Single-well Push-pull Research Articles

A novel four phase slug single-well push–pull test with regional flux: forward modeling and parameter estimation

The single-well push–pull (SWPP) test has been widely used for several applications, including determination of transport parameters such as regional groundwater flux velocity, effective porosity and dispersivity. A complete SWPP test usually includes four phases of injection, chase, rest, and extraction, while previous models mostly ignore the chase phase for the sake of simplification. Actually, the chase phase is important to minimize the impact of well skins and wellbore storage on the shape of the breakthrough curves (BTCs), thus is necessary and should be included in the model. To this end, two simplified methods are proposed in this study based on two different time tp and ta during a four-phase SWPP test to estimate the parameters related to transport, where tp and ta correspond to the time of the peak concentration during the injection phase and the time when the center of tracer mass returns to the wellbore during the extraction phase, respectively. Specifically, the tp method can be used to estimate the regional groundwater flow velocity and dispersivity (transverse and longitudinal), while the ta method is used to estimate the regional groundwater flow velocity and the effective porosity. Finite-element numerical modeling of the SWPP test based on the coupling of two-dimensional transient groundwater flow and solute transport is employed to test the applicability of the proposed methods. Results indicate that both the tp and ta methods proposed in this study perform satisfactorily in estimation of regional groundwater flow velocity, effective porosity and dispersivity, and these two methods require fewer observations than traditional curve fitting methods and do not require any optimization algorithms. We have also compared the proposed two methods with several previous methods, and conclude that these two new methods generally perform better than those of previous studies in terms of parameter estimation reliability.

Semi-analytical solution of single-well push–pull test under transient flow conditions

The single-well push–pull (SWPP) test has been extensively used in parameter estimation models. Numerous analytical solutions to the problem are available, all of which consider solute transport in a non-uniform flow field from steady radial flow created by the SWPP test. However, the non-uniform flow velocity is based on the Thiem equation and varies in a spatial manner, rather than a temporal manner. No analytical solution has been previously described for the solute transport equation of SWPP under fully transient flow. In this study, the generalized integral transform technique (GITT) was used to develop a new semi-analytical solution for solute transport in the SWPP test under fully transient flow. Four phases of the SWPP test were included: injection, chaser, rest, and extraction. With the proposed solution, the differences between a transient flow SWPP model solution and a piecewise steady-state flow SWPP model solution were non-negligible; such differences increased with decreases in the dimensionless parameter related to aquifer flow properties, which is proportional to transmissivity, aquifer thickness, and porosity, but inversely proportional to the storage coefficient and pumping rate. Additionally, long tails of BTCs are characteristic of type curves under the transient flow model. The long tails of BTCs under transient flow could result in overestimation of dispersivity if the SWPP model uses piecewise steady-state flow, rather than transient flow, for parameter estimation. The proposed semi-analytical solution provides a useful tool for curve-fitting during parameter estimation when the effects of transient flow are significant. The proposed solution can also serve as a performance comparison for testing numerical solutions to the SWPP test.

A modified and rapid method for the single-well push-pull (SWPP) test using SF6, Kr, and uranine tracers

In the present study, a single-well push-pull (SWPP) test was conducted with multi-component tracers, including inert gas (SF6 and Kr) and uranine (conservative), to understand the volatile/semi-volatile component transport characteristics in the groundwater system. In an SWPP test, it is essential to obtain an initial breakthrough curve (BTC) of the inert gas concentration at the beginning of the pulling stage to analyze the hydraulic properties of the groundwater system. As a result of the SWPP test using a proposed method in this study, physicochemical parameters of the groundwater and BTC of gas tracers and uranine were acquired simultaneously and successfully. In addition, on-site measurements of uranine, pCO2, and water quality data, such as electrical conductivity (EC), temperature, pH, and dissolved oxygen, were undertaken. Modification of an existing pCO2 measuring system allowed the gas samples to be collected, transported, and analyzed for inert gas components within a few hours. As a result, reliable and interpretable data with a recovery ratio of 26%, 85%, and 95% for SF6, Kr, and uranine, respectively, were obtained. The differences in the recovery ratio were utilized to identify the environmental system, whether it contains gas inside the isolated system (closed) or not (open), and to understand plume behavior characteristics in the experimental zone. By applying a two-dimensional advection-dispersion model to the acquired tracer test data and comparing the observed and computed tracer concentrations, helpful information was obtained on the hydraulic and transport characteristics of the targeted zone. This method can be extended to the design of dissolved CO2 transport monitoring of an aquifer above a CCS site.

