Lecture Packet #7: Transient Systems and Groundwater Storage

1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #7: Transient Systems and Groundwater Storage How does the mass of water stored in an aquifer change? Confined Aquifer – Compression of both the material and the water Unconfined Aquifer – Water drains out pores at the water table when the water table drops and fills pores when the water table rises. The change in store from compression is negligible. Return to the time derivative or change in storage term, which may be written as ∂n ∂ρ ∂ρn +ρ =n ∂t ∂t ∂t Porosity changes Density changes (compressibility) Overall definition: Ss, specific storage, (1/L) of a saturated aquifer is the volume that a unit volume of aquifer released from storage for a unit decline in head. (Volume per Volume per head change). Confined Aquifers Potentiometric Surface Unit Cross Section Unit decline in potentiometric surface b For a confined aquifer we must consider compressibility. How can a reduction in aquifer volume occur? • Compression of the individual grains or rock skeleton (assumed negligible – individual grains are incompressible) • Rearrangement of the grains – more compact • Compression of the water in the pores 1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet 7 Page 1 of 7 • We will get Ss = ρgα + ρgβn Stress at any depth is due to: Total stress acting downward on a plane σT = weight of rock and weight of water Some of the stress is borne by rock skeleton; some by water. σT = σT + P Æ effective stress (borne by rock) + fluid pressure When pumping an aquifer the change in stress is: d σT = dσT + dP Total Stress Fluid Pressure Effective Stress But the weight of overlying water and rock is essentially constant over time. So the change in total stress is zero. d σT = dσT + dP = 0 dσT = -dP Fluid pressure decreases, the stress on the grains becomes greater (imagine we took away the fluid). Fluid pressure controls the volumetric deformation. At a point, P = ρgh - ρgz = ρg(h – z) --- at a point, z is constant dP = dρgh and substituting (dρgz = 0 = derivative of a constant) dσe = -dP = -dρgh If pumping increases, head goes down, and the effective stress goes up. Consider what happens when σe goes up. 1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet 7 Page 2 of 7 Water Produced from Aquifer Compaction • Aquifer compressibility, α, [L2/M] is defined as follows (corresponds to shifting of grains and reduction in porosity. α= − (dVt ) /Vt dσ e Vt is the original volume (thickness) dVt is the change in volume (thickness) dσe is the change in effective stress Aquifer gets smaller with increase in effective stress. Consider a unit volume Vt = 1. αdσe = -dVt Volume of water pumped that comes from aquifer compaction: We also know that the change in volume of water produced by aquifer compaction must equal the volume of water and sand accounting for the reduction. dVw = -dVt substitution for the term on the right dVw = αdσe = -α(dρgh) = -αρg(dh) For a unit decrease in head due to pumping dh = -1 dVw = αρg because this is for a unit decrease in head, this is a component of our Ss, specific storage. Water Produced by the Expansion of Water We can also define fluid compressibility β= − (dVW ) /VW dP β is the fluid compressibility - compressibility of water n is the porosity (volume of water is total volume times n) so, for a unit total volume, nVt = n(1) = n dVW = -βndP = -βn(dρgh – dρgz) = -βnρgdh For a unit decline in head, dh = -1, dVw = +βnρg which is another component of Ss 1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet 7 Page 3 of 7 Specific Storage is sum: Ss = ρg(α + βn) It