Dual-phase flow in two-layer porous media: a radial composite Laplace domain approximation

The main purpose of this work is to present an interpretation method for injectivity test in a two-layer reservoir that can be extended to a multilayer approach, based on new analytical solutions to the well pressure response. The developed formulation uses a radially composite reservoir approach and considers that the water front propagation may be approximated by a piston-like flow displacement. The reservoir is assumed to be laterally infinite and properties such as permeability and porosity may be different in each layer. The solutions were developed in the Laplace domain and then inverted to real domain using the Stehfest Algorithm. The proposed formulation was then validated by comparison with a numerical flow simulator. Results showed a good agreement between the numerical simulator and the analytical model. Also, a sensitivity study was done by comparing the results of different scenarios varying oil viscosities and injection flow rate to assess how these properties affect the pressure and pressure derivative profiles.

Rate/Pressure Transient Analysis of a Variable Bottom Hole Pressure Multi-Well Horizontal Pad with Well Interference

Multi-well horizontal pads are common in unconventional reservoirs. With addition of infill wells and hydraulic fracturing, interference between multiple multi-fractured-horizontal wells (MFHWs) has become a serious issue. Current RTA workflows assume a single MFHW in the unconventional formation. This paper presents a new multi-MFHW solution and related analysis methodology to analyze targeted well rate performance in a multi-MFHW system. In this work, a semi-analytical equation describing multi-well pad in the Laplace domain with well interference is proposed. The proposed semi-analytical model can simulate the rate performance of a multi-well horizontal pad with variable BHP for a targeted well in the pad and different initial production durations for the offset well. From the constant BHP condition and Laplace transforms, we obtained multi-MFHW solutions for transient flow. We used superposition of various constant BHP solutions to study interference among various fractures and MFHWs. The variable BHP of the targeted well is achieved by a variable dimensionless BHP function in the Laplace domain without any convolution or deconvolution calculations. A systematic validation for the proposed method is conducted using a commercial numerical simulator for cases of different initial production times for offset MFHWs, multi-MFHWs with variable BHP. Through the total material balance of the multi-MFHW system, we can analyze a target well in the pad with this multi-MFHW analysis. Interference by offset wells often appears after pseudo-radial flow in the target well's hydraulic fracture. It causes the pressure derivative curve during elliptical and infinite-acting radial flow (IARF) to rise, as does the RNP derivative. The inverse semi-log derivative has the opposite trend. Well interference also makes the rate/pressure drop functions to deviate from initial straight lines in later stages. Sensitivity analysis of well spacing shows that "transition flow" will change from elliptical to formation linear flow between wells as well spacing increases and it can show the transitional flow characteristics in more common cases.

Frequency/Laplace Domain Methods for Computing Transient Responses of Fractional Oscillators

Although computing the transient response of fractional oscillators, characterized by second-order differential equations with fractional derivatives for the damping term, to external loadings has been studied, most existing methodologies have dealt with cases with either restricted fractional orders or simple external loadings. In this paper, considering complicated irregular loadings acting on oscillators with any fractional order between 0 and 1, efficient frequency/Laplace domain methods for getting transient responses are developed. The proposed methods are based on pole-residue operations. In the frequency domain approach, "artificial" poles located along the imaginary axis of complex plane are designated. In the Laplace domain approach, the "true" poles are extracted through two phases: (1) a discrete impulse response function (IRF) is produced by taking the inverse Fourier transform of the corresponding frequency response function (FRF) that is readily obtained from the exact TF, and (2) a complex exponential signal decomposition method, i.e., the Prony-SS method, is invoked to extract the poles and residues. Once the poles/residues of the system are known, those of the response can be determined by simple pole-residue operations. Sequentially, the response time history is readily obtained. Two fractional oscillators with rational and irrational derivatives, respectively, subjected to sinusoidal and complicated earthquake loading are presented to illustrate the procedure and verify the correctness of the proposed method. The verification is conducted by comparing the results from both the Laplace and the frequency domain approaches with those from the numerical Duhamel integral method.

Analysis of One-Dimensional Consolidation for Unsaturated Soils under Piecewise Cyclic Loading

This paper studies the one-dimensional (1D) consolidation behavior for unsaturated stratum subjected to piecewise cyclic loading. Combined with the widely accepted consolidation theory of unsaturated soils, a semianalytical method was employed to investigate the consolidation of unsaturated foundation considering piecewise cyclic loading in the Laplace domain. Furthermore, the reduced solutions were produced to perform the verification work accompanied by the results in the existing literature. Finally, a case study was conducted to explore the consolidation characteristics under piecewise cyclic loading (i.e., triangular and trapezoidal cyclic loadings). Parametric studies were carried out by variations of excess pore pressures and settlement against the ratio of air-water permeability coefficients, depth, and loading parameters. The research proposed in this paper can provide theoretical basis for the ground treatment of unsaturated soils, especially for rationally accelerating consolidation or avoiding sudden settlement.

Time-Dependent Wave-Structure Interaction Revisited: Thermo-Piezoelectric Scatterers

In this paper, we are concerned with a time-dependent transmission problem for a thermo-piezoelectric elastic body that is immersed in a compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided that an appropriate scaling factor is employed. As usual, based on estimates for solutions in the Laplace-transformed domain, we may obtain properties of corresponding solutions in the time-domain without having to perform the inversion of the Laplace-domain solutions.