8.1 General remarks
The importance of testing rock materials for permeability and the current methods being used have already been mentioned in section 4.9. The problem of interstitial water and how it influences rock masses is extremely complex.
It involves the chemical reactions of water on rock and rock on water; the physical characteristics of water; its behaviour in porous and in fissured rock; the formation of underground caverns, caves and rivers and also the stability of rock masses. Some of these points will be dealt with more fully in subsequent chapters on rock slopes, tunnels and dams. This chapter is a general summary on the behaviour of water in rock masses.
8.2 Some general equations on the flow of water in fissured rock
The flow of water in a porous solid, like concrete or rock, results from hydraulic gradients S from one point to the next:
S = grad (z + ply) = grad U, (1) z being the level and p the pressure at the point and y the specific weight of the water. U is the hydraulic potential. Changes in water pressure from one point to another create differentials in surface forces in the pores which are equivalent to body forces. It is normally accepted that the flow of water, and other fluids (oil) through pervious solids follows the Darcy law (Darcy, 1856; Muskat, 1937; C. Jaeger, 1956), which states that the velocity of percolation v is proportional to the gradient S of the hydraulic potential:
v = kS = k grad U, (2)
k being the permeability of the solid to water usually expressed as centimetres per second. This law has been generalized for the case of the flow of homo- geneous fluids in porous media as
&V KY_t ( 2 a )
where D is the effective diameter of the openings (pores or joints) of the solid, or a characteristic dimension of its texture; //, the dynamic viscosity;
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y the specific weight of the liquid and K = KD2 the physical coefficient of permeability, K, being a quantity without dimensions depending on the geometry of the pores.
If an array of fissures of opening, e, with parallel faces separated by a distance, d, is assumed, the following expression is obtained (Serafim & del Campo, 1965; Talobre, 1957) for the filtration through that array
and the flow through a single joint of constant thickness e is expressed as e2y
(The viscous flow in a circular tube is given by Poiseuille's formula, v = yR2S/S/i, with R = D/2 = radius of the tube.)
Some authors suggest differentiating between the two types of flow in fissured rock: the primary flow through the rock pores and the secondary flow through the rock fissures, assuming that the flow through the first is much slower than through the second. This may be correct for compact, widely fissured granite, but there are some porous rocks, with void index not higher than / = 5% to 10% for which the flow through the pores may be quite substantial. The degree of correlation between perviousness k and compressive stress depends on the shape of the voids and minute canals in the rock (sections 4.9 and 4.10). In most cases when permeability of the rock results from open fissures and fractures the factor k depends on the directions along which it is measured. If x, y and z are the three principal directions of the anisotropy (Serafim 1968):
{t>}=[tf]{grad£/}, (4) where [K] is a three-by-three matrix defined by nine numerical coefficients.
The three components of {v} are:
vx = kx du/dx
vy = k2 du/dy (5)
vz = k3 du/dz.
When analysing the steady flow in anisotropic rock it can be stated that the weight of a liquid which enters in a unit volume of the porous body in a unit of time is equal to the quantity which flows out of that volume. Therefore:
div (yv) = £ (yvx) + j (yvy) + ~ (yv2) = 0. (6)
General equations on the flow of water 195 Assuming the liquid (or gas) is not compressible we find that the two previous equations yield:
dktdu dk2du dk3du d>u d*u Pu_
+ +~ d F t e ' + ICl'dxi+ t'df* 'dz*'"' U)
when writing
x y
omitting the first term of the previous equation containing dkx dk2 A dk3
— , — and —-.
dx dy dz we obtain
S + $ + S = 0, o, ,A% = 0, (9)
In the case of a homogeneous field of percolation, with kx = k2 = k3 = k, the meaning of the last equation is obvious. The solution for such a Laplace equation is well known. Used in classical soil mechanics, it treats ground- water problems, percolation in loose soils and foundations. The methods of potential flow are used in most cases, or approximations based on the same method (Jaeger, 1949). They could therefore be used in rock mechanics whenever a homogenitic percolation is assumed. After using this method, Pacher & Yokota (1962/63) mention that in situ borehole measurements made at the Kurobe IV dam site correspond with those estimated from model tests and the electric analogue (chapter 11, sections 11.3, 4).
If during the flow, some water is retained or originates in a given unit volume around a point, unsteady flow results. In such a case the left member of equations (6) and (7) cannot be equal to zero and (7) can be written:
Jt ~ L * a ? + 2 df + fC3d? + Txte
dy dy dz dz r dz
where dV is a volume and y is the specific weight of the liquid, which is assumed to be constant. It can be assumed that the weight of water contained in the volume d F i s proportional to the percentage of the volume of voids or porosity n> less the percentage of the volume of air a. Then:
q = y{\ — d)n x dV.
If the liquid is compressible, its specific weight y at the pressure p is equal to y = y$p + yo,
where y0 is the specific weight at the pressure p0 = atmospheric and /? the volumetric compressibility. The theory can be developed further on such lines (Serafim, 1968).