup }\) :

Body forces due to the friction force per unit volume of medium (\(\hbox {Nm}^{3}\))
 \(\mathop {g}\limits ^{\rightharpoonup }\) :

Acceleration vector due to gravity (\(\hbox {ms}^{2}\))
 \(g\) :

Standard gravity (\(\hbox {ms}^{2}\))
 \(h\) :

Piezometric head (m)
 \(h_{0}\) :

Piezometric head at the well’s radius (m)
 \(h_{\mathrm{L}}\) :

Piezometric head in the laminar flow region (m)
 \(h_{\mathrm{T}}\) :

Piezometric head in the turbulent flow region (m)
 \(H\) :

Hydraulic head (m)
 \(\mathop {\iota }\limits ^{\rightharpoonup }\) :

Gradient vector of hydraulic head (–)
 \(J_{\mathrm{rock}}\) :

Shape (geometry) coefficient (–)
 \(K\) :

Intrinsic permeability (\(\hbox {m}^{2}\))
 \(k\) :

Hydraulic conductivity (\(\hbox {ms}^{1}\))
 \(L\) :

A characteristic length of the porous medium channel size (m)
 \(n\) :

Porosity of the medium (–)
 \(p\) :

Pressure (Pa)
 \(Q\) :

Discharge (\(\hbox {m}^{3}\,\hbox {s}^{1}\))
 \(q\) :

Flux (\(\hbox {ms}^{1}\))
 \(R_{\mathrm{h}}\) :

Mean hydraulic radius (m)
 \(r\) :

Radius (m)
 \(r_{\mathrm{c}}\) :

Critical radius (m)
 \(r_{\mathrm{well}}\) :

Radius of the well (m)
 \(Re\) :

Reynolds number (–)
 \(Re_{\mathrm{c}}\) :

Critical Reynolds number (–)
 \(T\) :

Time (s)
 \(\mathop {U}\limits ^{\rightharpoonup }\) :

Velocity vector (\(\hbox {ms}^{1}\))
 \(\mathop {V}\limits ^{\rightharpoonup }=\mathop {U}\limits ^{\rightharpoonup }/n\) :

Pore velocity (\(\hbox {ms}^{1}\))
 \(W\) :

Wilkin’s parameter representing the medium’s property (\(\hbox {s}^{\mathrm{a}}\,\hbox { m}^{0.5\mathrm{a}}\))
 \(z\) :

Vertical elevation (m)
 \(\alpha \) :

Forchheimer viscous term coefficient (\(\hbox {s\,m}^{1}\))
 \(\alpha _0\) :

Engelund equation constant (–)
 \(\beta \) :

Forchheimer dynamic term coefficient (\(\hbox {s}^{2}\,\hbox { m}^{2}\))
 \(\beta _0 \) :

Engelund equation constant (–)
 \(\rho \) :

Fluid’s density (\(\hbox {kg\,m}^{3}\))
 \(\mu \) :

Dynamic viscosity of the fluid (Pa s)
 \(\nu \) :

Fluid kinematic viscosity (\(\hbox {m}^{2}\,\hbox {s}^{1}\))
 \(\nabla \) :

Nabla operator (\(\hbox {m}^{1}\))
 \( \,\) :

Symbol for the magnitude (norm) of a vector (–)
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Acknowledgments
This study was conducted as part of a Ph.D. programme financed by the Swedish Hydropower Centre (SvensktVattenkraftcentrum, SVC), Stockholm. SVC was established by the Swedish Energy Agency, Elforsk and SvenskaKraftnät, together with Luleå University of Technology (LTU), the Royal Institute of Technology (KTH), Chalmers University of Technology and Uppsala University. http://www.svc.nu. Data for field test analysis were provided by SWECO consulting company from their field tests conducted at the Trängslet embankment dam in Sweden.
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Ferdos, F., Wörman, A. & Ekström, I. Hydraulic Conductivity of Coarse Rockfill used in Hydraulic Structures. Transp Porous Med 108, 367–391 (2015). https://doi.org/10.1007/s1124201504811
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DOI: https://doi.org/10.1007/s1124201504811
Keywords
 Embankment dam failure due to internal erosion
 Hydraulic conductivity
 Coarse rockfill
 Nonlinear flow law