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Analytical Treatment of a Push–Pull “Echo” Test

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Abstract

A semi-analytical streamline-based model, employing stratification and macro physics only, is developed and utilized to simulate injection/production phases of single-well push–pull tests. Modeling results are compared with experimental field data, giving an excellent match, without resorting to parameter fitting, simply by putting in known test-site properties, such as stratification data, hydraulic head gradients, and test parameters.

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Abbreviations

b j :

Height of layer j (m)

\({\mathcal C_j}\) :

\({\dfrac{\phi_A}{\phi_j} \dfrac{k_j}{k_A}}\)

C :

Injectant concentration in the producing well-stream

k :

Permeability (darcy)

P :

Pressure (Pa)

q 3D :

Volumetric rate (m3/day)

Q :

\({\dfrac{q_{\rm 3D}}{2\pi b\phi}}\) (m2/day)

r :

Polar coordinate, radial distance from Origo (m)

r w :

Well radius (m)

r max :

r(t = T i ), maximum radial tracer advancement (m)

t :

Absolute time, since the injection phase started (days)

T i :

Total duration of the injection phase (days)

T BT :

Time, since the production phase started, of breakthrough, may refer to overall production, production from a specific layer, or production from a specific streamline (days)

U u :

Natural, uniform groundwater velocity in the positive x-direction (m/day)

U θ :

\({r \cdot \dfrac{{\rm d} \theta}{{\rm d}t}}\) , tangential velocity (m · rad/day)

U r :

\({\dfrac{{\rm d}r}{{\rm d}t}}\) , radial velocity (m/day)

θ :

Polar coordinate, angle to the positive x-axis (rad)

\({\phi}\) :

Fractional porosity

ψ :

The Stream-function

i :

Denoting injection phase variables and parameters

j :

Index to denote layer number

n :

Accounting index to denote individual streamlines, \({n \in [1, N]}\) . n = 1 denotes the streamline with constant θ = 0, and n = N denotes the streamline with constant θ = π

p :

Denoting production phase variables and parameters

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Authors and Affiliations

  1. SINTEF-Materials and Chemistry, Trondheim, Norway

    Sverre G. Johnsen

  2. NTNU and Pera AS, Trondheim, Norway

    Curtis H. Whitson

Authors
  1. Sverre G. Johnsen
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  2. Curtis H. Whitson
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Correspondence to Sverre G. Johnsen.

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Johnsen, S.G., Whitson, C.H. Analytical Treatment of a Push–Pull “Echo” Test. Transp Porous Med 77, 399–415 (2009). https://doi.org/10.1007/s11242-008-9266-0

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