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for planning production work on geologists of the hydrodynamic model of the formation, mathematical experiments are effectively, which give good results

filtering, flyuid, configuration, unstuble regim, roof, sole, injection well

**УДК 1**

**Geldimyradov Arslan Geldimyradovich**

Doctor of Technical Science, assistant professor of Yagshygeldi Kakayev International University of Oil and Gas,

(Ashgabat, Turkmenistan)

* *

**DETERMINATION OF RESERVOIR CHARACTERISTICS BASED **

**ON STUDY OF PRESSURE CURVE UNDER THE CONDITIONS **

**OF TURKMENISTAN GAS WELLS**

*Abstract: **for planning production work on geologists of the hydrodynamic model of the formation, mathematical experiments are effectively, which give good results.*

* *

*Keywords:** filtering, flyuid, configuration, unstuble regim, roof, sole, injection well.*

* *

The filtration theory places emphasis on two main problem types - direct and inverse. Direct problems are widely applied for forecasting of the dynamics of operation parameters and field development, inclusive of known characteristics of a reservoir and properties of fluids therein. Inverse problems play an essential role, and are related to identification and solving of research problems of porosity & permeability properties of a reservoir based on specific analysis data or operation of one certain well, production well, observation well or group of injection wells. The first and the simplest contradictions ever are often attributed to interpretation of the results of sample analysis. Absolute permeability coefficient, anisotropy of permeability and reservoir thickness, dependence of correspondence permeability on saturation of single phases are determined following the extrusion of fluids inside samples through pores.

It is clear that specific data on operation of oil and gas fields contain information about porosity & permeability properties of a reservoir and dead-water surrounding a reservoir. That is why it is necessary to study interpretation of specific data on operation of wells for the purpose of “exact” determination of intended parameters of a reservoir.

We are aware that all inverse problems are improper problems. For example, small change of input parameters (dimensions) may result in large change in a solution (assessed reservoir properties). Effective and robust algorithms of solving detection problems shall satisfy the following conditions:

- adaptation processes shall be optimized for optimum solutions even when using the most unfavourable initial approximations;

- rule of determination of reservoir parameters shall be applied in case of rapid accumulation at iteration, as well as at minimal computing costs for finding a solution;

It is necessary to solve a problem of determination of the minimal quantitative rule and prevent a solution, which has no physical meaning, as well as decrease its inaccuracy.

Accuracy of problem detection depends on accuracy of measured indicators of operation of production and injection wells. Robust algorithms of a solution of inverse problems on the aforementioned reasons require that only certain (practicable) equivalents of a given set of reservoir parameters be obtained.

It is clear that accurate solutions to the problems important for us may be obtained only from test examples. Foremost, in such a case, “accurate” indicators of performance are obtained on a computer on the highest possible level. Secondly, we know for sure the required values such as reservoir performance of every element level. Thirdly, accuracy of the values of the top and bottom of a reservoir, as well as configuration and dimensions of the external boundary of a reservoir, is practically assured.

Through effective study, it is possible to explore that effective solutions may have positive impact on solving inverse problems with higher resolution.

This results in very important practical conclusion. If we require to increase accuracy of geology-mathematical reservoir model, we need to determine and implement the best method to “filtrate” examined production wells to better “perceive” properties of a reservoir and anisotropy of reservoir permeability [3].

It would be very helpful to perform mathematical experiments in geology-hydrodynamic model of a reservoir to plan such production activity and achieve positive results. It should be noted that a similar idea is based on common analysis methods of single wells in an unstable mode filter.

**Filtration equations of compressible fluids in a flexible reservoir.**

** **

**Units**

k - hydraulic filtration of a reservoir, hydraulic permeability, [cm^{2}], [d];

h – reservoir thickness, [cm];

* μ*– fluid viscosity under reservoir conditions, [cP];

*p(r,t)* - pressure at a distance from the axis of a well site in t, [kgf/cm^{2}];

*P*_{c} – bottom hole pressure, [kgf/cm^{2}];

*P _{пл}* - reservoir pressure, [kgf/cm

*r _{c}* - effective well radius (with consideration for skin effect), [cm];

*Q(t)* - well performance under reservoir conditions, [cm^{3}/s];

**- **hydraulic permeability of a reservoir, [dsm/cm];

- piezo-permeability parameter, [dsm^{ 2}/cP];

*m* - reservoir porosity;

*β _{ж}, β_{c}* - ratios of elastic volume and a reservoir accordingly [kgf/cm

*c* – adducted ratio of elastic volume of a reservoir, [kgf/cm^{2}];

cumulative index function.

Let us assume that one infinitely large reservoir with constant thickness, not crossing the top and bottom boundaries, from points (x,y) = (0,0) allows to produce from one well. Unless there is impact on a reservoir at initial point of time t= 0, then pressure distribution in the last moments will be symmetrical cylindrical p (x, y, t) = p (r, t):

** ** (1)

and it is written in a form of the heat-transfer equation [1,2]:

Where: - piezo-permeability ratio;

*m *– reservoir permeability;

* β _{ж}, β_{c}* – ratios of elastic volume and a reservoir.

*β* = mβ _{ж} + β_{c} *– adducted elastic volume ratio of a reservoir. Apart from the formula (1), pressure shall satisfy:

(2) initial and

** ** (3)

boundary conditions.

Where: *P _{пл} *– bottom hole pressure (pressure of non-developed reservoir);

*Q*(*t*) – well performance under reservoir conditions;

* r _{c}* - effective well radius (with consideration for skin effect).

- Masket and I.A. Charneyem [1,2] found a solution satisfying the above mentioned initial and boundary conditions of the equation (1).

