B aquifer constant, bbl/psi Cr rock compressibility, psi – 1Cw water compressibility, psi -ICwr effectivecompressibility of water and rock in aquiferf fraction of perimeter of circle that originaloil/water boundary constitutesrO radiusto perimeter of reservoir, ft t = time, daystD = dimensionless timetj = cumulative elapsed time at end of the interval, daysWe = cumulative water influx, bbl =aquifer porosity?p= pressure drop at OWCre = radius to perimeter of aquiferPD = dimensionless pressurePD’= first derivative ofdimensionless pressuretj=cumulative elapsed time at end of jth interval, days Water InfluxModelling using The van Everdingen-Hurst unsteady-state Model In 1949, one of the most significant solution for the waterinflux problem was established by van Everdingen and Hurst. Klinset al, (1988), they have developed mathematically the solutions to the radialdiffusivity equation for radial, unsteady state and single phase flow withrespect to the pressure dispersed in water aquifer. In addition, Utilizing van Everdingen-Hurstsolution to the diffusivity equation, as mentioned below, this is in case ofwater encroachment into the reservoir for radial aquifers. We(tDj)=BqD (tDj- tDk) Where: B=1.119hCwr ro2f?Pk= tDJ= The equations for hydrocarbon flowsystem into a wellbore which had expressed mathematically is the same asexpressed for those equations to define the flow of water from an aquifer intoa cylindrical reservoir. For instance, if a specific well isproducing at a constant flow rate (q) after a shut-in period, the pressurebehavior is primarily controlled by the unsteady state flowing behavior. Thisflowing behavior expressed as the time period during which the boundary has noeffect on the pressure behavior. Van Everdingen and Hurst’s constant-terminal pressuresolution, that has observed a significant value problems related to the water-encroachment.
If some average pressure is specified at theinterface over a given time, flow rate and hence water influx into thereservoir can be estimated. However, in case if pressure continues to drop atthe oil/water contact (OWC) over time, a number of constant-pressure steps canreplace this declining pressure and superposition can be used. In addition, the diffusivityequation which is considered as the dimensionless form is essentially thegeneral equation that is designed to model the unsteady state flow behavior inreservoirs. Thevan Everdingen-Hurst style and Carter-Tracy alteration gives the rigoroussolutions to the radiaI-diffusivity equation. In the other hand, the applicationof these solutions depends on the correct values of either dimensionlesspressure function which is the (PD) or the dimensionless rate influencefunction which is mentioned as (qD) Anyway, those Values of (PD) and (qD) are in general presentsand derived from tables provided in the official results and study of vanEverdingen and Hurst.
· Fetcovish,M. (1971). A simplified approach to water influx calculations. Journal of Petroleum Technology. · Klins, M.
and Bouchard, A. (1988). A Polynomial Approach to the van Everdingen-HurstDimensionless Variables for Water Encroachment. Society of Petroleum Engineering.Figure 1. Water influx into acylindrical reservoir.
However, in case of the constant terminalrate boundary, the rate of water encroachment is considered as a constant for thatgiven period and the pressure drop is calculated at the reservoir-aquiferboundary and then the water influx rate is determined. In the expression and explanationof the water encroachment from the water aquifer to the reservoir, there is significantchance that we can determine and calculate the encroachment rate rather thanthe pressure. Hurst and later Carter tried toutilize vanEverdingen Hurst constant-terminal-rate solution to enhance a new method and developit so that to approach to analyze water encroachment to the reservoir thateliminated the superposition. Eventually which is estimated by this equation below: In Addition, A whole and exact set toexchange the van Everdingen and Hurst tables adequately for both the terminal-pressureand terminal-rate, in result the radial flowcases finite and infinite aquifers is then applied. In case of the infinite aquifers, the value of (qD)as a function of dimensionless time is determined and calculated by theintegral which is mentioned below: However,in case of finite aquifers acting infinitely.
It is obvious that all aquifers willpresent as like they are infinite for tiny values of dimensionless time. Then forthe next time and during the times, the boundary affects will be observedgradually in time and finite aquifer actions strays consequently after that. The benefitof this section is to estimate for a given aquifer ratio the time at which the boundaryaffects are observed and known. Also oncethis crossover value of tD is predicted and known then after that the engineerthen can decide whether the case will be finite or infinite. Conclusion: Thesesimple equations mentioned values of PD and qD as exact as theoriginal van Everdingen and Hurst tables. However, for water encroachment process,these equations means and will represent the controllable replacement totabular guides for the van Everdingen and Hurst dimensionless functions.Anyway, the van Everdingen-Hurst style and Carter-Tracy alteration gives the rigoroussolutions to the radiaI-diffusivity equation.
In the other hand, the applicationof these solutions depends on the correct values of either dimensionlesspressure function which is the (PD) or the dimensionless rate influencefunction which is mentioned as (qD).