Formation permeability of hydraulically fractured reservoirs is pressure sensitive and spatially variant. Earlier analytical models describing pressure behavior of those reservoirs either account for spatial dependence or stress-dependence effect on permeability field. In this study, a novel analytical approach is presented to account for the combined effect of stress-sensitivity and spatial variability of formation permeability.
The model considers that the reservoir fluid flows linearly in the stimulated reservoir volume (SRV) and discharges to the infinitely conductive hydraulic fractures. A newly derived diffusivity equation for single porosity continuum is employed to seek solution for the constant terminal rate condition. As the hydraulic permeability of SRV is a function of space and pressure, the resulting diffusivity equation is highly nonlinear. After weakening the nonlinearity by applying Pedrosa's transformation, features of the perturbation technique and modified Bessel functions are implemented to determine the analytical solution in Laplace space. Finally, the real-time solution is obtained by inverting through Stehfest and GWR method.
The model captures the combined effect by incorporating two characteristic parameters: permeability modulus and threshold permeability. These parameters were turned off to compare and validate the obtained solutions with the corresponding conventional analytical solutions. It was found that the earlier models may underestimate the pressure behavior significantly throughout the reservoir lifetime if the temporal and spatial variability of permeability is disregarded. Moreover, it was observed that the permeability modulus has a comparatively more significant effect on pressure drawdown behavior than that of the stimulation ratio at the late times. Pressure propagation profiles for different sets of model parameters were generated. The effect of dimensionless permeability modulus and threshold permeability were analyzed to draw a contrast among constant, spatially varying and combined-effect case. Error propagation among those three conceptualized cases were studied.
The proposed analytical approach describes the dynamic behavior of SRV continuum more realistically by taking the spatial and pressure-dependent effect of permeability into account. Compared to the available analytical models, this approach can be employed to glean more accurate attributes of the hydraulically fractured horizontal well. The proposed concept could further be extended to develop solutions for the dual-porosity continuum.
Onur, Mustafa (The University of Tulsa) | Galvao, Mauricio (Petrobras) | Bircan, Davut Erdem (The University of Tulsa) | Carvalho, Marcio (Pontifical Catholic University of Rio de Janeiro) | Barreto, Abelardo (Pontifical Catholic University of Rio de Janeiro)
The objectives of this study are to (i) provide analytical transient coupled wellbore/reservoir model to interpret/analyze transient temperature drawdown/buildup data acquired at both the producing horizon (sandface) and a gauge depth above the producing horizon (wellbore) and (ii) delineate the information content of both transient sandface and wellbore temperature measurements. The analytical models consider flow of a slightly compressible, single-phase fluid in a homogeneous infinite-acting reservoir system and provide temperature-transient data for drawdown and buildup tests produced at constant rate at any gauge location along the wellbore including the sandface. The production in the wellbore is assumed to be from inside the production casing. The models account for Joule-Thomson (J-T), adiabatic fluidexpansion, conduction and convection effects as well as nearby wellbore damage effects. The well/reservoir system considered is a fully penetrating vertical well in a two-zone radial composite reservoir system. The inner zone may represent a damaged (skin) zone, and the outer (non-skin) zone represents an infinitely extended reservoir. The analytical solutions for the sandface transient temperatures are obtained by solving the decoupled isothermal (pressure) diffusivity and temperature differential equations for the inner and outer zones with the Boltzmann transformation, and the coupled wellbore differential equation is solved by Laplace transformation. The developed solution compares well with the results of a rigorous thermal numerical simulator and determines the information content of the sandface and wellbore temperature data including skin zone effects. The analytical models can be used as forward models for estimating the parameters of interest by nonlinear regression built on any gradient-based estimation method such as the maximum likelihood estimation (MLE).
There are many advantages of developing transient flow solutions in the Laplace transform domain. For example, in the Laplace transform domain, Duhamel's theorem provides a convenient means of developing transient flow solutions for variable rate production problems using the solutions for the corresponding constant rate production problem. Applying the Laplace transform converts the convolution integral in Eq. 1 to an algebraic expression, and Duhamel's theorem is given in the Laplace transform domain as The simplicity of the expression given in Eq. 2 explains our interest in obtaining transient-flow solutions in the Laplace transform domain. Another example to explain the convenience of the Laplace domain solutions is for the naturally fractured reservoirs. The general solutions for Eqs. 3 and 4 are given, respectively, by This discussion demonstrates that it is possible to derive transient flow solutions for naturally fractured reservoirs by following the same lines as those for the homogeneous reservoirs. Furthermore, if the solution for the corresponding homogeneous reservoir system is known in the Laplace transform domain, then the solution for the naturally fractured reservoir problem may be directly obtained from Eq. 9. Obtaining the Laplace transforms of the Green's and source function solutions developed in the time domain with the methods explained on the Source function solutions of the diffusion equation and Solving unsteady flow problems with Green's and source functions pages usually poses a difficult problem.