has units [L-1]. It is volume produced per aquifer volume per head decline. • First term is from aquifer compressibility. • Second term is from water expansion. Let’s return to flow equation (time derivative term): ∂n ∂ρ ∂ρn +ρ =n ∂t ∂t ∂t The total change in mass with time is due to • • A change in density with time term associated with water compressibility. A change in porosity with time term associated with the compressibility of the aquifer. Our analysis of “where water from storage comes from” gives a term involving Ss = ρg(α + βn) to replace the change in mass with time term above. Going back to the continuity equation: [ ] ⎡ ∂[ρq x ] ∂ ρq y ∂[ρq z ]⎤ ∂ρn = −⎢ + + ⎥ ∂t dx dy dz ⎦ ⎣ Rate of Change of Mass stored: [ ] ⎡ ∂[ρq x ] ∂ ρq y ∂[ρq z ]⎤ ∂ρn ∂ρ ∂n =n +ρ = −⎢ + + ⎥ ∂t ∂t ∂t dy dz ⎦ ⎣ dx Substituting in for Darcy’s law, pulling out K (homogeneous, optional), and substituting for the storage terms, we obtain: ⎡∂ 2h ∂ 2h ∂ 2h ⎤ ∂h ∂h ρS s βnρg = ρK ⎢ 2 + 2 + 2 ⎥ =ρ ∂t ∂t dy dz ⎦ ⎣ dx Note storage term must convert a volume produced to a mass produced, so multiply by ρ. Cancel first ρ, giving our final confined flow equation as: Ss ⎡∂ 2h ∂ 2h ∂ 2h ⎤ ∂h ∂h = ρg ( βn + α ) = K ⎢ 2 + 2 + 2 ⎥ ∂t ∂t dy dz ⎦ ⎣ dx 1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet 7 Page 4 of 7 Summary of Storage Mechanisms – Confined Aquifer Water is released from storage during a decrease in h by two mechanisms: • Compaction of aquifer caused by increasing effective stress and rearrangement of grains. • Expansion of water caused by decreasing fluid pressure. This gives rise to a specific storage coefficient with two terms and a function of α and β. Material Compressibility [m2/N or 1/Pa] Clay 10-6 – 10-8 Sand 10-7 – 10-9 Gravel 10-8 – 10-10 Sound Rock 10-9 – 10-11 Water 4.4 x 10-10 Where does stored water come from in confined aquifers? Assume: Ss = ρg(α + βn) • 10 ft of drawdown (reduction in head) • Porosity of 30% Percent of water from storage Compressibility (m2/N) 10-6 Clay Sand 10-8 Granite 10-10 Water 4.4 x 10-10 Water from Storage, Ss x ∆H 0.0303 0.000694 0.000398 100 80 From expansion of water 60 From media compression 40 20 clay gravel granite With more realistic porosities, results are similar Clay Gravel Granite Porosity 45% 35% 25% 1.72, Groundwater Hydrology Prof. Charles Harvey Compression 98.06% 39.37% 0.90% Expansion 1.94% 60.63% 99.10% Total Water 0.0305 0.00759 0.00332 Lecture Packet 7 Page 5 of 7 Key Equations for a confined aquifer • Ss ⎡∂ 2h ∂ 2h ∂ 2h ⎤ ∂h = K⎢ 2 + 2 + 2 ⎥ ∂t dy dz ⎦ ⎣ dx • Ss or S s ∂h = K∇ 2 h ∂t 3D Heterogeneous, Anisotropic ∂h ⎡ ∂ ∂h ∂ ∂h ∂ ∂h ⎤ = ⎢ Kx + Ky + Kz ⎥ ∂t ⎣ dx dx dy dy dz dz ⎦ • Ss 3D flow equation, Homogeneous Isotropic 3D Heterogeneous, Isotropic ∂h ⎡ ∂ ⎛ ∂h ⎞ ∂ ⎛ ∂h ⎞ ∂ ⎛ ∂h ⎞⎤ = ⎢ ⎜ K ⎟ + ⎜ K ⎟ + ⎜ K ⎟⎥ ∂t ⎣ dx ⎝ dx ⎠ dy ⎜⎝ dy ⎟⎠ dz ⎝ dz ⎠⎦ For Steady State, S s ∂h = 0 for any equation. ∂t For steady state: • No water from storage • S values doesn’t matter 1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet 7 Page 6 of 7 2D Flow Equation • S Confined aquifer, homogeneous, isotropic. ⎡∂ 2h ∂ 2h ⎤ ∂h =T⎢ 2 + 2 ⎥ ∂t dy ⎦ ⎣ dx S = Ssb = Storativity [L3/ L3] Å Values like 10-2 to 10-6 Compare values to Sy ! T = Kb = Transmissivity [L2/T] B = aquifer thickness [L] • S ∂h ⎡ ∂ ∂h ∂ ∂h ⎤ = ⎢ Tx + Ty ⎥ ∂t ⎣ dx dx dy dy ⎦ • S Confined aquifer, Heterogeneous, Anisotropic Confined aquifer, Heterogeneous, Isotropic ∂h ⎡ ∂ ⎛ ∂h ⎞ ∂ ⎛ ∂h ⎞⎤ = ⎢ ⎜ T ⎟ + ⎜T ⎟⎥ ∂t ⎣ dx ⎝ dx ⎠ dy ⎜⎝ dy ⎟⎠⎦ • Confined aquifer, Homogeneous, Anisotropic ∂h ⎡ ∂ 2 h ∂ 2h ⎤ S = ⎢Tx + Ty 2 ⎥ ∂t ⎣ dx 2 dy ⎦ 1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet 7 Page 7 of 7