The effect of boundary conditions on the single-well push-pull test in a partially penetrated well

Boundary conditions of well screen are critical formodeling reactive transport around a partially penetrated well in a single-well push–pull (SWPP) test. Previous SWPP test models often assumed that water flux was uniformly distributed along the well screen of the partially penetrated well, named the uniformly distributed flux boundary condition (UFBC). To investigate the influence of such an assumption on reactive transport in the SWPP test, we develop a new finite-difference numerical model under a flow field with a finite hydraulic diffusivity in this study, considering the wellbore storage. The uniformly distributed head boundary condition (UHBC) is employed for both groundwater flow and reactive transport across the well screen. The dispersion coefficients are in a tensor form, depending on both horizontal and vertical flow components. The results show that the UFBC model may cause non-negligible errors when modeling reactive transport in the SWPP test with a partially penetrated well. Such errors are obvious at the early stage of the extraction phase, but become negligible at late stage of the extraction phase. The UHBC solution for reactive transport is smaller than the UFBC solution at the middle of the well screen, but greater than the UFBC solution at the top or bottom of the well screen. Using the SWPP test to investigate the reactive transport in subsurface, in addition to the breakthrough curves (BTCs), the information of the flow field (like well geometry information and the observed time-drawdown data) is necessary and could be used to calibrate the model. Field application demonstrates that the UHBC model performs better than the UFBC model when interpreting BTCs of the SWPP test in the extraction phase. The findings of this study can help for better understanding of reactive transport in the SWPP test.

Revisiting simplified model of a single-well push–pull test for estimating regional flow velocity

Simplified analytical models of the single-well push–pull (SWPP) test have been widely used to estimate the regional flow velocity and the aquifer porosity, where the simplifications refer to that the dispersion effect is negligible, and the flow is steady state during the entire SWPP test. However, the errors caused by such assumptions have not been investigated thoroughly. In this study, numerical modeling of the SWPP test without above-mentioned simplifications and the field experimental data will be employed to test the applicability of the simplified analytical models. The numerical simulation is based on a mathematical model composed of a two-dimensional transient flow and transport model, which will be solved by the finite-element method. The results indicate that errors increase with the increasing dispersivity when using the simplified analytical models to estimate both regional flow velocity and porosity. The errors refer to the relative difference between the input parameters of the numerical modeling and the estimated parameters by the simplified models. Such errors also depend on the input values of regional groundwater velocity, drift time of the SWPP test, and aquifer porosity. Specifically, the errors caused by the simplified models of estimating regional groundwater velocity decrease with increasing input regional groundwater velocity, increasing input drift time, and decreasing input porosity. The errors produced by the simplified models of estimating porosity decrease with increasing input regional groundwater velocity, increasing input drift time, and decreasing input porosity. Field in-situ experiments show that the fitness of the observed breakthrough curves (BTCs) by the numerical modeling is far better than the simplified analytical models. The estimated regional flow velocity by the numerical modeling of this study is closer to the estimated values by previous other studies.

Estimation of very slow groundwater movement using a Single-Well Push-Pull test

An estimation method for evaluating groundwater velocity by means of the Single-Well Push-Pull (SWPP) test by Leap and Kaplan (1988) was applied to a confined aquifer at 100 m depth with a very slow groundwater movement. The SWPP test consists of a push phase, a drift phase, and a pull phase, and the groundwater velocity is evaluated based on the Breakthrough Curves obtained from the SWPP test. In this study, the accuracy of the SWPP test system was evaluated by performing a SWPP test without drift time four times in total. Also, four types of SWPP tests with different drift times from 10 to 1802 days were conducted to obtain an estimate of the regional groundwater velocity for a maximum of approximately five years. Furthermore, the stable isotope of water (δ18O and δD) and uranine, which are known as conservative tracers, were used as tracers in the SWPP tests so as to evaluate their availabilities.The SWPP test results showed a higher reproducibility of the SWPP test using δ18O and δD than that using uranine. The sorption of uranine in the aquifer was suggested based on the SWPP test results, indicating that the hydrological properties can be obtained by comparing the SWPP test results using δ18O, δD and uranine as tracers. The SWPP test yielded an actual groundwater velocity of 0.37 m/year, which is similar to approximately 0.1 m/year order of the estimated groundwater velocity based on the results of past studies. Moreover, setting the appropriate correction and drift time in the SWPP test played an important role in investigating the very slow groundwater flow in the aquifer. When a long-term drift time was set, the recovery rate of the tracer and the chaser of δ18O and δD was decreased by diffusion in the aquifer’s silt and clay layer.