Integral transforms are useful in solving differential equations. A special form of the linear integral transforms, known as the Laplace transformation, is particularly useful in the solution of the diffusion equation in transient flow. The following fundamental properties of the Laplace transformation are useful in the solution of common transient flow problems. For the Laplace transform to be useful, the inverse Laplace transformation must be uniquely defined. In this operation, p(t) represents the inverse (transform) of the Laplace domain function, .
Rate and pressure transient analysis is considered a routine process that has been developed and refined over many years. The underlying assumptions of linearity justify the use of superposition (in time and space), convolution and deconvolution. The reality of non-linearities are handled on a case by case basis depending on their source (fluid, well or reservoir). Shale gas wells are subject to significant non-linearity over their producing life.
We review some of the fundamental equations that govern pressure and rate transient behavior, introduce several new techniques which are suited to the analysis of data from producing wells and apply them to a synthetic example of a shale gas well.
First, we use simple calculus to show how the convolution integral is derived from standard multi-rate superposition. Then, from the convolution integral, we derive an equation that describes the pressure response due to a step-ramp rate (i.e. an instantaneous rate change from initial conditions followed by a linear variation in rate). It results in a combination of the pressure change due to a constant rate and it's integral. Applying superposition to this equation allows any rate variation to be approximated by a sequence of ramps with far fewer points than those required to achieve the same level of accuracy using standard constant step rate superposition.
Second, we re-write multi-rate superposition functions allowing for stepwise linear variable rate which, when applied to flowing data and used to calculate the pressure derivative, can result in a much smoother response and hence an overall improvement in the analysis of rate and pressure transients recorded from producing wells.
Third, we review the use of the Laplace transform and how it can be applied to discrete data with a view to deconvolving rate transient data.
Finally, we demonstrate how data de-trending can remove the impact of long term non-linearities and apply the methods mentioned above to a synthetic dataset based on a typical shale gas well production profile.
We illustrate the advantages of the newly introduced superposition functions compared to conventional analysis methods when applied to the pressure transients of wells flowing at variable rate.
As an example, we have simulated the production of two shale gas wells over twenty years. Both have the same production profile, but one includes pressure dependent permeability. At various intervals during the life of the well, we introduce a relatively short well test which imposes a small variation in rate but does not include a shut-in. We de-trend the rate transients and then apply the techniques described above to analyse the resulting data. The interpretation allows us to identify non-linearities that may be influencing well productivity over time and to obtain a better understanding of the physics of shale gas production.
The mathematics documented in the paper provides a useful overview of how convolution, superposition, deconvolution and Laplace transforms provide the means to analyse pressure and rate transients for linear systems.
Data de-trending removes the impact of long term non-linearities on shorter transient test periods.
We develop and demonstrate some new and improved techniques for rate and pressure transient analysis, and we illustrate how these can provide insight into the non-linearities affecting shale gas production.
This paper suggests the simplified correlation ratios, that are an approximate analytical representation of exact numerical solution, for evaluation of fracturing characteristics of the fractured reservoirs with the use of decline curves. Formalization and automation methods are suggested for finding fractured reservoir approximate parameters and range of uncertainty under the Warren-Root model.
Pressure-transient models are presented for evaluation of the behavior of vertical, vertically fractured, and horizontal wells in radial and linear (three-region) composite reservoirs with moving fluid fronts. The Laplace Transform Finite Difference (LTFD) numerical solution methodology combined with the well- known Buckley-Leverett (BL) frontal-advance equation have been used to develop solutions for the moving boundary problem. Hybrid semi-analytic and numerical solutions have been constructed for finite-conductivity vertical fractures and infinite-conductivity horizontal wellbores. Complete descriptions of the mathematical models are presented; the pressure-transient solutions, frontal position and velocity, and saturation distributions. Indications are that monitoring of the transient behavior permits detection of water encroachment to the producing well, prior to breakthrough. This enables modifications in the production operations to be taken proactively to delay breakthrough.
The results of the pressure-transient models reported in this paper have been compared with the available moving boundary radial composite solutions in the literature and the results of the vertically fractured and horizontal well solutions have been validated using analytic and numerical reservoir simulation. Six well and reservoir model combinations have been considered in this investigation for which oilfield applications exist for each composite system considered. These include solutions of the pressure-transient behavior of an unfractured vertical well in a radial composite reservoir, a vertically fractured well in linear and radial composite systems, a horizontal well in radial and linear composite systems, and a vertical fracture intersected by a horizontal well in a linear composite system. Extension of the solution methodology used in this study for evaluating the pressure-transient behavior of a selectively completed horizontal wellbore in a cylindrical composite reservoir has also been considered in this study. Each of these solutions include moving fluid fronts, whose position and velocity are determined from the frontal advance model and fractional flow theory. General fractional flow solutions have been implemented that utilize conventional laboratory relative permeability measurements.
