Method for dynamic measurement of thermal diffusivity during laser soldering of biological tissues

In this paper, we propose a method for dynamic measurement of the thermal diffusivity coefficient during laser soldering of biological tissues. The method is based on modelling the function of temperature dependence on time during cooling of biological tissue after exposure to laser radiation. The simulation is carried out by solving the heat equation for a homogeneous biological tissue and the absence of external heat sources. The desired value of the thermal diffusivity coefficient was determined by optimizing the residual functional of the temperature functions from time obtained experimentally and by solving the thermal diffusivity equation. Experiments were carried out to measure the thermal diffusivity coefficient by the proposed method for myocardial and skin tissues at maximum heating temperatures of 40, 50, 60 °C. The measured values of the thermal diffusivity coefficient for the myocardium are in the range from 2.3 to 2.7 m2/s*10-6, and for the skin from 1.5 to 1.7 m2/s*10-6.

Pressures and Pressure Derivatives of a Vertical Well Located Within Two Inclined Faults: Case Study of Basic Angles and Unequal Well Distances from Faults

Pressure transient analysis has been used to evaluate performance of a vertical well located within two intersecting sealing faults. The nature and types of boundary affect productivity in bounded reservoirs. Well performance is strongly affected by well location with respect to the boundary, be it single, paired and parallel or paired and inclined. The goal of this research was to study pressure behavior as well as performance of a vertical well located within two intersecting sealing faults inclined at various angles θ and at unequal distances to faults. Unlike similar works previously carried out, this work can be used to study or predict pressure distribution of a well in a wedge system located at unequal distances to faults. Using the concept of images, the study proposed new models for estimating distances between image well(s) and active well. These models were applied in the solution to the dimensionless diffusivity equation to characterize pressure transient behavior of a well located at unequal distances to the inclined faults. These pressures and pressure derivatives were computed from the total pressure drop expression summing all the image wells by the principle of superposition. The MATLAB, Python and Excel software were deployed to compute all the dimensionless pressures for the different well designs. The results obtained show that 1) the proposed models give accurate estimation of active well distances to image wells; 2) the models show that the distance between the active and image wells d0,i increases for the range of values of angles 0°< θ0,i ≤ 180° and decreases for the range 180° < θ0,i < 360°; 3) the relationship between unequal well distances and productivity has a maximum point; 4) beyond this point, the well ceases to be productive and; 5) this maximum point is at equal distances of the well from both faults, in this case, 15 ft. Larger magnitudes of dimensionless pressure derivatives would indicate higher oil production for any well design and inclination of the boundaries. Worthy of future works are similar studies on 1) horizontal wells and 2) mixed boundaries, that is, one sealing fault and one constant pressure boundary.

Semi-Analytical Solution to Assess CO2 Leakage in the Subsurface through Abandoned Wells

Geological carbon storage is an effective method capable of reducing carbon dioxide (CO2) emissions at significant scales. Subsurface reservoirs with sealing caprocks can provide long-term containment for the injected fluid. Nevertheless, CO2 leakage is a major concern. The presence of abandoned wells penetrating the reservoir caprock may cause leakage flow-paths for CO2 to the overburden. Assessment of time-varying leaky wells is a need. In this paper, we propose a new semi-analytical approach based on pressure-transient analysis to model the behavior of CO2 leakage and corresponding pressure distribution within the storage site and the overburden. Current methods assume instantaneous leakage of CO2 occurring with injection, which is not realistic. In this work, we employ the superposition in time and space to solve the diffusivity equation in 2D radial flow to approximate the transient pressure in the reservoirs. Fluid and rock compressibilities are taken into consideration, which allow calculating the breakthrough time and the leakage rate of CO2 to the overburden accurately. We use numerical simulations to verify the proposed time-dependent semi-analytical solution. The results show good agreement in both pressure and leakage rates. Sensitivity analysis is then conducted to assess different CO2 leakage scenarios to the overburden. The developed semi-analytical solution provides a new simple and practical approach to assess the potential of CO2 leakage outside the storage site. This approach is an alternative to numerical methods when detailed simulations are not feasible. Furthermore, the proposed solution can also be used to verify numerical codes, which often exhibit numerical artifacts.

