Analysis of the Transient Production of a Thermally Stimulated Well
- Susana M. Bidner (U. of Buenos Aires) | Moraclo M. Kostiria (U. Nacional de la Plata Argentina) | Victoria C. Vampa (U. Nacional de la Plata Argentina)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1990
- Document Type
- Journal Paper
- 539 - 543
- 1990. Society of Petroleum Engineers
- 5.8.5 Oil Sand, Oil Shale, Bitumen, 5.4.10 Microbial Methods, 4.1.2 Separation and Treating, 5.2.2 Fluid Modeling, Equations of State, 5.2 Reservoir Fluid Dynamics, 2.4.3 Sand/Solids Control, 5.3.4 Reduction of Residual Oil Saturation, 5.2.1 Phase Behavior and PVT Measurements, 5.4.6 Thermal Methods, 5.3.4 Integration of geomechanics in models, 4.1.5 Processing Equipment, 5.4.2 Gas Injection Methods
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Summary. This paper presents a simple mathematical model to predict the oil production rate from a cyclic-steam-stimulated well. The oil volume flowing from the cold zone is estimated by numerically solving the diffusivity equation governing the radial transient flow of a fluid with a small and constant compressibility for three outer boundary conditions-infinite medium, constant pressure, and no flow-corresponding to different reservoir types. At the interface between the cold and hot zones, pressure is calculated by means of the superposition principle. Numerical solutions give pressure distribution as a function of radius and time. They also allow assessment of the effects of transient flow and outer boundary conditions on oil production rate. The numerical response of a cycle of steam injection strongly depends on the outer boundary condition. Excellent agreement is found between model-calculated and observed field rates.
Steam stimulation aids oil production through several mechanisms: (l) oil-viscosity reduction, (2) thermal expansion of reservoir fluids, (3) reduction in residual oil saturation, (4) formation compaction, (5) steam distillation effects, and (6) reduction of interfacial tension and capillary forces. The injected steam also imparts exclusive energy through compression of formation fluids.
To account for these physical phenomena, a numerical, 3D, multiphase compositional simulator is required. Different physical and physical chemical parameters must be defined and measured independently or assessed according to the production history.
To provide a first assessment of the performance of a thermally stimulated well, models have been developed that simplify the physical reality to some extent to obtain analytical solutions. Such solutions are useful to predict the oil recovery by steam stimulation and to study the effects of physical parameters. Probably the most widely applied analytical model is that of Boberg and Lantz.
Boberg and Lantz considered that the success of steam soak depends essentially on reducing oil viscosity around the wellbore by heating. They assumed that the temperature distribution during injection can be described by a step function so that for rw is less than rh the temperature is uniform and equal to the condensing steam temperature at the sandface pressure, where for r is greater than rh the original reservoir temperature remains unchanged. The average temperature in the oil sand for the region where r is less than rh is computed as a function of time by an energy balance after the end of steam injection. The oil viscosity in the region is determined from this average temperature. Finally, the oil production rate is estimated by a steady- or pseudosteady-state, radial-flow approximation that takes into account the reduced oil viscosity at r is less than h.
The main drawback of Boberg and Lantz's estimations is the assumption that steady-state conditions exist from the start of production. At an early stage, transient flow prevails. Therefore, style state models predict pessimistic oil production at early time. With this in mind, Bentsen and Donohue estimated the transient-flow oil rates, assuming that the outer boundary conditions are those for an infinite reservoir. They also assumed that transient flow occurs until the rate predicted by the steady-state model is greater than that predicted by the transient model. After this time, the flow is assumed to be steady state. This assumption arises from the fact that the infinite boundary condition is realistic immediately upon opening the well on production. van Everdingen and Hurst provided the solution for an infinite medium as a set of tables.
On the otherhand, Bentsen and Donohue follow the Boberg and Lantz model in the treatment of the energy balances. Dake describes Bentsen and Donohue's model in more detail.
A modification of Bentsen and Donohue's model is described that consists of numerically solving the diffusivity equation for transient radial flow for three outer boundary conditions. it has the following advantages: (1) arbitrary definitions dividing transient and steady-state flow are not necessary, (2) van Everdingen and Hurst's tables can be avoided, and (3) the pressure distribution as a function of radius and time can be found. The method's main application is the calculation of oil production for different flowing well conditions (i.e., inner boundary conditions) in terms of bottomhole flowing pressure (BHFP) or flow rate as functions of time and for different boundary conditions at the outer drainage radius, (i.e., constant pressure, no flow, and infinite medium.)
This mathematical model is summarized below.
1. Saturated steam is injected into the wellbore. The stun quality and heat loss in the well are estimated according to Kostiria and Bidner's approach.
2. The reservoir is divided into two concentric cylinders around the wellbore. The inner cylinder, which extends from the wellbore radius to the heated radius, is called the hot zone. The outer cylinder, from the heated radius to the drainage radius, is called the cold zone.
3. During the injection period, the average temperature in the hot zone can be determined with Marx and Langennheim's equation.
4. During the soak period, the average temperature in the hot zone is estimated by solving the differential equation of 2D transient heat conduction, which accounts for heat losses to cap and base rock.
5. During the production period, a correction term that accounts for the energy removed from the oil sand by the produced oil, gas, and water is added to the equations used in Step 4. Boberg and Lantz proposed the solution of this 2D transient convection/ conduction equation. It estimates the average temperature, which in turn determines the oil viscosity in the hot zone, a time-dependent value. Steps 3 through 5 involve energy balances during injection, soak, and production.
6. The flow toward the well is radial and transient during pro duction. Row in the hot and cold zones is estimated separately, although both become equal at the interface.
7. The cold-zone flow is governed by the transient diffusivity equation obtained by combining the continuity equation with Darcy's law and the equations of state for small- and constant-compressibility fluids. The equation is solve numerically for three different conditions at the outer boundary: constant pressure, no flow, and infinite medium (Appendix A). Constant pressure is the inner boundary condition for the diffusivity equation. The solution of the transient diffusivity equation gives pressure distribution as a function of time and radius. Therefore, oil production rate and cumulative oil production can be determined inside the cold zone, and in particular, at the interface between the cold and the hot zone.
8. At the coldhot interface, the variation of pressure with time is simulated by applying the superposition principle. For each timestep; pressure at the interface is unknown. it is determined by equating the volume of oil produced at the well with the volume of oil flowing from the cold zone.
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