New model of reactive transport in a single-well push–pull test with aquitard effect and wellbore storage

Abstract. The model of single-well push–pull (SWPP) test has been widely used to investigate reactive radial dispersion in remediation or parameter estimation of in situ aquifers. Previous analytical solutions only focused on a completely isolated aquifer for the SWPP test, excluding any influence of aquitards bounding the tested aquifer, and ignored the wellbore storage of the chaser and rest phases in the SWPP test. Such simplification might be questionable in field applications when test durations are relatively long because solute transport in or out of the bounding aquitards is inevitable due to molecular diffusion and cross-formational advective transport. Here, a new SWPP model is developed in an aquifer–aquitard system with wellbore storage, and the analytical solution in the Laplace domain is derived. Four phases of the test are included: the injection phase, the chaser phase, the rest phase and the extraction phase. As the permeability of the aquitard is much smaller than the permeability of the aquifer, the flow is assumed to be perpendicular to the aquitard; thus only vertical dispersive and advective transports are considered for the aquitard. The validity of this treatment is tested against results grounded in numerical simulations. The global sensitivity analysis indicates that the results of the SWPP test are largely sensitive (i.e., influenced by) to the parameters of porosity and radial dispersion of the aquifer, whereas the influence of the aquitard on results could not be ignored. In the injection phase, the larger radial dispersivity of the aquifer could result in the smaller values of breakthrough curves (BTCs), while there are greater BTC values in the chaser and rest phases. In the extraction phase, it could lead to the smaller peak values of BTCs. The new model of this study is a generalization of several previous studies, and it performs better than previous studies ignoring the aquitard effect and wellbore storage for interpreting data of the field SWPP test reported by Yang et al. (2014).

New Simplified Models of Single‐Well Push‐Pull Tests With Mixing Effect

AbstractWhen using single‐well push‐pull (SWPP) tests to characterize in situ aquifer related to reactive transport, approximate analytical solutions were preferred to determine the aquifer dispersivity and other parameters, due to its computational efficiency. Simplified models were proposed in this study to estimate the retardation factor for a SWPP test with a reactive tracer, where the dispersivity had to be determined firstly using a SWPP test with a conservative tracer. Here, we find that previous models associated with SWPP tests contain an untested assumption when dealing with solute transport in the wellbore, probably causing great errors when interpreting the test data. This assumption states that the wellbore mixing effect is negligible in both injection and extraction phases, where the mixing effect refers to the mixing process between the native wellbore water and solute introduced into the wellbore. In this study, new approximate analytical solutions of the SWPP test are proposed for a fully penetrating well and a point‐source well by specifically considering the wellbore mixing effect. The new models of this study are tested by semianalytical solutions in Laplace domain and finite difference solutions. Results indicate that the new models perform substantially better than previous models of ignoring the wellbore mixing effect, which should not be overlooked when interpreting SWPP test data.

A new solution to transient single-well push-pull test with low-permeability non-Darcian leakage effects

An accurate solute transport model is critical to the interpretation of single-well push-pull (SWPP) test. Previous studies of SWPP test generally consider solitary aquifer that is confined by impermeable layers. Also, existing solutions for solute transport in aquifer-aquitard systems only consider the injection phase and over-simplify the flow field by assuming uniformly distributed velocity in the aquitard. In this study, we developed a numerical model with Dirichlet boundary condition for SWPP test affected by leakage described by a low-permeability non-Darcian expression involving a threshold pressure gradient (I0). Our SWPP test model considered transient flow in multi-phases, which include injection, chase, rest and extraction phases. Finite-difference scheme was adopted to solve the models of flow and solute transport. The results indicate that an increasing hydraulic diffusivity leads to a greater peak value of breakthrough curve (BTC) while a medium with larger grain size results in less estimation error when using steady-state flow model to interpret the transient SWPP test. Additionally, A greater I0 makes the solute stored in aquitard more difficult to be extracted out due to dispersion dominance, which results in higher BTC values. For the purpose of application, a lumped dimensionless index called the non-Darcian index (NDI) was proposed to quantify the overestimation degree by neglecting leakage, and the underestimation degree by accounting for Darcian leakage, when interpreting the SWPP test with low-permeability non-Darcian leakage. The long-term slope of breakthrough curve coupled with the NDI can be employed to determine the cases in which the low-permeability non-Darcian leakage should be considered.