A Complement to Decline Curve Analysis

Summary Decline curve analysis (DCA) has been the mainstay in unconventional reservoir evaluation. Because of the extremely low matrix permeability, each well is evaluated economically for ultimate recovery as if it were its own reservoir. Classification and normalization of well potential is difficult because of ever-changing stimulation total contact area and a hyperbolic curve fit parameter that is disconnected from any traditional reservoir characterization descriptor. A new discrete fracture model approach allows direct modeling of inflow performance in terms of fracture geometry, drainage volume shape, and matrix permeability. Running such a model with variable geometrical input to match the data in lieu of standard regression techniques allows extraction of a meaningful parameter set for reservoir characterization, an expected outcome from all conventional well testing. Because the entirety of unconventional well operation is in transient mode, the discrete fractured well solution to the diffusivity equation is used to model temporal well performance. The analytical solution to the diffusivity equation for a line source or a 2D fracture operating under constrained bottomhole pressure consists of a sum of terms, each with exponential damping with time. Each of these terms has a relationship with the constant rate, semisteady-state solution for inflow, although the well is not operated with constant rate, nor will this flow regime ever be realized. The new model is compared with known literature models, and sensitivity analyses are presented for variable geometry to illustrate the depiction of different time regimes naturally falling out of the unified diffusivity equation solution for discrete fractures. We demonstrate that apparent hyperbolic character transitioning to exponential decline can be modeled directly with this new methodology without the need to define any crossover point. The mathematical solution to the physical problem captures the rate transient functionality and any and all transitions. Each exponential term in the model is related to the various possible interferences that may develop, each occurring at a different time, thus yielding geometrical information about the drainage pattern or development of fracture interference within the context of ultralow matrix permeability. Previous results analyzed by traditional DCA can be reinterpreted with this model to yield an alternate set of descriptors. The approach can be used to characterize the efficacy of evolving stimulation practices in terms of geometry within the same field and thus contribute to the current type curve analyses subject to binning. It enables the possibility of intermixing of vertical and horizontal well performance information as simply gathering systems of different geometry operating in the same reservoir. The new method will assist in reservoir characterization and evaluation of evolving stimulation technologies in the same field and allow classification of new type curves.

Pressure behaviour of a horizontal well sandwiched between two parallel sealing faults

Generally, reservoir fluid flow is governed by diffusivity equation and solution to this equation helps to investigate pressure behaviour under certain reservoir and wellbore boundary conditions. In this paper however, the analytical solution method of Green and Source function is deployed to determine the performance of a horizontal well located between two parallel sealing faults, assuming simple rectangular reservoir geometry. Also, the dimensionless pressure and derivative approach is applied for all computations as it prevents the problem of unit conversions, reduces longer expressions and it helps to handle numerical values. The pressure expression derived from this work reveals that a maximum of two flow periods occur for the stated reservoir model. It was found out that an inverse relationship exists between dimensionless pressure and dimensionless length while pressure increased with thickness. Also high vertical permeability shortens the effect of the early radial flow period experienced by the horizontal well, thereby increasing productivity index. Finally, it was discovered that increased perforation length reduces the production potential of the horizontal well. Keywords: Dimensionless pressure, pressure derivatives, heterogeneity, pressure performance, reservoir and wellbore characterization.

Integration of Pressure Transient Data into Reservoir Models Using the Fast Marching Method

Summary Calibration of reservoir model properties by integration of well-test data remains an important research topic. Well-test data have been recognized as an effective tool to describe transient flow behavior in petroleum reservoirs. It is also closely related to the drainage volume of the well and the pressure-front propagation in the subsurface. Traditional analytic means of estimating reservoir permeability relies on an interpretation of the diagnostic plot of the well pressure and production data, which usually leads to a bulk average estimation of the reservoir permeability. When more detailed characterization is needed, a forward model that is sensitive to the reservoir heterogeneity needs to be established, and a numerical inversion technique is required. We use the concept of the diffusive time of flight (DTOF) to formulate an asymptotic solution of the diffusivity equation that describes transient flow behavior in heterogeneous petroleum reservoirs. The DTOF is obtained from the solution of the Eikonal equation using the fast marching method (FMM). It can be used as a spatial coordinate that reduces the 3D diffusivity equation to an equivalent 1D formulation. We investigate the drainage-volume evolution as a function of time in terms of the DTOF. The drainage volume might be directly related to the well-test derivative, which can be used in an inversion calculation to calibrate reservoir model parameters. The analytic sensitivity coefficients of the well-test derivative with respect to reservoir permeability are derived and incorporated into an objective function to perform model calibration. The key to formulating the sensitivity coefficients is to use the functional derivative of the Eikonal equation to derive the analytic sensitivity of the DTOF to reservoir permeability. Its solution is implemented by tracking the characteristic trajectory of the local Eikonal solver within the FMM. The major advantage of calculating the sensitivity coefficients using the FMM is its significant computational efficiency during the iterative inversion process. This inverse-modeling approach is tested on a 2D synthetic heterogeneous reservoir model and then applied to the 3D Brugge Field, where a single well with constant flow rate is simulated. The well-test derivative is shown to be inversely proportional to the drainage volume and is treated as the objective function for inversion. With an additional constraint to honor the prior model, our inverse-modeling approach will adjust the reservoir model to obtain permeability as a function of distance from the well within the drainage volume. It provides a modification of reservoir permeability both within and beyond the depth of investigation (DOI).