Single-well tracer tests, or push-pull tests, typically involve the forced-gradient injection of an aqueous solution containing one or more non-reactive solute tracers, followed by a forced-gradient extraction of the mixture of the injection and aquifer fluids. The extraction fluid is analyzed for the concentration of the solute tracers over time and a concentration versus time profile, or breakthrough curve, is generated. The breakthrough curve data can be analyzed and/or modeled to estimate aquifer properties such as groundwater velocity (v) and aquifer dispersivity (α) as well as contaminant properties such as sorption (R) and reaction rates (k). For a complete review of single-well tracer tests, see the textbook by Istok et al., (2012) titled, Push-Pull Tests for Site Characterization.

These tests have proven to be very useful tools for aquifer and contaminant characterization but there remains a need and a potential to improve the field-based methodology and the data analysis/modeling. For example, a single-well tracer test may be conducted without the forced-gradient extraction phase, instead relying on the natural-gradient to transport the injection fluid back through the well. This modification would eliminate the production and disposal of relatively large amounts of potentially contaminated wastewater but with two important caveats: 1) the up-gradient portion of the tracer must drift back through the well and 2) the natural groundwater velocity must be sufficiently fast to complete the test in a timely manner. In theory, this type of test, herein referred to as a single-well injection-drift, or SWID, test, can generate a data set that can be modeled for two fundamental flow parameters, hydraulic conductivity (K) and specific storage (Ss), and two fundamental transport parameters, groundwater velocity (v) and dispersity (α).

The flow during a SWIFT test can be modeled numerically using a finite-difference approach as follows:

Governing Flow Equation: d^2h/dx^2+d^2h/dy^2=S/T*(dh/dt)-R/T

Initial Condition: h(x,y,t=0)=-i*x+ho

Boundary Conditions:

Stress Period 1: d^2h/dx^2+d^2h/dy^2=0

h(x=0,y,0<t<=t1)=ho

h(x=inf,y, 0<t<=t1)=-i*inf+ho

h(x,y=0, 0<t<=t1)=-i*x+ho

h(x,y=inf, 0<t<=t1)=-i*x+ho

Stress Period 2: d^2h/dx^2+d^2h/dy^2=S/T*(dh/dt)-R/T

Same as Stress Period 1 plus: R(x=well,y=well,t1<t<=t2)=Q/(dx*dy)

Stress Period 3: d^2h/dx^2+d^2h/dy^2=S/T*(dh/dt)

Same as Stress Period 2

Finite Difference Equation: h[i,j,n+1]=1/A*(A*h[i,j,n]+(hbar-h[i,j,n])+(delx*dely*R)/(4*T))

Where A=(delx*dely*S)/(4*T*delt) & hbar=(h[i+1,j,n]+h[i-1,j,n]+h[i,j+1,n]+h[i,j-1,n])/4

Note 1: i is x index, j is y index, n is time (t) index

Note 2: (T*delt)/(S*delx*dely)<0.25 stable fully explicit scheme

Note 3: Stress period 1 is over relaxed at 1.5 for more efficient Gauss-Seidel iteration

Variables from Hoss thesis: i=-0.03, Q=42.8ft^3/day, ti=0.3125days, K=28.3ft/day, b=5ft, S=0.25, ho[well]=100ft

The numerical approach shown above can be simulated using the Fortran code in this text file: SWID Flow

The output from the code above can be visualized below in three graphs that show head versus time and space during a simulated SWID test.

The numerical model can be compared to an analytical model (Cooper-Jacob) under simpler conditions that include no regional flow and a static head of 100.75 ft, as shown below.

The transport during a SWID test can be modeled numerically using a finite-difference approach, as follows: (working on this now..)