New model of Single-Well Push-Pull thermal test in a Fracture-Matrix system

Single-well push pull (SWPP) thermal tests have been widely used to characterize fracture-matrix systems for storing and/or exploring geothermal energy. Two important assumptions in previous mathematical models of the SWPP thermal test are found to be problematic: the thermal mixing effect in the wellbore is ignored and the fracture thermal conductance is omitted. These two assumptions are not correct in many applications and have to be revoked. Meanwhile, solutions associated with previous three-stage (injection, resting and extraction) models of SWPP tests are computationally very expensive to obtain and need improvement for practical usage. In this study, a new three-stage model is developed considering both the wellbore thermal mixing effect and the fracture thermal conductance, and an analytical solution has been developed using Laplace transform method and Green’s function method. To test the new model, a high-resolution finite-difference numerical solution is established, and the results show that the analytical and numerical solutions agree with each other very well. Comparing with previous three-stage analytical solution, the computational cost is much less for the new solution of this study. We find that breakthrough curves (BTCs) in the wellbore for the extraction stage are not only dependent on the values of the injecting and pumping rates, but also on the ratio between them. The new model is most sensitive to fracture aperture, and least sensitive to flow velocity in the fracture. The previous models of ignoring the thermal mixing effect and/or the fracture thermal conductance may cause great errors and should be avoided.

Skin effect on single-well push-pull tests with the presence of regional groundwater flow

Single-well push-pull (SWPP) test is one of the most important ways to estimate flow and transport parameters, e.g. porosity, dispersivity, regional groundwater flow velocity. Usually the wellbore is surrounded by a finite-thickness skin, such as a gravel pack. The positive skin has a smaller hydraulic conductivity than the aquifer formation zone, and the negative skin has a greater one. In this study, a numerical model of a SWPP test considering skin effects was established using the finite-element COMSOL Multiphysics program to estimate aquifer parameters. Several important results were obtained. Firstly, regional groundwater velocity affects the types of breakthrough curves (BTCs) through changing the flow pattern. Secondly, a positive (or negative) skin leads to a slower (or faster) tracer transport process, and a smaller ratio between the hydraulic conductivity of the skin and that of the aquifer formation zone results in greater solute plume retardation in the skin zone. Thirdly, a larger thickness of the positive skin leads to a higher tracer concentration around the well, and the opposite is true if the skin is negative. Besides, the model of Leap and Kaplan (1988) underestimates the regional groundwater velocity for the second type of BTCs, and a larger longitudinal dispersivity can lead to a greater error, where the second type of BTCs has a rising limb at early stage followed by a falling limb at late stage. The general conclusion is that the estimations of groundwater velocity from SWPP tests are also affected by skin effects.

Reactive transport with wellbore storages in a single-well push–pull test

Abstract. Using the single-well push–pull (SWPP) test to determine the in situ biogeochemical reaction kinetics, a chase phase and a rest phase were recommended to increase the duration of reaction, besides the injection and extraction phases. In this study, we presented multi-species reactive models of the four-phase SWPP test considering the wellbore storages for both groundwater flow and solute transport and a finite aquifer hydraulic diffusivity, which were ignored in previous studies. The models of the wellbore storage for solute transport were proposed based on the mass balance, and the sensitivity analysis and uniqueness analysis were employed to investigate the assumptions used in previous studies on the parameter estimation. The results showed that ignoring it might produce great errors in the SWPP test. In the injection and chase phases, the influence of the wellbore storage increased with the decreasing aquifer hydraulic diffusivity. The peak values of the breakthrough curves (BTCs) increased with the increasing aquifer hydraulic diffusivity in the extraction phase, and the arrival time of the peak value became shorter with a greater aquifer hydraulic diffusivity. Meanwhile, the Robin condition performed well at the rest phase only when the chase concentration was zero and the solute in the injection phase was completely flushed out of the borehole into the aquifer. The Danckwerts condition was better than the Robin condition even when the chase concentration was not zero. The reaction parameters could be determined by directly best fitting the observed data when the nonlinear reactions were described by piece-wise linear functions, while such an approach might not work if one attempted to use nonlinear functions to describe such nonlinear reactions. The field application demonstrated that the new model of this study performed well in interpreting BTCs of a SWPP test.