** **(4)

** **

Where: *I*_{0} – Bessel function of the first and zero orders with assumed parameter. Given that the well start-up point (*r _{c}*

** **(5)

The last equation is the main equation for interpretation of the pressure curve (PC).

** **

Let us consider several examples of practical importance.

**Example 1.** Let *Q*(*t*) = *Q*_{0} = const. is the constant yield of a well. Then:

This equation may be written in the following way:

** **(6)

** **Where:

is the cumulative index function.

Under small parameter values, the cumulative index function has the following asymptotics:

** **(7)

Herewith, introduced error at *x* < 0.01 does not exceed 0.25%, at *x *< 0.03 does not exceed 1%, at *x *< 0.1 does not exceed 5,7%, at* x *< 0,14 does not exceed 9,7%.

*Figure №1. Comparison of the cumulative index function expint (x) = - Ei (-x) (continuous curve) and its approximation ln 1x - 0,57722 (discontinuous curve)*

* *

*Figure № 2 Compression curve at the bottom of a well over time: continuous curve - according to the equation (6), discontinuous (dash) curve - according to the equation (8)*

** **

at, ** **(for sufficiently large time period) the bottom hole pressure can be expressed as

** **(8)

** **The Figure №2 shows the bottom hole pressure - time curves obtained according to the equations (6) (continuous curve) and (8) (discontinuous dash curve) for the following reference values: *χ* =1000 cm^{2}/с, *r _{c}*=10 cm

**Example 2. **Let at 0 ≤ *t* ≤ *T,* *Q*(*t*) = *Q*_{0}, at *t* >*T,* *Q*(*t*) = *Q*_{1}. Then: Pressure *t* ≤*T* is determined by the equation from example №1, yet when t>T, we obtain:

According to determination of the cumulative index function:

** **(9)

Consequently:

In case of *t>T*, to determine pressure in the bottom hole, the following equation is justified:

** **(10)

Particularly, *Q*_{1}=0, we obtain formerly known equation for bottom hole pressure recovery curve of a well, which was suspended after long operational period:

** **(11)

when for the cumulative index function (7) applying approximation we obtain the final equation as follows:

** **(12)

which can be written in the following form:

Similarly, for a general case, even if *Q*_{1} as opposed to zero, except for the equation (10) the following more simple approximation may be applied:

** **** **(13)

The following example summarized the above mentioned examples.

**Example №3.** Let 0 = *t*_{0} < *t*_{1} < ... < *t _{n}* - time sequence, fluid outflow at the point of time

** **** **(14)

Then: Pressure distribution over time t corresponding the time period t_{i} = t < t_{i + 1} according to the equation (5) has the following form:

(15)

If t>tn, then we obtain the similar.

**Example №4.** Let T> 0 and

** **(16)

In such a case the pressure equation at t<T corresponds to the equation in Example №1. If t> T, then according to the equation (5) we obtain:

Therefore, at *t* > *T* and ** **according to the equation, we obtain:

** **(17)

**Example №5. **Let *δT >* 0 and

** **** **** **(18)

Then: at *t* > *δT* there will be:

Let us calculate each distance separately. For a general case, when 0 ≤ *t*_{1} ≤ *t*_{2} ≤ t we obtain:

** **(19)

** **(20)

Consequently:

** **

Therefore, at t>T we obtain the following equation for outflow (18):

** **(21а)

If *t* ≤ *δT* then similarly to the previous equation, we obtain:

** **** **(21b)

**Example №6.** Let *t*_{1} < *t*_{2}, and

** **** **(22)

Then: at *t* ≤ *t*_{1} *p*(*r*,*t*) = *p _{пл}*. If

by introducing the values, we obtain:

** **

Now, using the equations (19) и (20), we obtain for time t1 ≤ t ≤ t2:

** **** **(23)

If t_{1} > t_{2} then we obtain the above indicated analogy:

**Example №7.** Let *T* > *δT*_{1} > 0, and *δT*_{2} > 0,

** **(24)

Obtained schematic diagram of outflow change based on the calculations is shown on Figure №3.

.

*Figure №3. Outflow diagram*

** **

**Determination (identification) of parameters of gas reservoirs**

When filtrating the ideal gas, the pressure equation differs from the fluid equation (1) and the difference is pressure square instead of pressure, and piezo-permeability ** **will be replaced by the parameter (see [2], p. 172):

** **(25)

Where:

*m*_{0} – porosity factor;

*k *– reservoir permeability;

- – gas viscosity;

*р _{пл} *– pressure in non-developed reservoir.

In such a case the initial condition shall be written as follows:

(26)

With respect to the ideal gas, the following ratio ([2], p. 173): is used instead of lateral condition (3):

** **** **(27)

Where:

*p _{am} *– atmospheric pressure;

*Q _{am} *– outflow to the wellhead. Boundary condition of infinity will be transformed as follows:

** ** ** **(28)

Therefore, when determining parameters of the ideal gas reservoir based on hydraulic dynamic analysis of wells, one can use the described theory of elastic fluids, but instead of pressure, it is necessary to apply pressure square, and instead of outflow from the well bore, it is necessary to use 2 · *p _{am}* ·

** **

**REFERENCES:**

** **

Азиз Х., Сеттари Э. Математическое моделирование пластовых систем. – М.: Недра, 1982, стр.

Басниев К.С., Власов А.М., Кочина И.Н. Максимов В.М. Подземная гидравлика. – М.: Недра, 1986, стр. 303.

Гурбанмырадов О.А., Эседуллаев Р.Э., Дурдыев Н.Т., А.Р. Деряев. Гидродинамические проблемы, возникающие при бурении нефтегазовых скважин и способы их решения. – A.: Наука, 2018, стр.336.

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