12:166, Zhang

Spring, 2002

Lecture on

Darcy’s Law and Hydraulic Conductivity

 

·    Darcy’s Law (established in 1856 by Henry Darcy)

 

Where    Q is the volumetric flow rate (L3 / T, cm3/sec, gal/min)

                                K is the hydraulic conductivity (L / T, cm/sec, gal/day/ft2)

                                A is the cross-sectional area (L2, cm2, ft2)

                                i is the hydraulic gradient and 

 

                               

 

q is the specific discharge or Darcy’s velocity or Darcy’s flux (L / T, cm/sec);

 

·         Hydraulic Conductivity (K) and Permeability (k)

 

where  

                        k  is the intrinsic permeability or just permeability (L2, cm2, ft2)

                        C is a shape factor, d is the mean grain diameter of the sand

                        r is the fluid density, and  m is the viscosity

 

            Note:   (1) K is also called the coefficient of permeability in older texts;

                        (2) K is a function of both the medium and fluid but

                        (3) k is a function of only the medium;

                        (4) The units of k are m2, cm2, and darcy (1 darcy = 10-8 cm2);

                        (5) The ranges of K vary over many orders of magnitude;

                        (6) K varies over space in aquifers and is log-normally distributed.

 

·    Estimation of Hydraulic Conductivity

o       Hazen method based on grain-size distribution:

o       Permeameter

§         Constant-head

§         Falling-head

 

·    Transmissivity (T)

            For a confined aquifer:              T = Kb                        which does not change with time

            For a unconfined aquifer:                       T = Kh                        which may change with time

 

·        Storativity or Storage Coefficient:          S = Ss b = rgb(a + nb )

Remark: In a confined aquifer, water is released from storage due to two mechanisms:

(1)   S = 0.05 – 0.00005

(2)   Sy = 0.01 – 0.3

(3)   Sy >> S

 

 

·    Specific Storage:              Ss = rg(a + nb )        

            Where a is the compressibility of porous media and      b is the compressibility of water

            Remark: In a confined aquifer, water is released from storage due to two mechanisms:

(1)   Compaction of aquifer materials

(2)   Expansion of water

 

           

·    Heterogeneity and Anisotropy

                        A formation is homogeneous if K does not vary over space;

                        A formation is heterogeneous if K varies over space;

                        A formation is isotropic if K does not vary with the direction of measurement;                           A formation  is anisotropic if K varies with the direction of measurement;

 

·   Equivalent Hydraulic Conductivity

The effective K is the hydraulic conductivity for a homogeneous and anisotropic

formation which is hydraulically equivalent to a system of heterogeneous and isotropic formations.

Example: In a layered formation

            (1) When flow is parallel to the layers;  (2) When flow is normal to the layers

 

 

 

                       

----------------------------------------------------------------------------------------------------------------------------------------

Reading:         Section 3.4 and 3.11 (p.81- 104)

Homework:     Problems 6, 8, 10, 16, and 20 at the end of Chpt. 3 (p.110). Due on Mon., Feb.11.