Several Considerations on Well Treatment in a Numerical Model for Single-Well Push-Pull Tests in Carbon Dioxide Injection Simulation

Several Considerations on Well Treatment in a Numerical Model for Single-Well Push-Pull Tests in Carbon Dioxide Injection Simulation

Models of Single‐Well Push‐Pull Test With Mixing Effect in the Wellbore

AbstractThe mechanism of solute transport around the wellbore was found to play an important role in the single‐well push‐pull (SWPP) test, but it was grossly overlooked in previous studies. For instance, the mixing effect of injected tracer with water in the wellbore was ignored in analyzing both injection and extraction phases of SWPP. In this study, new models were developed by including such a mixing effect in the wellbore. Two types of boundary conditions at the wellbore were considered: the resident concentration continuity and the flux concentration continuity. To test the assumptions used in the mathematical model, the stochastic modeling, the numerical simulation, and the laboratory‐controlled experiment were executed. Results showed that the SWPP test was sensitive to the mixing effect in both injection and extraction phases. A larger wellbore volume could result in a smaller concentration at the late stage of the extraction phase. Flux concentration continuity was more reasonable in describing solute transport at the wellbore‐aquifer interface than resident concentration continuity, and the difference between them decreased with decreasing radial dispersivity. The MODFLOW/MT3DMS package contained an invalid assumption on the mixing effect for the SWPP test. Stochastic modeling demonstrated that the homogeneous assumption was a good approximation for the reality when the variance of natural logarithm of the autocorrelated hydraulic conductivity field was less than 0.25 ( ). The laboratory‐controlled experiment showed that the radial advection‐dispersion equation model of this study worked well for the well‐sorted sand aquifer.

Fractional Models Simulating Non‐Fickian Behavior in Four‐Stage Single‐Well Push‐Pull Tests

AbstractFour‐stage single‐well push‐pull (SWPP) tracer tests, including injection, chasing, resting, and pumping, were conducted in a fractured aquifer at Newark basin. An anomalous transport phenomenon observed in the SWPP tests is the linear decline of breakthrough curves (BTCs) at late time with slope of −1.8 in log‐log plots. A time‐dependent fractional model is developed to interpret the anomalous transport behavior. This model considers a time‐dependent power law memory function and a time‐dependent fractional advection‐dispersion operator. The fractional advection‐dispersion equations (fADE) are solved in a radial coordinate system using the implicit Euler method. A semi‐analytical solution of the first‐order rate‐limited mobile‐immobile model (FORMIM) is derived for comparison. It is found that both the nonlocal transport in time and space can produce the long‐tailed BTC. A smaller time‐fractional or space‐fractional index leads to a lower peak concentration and a larger late‐time slope. The mass distribution of the fractional‐in‐space (FS) model exhibits power law decline at the leading plume edge. Early breakthrough during pumping is not observed because the mobile mass at the start of pumping is nonzero and more concentrated near the wellbore. The capacity ratio is an important factor that affects the peak concentration. A larger capacity ratio leads to greater peak concentration. A smaller time‐fractional index in the injection, chasing, or resting stage will move the BTC downward and the slope of the late time BTC is determined by the space‐fractional index over all stages and the time‐fractional index in the pumping stage. The capability of the existing models to recover the BTC of the SWPP test is discussed and some guidelines for how to choose the appropriate model to interpret the SWPP test data are proposed.

Single-well push-pull test in transient Forchheimer flow field

Using the single-well push-pull (SWPP) test to quantify in situ aquifer characteristics associated with solute transport (dispersion coefficient, geobiochemical reaction rates), the accuracy of parameter estimation was not only dependent on the solute transport models but also the groundwater flow models. However, many previous studies on the SWPP test were based on assumptions over-simplifying the flow field, namely, groundwater flow followed Darcy’s law; flow was in the steady state during the entire test duration; the wellbore storage could be negligible. In this study, we have carefully examined such assumptions by developing a new finite-difference model of the SWPP test under the transient Forchheimer flow condition, considering the wellbore storage. The SWPP test included an injection phase, a chaser phase, a rest phase, and an extraction phase. The results showed that the concentration of the steady-state flow solution was greater than that of the transient flow solution at the beginning, and its peak value was also greater than that of the transient flow solution. The difference between the breakthrough curves (BTCs) of the transient flow SWPP model and the steady-state flow SWPP model was not negligible, and such a difference increased with the decreasing specific storage. We also found that BTCs were not sensitive to the inertial force coefficient, while they were sensitive to the wellbore storage. BTCs with different radius of the wellbore (rw) were clearly different from each other, and a larger rw resulted in a greater concentration at the well during the extraction phase.

On the Loss of Injectant and Apparent Dispersivity in Single-Well Push-Pull Tests Influenced by Natural Drift

In a previous article, a semi-analytical stream-line based model was developed for single-well push-pull tests in stratified reservoirs subject to natural drift. Based on that model, this article gives the theoretical maximum injection phase duration giving conservative injectant production, in a single-layer reservoir with natural drift. In addition, an analytical expression for the single-layer apparent dispersivity due to natural drift is given. Finally, it is shown how the large-scale apparent dispersivities observed in experimental data from a stratified aquifer may be predicted by constructing a production profile using the estimated single-layer apparent dispersivities.