Kwon, Jungmin (Seoul National University) | Song, Hyeonjun (Seoul National University) | Shin, Changsoo (Seoul National University) | Jang, U Geun (Korea Polar Research Institute) | Jun, Hyunggu (Korea Institute of Ocean Science and Technology) | Whang, Hyunseok (Seoul National University)
Laplace domain FWI has advantages due to its insensitivity of the bandwidth with respect to the source wavelet. However, Laplace domain FWI has an inaccuracy problem from its unwanted cross-correlation terms between residuals of first arrival travel times and apparent amplitudes. In this paper, using a new form of objective function (shifted Laplace domain wave field), we obtained improved data resulting from suppressing unwanted cross-correlation terms, crucial factors that make inaccuracy of an inversion result, of residual travel times and apparent amplitudes. To verify our theory, we implemented this theory to a BP 2004 benchmark model and obtained an improved result compared to a method of conventional Laplace domain FWI.
Presentation Date: Tuesday, September 26, 2017
Start Time: 1:50 PM
Location: Exhibit Hall C, E-P Station 3
Presentation Type: EPOSTER
In recent years, ‘fracture hits’ are frequently observed for multi-fractured horizontal wells drilled from pads and completed in tight/shale reservoirs. These occurrences have increased with increased well density. One of the suggested common causes is communication of multiple wells (at least two wells) through primary and secondary fractures created during hydraulic fracture stimulation treatment. The flow signatures of communicating wells differ substantially from that of a single isolated well. The main purpose of this paper is therefore to quantify flow characteristics of multiple pad wells completed in a tight oil reservoir and communicating through primary and secondary fractures using a Laplace domain hybrid solution.
In the hybrid model, fluid flow is assumed to occur as a combination of fracture flow and matrix flow. The Laplace-transform finite-difference (LTFD) method is used to numerically model fracture flow, with sufficient flexibility to consider arbitrary fracture geometries and fracture conductivity distributions. The analytical matrix flow model, derived using the line-source function in the Laplace domain, is dynamically coupled with the fracture flow model, by imposing the continuity of pressure and flux on the fracture surface. The main advantage of the solution occurring in the Laplace domain is that computations can be performed at predetermined, discrete times, and with grids only for fractures. Thus, stability and convergence problems caused by time discretization are avoided and the burden of gridding and computation is decreased without loss of important fracture characteristics. A non-uniform distribution of initial pressure in fractures, and variable BHPs of wells, are considered in an updated version of the hybrid model presented in this work to consider practical cases. The model is validated through comparison with a fully numerical simulator.
Forecast sensitivities performed using the new hybrid model suggest that pressure interference caused by communication between wells significantly affects the flow signature of each well. The non-uniform distribution of initial pressure in fractures mainly influences the early-time flow behavior. With continued production, the matrix will dominate the system; thus, the production profiles of non-uniform pressure case merge with those of the uniform pressure case. For case of variable well BHPs, well interference effects become weaker with a decrease in BHP differences between wells.
An anomalous flowrate feature (often a "hump" or even a "spike") is characteristically observed at early-times during flowback performance in multi-fractured horizontal wells (MFHW) completed in ultra-low permeability (shale) reservoirs prior to the onset of a characteristic reservoir flow regime (i.e., linear or bilinear flow). The flowrate feature tends to occur in all fluid phases and this feature is thought to be attributed to the "clean-up" behavior following well stimulation and/or the phase behavior of the fluid as it flows along the well path.
The guiding principle of this work is that this anomalous flowrate feature can be represented by decaying skin effects, a changing wellbore storage effect, or a combination of both decaying skin effects and changing wellbore storage effects. The goal of this work is to provide a proof-of-concept which considers the simplified case of a vertical well with a single vertical fracture to develop a series of time-dependent skin and wellbore storage models that can effectively be used to characterize the early-time flowrate behavior observed in practice. For this study, we forced a constant wellbore flowing pressure constraint, and while we recognize that this constraint is not truly met in practice, we believe that this approach can serve as a base model for diagnostics/interpretative analyses.
Based on the work developed by Fair (1981) and Larsen and Kviljo (1990), our procedure is to couple a series of time-dependent wellbore storage and skin effect models with a set of "power law" reservoir flow models (i.e., linear flow, bilinear flow and a generalized power-law flow model). Specifically, we combine the time-dependent wellbore storage and skin effect models with the constant rate solution reservoir flow models, then apply the convolution integral to produce the constant pressure condition — all in the Laplace domain. In order to generate various scenarios of production performance, we use the Gaver-Wynn-Rho Algorithm implemented in Mathematica to numerically invert the Laplace domain solutions into the real time domain. A generalized workflow is provided to demonstrate the addition of time-dependent wellbore storage and skin effects to any prescribed reservoir model.
Using the various wellbore storage and skin time-dependent models proposed in this work, we observe that each of these models, individually and in combination, provide behavior indicative of early-time flowrates observed in the field. In short, we demonstrate that each time-dependent model has unique characteristics, which could, in concept, allow for characterization of flow behavior in the fracture prior to the onset of an undistorted "reservoir" flow geometry (i.e., formation linear or bilinear flow).